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This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
Copyright (C) 2014-2018 Sylvie Boldo
Copyright (C) 2014-2018 Guillaume Melquiond
Copyright (C) 2014-2018 Pierre Roux
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
Copyright (C) 2014-2018 Guillaume Melquiond
Copyright (C) 2014-2018 Pierre Roux
Require Import Psatz. Require Import Raux Defs Generic_fmt Operations Ulp FLX FLT FTZ. Open Scope R_scope. Section Double_round. Variable beta : radix. Notation bpow e := (bpow beta e). Notation mag x := (mag beta x). Definition round_round_eq fexp1 fexp2 choice1 choice2 x := round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) x.
A little tactic to simplify terms of the form bpow a × bpow b.
Ltac bpow_simplify := (* bpow ex * bpow ey ~~> bpow (ex + ey) *) repeat match goal with | |- context [(Raux.bpow _ _ * Raux.bpow _ _)] => rewrite <- bpow_plus | |- context [(?X1 * Raux.bpow _ _ * Raux.bpow _ _)] => rewrite (Rmult_assoc X1); rewrite <- bpow_plus | |- context [(?X1 * (?X2 * Raux.bpow _ _) * Raux.bpow _ _)] => rewrite <- (Rmult_assoc X1 X2); rewrite (Rmult_assoc (X1 * X2)); rewrite <- bpow_plus end; (* ring_simplify arguments of bpow *) repeat match goal with | |- context [(Raux.bpow _ ?X)] => progress ring_simplify X end; (* bpow 0 ~~> 1 *) change (Raux.bpow _ 0) with 1; repeat match goal with | |- context [(_ * 1)] => rewrite Rmult_1_r end. Definition midp (fexp : Z -> Z) (x : R) := round beta fexp Zfloor x + / 2 * ulp beta fexp x. Definition midp' (fexp : Z -> Z) (x : R) := round beta fexp Zceil x - / 2 * ulp beta fexp x.beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < bpow (mag x) - / 2 * ulp beta fexp2 x -> x < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < bpow (mag x) - / 2 * ulp beta fexp2 x -> x < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx < midp fexp1 x - / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:Rx < midp fexp1 x - / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xnow apply (Rplus_lt_reg_r (round beta fexp1 Zfloor x)); ring_simplify.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:Rround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RRabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RRabs (x'' - x) <= / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:R/ 2 * ulp beta fexp2 x <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:R/ 2 * ulp beta fexp2 x <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))x' <= xexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x + (x - x')) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x) + Rabs (x - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x) + Rabs (x - x') < / 2 * bpow (fexp2 (mag x)) + / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x - x') < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))exact Hr1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))now rewrite Rabs_right; [|now apply Rle_ge]; apply Hx2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Rabs (x - x') < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 00 = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 00 = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - 0 * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - 0) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x - 0) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (0 - x) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0/ 2 * bpow (fexp2 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0bpow (fexp2 (mag x)) < bpow (fexp1 (mag x))omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0(fexp2 (mag x) < fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0mag x'' = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0mag x'' = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x'' <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x <= mag x'')%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x'' <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs x'' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs (x'' - x + x) < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs (x'' - x) + Rabs x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs (x'' - x) + Rabs x < / 2 * bpow (fexp2 (mag x)) + (bpow (mag x) - / 2 * bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))now rewrite Rabs_right; [|apply Rle_ge; apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)x':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x <= mag x'')%Znow apply mag_round_ge; [|apply valid_rnd_N|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':round beta fexp2 (Znearest choice2) x <> 0(mag x <= mag (round beta fexp2 (Znearest choice2) x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x'')))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs ((x * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xx - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag x/ 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x)) <= / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xbpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x) <= bpow (fexp1 (mag x))unfold ulp, cexp; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag x0 <= bpow (fexp2 (mag x)) -> bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs ((x'' * bpow (- fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))now bpow_simplify. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:x < bpow (mag x) - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))x'':=round beta fexp2 (Znearest choice2) x:RHr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Pxx':0 <= x - x'Hr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zx < midp fexp1 x - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xnow apply (Rplus_lt_reg_r (round beta fexp1 Zfloor x)); ring_simplify.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xHx2:x - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx - round beta fexp1 Zfloor x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:Rx - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x' <= xexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'H:x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))x < bpow (mag x) - / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * (ulp beta fexp1 x - ulp beta fexp2 x) <= bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * ulp beta fexp1 x - / 2 * ulp beta fexp2 x <= bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * bpow (cexp beta fexp1 x) - / 2 * bpow (cexp beta fexp2 x) <= bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * bpow (cexp beta fexp1 x) <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * bpow (cexp beta fexp1 x) * bpow (- mag x) <= bpow (mag x) * bpow (- mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0/ 2 * bpow (fexp1 (mag x) - mag x) <= 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 02 * (/ 2 * bpow (fexp1 (mag x) - mag x)) <= 2 * 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0bpow (fexp1 (mag x) - mag x) <= 2 * 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0bpow (fexp1 (mag x) - mag x) <= 1omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Zx':x' = 0(fexp1 (mag x) - mag x <= 0)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 < x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 < x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= round beta fexp1 Zfloor xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 * bpow (- fexp1 (mag x)) <= round beta fexp1 Zfloor x * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= round beta fexp1 Zfloor x * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= IZR (Zfloor (scaled_mantissa beta fexp1 x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0(0 <= Zfloor (scaled_mantissa beta fexp1 x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= scaled_mantissa beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= scaled_mantissa beta fexp1 (Rabs x)apply Rabs_pos.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 00 <= Rabs (scaled_mantissa beta fexp1 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' <= bpow (mag x) - ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' <= bpow (mag x) - ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' + ulp beta fexp1 x <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' + ulp beta fexp1 (round beta fexp1 Zfloor x) <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'0 <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' + ulp beta fexp1 (round beta fexp1 Zfloor x) <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' + ulp beta fexp1 x' <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'0 < x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'generic_format beta fexp1 x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' < bpow (mag x)exact Px'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'0 < x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'generic_format beta fexp1 x'now apply generic_format_round; [|apply valid_rnd_DN].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'generic_format beta fexp1 (round beta fexp1 Zfloor x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x < bpow (mag x)now apply round_DN_pt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x' <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'x < bpow (mag x)apply bpow_mag_gt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Rabs x < bpow (mag x)exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Valid_exp fexp1now apply Rlt_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'0 <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 x/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xH:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 x/ 2 * ulp beta fexp1 x <= 1 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xH:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 x0 <= ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xH:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xH:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - ulp beta fexp1 xH:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - bpow (cexp beta fexp1 x)H:/ 2 * ulp beta fexp1 x <= ulp beta fexp1 xx < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)lra. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':x < midp fexp1 x - / 2 * ulp beta fexp2 xx':=round beta fexp1 Zfloor x:RHx2:x - x' < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))Pxx':0 <= x - x'Nzx':x' <> 0Px':0 < x'Hx':x' <= bpow (mag x) - bpow (cexp beta fexp1 x)H:/ 2 * bpow (cexp beta fexp1 x) <= bpow (cexp beta fexp1 x)x < bpow (mag x) - / 2 * bpow (cexp beta fexp2 x)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp2 (mag x) = fexp1 (mag x) -> x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp2 (mag x) = fexp1 (mag x) -> x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xHx:x - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply (Rplus_lt_reg_r (round beta fexp1 Zfloor x)); ring_simplify.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xHx:x - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx - round beta fexp1 Zfloor x < / 2 * ulp beta fexp1 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:Rx - x' < / 2 * ulp beta fexp1 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 x0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 x0 <= x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xx' <= xexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xValid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs ((x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'Rabs (x - IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < x - IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < x - x' < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'x - x' < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < x - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'- (/ 2 * bpow (fexp1 (mag x))) < - 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'0 < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'/ 2 * 0 < / 2 * bpow (fexp1 (mag x))apply bpow_gt_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'0 < bpow (fexp1 (mag x))rewrite ulp_neq_0 in Hx;try apply Rgt_not_eq; assumption.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'x - x' < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (F2R {| Fnum := Znearest choice2 (scaled_mantissa beta fexp2 x); Fexp := cexp beta fexp2 x |}) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (IZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (IZR (Znearest choice2 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2round beta fexp1 (Znearest choice1) (IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2Valid_rnd (Znearest choice1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2generic_format beta fexp1 (IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = round beta fexp1 (Znearest choice1) xnow rewrite (Znearest_imp _ _ (Zfloor (x * bpow (- fexp1 (mag x))))).beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2Valid_rnd (Znearest choice1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2generic_format beta fexp1 (IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2generic_format beta fexp1 (IZR (Zfloor (scaled_mantissa beta fexp1 x)) * bpow (cexp beta fexp1 x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2generic_format beta fexp1 (round beta fexp1 Zfloor x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2Valid_rnd Zfloornow apply valid_rnd_DN. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':x < midp fexp1 xx':=round beta fexp1 Zfloor x:RHx:x - x' < / 2 * ulp beta fexp1 xPxx':0 <= x - x'H:Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zfloor (x * bpow (- fexp1 (mag x))))) < / 2Valid_rnd Zfloorbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x))%Z -> (fexp1 (mag x) <= mag x)%Z -> x < midp fexp1 x -> ((fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x))%Z -> (fexp1 (mag x) <= mag x)%Z -> x < midp fexp1 x -> ((fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp1 (mag x) <= fexp2 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp2 (mag x) < fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp1 (mag x) <= fexp2 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_lt_mid_same_place.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp1 (mag x) <= fexp2 (mag x))%ZHf2'':fexp2 (mag x) = fexp1 (mag x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp2 (mag x) < fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp2 (mag x) < fexp1 (mag x))%ZHf2'':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_lt_mid_further_place. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:x < midp fexp1 xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x < midp fexp1 x - / 2 * ulp beta fexp2 xHf2':(fexp2 (mag x) < fexp1 (mag x))%ZHf2'':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx'':x < midp fexp1 x - / 2 * ulp beta fexp2 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> round beta fexp2 (Znearest choice2) x < bpow (mag x) -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> round beta fexp2 (Znearest choice2) x < bpow (mag x) -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zround beta fexp2 (Znearest choice2) x < bpow (mag x) -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)now unfold midp' in Hx2'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - / 2 * ulp beta fexp1 x + / 2 * ulp beta fexp2 x < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx1:round beta fexp2 (Znearest choice2) x < bpow (mag x)Hx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp2 (Znearest choice2) x < bpow (mag x) -> round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp2 (Znearest choice2) x < bpow (mag x) -> round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rround beta fexp2 (Znearest choice2) x < bpow (mag x) -> x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:Rx'' < bpow (mag x) -> x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Rabs (x'' - x) <= / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)/ 2 * ulp beta fexp2 x <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)/ 2 * ulp beta fexp2 x <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))0 <= x' - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))0 <= x' - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))x <= x'exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x + (x - x')) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x) + Rabs (x - x') < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x) + Rabs (x - x') < / 2 * bpow (fexp2 (mag x)) + / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x - x') < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))exact Hr1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x - x') < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x' - x) < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))now rewrite Rabs_right; [|now apply Rle_ge]; apply Hx2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x))Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xRabs (x' - x) < / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0round beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 00 = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 00 = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) - 0) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - 0 * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - 0) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (x - 0) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0Rabs (0 - x) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0/ 2 * bpow (fexp2 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0bpow (fexp2 (mag x)) < bpow (fexp1 (mag x))omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (0 - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Zx'':x'' = 0(fexp2 (mag x) < fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0mag x'' = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0mag x'' = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x'' <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x <= mag x'')%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x'' <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Rabs x'' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 00 <= x''beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Valid_exp fexp2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0generic_format beta fexp2 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 00 <= xexact Vfexp2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Valid_exp fexp2now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Valid_rnd (Znearest choice2)apply generic_format_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0generic_format beta fexp2 0now apply Rlt_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 00 <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0(mag x <= mag x'')%Znow apply mag_round_ge; [|apply valid_rnd_N|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':round beta fexp2 (Znearest choice2) x <> 0(mag x <= mag (round beta fexp2 (Znearest choice2) x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x'')))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs ((x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x - IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) - x) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xIZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) - x < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag x/ 2 * (ulp beta fexp1 x - ulp beta fexp2 x) <= / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xulp beta fexp1 x - ulp beta fexp2 x <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag x0 <= bpow (fexp2 (mag x)) -> ulp beta fexp1 x - ulp beta fexp2 x <= bpow (fexp1 (mag x))unfold cexp; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag x0 <= bpow (fexp2 (mag x)) -> bpow (cexp beta fexp1 x) - bpow (cexp beta fexp2 x) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs ((x'' * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))now bpow_simplify. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zceil x:Rx'':=round beta fexp2 (Znearest choice2) x:RHx1:x'' < bpow (mag x)Hx2:x' - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)Hr1:Rabs (x'' - x) <= / 2 * bpow (fexp2 (mag x))Px'x:0 <= x' - xHr2:Rabs (x'' - x') < / 2 * bpow (fexp1 (mag x))Nzx'':x'' <> 0Lx'':mag x'' = mag xRabs (x'' * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)now unfold midp' in Hx2'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - / 2 * ulp beta fexp1 x + / 2 * ulp beta fexp2 x < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:Rround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:Rround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) x(* bpow (mag x) <= x'' *)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' < bpow (mag x) + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xx'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' < bpow (mag x) + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' <= x + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x + / 2 * ulp beta fexp2 x < bpow (mag x) + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' <= x + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x'' - x <= / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Rabs (x'' - x) <= / 2 * ulp beta fexp2 xexact Vfexp2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Valid_exp fexp2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x + / 2 * ulp beta fexp2 x < bpow (mag x) + / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''x < bpow (mag x)apply bpow_mag_gt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Rabs x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xx'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xx'' <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x)) <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x)) * bpow (- fexp2 (mag x)) <= bpow (mag x) * bpow (- fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) <= bpow (mag x - fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) <= IZR (beta ^ (mag x - fexp2 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x(Znearest choice2 (x * bpow (- fexp2 (mag x))) <= beta ^ (mag x - fexp2 (mag x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x(Znearest choice2 (x * bpow (- fexp2 (mag x))) < beta ^ (mag x - fexp2 (mag x)) + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) < IZR (beta ^ (mag x - fexp2 (mag x)) + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) < bpow (mag x - fexp2 (mag x)) + 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x)) < (bpow (mag x - fexp2 (mag x)) + 1) * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x)) < bpow (mag x - fexp2 (mag x)) * bpow (fexp2 (mag x)) + bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xIZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x)) < bpow (mag x) + bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 xbpow (mag x) + / 2 * ulp beta fexp2 x <= bpow (mag x) + bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x/ 2 * ulp beta fexp2 x <= bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x/ 2 * ulp beta fexp2 x <= 1 * bpow (fexp2 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x/ 2 * bpow (cexp beta fexp2 x) <= 1 * bpow (fexp2 (mag x))lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''H'x'':x'' < bpow (mag x) + / 2 * ulp beta fexp2 x/ 2 <= 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Rabs (x - x'') < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Rabs (x - x'') < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Rabs (x - x'') <= / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)/ 2 * ulp beta fexp2 x < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Rabs (x - x'') <= / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Rabs (x'' - x) <= / 2 * ulp beta fexp2 xexact Vfexp2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Valid_exp fexp2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)/ 2 * ulp beta fexp2 x < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)ulp beta fexp2 x < ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)bpow (cexp beta fexp2 x) < bpow (cexp beta fexp1 x)omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)(fexp2 (mag x) < fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xround beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x'')))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 x(0 <= mag x - fexp1 (mag x''))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x'')))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 x(0 <= mag x - fexp1 (mag x''))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 x(0 <= mag x - fexp1 (mag (bpow (mag x))))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 x(0 <= mag x - fexp1 (mag x + 1))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 x(fexp1 (mag x + 1) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHle:(mag x <= fexp1 (mag x))%Z(fexp1 (mag x + 1) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHle:(mag x <= fexp1 (mag x))%ZH:mag x = fexp1 (mag x)(fexp1 (mag x + 1) <= mag x)%Znow apply Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHle:(mag x <= fexp1 (mag x))%ZH:mag x = fexp1 (mag x)(fexp1 (fexp1 (mag x) + 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (Znearest choice1 (x'' * bpow (- fexp1 (mag x'')))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (beta ^ (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) - IZR (beta ^ (mag x - fexp1 (mag x'')))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (beta ^ (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x'')) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (beta ^ (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x'')) = IZR (beta ^ (mag x - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) - IZR (beta ^ (mag x - fexp1 (mag x)))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZIZR (beta ^ (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x'')) = IZR (beta ^ (mag x - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%Zbpow (mag x - fexp1 (mag x'')) * bpow (fexp1 (mag x'')) = IZR (beta ^ (mag x - fexp1 (mag x))) * bpow (fexp1 (mag x))now bpow_simplify.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%Zbpow (mag x - fexp1 (mag x'')) * bpow (fexp1 (mag x'')) = bpow (mag x - fexp1 (mag x)) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) - IZR (beta ^ (mag x - fexp1 (mag x)))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) - bpow (mag x - fexp1 (mag x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) - bpow (mag x - fexp1 (mag x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) - bpow (mag x - fexp1 (mag x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs ((x * bpow (- fexp1 (mag x)) - bpow (mag x - fexp1 (mag x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) - bpow (mag x - fexp1 (mag x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x - bpow (mag x)) < / 2 * bpow (fexp1 (mag x))rewrite <- Hx''pow; exact Hr.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * bpow (cexp beta fexp1 x)Hf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x - bpow (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) - IZR (beta ^ (mag x - fexp1 (mag x'')))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) - bpow (mag x - fexp1 (mag x''))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) - bpow (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x'')) < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) - bpow (mag x - fexp1 (mag x''))) * Rabs (bpow (fexp1 (mag x''))) < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs ((x'' * bpow (- fexp1 (mag x'')) - bpow (mag x - fexp1 (mag x''))) * bpow (fexp1 (mag x''))) < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' * bpow (- fexp1 (mag x'')) * bpow (fexp1 (mag x'')) - bpow (mag x - fexp1 (mag x'')) * bpow (fexp1 (mag x''))) < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs (x'' - bpow (mag x)) < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%ZRabs 0 < / 2 * bpow (fexp1 (mag x''))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%Z0 < / 2 * bpow (fexp1 (mag x''))apply Rmult_lt_compat_l; [lra|apply bpow_gt_0]. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%ZHx2':midp' fexp1 x + / 2 * ulp beta fexp2 x < xx':=round beta fexp1 Zfloor x:RHx2:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)x'':=round beta fexp2 (Znearest choice2) x:RHx'':bpow (mag x) <= x''Hx''pow:x'' = bpow (mag x)Hr:Rabs (x - x'') < / 2 * ulp beta fexp1 xHf:(0 <= mag x - fexp1 (mag x''))%Z/ 2 * 0 < / 2 * bpow (fexp1 (mag x''))beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp2 (mag x) = fexp1 (mag x) -> midp' fexp1 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp2 (mag x) = fexp1 (mag x) -> midp' fexp1 x < x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xround beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xround beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xnow unfold midp' in Hx'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xround beta fexp1 Zceil x - / 2 * ulp beta fexp1 x < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) * bpow (fexp1 (mag x)) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) * Rabs (bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs ((IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x))) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xRabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - x) < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x- (/ 2 * bpow (fexp1 (mag x))) < IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - x < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x- (/ 2 * bpow (fexp1 (mag x))) < IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xIZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - x < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x- (/ 2 * bpow (fexp1 (mag x))) < IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x- (/ 2 * bpow (fexp1 (mag x))) < 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x0 <= IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x- (/ 2 * bpow (fexp1 (mag x))) < 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x0 < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x/ 2 * 0 < / 2 * bpow (fexp1 (mag x))apply bpow_gt_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x0 < bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 x0 <= IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xx <= IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xValid_exp fexp1rewrite ulp_neq_0 in Hx;[exact Hx|now apply Rgt_not_eq].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xIZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) - x < / 2 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round beta fexp1 (Znearest choice1) (F2R {| Fnum := Znearest choice2 (scaled_mantissa beta fexp2 x); Fexp := cexp beta fexp2 x |}) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round beta fexp1 (Znearest choice1) (IZR (Znearest choice2 (x * bpow (- fexp2 (mag x)))) * bpow (fexp2 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round beta fexp1 (Znearest choice1) (IZR (Znearest choice2 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round beta fexp1 (Znearest choice1) (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2round beta fexp1 (Znearest choice1) (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x))) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Valid_rnd (Znearest choice1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2generic_format beta fexp1 (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) = round beta fexp1 (Znearest choice1) xnow rewrite (Znearest_imp _ _ (Zceil (x * bpow (- fexp1 (mag x))))); [|rewrite Rabs_minus_sym].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2IZR (Zceil (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = IZR (Znearest choice1 (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x))now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Valid_rnd (Znearest choice1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2generic_format beta fexp1 (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2generic_format beta fexp1 (IZR (Zceil (scaled_mantissa beta fexp1 x)) * bpow (cexp beta fexp1 x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2generic_format beta fexp1 (round beta fexp1 Zceil x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Valid_rnd Zceilnow apply valid_rnd_UP.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Valid_rnd Zceilnow rewrite Rabs_minus_sym. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:fexp2 (mag x) = fexp1 (mag x)Hx':midp' fexp1 x < xHx:round beta fexp1 Zceil x - x < / 2 * ulp beta fexp1 xH:Rabs (IZR (Zceil (x * bpow (- fexp1 (mag x)))) - x * bpow (- fexp1 (mag x))) < / 2Rabs (x * bpow (- fexp1 (mag x)) - IZR (Zceil (scaled_mantissa beta fexp1 x))) < / 2beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x))%Z -> (fexp1 (mag x) <= mag x)%Z -> midp' fexp1 x < x -> ((fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x))%Z -> (fexp1 (mag x) <= mag x)%Z -> midp' fexp1 x < x -> ((fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp1 (mag x) <= fexp2 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp2 (mag x) < fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp1 (mag x) <= fexp2 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_gt_mid_same_place.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp1 (mag x) <= fexp2 (mag x))%ZHf2'':fexp2 (mag x) = fexp1 (mag x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp2 (mag x) < fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp2 (mag x) < fexp1 (mag x))%ZHf2'':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_gt_mid_further_place. Qed. Section Double_round_mult.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x))%ZHf1:(fexp1 (mag x) <= mag x)%ZHx:midp' fexp1 x < xHx':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> midp' fexp1 x + / 2 * ulp beta fexp2 x < xHf2':(fexp2 (mag x) < fexp1 (mag x))%ZHf2'':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHx'':midp' fexp1 x + / 2 * ulp beta fexp2 x < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall x y : R, x <> 0 -> y <> 0 -> mag (x * y) = (mag x + mag y - 1)%Z \/ mag (x * y) = (mag x + mag y)%Zbeta:radixforall x y : R, x <> 0 -> y <> 0 -> mag (x * y) = (mag x + mag y - 1)%Z \/ mag (x * y) = (mag x + mag y)%Zbeta:radixx, y:RZx:x <> 0Zy:y <> 0mag (x * y) = (mag x + mag y - 1)%Z \/ mag (x * y) = (mag x + mag y)%Zomega. Qed. Definition round_round_mult_hyp fexp1 fexp2 := (forall ex ey, (fexp2 (ex + ey) <= fexp1 ex + fexp1 ey)%Z) /\ (forall ex ey, (fexp2 (ex + ey - 1) <= fexp1 ex + fexp1 ey)%Z).beta:radixx, y:RZx:x <> 0Zy:y <> 0H:(mag x + mag y - 1 <= mag (x * y))%ZH0:(mag (x * y) <= mag x + mag y)%Zmag (x * y) = (mag x + mag y - 1)%Z \/ mag (x * y) = (mag x + mag y)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, round_round_mult_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x * y)beta:radixforall fexp1 fexp2 : Z -> Z, round_round_mult_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 (0 * y)now apply generic_format_0.beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 0beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y = 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y <> 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y = 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y = 0generic_format beta fexp2 (x * 0)now apply generic_format_0.beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y = 0generic_format beta fexp2 0beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x <> 0Zy:y <> 0generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x * y)beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0x = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp1 x); Fexp := cexp beta fexp1 x |} -> y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp1 y); Fexp := cexp beta fexp1 y |} -> x * y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp2 (x * y)); Fexp := cexp beta fexp2 (x * y) |}beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0x = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp1 x); Fexp := fexp1 (mag x) |} -> y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp1 y); Fexp := fexp1 (mag y) |} -> x * y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp2 (x * y)); Fexp := fexp2 (mag (x * y)) |}beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zx = F2R {| Fnum := mx; Fexp := fexp1 (mag x) |} -> y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp1 y); Fexp := fexp1 (mag y) |} -> x * y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp2 (x * y)); Fexp := fexp2 (mag (x * y)) |}beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):Zx = F2R {| Fnum := mx; Fexp := fexp1 (mag x) |} -> y = F2R {| Fnum := my; Fexp := fexp1 (mag y) |} -> x * y = F2R {| Fnum := Ztrunc (scaled_mantissa beta fexp2 (x * y)); Fexp := fexp2 (mag (x * y)) |}beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR my * bpow (fexp1 (mag y)) -> x * y = IZR (Ztrunc (scaled_mantissa beta fexp2 (x * y))) * bpow (fexp2 (mag (x * y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))x * y = IZR (Ztrunc (scaled_mantissa beta fexp2 (x * y))) * bpow (fexp2 (mag (x * y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = IZR (Ztrunc (scaled_mantissa beta fexp2 (x * y))) * bpow (fexp2 (mag (x * y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = F2R fxybeta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxyx * y = IZR (Ztrunc (scaled_mantissa beta fexp2 (x * y))) * bpow (fexp2 (mag (x * y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = F2R fxybeta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = IZR (mx * my) * bpow (fexp1 (mag x) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = IZR (mx * my) * (bpow (fexp1 (mag x)) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betax * y = IZR mx * IZR my * (bpow (fexp1 (mag x)) * bpow (fexp1 (mag y)))ring.beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaIZR mx * bpow (fexp1 (mag x)) * (IZR my * bpow (fexp1 (mag y))) = IZR mx * IZR my * (bpow (fexp1 (mag x)) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxyx * y = IZR (Ztrunc (scaled_mantissa beta fexp2 (x * y))) * bpow (fexp2 (mag (x * y)))beta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxyx * y <> 0 -> (cexp beta fexp2 (x * y) <= Fexp fxy)%Zbeta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxy(cexp beta fexp2 (x * y) <= Fexp fxy)%Zbeta:radixfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxy(fexp2 (mag (x * y)) <= fexp1 (mag x) + fexp1 (mag y))%Znow destruct (mag_mult_disj x y Zx Zy) as [Lxy|Lxy]; rewrite Lxy. Qed. Variable rnd : R -> Z. Context { valid_rnd : Valid_rnd rnd }.beta:radixfexp1, fexp2:Z -> ZHfexp1:forall ex ey : Z, (fexp2 (ex + ey) <= fexp1 ex + fexp1 ey)%ZHfexp2:forall ex ey : Z, (fexp2 (ex + ey - 1) <= fexp1 ex + fexp1 ey)%Zx, y:RZx:x <> 0Zy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx * my; Fexp := fexp1 (mag x) + fexp1 (mag y) |}:float betaHxy:x * y = F2R fxy(fexp2 (mag (x * y)) <= fexp1 (mag x) + fexp1 (mag y))%Zbeta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndforall fexp1 fexp2 : Z -> Z, round_round_mult_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round beta fexp1 rnd (round beta fexp2 rnd (x * y)) = round beta fexp1 rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndforall fexp1 fexp2 : Z -> Z, round_round_mult_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round beta fexp1 rnd (round beta fexp2 rnd (x * y)) = round beta fexp1 rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 rnd (round beta fexp2 rnd (x * y)) = round beta fexp1 rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp2 rnd (x * y) = x * ybeta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHxy:round beta fexp2 rnd (x * y) = x * yround beta fexp1 rnd (round beta fexp2 rnd (x * y)) = round beta fexp1 rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp2 rnd (x * y) = x * ynow apply (round_round_mult_aux fexp1 fexp2).beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x * y)now rewrite Hxy at 1. Qed. Section Double_round_mult_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndfexp1, fexp2:Z -> ZHfexp:round_round_mult_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHxy:round beta fexp2 rnd (x * y) = x * yround beta fexp1 rnd (round beta fexp2 rnd (x * y)) = round beta fexp1 rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round beta (FLX_exp prec) rnd (round beta (FLX_exp prec') rnd (x * y)) = round beta (FLX_exp prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round beta (FLX_exp prec) rnd (round beta (FLX_exp prec') rnd (x * y)) = round beta (FLX_exp prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround beta (FLX_exp prec) rnd (round beta (FLX_exp prec') rnd (x * y)) = round beta (FLX_exp prec) rnd (x * y)unfold round_round_mult_hyp; split; intros ex ey; unfold FLX_exp; omega. Qed. End Double_round_mult_FLX. Section Double_round_mult_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_mult_hyp (FLX_exp prec) (FLX_exp prec')beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= 2 * emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round beta (FLT_exp emin prec) rnd (round beta (FLT_exp emin' prec') rnd (x * y)) = round beta (FLT_exp emin prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= 2 * emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round beta (FLT_exp emin prec) rnd (round beta (FLT_exp emin' prec') rnd (x * y)) = round beta (FLT_exp emin prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= 2 * emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround beta (FLT_exp emin prec) rnd (round beta (FLT_exp emin' prec') rnd (x * y)) = round beta (FLT_exp emin prec) rnd (x * y)unfold round_round_mult_hyp; split; intros ex ey; unfold FLT_exp; generalize (Zmax_spec (ex + ey - prec') emin'); generalize (Zmax_spec (ex + ey - 1 - prec') emin'); generalize (Zmax_spec (ex - prec) emin); generalize (Zmax_spec (ey - prec) emin); omega. Qed. End Double_round_mult_FLT. Section Double_round_mult_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= 2 * emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_mult_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= 2 * emin + prec)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round beta (FTZ_exp emin prec) rnd (round beta (FTZ_exp emin' prec') rnd (x * y)) = round beta (FTZ_exp emin prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= 2 * emin + prec)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round beta (FTZ_exp emin prec) rnd (round beta (FTZ_exp emin' prec') rnd (x * y)) = round beta (FTZ_exp emin prec) rnd (x * y)beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= 2 * emin + prec)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround beta (FTZ_exp emin prec) rnd (round beta (FTZ_exp emin' prec') rnd (x * y)) = round beta (FTZ_exp emin prec) rnd (x * y)unfold round_round_mult_hyp; split; intros ex ey; unfold FTZ_exp; unfold Prec_gt_0 in *; destruct (Z.ltb_spec (ex + ey - prec') emin'); destruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ey - prec) emin); destruct (Z.ltb_spec (ex + ey - 1 - prec') emin'); omega. Qed. End Double_round_mult_FTZ. End Double_round_mult. Section Double_round_plus.beta:radixrnd:R -> Zvalid_rnd:Valid_rnd rndemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= 2 * emin + prec)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_mult_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixforall x y : R, 0 < y -> y <= x -> mag (x + y) = mag x \/ mag (x + y) = (mag x + 1)%Zbeta:radixforall x y : R, 0 < y -> y <= x -> mag (x + y) = mag x \/ mag (x + y) = (mag x + 1)%Zbeta:radixx, y:RPy:0 < yHxy:y <= xmag (x + y) = mag x \/ mag (x + y) = (mag x + 1)%Zomega. Qed.beta:radixx, y:RPy:0 < yHxy:y <= xH:(mag x <= mag (x + y))%ZH0:(mag (x + y) <= mag x + 1)%Zmag (x + y) = mag x \/ mag (x + y) = (mag x + 1)%Zbeta:radixforall (fexp : Z -> Z) (x y : R), 0 < x -> 0 <= y -> generic_format beta fexp x -> (mag y <= fexp (mag x))%Z -> mag (x + y) = mag xbeta:radixforall (fexp : Z -> Z) (x y : R), 0 < x -> 0 <= y -> generic_format beta fexp x -> (mag y <= fexp (mag x))%Z -> mag (x + y) = mag xbeta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%Zmag (x + y) = mag xbeta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%Z0 <= y < ulp beta fexp xbeta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%Zy < ulp beta fexp xbeta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%ZRabs y < ulp beta fexp xbeta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%Zbpow (mag y) <= ulp beta fexp xnow apply bpow_le. Qed.beta:radixfexp:Z -> Zx, y:RPx:0 < xNny:0 <= yFx:generic_format beta fexp xHsep:(mag y <= fexp (mag x))%Zbpow (mag y) <= bpow (cexp beta fexp x)beta:radixforall x y : R, 0 < x -> 0 < y -> (mag y <= mag x - 2)%Z -> mag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zbeta:radixforall x y : R, 0 < x -> 0 < y -> (mag y <= mag x - 2)%Z -> mag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zbeta:radixx, y:RPx:0 < xPy:0 < yHln:(mag y <= mag x - 2)%Zmag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zbeta:radixx, y:RPx:0 < xPy:0 < yHln:(mag y <= mag x - 2)%ZHxy:y < xmag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zbeta:radixx, y:RPx:0 < xPy:0 < yHln:(mag y <= mag x - 2)%ZHxy:y < xHln2:(mag (x - y) <= mag x)%Zmag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zomega. Qed.beta:radixx, y:RPx:0 < xPy:0 < yHln:(mag y <= mag x - 2)%ZHxy:y < xHln2:(mag (x - y) <= mag x)%ZHln3:(mag x - 1 <= mag (x - y))%Zmag (x - y) = mag x \/ mag (x - y) = (mag x - 1)%Zbeta:radixforall fexp : Z -> Z, Valid_exp fexp -> forall x y : R, 0 < x -> 0 < y -> y < x -> bpow (mag x - 1) < x -> generic_format beta fexp x -> (mag y <= fexp (mag x))%Z -> mag (x - y) = mag xbeta:radixforall fexp : Z -> Z, Valid_exp fexp -> forall x y : R, 0 < x -> 0 < y -> y < x -> bpow (mag x - 1) < x -> generic_format beta fexp x -> (mag y <= fexp (mag x))%Z -> mag (x - y) = mag xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zmag (x - y) = mag xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag x - 1) <= Rabs (x - y) < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag x - 1) <= Rabs (x - y)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs (x - y) < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag x - 1) <= Rabs (x - y)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag x - 1) <= x - ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zy < ulp beta fexp (bpow (mag x - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) <= x - ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zy < ulp beta fexp (bpow (mag x - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zy < bpow (fexp (mag x - 1 + 1)%Z)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zy < bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs y < bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs y < bpow (mag y)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag y) <= bpow (fexp (mag x))apply bpow_mag_gt.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs y < bpow (mag y)now apply bpow_le.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zbpow (mag y) <= bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) <= x - ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) + y <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) + y <= bpow (mag x - 1) + ulp beta fexp (bpow (mag x - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) + ulp beta fexp (bpow (mag x - 1)) <= xnow apply Rplus_le_compat_l; apply Rlt_le.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) + y <= bpow (mag x - 1) + ulp beta fexp (bpow (mag x - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))bpow (mag x - 1) + ulp beta fexp (bpow (mag x - 1)) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))succ beta fexp (bpow (mag x - 1)) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))generic_format beta fexp (bpow (mag x - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))(fexp (mag x - 1 + 1) <= mag x - 1)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))(fexp (mag x) <= mag x - 1)%Znow apply mag_generic_gt; [|now apply Rgt_not_eq|].beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZHy:y < ulp beta fexp (bpow (mag x - 1))(fexp (mag x) < mag x)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs (x - y) < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y >= 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y < xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx < bpow (mag x)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y < xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y < x + 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Z- y < 0now apply Ropp_lt_contravar.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Z- y < - 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx < bpow (mag x)apply bpow_mag_gt.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%ZRabs x < bpow (mag x)lra. Qed. Definition round_round_plus_hyp fexp1 fexp2 := (forall ex ey, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z).beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPx:0 < xPy:0 < yYltx:y < xXgtpow:bpow (mag x - 1) < xFx:generic_format beta fexp xLy:(mag y <= fexp (mag x))%Zx - y >= 0beta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp1 (mag x) <= fexp1 (mag y))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp1 (mag x) <= fexp1 (mag y))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 (x + y)now apply (generic_inclusion_mag beta fexp1).beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag y) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 ybeta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 (x + y)now apply (generic_inclusion_mag beta fexp1).beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag x) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0x = IZR (Ztrunc (scaled_mantissa beta fexp1 x)) * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (scaled_mantissa beta fexp1 y)) * bpow (fexp1 (mag y)) -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (scaled_mantissa beta fexp1 y)) * bpow (fexp1 (mag y)) -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR my * bpow (fexp1 (mag y)) -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betageneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = F2R fxybeta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betaHxy:x + y = F2R fxygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = F2R fxybeta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = IZR (mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = (IZR mx + IZR (my * beta ^ (fexp1 (mag y) - fexp1 (mag x)))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = IZR mx * bpow (fexp1 (mag x)) + IZR (my * beta ^ (fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = x + IZR (my * beta ^ (fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = x + IZR my * IZR (beta ^ (fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = x + IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) * bpow (fexp1 (mag x))now rewrite <- Fy.beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betax + y = x + IZR my * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betaHxy:x + y = F2R fxygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betaHxy:x + y = F2R fxyx + y <> 0 -> (cexp beta fexp2 (x + y) <= Fexp fxy)%Znow unfold cexp, fxy; simpl. Qed.beta:radixfexp1, fexp2:Z -> Zx, y:ROxy:(fexp1 (mag x) <= fexp1 (mag y))%ZHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZNzx:x <> 0Nzy:y <> 0mx:=Ztrunc (scaled_mantissa beta fexp1 x):Zmy:=Ztrunc (scaled_mantissa beta fexp1 y):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))fxy:={| Fnum := mx + my * beta ^ (fexp1 (mag y) - fexp1 (mag x)); Fexp := fexp1 (mag x) |}:float betaHxy:x + y = F2R fxy(cexp beta fexp2 (x + y) <= Fexp fxy)%Zbeta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp2 (mag (x + y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp2 (mag (x + y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x + y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHle:(fexp1 (mag x) <= fexp1 (mag y))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHgt:(fexp1 (mag y) < fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)now apply (round_round_plus_aux0_aux_aux fexp1).beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHle:(fexp1 (mag x) <= fexp1 (mag y))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHgt:(fexp1 (mag y) < fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)now apply (round_round_plus_aux0_aux_aux fexp1); [omega| | | |]. Qed. (* fexp1 (mag x) - 1 <= mag y : * addition is exact in the largest precision (fexp2). *)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (y + x)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (y + x)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHgt:(fexp1 (mag y) < fexp1 (mag x))%Zgeneric_format beta fexp2 (y + x)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> y <= x -> (fexp1 (mag x) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> y <= x -> (fexp1 (mag x) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag xgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Znow apply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Znow apply Hexp3; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Znow apply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:(mag y <= mag x)%ZHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x + 1 - 1) + 1 <= mag x)%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:(mag y <= mag x)%ZHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x) + 1 <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag y))%Znow apply Hexp3; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x + 1 - 1) + 1 <= mag y)%Zomega. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHyx:y <= xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x) + 1 <= mag y)%Zbeta:radixforall k : Z, (0 < k)%Z -> forall (fexp : Z -> Z) (x y : R), 0 < x -> 0 < y -> (mag y <= fexp (mag x) - k)%Z -> mag (x + y) = mag x -> generic_format beta fexp x -> 0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radixforall k : Z, (0 < k)%Z -> forall (fexp : Z -> Z) (x y : R), 0 < x -> 0 < y -> (mag y <= fexp (mag x) - k)%Z -> mag (x + y) = mag x -> generic_format beta fexp x -> 0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radix(2 <= beta)%Zbeta:radixHbeta:(2 <= beta)%Zforall k : Z, (0 < k)%Z -> forall (fexp : Z -> Z) (x y : R), 0 < x -> 0 < y -> (mag y <= fexp (mag x) - k)%Z -> mag (x + y) = mag x -> generic_format beta fexp x -> 0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radix(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixbeta_val:Zbeta_prop:(2 <=? beta_val)%Z = true(2 <= {| radix_val := beta_val; radix_prop := beta_prop |})%Zbeta:radixHbeta:(2 <= beta)%Zforall k : Z, (0 < k)%Z -> forall (fexp : Z -> Z) (x y : R), 0 < x -> 0 < y -> (mag y <= fexp (mag x) - k)%Z -> mag (x + y) = mag x -> generic_format beta fexp x -> 0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xFx:generic_format beta fexp x0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xgeneric_format beta fexp x -> 0 < x + y - round beta fexp Zfloor (x + y) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xx = IZR (Ztrunc (x * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) -> 0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag (x + y))))) * bpow (fexp (mag (x + y))) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xx = IZR (Ztrunc (x * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) -> 0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):Zx = IZR mx * bpow (fexp (mag x)) -> 0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))(IZR mx * bpow (fexp (mag x)) + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))now bpow_simplify.beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))IZR mx * bpow (fexp (mag x)) * bpow (- fexp (mag x)) + y * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))0 < x + y - IZR (Zfloor ((x + y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))0 < x + y - IZR (Zfloor (IZR mx + y * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))0 < y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))0 < x + y - IZR (Zfloor (IZR mx + y * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))0 < y * bpow (- fexp (mag x))now apply Rmult_lt_compat_l; [|apply bpow_gt_0].beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))y * 0 < y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))0 < x + y - IZR (Zfloor (IZR mx + y * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))y * bpow (- fexp (mag x)) < / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR (Zfloor (IZR mx + y * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))y * bpow (- fexp (mag x)) < / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))y * bpow (- fexp (mag x)) < bpow (mag y) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (mag y) * bpow (- fexp (mag x)) <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))y * bpow (- fexp (mag x)) < bpow (mag y) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))y < bpow (mag y)apply bpow_mag_gt.beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))Rabs y < bpow (mag y)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (mag y) * bpow (- fexp (mag x)) <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (mag y) * bpow (- fexp (mag x)) <= bpow (fexp (mag x) - k) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (fexp (mag x) - k) * bpow (- fexp (mag x)) <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (mag y) * bpow (- fexp (mag x)) <= bpow (fexp (mag x) - k) * bpow (- fexp (mag x))now apply bpow_le.beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (mag y) <= bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (fexp (mag x) - k) * bpow (- fexp (mag x)) <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))bpow (- k) <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow k <= / IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%ZHk:(0 < 0)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - 0)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow 0 <= / IZR (beta ^ 0)beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.pos p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.pos p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow (Z.pos p) <= / IZR (beta ^ Z.pos p)beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.neg p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.neg p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow (Z.neg p) <= / IZR (beta ^ Z.neg p)omega.beta:radixHbeta:(2 <= beta)%ZHk:(0 < 0)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - 0)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow 0 <= / IZR (beta ^ 0)beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.pos p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.pos p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow (Z.pos p) <= / IZR (beta ^ Z.pos p)now apply Rle_refl.beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.pos p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.pos p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ IZR (Pos.iter (Z.mul beta) 1%Z p) <= / IZR (Pos.iter (Z.mul beta) 1%Z p)beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.neg p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.neg p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))/ bpow (Z.neg p) <= / IZR (beta ^ Z.neg p)beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.neg p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.neg p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))(0 < 0)%Zapply Zlt_neg_0.beta:radixHbeta:(2 <= beta)%Zp:positiveHk:(0 < Z.neg p)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - Z.neg p)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))(Z.neg p < 0)%Zbeta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR (Zfloor (IZR mx + y * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR mx * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx <= IZR mx + y * bpow (- fexp (mag x)) < IZR (mx + 1)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR mx * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR mx * bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)x + y - IZR mx * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < x + y - IZR mx * bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 * bpow (- fexp (mag x)) < (x + y - IZR mx * bpow (fexp (mag x))) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < (x + y) * bpow (- fexp (mag x)) - IZR mx * bpow (fexp (mag x)) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < (x + y) * bpow (- fexp (mag x)) - IZR mxnow apply Rmult_lt_0_compat; [|apply bpow_gt_0].beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 < y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)x + y - IZR mx * bpow (fexp (mag x)) < bpow (fexp (mag x) - k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)(x + y - IZR mx * bpow (fexp (mag x))) * bpow (- fexp (mag x)) < bpow (fexp (mag x) - k) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)(x + y) * bpow (- fexp (mag x)) - IZR mx * bpow (fexp (mag x)) * bpow (- fexp (mag x)) < bpow (fexp (mag x) - k) * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)(x + y) * bpow (- fexp (mag x)) - IZR mx < bpow (- k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y * bpow (- fexp (mag x)) < bpow (- k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)/ IZR (beta ^ k) <= bpow (- k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)/ IZR (beta ^ k) <= / bpow know rewrite IZR_Zpower; [right|omega].beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)bpow k <= IZR (beta ^ k)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx <= IZR mx + y * bpow (- fexp (mag x)) < IZR (mx + 1)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx <= IZR mx + y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx + y * bpow (- fexp (mag x)) < IZR (mx + 1)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx <= IZR mx + y * bpow (- fexp (mag x))now apply Rlt_le.beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)0 <= y * bpow (- fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)IZR mx + y * bpow (- fexp (mag x)) < IZR (mx + 1)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y * bpow (- fexp (mag x)) < 1beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y * bpow (- fexp (mag x)) * bpow (fexp (mag x)) < 1 * bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y * bpow (- fexp (mag x)) * bpow (fexp (mag x)) < bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y < bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y < bpow (mag y)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)bpow (mag y) < bpow (fexp (mag x))beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)y < bpow (mag y)apply bpow_mag_gt.beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)Rabs y < bpow (mag y)apply bpow_lt; omega. Qed. (* mag y <= fexp1 (mag x) - 2 : round_round_lt_mid applies. *)beta:radixHbeta:(2 <= beta)%Zk:ZHk:(0 < k)%Zfexp:Z -> Zx, y:Rdefinitions.RPx:0 < xPy:0 < yHln:(mag y <= fexp (mag x) - k)%ZHlxy:mag (x + y) = mag xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))R:(x + y) * bpow (- fexp (mag x)) = IZR mx + y * bpow (- fexp (mag x))LB:0 < y * bpow (- fexp (mag x))UB:y * bpow (- fexp (mag x)) < / IZR (beta ^ k)bpow (mag y) < bpow (fexp (mag x))beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 2)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 2)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(2 <= beta)%Zbeta:radixHbeta:(2 <= beta)%Zforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 2)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixbeta_val:Zbeta_prop:(2 <=? beta_val)%Z = true(2 <= {| radix_val := beta_val; radix_prop := beta_prop |})%Zbeta:radixHbeta:(2 <= beta)%Zforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 2)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zbpow (-2) <= / 2 * / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zbpow (-2) <= / 2 * / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zbpow (-2) <= / 4beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z/ bpow 2 <= / 4beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z4 <= bpow 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z(2 * 2 <= beta * (beta * 1))%Znow apply Zmult_le_compat; omega.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z(2 * 2 <= beta * beta)%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZValid_exp fexp1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZValid_exp fexp2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z0 < x + ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp1 (mag (x + y)) <= mag (x + y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zx + y < midp fexp1 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * ulp beta fexp2 (x + y)exact Vfexp1.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZValid_exp fexp1exact Vfexp2.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZValid_exp fexp2lra.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z0 < x + ynow rewrite Lxy.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp1 (mag (x + y)) <= mag (x + y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp1 (mag x) <= mag x)%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zx + y < midp fexp1 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zx + y < round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zx + y + - round beta fexp1 Zfloor (x + y) < round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y) + - round beta fexp1 Zfloor (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (fexp1 (mag x) - 2) <= round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y) + - round beta fexp1 Zfloor (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (fexp1 (mag x) - 2) <= / 2 * ulp beta fexp1 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (fexp1 (mag x) - 2) <= / 2 * bpow (cexp beta fexp1 (x + y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (fexp1 (mag x) - 2) <= / 2 * bpow (fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (fexp1 (mag x) - 2) * bpow (- fexp1 (mag x)) <= / 2 * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Zbpow (-2) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z/ 2 * / 2 <= / 2apply Rmult_le_compat_l; lra.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z/ 2 * / 2 <= / 2 * 1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * ulp beta fexp2 (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * bpow (cexp beta fexp2 (x + y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%Z(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z(x + y) * bpow (- fexp1 (mag x)) < (midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z(x + y) * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) < (midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y + - round beta fexp1 Zfloor (x + y) < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x)) + - round beta fexp1 Zfloor (x + y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y - round beta fexp1 Zfloor (x + y) < / 2 * ulp beta fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 2) <= / 2 * ulp beta fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 2) * bpow (- fexp1 (mag x)) <= (/ 2 * ulp beta fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 2) * bpow (- fexp1 (mag x)) <= (/ 2 * bpow (cexp beta fexp1 (x + y)) - / 2 * bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (-2) <= / 2 - / 2 * bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 2 * / 2 <= / 2 - / 2 * bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 2 * / 2 <= / 2 * (1 - bpow (fexp2 (mag x) - fexp1 (mag x)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 2 <= 1 - bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 2 - 1 <= - bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z- / 2 <= - bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp2 (mag x) - fexp1 (mag x)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp2 (mag x) - fexp1 (mag x)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp2 (mag x) - fexp1 (mag x)) <= bpow (-1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (-1) <= / 2apply bpow_le; omega.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp2 (mag x) - fexp1 (mag x)) <= bpow (-1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (-1) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ IZR (beta * 1) <= / 2apply IZR_le; omega. } Qed. (* round_round_plus_aux{0,1} together *)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 2)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow2:bpow (-2) <= / 2 * / 2P2:(0 < 2)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z2 <= IZR (beta * 1)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(mag y <= fexp1 (mag x) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)now apply round_round_plus_aux1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(mag y <= fexp1 (mag x) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x + y) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zgeneric_format beta fexp2 (x + y)reflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x + y) = round beta fexp1 (Znearest choice1) (x + y)now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%Zgeneric_format beta fexp2 (x + y)now apply (round_round_plus_aux0 fexp1). Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHly:(fexp1 (mag x) - 2 < mag y)%ZHf1:(fexp1 (mag x) - 1 <= mag y)%Zgeneric_format beta fexp2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) y) = round beta fexp1 (Znearest choice1) ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) y = round beta fexp1 (Znearest choice1) ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 yreflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) y = round beta fexp1 (Znearest choice1) ynow apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 yexact Fy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xreflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xnow apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xexact Fx.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)now apply round_round_plus_aux2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y + x)) = round beta fexp1 (Znearest choice1) (y + x)now apply round_round_plus_aux2. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp2 (mag (x - y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x - y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixforall (fexp1 fexp2 : Z -> Z) (x y : R), (fexp2 (mag (x - y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x - y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> Zx, y:R(fexp2 (mag (x - y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x - y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> Zx, y:R(fexp2 (mag (x + - y)) <= fexp1 (mag x))%Z -> (fexp2 (mag (x + - y)) <= fexp1 (mag y))%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + - y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + - y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + - y)) <= fexp1 (mag y))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + - y)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + - y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + - y)) <= fexp1 (mag (- y)))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + - y)now apply (round_round_plus_aux0_aux fexp1). Qed. (* fexp1 (mag x) - 1 <= mag y : * substraction is exact in the largest precision (fexp2). *)beta:radixfexp1, fexp2:Z -> Zx, y:RHlnx:(fexp2 (mag (x + - y)) <= fexp1 (mag x))%ZHlny:(fexp2 (mag (x + - y)) <= fexp1 (mag (- y)))%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)generic_format beta fexp2 (x + - y)beta:radixforall fexp1 fexp2 : Z -> Z, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (fexp1 (mag x) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (fexp1 (mag x) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHor:mag y = mag x \/ mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x - 1))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) - 1 <= mag x - 1)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zomega.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(mag (x - y) - 1 <= mag x)%Znow rewrite Lxmy; apply Hexp3.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x - 1 + 1) - 1 <= mag x)%Znow apply Z.le_trans with (mag y).beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x) - 1 <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag y))%Znow replace (_ + _)%Z with (mag x : Z); [|ring]. Qed. (* mag y <= fexp1 (mag x) - 2, * fexp1 (mag (x - y)) - 1 <= mag y : * substraction is exact in the largest precision (fexp2). *)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x - 1 + 1) - 1 <= mag y)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (fexp1 (mag (x - y)) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (fexp1 (mag (x - y)) - 1 <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp1 (mag (x - y)) <= fexp1 (mag x))%Zapply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp1 (mag (x - y)) <= fexp1 (mag x))%Znow apply Hexp3. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 2)%ZHln':(fexp1 (mag (x - y)) - 1 <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixforall fexp : Z -> Z, Valid_exp fexp -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp (mag x) - 1)%Z -> generic_format beta fexp x -> generic_format beta fexp y -> round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixforall fexp : Z -> Z, Valid_exp fexp -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp (mag x) - 1)%Z -> generic_format beta fexp x -> generic_format beta fexp y -> round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFx:generic_format beta fexp xFy:generic_format beta fexp yround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFx:generic_format beta fexp xFy:generic_format beta fexp yPx:0 < xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xgeneric_format beta fexp x -> round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xx = IZR (Ztrunc (x * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) -> round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):Zx = IZR mx * bpow (fexp (mag x)) -> round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%Zround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%Zround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xround beta fexp Zceil (x - y) = xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xRxy:round beta fexp Zceil (x - y) = xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xround beta fexp Zceil (x - y) = xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (Zceil ((x - y) * bpow (- fexp (mag (x - y))))) * bpow (fexp (mag (x - y))) = xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (Zceil ((x - y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) = xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (Zceil ((x - y) * bpow (- fexp (mag x)))) * bpow (fexp (mag x)) = IZR mx * bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (Zceil ((x - y) * bpow (- fexp (mag x)))) = IZR mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xZceil ((x - y) * bpow (- fexp (mag x))) = mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xZceil ((IZR mx * bpow (fexp (mag x)) - y) * bpow (- fexp (mag x))) = mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xZceil (IZR mx * bpow (fexp (mag x)) * bpow (- fexp (mag x)) - y * bpow (- fexp (mag x))) = mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xZceil (IZR mx - y * bpow (- fexp (mag x))) = mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (mx - 1) < IZR mx - y * bpow (- fexp (mag x)) <= IZR mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (mx - 1) < IZR mx - y * bpow (- fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR mx - y * bpow (- fexp (mag x)) <= IZR mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (mx - 1) < IZR mx - y * bpow (- fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR mx + IZR (- (1)) < IZR mx - y * bpow (- fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR (- (1)) < - (y * bpow (- fexp (mag x)))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy * bpow (- fexp (mag x)) < IPR 1beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy * bpow (- fexp (mag x)) * bpow (fexp (mag x)) < IPR 1 * bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy < bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy < bpow (mag y)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xbpow (mag y) <= bpow (fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy < bpow (mag y)apply bpow_mag_gt.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xRabs y < bpow (mag y)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xbpow (mag y) <= bpow (fexp (mag x))omega.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag x(mag y <= fexp (mag x))%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR mx - y * bpow (- fexp (mag x)) <= IZR mxbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xIZR mx - y * bpow (- fexp (mag x)) <= IZR mx + 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag x- (y * bpow (- fexp (mag x))) <= 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag x0 <= y * bpow (- fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xy * 0 <= y * bpow (- fexp (mag x))now apply bpow_ge_0.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag x0 <= bpow (- fexp (mag x))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xRxy:round beta fexp Zceil (x - y) = xround beta fexp Zceil (x - y) - (x - y) <= yapply Rle_refl.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:bpow (mag x - 1) < xLxy:mag (x - y) = mag xRxy:round beta fexp Zceil (x - y) = xy <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)x = bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)x = bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)bpow (mag x - 1) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHly:(mag y <= fexp (Build_mag_prop beta x ex Hex) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (Build_mag_prop beta x ex Hex))):ZFx:x = IZR mx * bpow (fexp (Build_mag_prop beta x ex Hex))Hfx:(fexp (Build_mag_prop beta x ex Hex) < Build_mag_prop beta x ex Hex)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (Build_mag_prop beta x ex Hex - 1)bpow (ex - 1) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHly:(mag y <= fexp (Build_mag_prop beta x ex Hex) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (Build_mag_prop beta x ex Hex))):ZFx:x = IZR mx * bpow (fexp (Build_mag_prop beta x ex Hex))Hfx:(fexp (Build_mag_prop beta x ex Hex) < Build_mag_prop beta x ex Hex)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (Build_mag_prop beta x ex Hex - 1)bpow (ex - 1) <= Rabs xnow apply Rgt_not_eq.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHly:(mag y <= fexp (Build_mag_prop beta x ex Hex) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (Build_mag_prop beta x ex Hex))):ZFx:x = IZR mx * bpow (fexp (Build_mag_prop beta x ex Hex))Hfx:(fexp (Build_mag_prop beta x ex Hex) < Build_mag_prop beta x ex Hex)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (Build_mag_prop beta x ex Hex - 1)x <> 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)round beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)mag (x - y) = (mag x - 1)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%Zround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)mag (x - y) = (mag x - 1)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)(mag (x - y) <= mag x - 1)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)(mag x - 1 <= mag (x - y))%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)(mag (x - y) <= mag x - 1)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)x - y <> 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Rabs (x - y) < bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)x - y <> 0now intro Hx'; rewrite Hx' in Hxy; apply (Rlt_irrefl y).beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)x <> yrewrite Rabs_right; lra.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Rabs (x - y) < bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)(mag x - 1 <= mag (x - y))%Zomega.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)(mag y <= mag x - 2)%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%Zround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zround beta fexp Zceil (x - y) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZRxy:round beta fexp Zceil (x - y) <= xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zround beta fexp Zceil (x - y) <= xbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zround beta fexp Zceil (x - y) <= bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZIZR (Zceil ((x - y) * bpow (- fexp (mag (x - y))))) * bpow (fexp (mag (x - y))) <= bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZIZR (Zceil ((x - y) * bpow (- fexp (mag x - 1)%Z))) * bpow (fexp (mag x - 1)%Z) <= bpow (mag x - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZIZR (Zceil ((x - y) * bpow (- fexp (mag x - 1)%Z))) * bpow (fexp (mag x - 1)%Z) * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - 1) * bpow (- fexp (mag x - 1)%Z)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZIZR (Zceil ((x - y) * bpow (- fexp (mag x - 1)%Z))) <= bpow (mag x - fexp (mag x - 1)%Z - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZIZR (Zceil ((x - y) * bpow (- fexp (mag x - 1)%Z))) <= IZR (beta ^ (mag x - fexp (mag x - 1)%Z - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z(Zceil ((x - y) * bpow (- fexp (mag x - 1))) <= beta ^ (mag x - fexp (mag x - 1) - 1))%Zbeta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z(x - y) * bpow (- fexp (mag x - 1)%Z) <= IZR (beta ^ (mag x - fexp (mag x - 1)%Z - 1))beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z(x - y) * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - fexp (mag x - 1)%Z - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z(bpow (mag x - 1) - y) * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - fexp (mag x - 1)%Z - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zbpow (mag x - 1) * bpow (- fexp (mag x - 1)%Z) - y * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - fexp (mag x - 1)%Z - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zbpow (mag x - fexp (mag x - 1)%Z - 1) - y * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - fexp (mag x - 1)%Z - 1)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zbpow (mag x - fexp (mag x - 1)%Z - 1) - y * bpow (- fexp (mag x - 1)%Z) <= bpow (mag x - fexp (mag x - 1)%Z - 1) + 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z- (y * bpow (- fexp (mag x - 1)%Z)) <= 0beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z0 <= y * bpow (- fexp (mag x - 1)%Z)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Zy * 0 <= y * bpow (- fexp (mag x - 1)%Z)now apply bpow_ge_0.beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%Z0 <= bpow (- fexp (mag x - 1)%Z)lra. Qed. (* mag y <= fexp1 (mag x) - 2 : * mag y <= fexp1 (mag (x - y)) - 2 : * round_round_gt_mid applies. *)beta:radixfexp:Z -> ZVfexp:Valid_exp fexpx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp (mag x) - 1)%ZFy:generic_format beta fexp yPx:0 < xmx:=Ztrunc (x * bpow (- fexp (mag x))):ZFx:x = IZR mx * bpow (fexp (mag x))Hfx:(fexp (mag x) < mag x)%ZHfy:(fexp (mag y) < mag y)%ZHx:x <= bpow (mag x - 1)Xpow:x = bpow (mag x - 1)Lxy:mag (x - y) = (mag x - 1)%ZHfx1:(fexp (mag x - 1) < mag x - 1)%ZRxy:round beta fexp Zceil (x - y) <= xround beta fexp Zceil (x - y) - (x - y) <= ybeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (mag y <= fexp1 (mag (x - y)) - 2)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (mag y <= fexp1 (mag (x - y)) - 2)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(2 <= beta)%Zbeta:radixHbeta:(2 <= beta)%Zforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (mag y <= fexp1 (mag (x - y)) - 2)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixbeta_val:Zbeta_prop:(2 <=? beta_val)%Z = true(2 <= {| radix_val := beta_val; radix_prop := beta_prop |})%Zbeta:radixHbeta:(2 <= beta)%Zforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 2)%Z -> (mag y <= fexp1 (mag (x - y)) - 2)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zbpow (-2) <= / 2 * / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zbpow (-2) <= / 2 * / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zbpow (-2) <= / 4beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z/ bpow 2 <= / 4beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z4 <= bpow 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z(2 * 2 <= beta * (beta * 1))%Znow apply Zmult_le_compat; omega.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z(2 * 2 <= beta * beta)%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2y < bpow (mag y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2y < bpow (mag y)apply bpow_mag_gt.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Rabs y < bpow (mag y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Valid_exp fexp1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Valid_exp fexp2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag (x - y)) <= mag (x - y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)midp' fexp1 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z -> midp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - yexact Vfexp1.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Valid_exp fexp1exact Vfexp2.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Valid_exp fexp2lra.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 < x - yapply Hexp4; omega.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag (x - y)) <= mag (x - y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag (x - y)) < mag (x - y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag x - 1) < mag x - 1)%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(mag x - 1 <= mag (x - y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag x - 1) < mag x - 1)%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp1 (mag y) < mag y)%Znow apply mag_minus_lb; [| |omega].beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(mag x - 1 <= mag (x - y))%Zbeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)midp' fexp1 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + (/ 2 * ulp beta fexp1 (x - y) - (x - y)) < x - y + (/ 2 * ulp beta fexp1 (x - y) - (x - y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - x + y < / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < bpow (fexp1 (mag (x - y)) - 2)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 2) <= / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < bpow (fexp1 (mag (x - y)) - 2)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)y < bpow (fexp1 (mag (x - y)) - 2)now apply bpow_le.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (mag y) <= bpow (fexp1 (mag (x - y)) - 2)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 2) <= / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 2) <= / 2 * bpow (cexp beta fexp1 (x - y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 2) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1 - 1) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) * bpow (- (1)) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (- (1)) * bpow (fexp1 (mag (x - y)) - 1) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 <= bpow (- (1))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (- (1)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= bpow (fexp1 (mag (x - y)))now apply bpow_ge_0.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 <= bpow (- (1))now apply bpow_ge_0.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (- (1)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)/ IZR (beta * 1) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)0 < 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)2 <= IZR betanow apply IZR_le.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)2 <= IZR betaapply bpow_le; omega.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z -> midp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zmidp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) + (/ 2 * ulp beta fexp1 (x - y) - (x - y) - / 2 * ulp beta fexp2 (x - y)) < x - y + (/ 2 * ulp beta fexp1 (x - y) - (x - y) - / 2 * ulp beta fexp2 (x - y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - x + y < / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - (x - y) < / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zy < / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (mag y) <= / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 2) <= / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1 - 1) <= / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) * bpow (- (1)) <= / 2 * ulp beta fexp1 (x - y) - / 2 * ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) * bpow (- (1)) <= / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 <= bpow (- (1))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y)apply bpow_ge_0.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 <= bpow (- (1))apply bpow_ge_0.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) <= / 2beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR (beta * 1) <= / 2now apply IZR_le.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 <= IZR betabeta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= bpow (cexp beta fexp1 (x - y)) - bpow (cexp beta fexp2 (x - y))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= bpow (fexp1 (mag (x - y))) - bpow (fexp2 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) + bpow (fexp2 (mag (x - y))) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) + bpow (fexp2 (mag (x - y))) <= 2 * bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 * bpow (fexp1 (mag (x - y)) - 1) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) + bpow (fexp2 (mag (x - y))) <= 2 * bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) + bpow (fexp2 (mag (x - y))) <= bpow (fexp1 (mag (x - y)) - 1) + bpow (fexp1 (mag (x - y)) - 1)now apply bpow_le.beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp2 (mag (x - y))) <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 * bpow (fexp1 (mag (x - y)) - 1) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 * (bpow (fexp1 (mag (x - y))) * bpow (- (1))) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y))) * (bpow (- (1)) * 2) <= bpow (fexp1 (mag (x - y)))beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y))) * (bpow (- (1)) * 2) <= bpow (fexp1 (mag (x - y))) * 1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) * 2 <= 1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR (beta * 1) * 2 <= 1beta:radixHbeta:(2 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR beta * 2 <= 1beta:radixHbeta:/ IZR beta <= / 2fexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR beta * 2 <= 1beta:radixHbeta:2 <= IZR betafexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 < 2beta:radixHbeta:/ IZR beta <= / 2fexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR beta * 2 <= 1beta:radixHbeta:2 <= IZR betafexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 < 2apply Rlt_0_2. Qed. (* round_round_minus_aux{0,1,2} together *)beta:radixHbeta:2 <= IZR betafexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow2:bpow (-2) <= / 2 * / 2Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z0 < 2beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) 0) = round beta fexp1 (Znearest choice1) 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xValid_rnd (Znearest choice2)reflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)now apply round_round_minus_aux2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(mag y <= fexp1 (mag (x - y)) - 2)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)reflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)now apply (round_round_minus_aux1 fexp1). }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 2)%ZHly':(fexp1 (mag (x - y)) - 2 < mag y)%ZHf1:(fexp1 (mag (x - y)) - 1 <= mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)reflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)now apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%ZValid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)now apply (round_round_minus_aux0 fexp1). Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 2 < mag y)%ZHf1:(fexp1 (mag x) - 1 <= mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- y)) = round beta fexp1 (Znearest choice1) (- y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) y) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) y = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 yreflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) y = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ynow apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 yexact Fy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xreflexivity.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xnow apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xexact Fx.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (y - x))) = round beta fexp1 (Znearest choice1) (- (y - x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= y- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (y - x)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (y - x)now apply round_round_minus_aux3.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (y - x)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (y - x)now apply round_round_minus_aux3. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (- x - y))) = round beta fexp1 (Znearest choice1) (- (- x - y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)now apply round_round_plus_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 (- y)Sx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yPx:0 <= - xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)now apply round_round_minus_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yPx:0 <= - xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0Py:0 <= - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)now apply round_round_minus_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)Sx:0 <= xSy:y < 0Py:0 <= - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)now apply round_round_plus_aux. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + - y)now apply round_round_plus. Qed. Section Double_round_plus_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)round_round_eq fexp1 fexp2 choice1 choice2 (x + - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (ex - prec' <= ey - prec)%Zomega. Qed.beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (ex - prec' <= ey - prec)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ynow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')now apply FLX_round_round_plus_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xnow apply generic_format_FLX. Qed.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ybeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ynow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')now apply FLX_round_round_plus_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xnow apply generic_format_FLX. Qed. End Double_round_plus_FLX. Section Double_round_plus_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex + 1 - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex + 1 - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(Z.max (ex - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ynow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')now apply FLT_round_round_plus_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xnow apply generic_format_FLT. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ynow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')now apply FLT_round_round_plus_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xnow apply generic_format_FLT. Qed. End Double_round_plus_FLT. Section Double_round_plus_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_plus_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex + 1 - prec <? emin then emin + prec - 1 else ex + 1 - prec) - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex - 1 - prec <? emin then emin + prec - 1 else ex - 1 - prec) + 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex + 1 - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex + 1 - prec <? emin then emin + prec - 1 else ex + 1 - prec) - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - 1 - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex - 1 - prec <? emin then emin + prec - 1 else ex - 1 - prec) + 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ynow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_plus_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xnow apply generic_format_FTZ. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ynow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_plus_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xnow apply generic_format_FTZ. Qed. End Double_round_plus_FTZ. Section Double_round_plus_radix_ge_3. Definition round_round_plus_radix_ge_3_hyp fexp1 fexp2 := (forall ex ey, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z) /\ (forall ex ey, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z). (* fexp1 (mag x) <= mag y : * addition is exact in the largest precision (fexp2). *)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec + 1 <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ybeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> (fexp1 (mag x) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> (fexp1 (mag x) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= ygeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag xgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Znow apply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Znow apply Hexp3; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHle:(mag y <= fexp1 (mag x))%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Zgeneric_format beta fexp2 (x + y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Znow apply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:(mag y <= mag x)%ZHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x + 1 - 1) + 1 <= mag x)%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:(mag y <= mag x)%ZHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x) + 1 <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%Z(fexp2 (mag (x + y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag y))%Znow apply Hexp3; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = mag x(fexp2 (mag x) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp2 (mag x + 1) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x + 1 - 1) + 1 <= mag y)%Zomega. Qed. (* mag y <= fexp1 (mag x) - 1 : round_round_lt_mid applies. *)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Hexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y <= xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xNny:0 <= yHgt:(fexp1 (mag x) < mag y)%ZLxy:mag (x + y) = (mag x + 1)%Z(fexp1 (mag x) + 1 <= mag y)%Zbeta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 1)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> (mag y <= fexp1 (mag x) - 1)%Z -> generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zbpow (-1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zbpow (-1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z/ IZR (beta * 1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z/ IZR beta <= / 3now apply IZR_le.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Z3 <= IZR betabeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZValid_exp fexp1beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZValid_exp fexp2beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z0 < x + ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp1 (mag (x + y)) <= mag (x + y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zx + y < midp fexp1 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * ulp beta fexp2 (x + y)exact Vfexp1.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZValid_exp fexp1exact Vfexp2.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZValid_exp fexp2lra.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z0 < x + ynow rewrite Lxy.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp1 (mag (x + y)) <= mag (x + y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp1 (mag x) <= mag x)%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zx + y < midp fexp1 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zx + y < round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zx + y + - round beta fexp1 Zfloor (x + y) < round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y) + - round beta fexp1 Zfloor (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (fexp1 (mag x) - 1) <= round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y) + - round beta fexp1 Zfloor (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * ulp beta fexp1 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * bpow (cexp beta fexp1 (x + y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * bpow (fexp1 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (fexp1 (mag x) - 1) * bpow (- fexp1 (mag x)) <= / 2 * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x))apply (Rle_trans _ _ _ Bpow3); lra.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Zbpow (- (1)) <= / 2beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * ulp beta fexp2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag (x + y)) <= fexp1 (mag (x + y)) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * bpow (cexp beta fexp2 (x + y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%Z(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> x + y < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y < midp fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y < round beta fexp1 Zfloor (x + y) + / 2 * ulp beta fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y - round beta fexp1 Zfloor (x + y) < / 2 * ulp beta fexp1 (x + y) - / 2 * bpow (fexp2 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx + y - round beta fexp1 Zfloor (x + y) < / 2 * (ulp beta fexp1 (x + y) - bpow (fexp2 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * (ulp beta fexp1 (x + y) - bpow (fexp2 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * (bpow (cexp beta fexp1 (x + y)) - bpow (fexp2 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) <= / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) * bpow (- fexp1 (mag x)) <= / 2 * (bpow (fexp1 (mag x)) - bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) * bpow (- fexp1 (mag x)) <= / 2 * ((bpow (fexp1 (mag x)) - bpow (fexp2 (mag x))) * bpow (- fexp1 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp1 (mag x) - 1) * bpow (- fexp1 (mag x)) <= / 2 * (bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) - bpow (fexp2 (mag x)) * bpow (- fexp1 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (- (1)) <= / 2 * (1 - bpow (fexp2 (mag x) - fexp1 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 3 <= / 2 * (1 - bpow (fexp2 (mag x) - fexp1 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z/ 2 * (2 / 3) <= / 2 * (1 - bpow (fexp2 (mag x) - fexp1 (mag x)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z2 / 3 <= 1 - bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z2 / 3 - 1 <= - bpow (fexp2 (mag x) - fexp1 (mag x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z- / 3 <= - bpow (fexp2 (mag x) - fexp1 (mag x))now apply Rle_trans with (bpow (- 1)); [apply bpow_le; omega|]. Qed. (* round_round_plus_radix_ge_3_aux{0,1} together *)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPx:0 < xPy:0 < yHly:(mag y <= fexp1 (mag x) - 1)%ZFx:generic_format beta fexp1 xLxy:mag (x + y) = mag xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZBpow3:bpow (-1) <= / 3P1:(0 < 1)%ZHf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zbpow (fexp2 (mag x) - fexp1 (mag x)) <= / 3beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(mag y <= fexp1 (mag x) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)now apply round_round_plus_radix_ge_3_aux1.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(mag y <= fexp1 (mag x) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x + y) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zgeneric_format beta fexp2 (x + y)reflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x + y) = round beta fexp1 (Znearest choice1) (x + y)now apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%Zgeneric_format beta fexp2 (x + y)now apply (round_round_plus_radix_ge_3_aux0 fexp1). Qed.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHly:(fexp1 (mag x) - 1 < mag y)%ZHf1:(fexp1 (mag x) <= mag y)%Zgeneric_format beta fexp2 (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) y) = round beta fexp1 (Znearest choice1) ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) y = round beta fexp1 (Znearest choice1) ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 yreflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) y = round beta fexp1 (Znearest choice1) ynow apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 yexact Fy.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xreflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xnow apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xexact Fx.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)now apply round_round_plus_radix_ge_3_aux2.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y + x)) = round beta fexp1 (Znearest choice1) (y + x)now apply round_round_plus_radix_ge_3_aux2. Qed. (* fexp1 (mag x) <= mag y : * substraction is exact in the largest precision (fexp2). *)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixforall fexp1 fexp2 : Z -> Z, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (fexp1 (mag x) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (fexp1 (mag x) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHor:mag y = mag x \/ mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeq:mag y = mag x(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) - 1 <= mag x)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x - 1))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) - 1 <= mag x - 1)%Znow apply mag_minus.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHlt:(mag x - 2 < mag y)%ZHeqm1:mag y = (mag x - 1)%Z(mag (x - y) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zomega.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(mag (x - y) - 1 <= mag x)%Znow rewrite Lxmy; apply Hexp3.beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = mag x(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x - 1 + 1) <= mag x)%Znow apply Z.le_trans with (mag y).beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x) <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp2 (mag x - 1) <= fexp1 (mag y))%Znow replace (_ + _)%Z with (mag x : Z); [|ring]. Qed. (* mag y <= fexp1 (mag x) - 1, * fexp1 (mag (x - y)) <= mag y : * substraction is exact in the largest precision (fexp2). *)beta:radixfexp1, fexp2:Z -> ZHexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(fexp1 (mag x) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHge:(mag y <= mag x - 2)%ZLxmy:mag (x - y) = (mag x - 1)%Z(fexp1 (mag x - 1 + 1) <= mag y)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag (x - y)) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag (x - y)) <= mag y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ygeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp1 (mag (x - y)) <= fexp1 (mag x))%Zapply Hexp4; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp1 (mag (x - y)) <= fexp1 (mag x))%Znow apply Hexp3. Qed. (* mag y <= fexp1 (mag x) - 1 : * mag y <= fexp1 (mag (x - y)) - 1 : * round_round_gt_mid applies. *)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp1:forall ex ey : Z, (fexp1 (ex + 1) <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp2:forall ex ey : Z, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp3:forall ex ey : Z, (fexp1 ex <= ey)%Z -> (fexp2 ex <= fexp1 ey)%ZHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHyx:y < xHln:(mag y <= fexp1 (mag x) - 1)%ZHln':(fexp1 (mag (x - y)) <= mag y)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xLyx:(mag y <= mag x)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z(fexp2 (mag (x - y)) <= fexp1 (mag y))%Zbeta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 1)%Z -> (mag y <= fexp1 (mag (x - y)) - 1)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y < x -> (mag y <= fexp1 (mag x) - 1)%Z -> (mag y <= fexp1 (mag (x - y)) - 1)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%Zround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zbpow (-1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Zbpow (-1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z/ IZR (beta * 1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z/ IZR beta <= / 3now apply IZR_le.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%Z3 <= IZR betabeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3y < bpow (mag y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3y < bpow (mag y)apply bpow_mag_gt.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Rabs y < bpow (mag y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Valid_exp fexp1beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Valid_exp fexp2beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)0 < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag (x - y)) <= mag (x - y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)midp' fexp1 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z -> midp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - yexact Vfexp1.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Valid_exp fexp1exact Vfexp2.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Valid_exp fexp2lra.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)0 < x - yapply Hexp4; omega.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag (x - y)) <= mag (x - y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag (x - y)) < mag (x - y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag x - 1) < mag x - 1)%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(mag x - 1 <= mag (x - y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag x - 1) < mag x - 1)%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp1 (mag y) < mag y)%Znow apply mag_minus_lb; [| |omega].beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(mag x - 1 <= mag (x - y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)midp' fexp1 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + (/ 2 * ulp beta fexp1 (x - y) - (x - y)) < x - y + (/ 2 * ulp beta fexp1 (x - y) - (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - x + y < / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)round beta fexp1 Zceil (x - y) - (x - y) < bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)y < bpow (fexp1 (mag (x - y)) - 1)now apply bpow_le.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (mag y) <= bpow (fexp1 (mag (x - y)) - 1)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= / 2 * ulp beta fexp1 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= / 2 * bpow (cexp beta fexp1 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y)) - 1) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (fexp1 (mag (x - y))) * bpow (- (1)) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (- (1)) * bpow (fexp1 (mag (x - y))) <= / 2 * bpow (fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)bpow (- (1)) <= / 2beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)/ IZR (beta * 1) <= / 2now apply IZR_le; omega.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)2 <= IZR betabeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z -> midp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zmidp' fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) < x - ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - / 2 * ulp beta fexp1 (x - y) + / 2 * ulp beta fexp2 (x - y) + (/ 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y)) - (x - y)) < x - y + (/ 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y)) - (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - x + y < / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 Zceil (x - y) - (x - y) < / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zy < / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (mag y) <= / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= / 2 * (ulp beta fexp1 (x - y) - ulp beta fexp2 (x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= / 2 * (bpow (cexp beta fexp1 (x - y)) - bpow (cexp beta fexp2 (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) <= / 2 * (bpow (fexp1 (mag (x - y))) - bpow (fexp2 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) * bpow (- fexp1 (mag (x - y))) <= / 2 * (bpow (fexp1 (mag (x - y))) - bpow (fexp2 (mag (x - y)))) * bpow (- fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) * bpow (- fexp1 (mag (x - y))) <= / 2 * ((bpow (fexp1 (mag (x - y))) - bpow (fexp2 (mag (x - y)))) * bpow (- fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp1 (mag (x - y)) - 1) * bpow (- fexp1 (mag (x - y))) <= / 2 * (bpow (fexp1 (mag (x - y))) * bpow (- fexp1 (mag (x - y))) - bpow (fexp2 (mag (x - y))) * bpow (- fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) <= / 2 * (1 - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ 3 <= / 2 * (1 - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (- (1)) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR (beta * 1) <= / 3now apply IZR_le.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z3 <= IZR betabeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ 3 <= / 2 * (1 - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ 2 * (2 / 3) <= / 2 * (1 - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 / 3 <= 1 - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z2 / 3 - 1 <= - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z- / 3 <= - bpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))) <= bpow (-1)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (-1) <= / 3apply bpow_le; omega.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (fexp2 (mag (x - y)) - fexp1 (mag (x - y))) <= bpow (-1)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Zbpow (-1) <= / 3beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z/ IZR (beta * 1) <= / 3now apply IZR_le. Qed. (* round_round_minus_aux{0,1,2} together *)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RPy:0 < yHxy:y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%ZFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x))%ZHfx:(fexp1 (mag x) < mag x)%ZBpow3:bpow (-1) <= / 3Ly:y < bpow (mag y)Hf2':(fexp2 (mag (x - y)) <= fexp1 (mag (x - y)) - 1)%Z3 <= IZR betabeta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 < y -> y <= x -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) 0) = round beta fexp1 (Znearest choice1) 0beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xValid_rnd (Znearest choice2)reflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xround beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0now apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y = xValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)now apply round_round_minus_radix_ge_3_aux2.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(mag y <= fexp1 (mag (x - y)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zgeneric_format beta fexp2 (x - y)reflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)now apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%Zgeneric_format beta fexp2 (x - y)now apply (round_round_minus_radix_ge_3_aux1 fexp1). }beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(mag y <= fexp1 (mag x) - 1)%ZHly':(fexp1 (mag (x - y)) - 1 < mag y)%ZHf1:(fexp1 (mag (x - y)) <= mag y)%Zgeneric_format beta fexp2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zgeneric_format beta fexp2 (x - y)reflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zround beta fexp1 (Znearest choice1) (x - y) = round beta fexp1 (Znearest choice1) (x - y)now apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%ZValid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%Zgeneric_format beta fexp2 (x - y)now apply (round_round_minus_radix_ge_3_aux0 fexp1). Qed.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RPy:0 < yHyx:y <= xFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:0 < xHy:y <> xHyx':y < xHly:(fexp1 (mag x) - 1 < mag y)%ZHf1:(fexp1 (mag x) <= mag y)%Zgeneric_format beta fexp2 (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, 0 <= x -> 0 <= y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- y)) = round beta fexp1 (Znearest choice1) (- y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) y) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) y = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 yreflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) y = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) ynow apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0Valid_rnd (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z)))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp2 ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0y <> 0 -> (fexp2 (mag y) <= fexp1 (mag y))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 yexact Fy.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0generic_format beta fexp1 ybeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xreflexivity.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0round beta fexp1 (Znearest choice1) x = round beta fexp1 (Znearest choice1) xnow apply valid_rnd_N.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp2 xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp4:forall ex ey : Z, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Zx, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0x <> 0 -> (fexp2 (mag x) <= fexp1 (mag x))%Zbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xexact Fx.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Zy:y = 0generic_format beta fexp1 xbeta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x < yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (y - x))) = round beta fexp1 (Znearest choice1) (- (y - x))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= y- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (y - x)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (y - x)now apply round_round_minus_radix_ge_3_aux3.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:x <= yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (y - x)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (y - x)now apply round_round_minus_radix_ge_3_aux3. Qed.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RNnx:0 <= xNny:0 <= yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNzx:x <> 0Nzy:y <> 0Px:0 < xPy:0 < yH:y <= xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - y)) = round beta fexp1 (Znearest choice1) (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (- x - y))) = round beta fexp1 (Znearest choice1) (- (- x - y))beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 ySx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)now apply round_round_plus_radix_ge_3_aux.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 (- y)Sx:x < 0Sy:y < 0Px:0 <= - xPy:0 <= - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yPx:0 <= - xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)now apply round_round_minus_radix_ge_3_aux.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 ySx:x < 0Sy:0 <= yPx:0 <= - xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (y - - x)) = round beta fexp1 (Znearest choice1) (y - - x)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:y < 0Py:0 <= - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)now apply round_round_minus_radix_ge_3_aux.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)Sx:0 <= xSy:y < 0Py:0 <= - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x - - y)) = round beta fexp1 (Znearest choice1) (x - - y)now apply round_round_plus_radix_ge_3_aux. Qed.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 ySx:0 <= xSy:0 <= yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x + y)) = round beta fexp1 (Znearest choice1) (x + y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radix(3 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_plus_radix_ge_3_hyp fexp1 fexp2 -> forall x y : R, generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x - y)beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x + - y)now apply round_round_plus_radix_ge_3. Qed. Section Double_round_plus_radix_ge_3_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixHbeta:(3 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_plus_radix_ge_3_hyp fexp1 fexp2x, y:RFx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)round_round_eq fexp1 fexp2 choice1 choice2 (x + - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (ex - prec' <= ey - prec)%Zomega. Qed.beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (ex - prec' <= ey - prec)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec y(3 <= beta)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) yexact Hbeta.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec y(3 <= beta)%Znow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')now apply FLX_round_round_plus_radix_ge_3_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xnow apply generic_format_FLX. Qed.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ybeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec <= prec')%Z -> forall x y : R, FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x - y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec y(3 <= beta)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) yexact Hbeta.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec y(3 <= beta)%Znow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')now apply FLX_round_round_plus_radix_ge_3_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec yround_round_plus_radix_ge_3_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xnow apply generic_format_FLX. Qed. End Double_round_plus_radix_ge_3_FLX. Section Double_round_plus_radix_ge_3_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec <= prec')%Zx, y:RFx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin)%Z -> (2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin)%Z -> (2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex + 1 - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex + 1 - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex + 1 - prec >= emin)%Z /\ Z.max (ex + 1 - prec) emin = (ex + 1 - prec)%Z \/ (ex + 1 - prec < emin)%Z /\ Z.max (ex + 1 - prec) emin = emin -> (Z.max (ex + 1 - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 - prec >= emin)%Z /\ Z.max (ex - 1 - prec) emin = (ex - 1 - prec)%Z \/ (ex - 1 - prec < emin)%Z /\ Z.max (ex - 1 - prec) emin = emin -> (Z.max (ex - 1 - prec) emin + 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (ex - 1 <= ey)%Z -> (Z.max (ex - prec') emin' <= Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec y(3 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) yexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec y(3 <= beta)%Znow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')now apply FLT_round_round_plus_radix_ge_3_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xnow apply generic_format_FLT. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' <= emin)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec y(3 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) yexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec y(3 <= beta)%Znow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')now apply FLT_round_round_plus_radix_ge_3_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_plus_radix_ge_3_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xnow apply generic_format_FLT. Qed. End Double_round_plus_radix_ge_3_FLT. Section Double_round_plus_radix_ge_3_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' <= emin)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> round_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zround_round_plus_radix_ge_3_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex + 1 - prec <? emin then emin + prec - 1 else ex + 1 - prec) <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - 1 - prec <? emin then emin + prec - 1 else ex - 1 - prec) + 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex + 1 - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex + 1 - prec <? emin then emin + prec - 1 else ex + 1 - prec) <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - 1 - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - 1 - prec <? emin then emin + prec - 1 else ex - 1 - prec) + 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ey - prec) emin); omega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - 1 <= ey)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x + y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec y(3 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) yexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec y(3 <= beta)%Znow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_plus_radix_ge_3_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xnow apply generic_format_FTZ. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(3 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec y(3 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) yexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec y(3 <= beta)%Znow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_plus_radix_ge_3_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_plus_radix_ge_3_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xnow apply generic_format_FTZ. Qed. End Double_round_plus_radix_ge_3_FTZ. End Double_round_plus_radix_ge_3. End Double_round_plus.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(3 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin' + prec' <= emin + 1)%ZHprec:(2 * prec <= prec')%Zx, y:RFx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ybeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> (Rabs (x - midp fexp1 x) <= / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> (Rabs (x - midp fexp1 x) <= / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Z(Rabs (x - midp fexp1 x) <= / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Z(Rabs (x - (round beta fexp1 Zfloor x + / 2 * ulp beta fexp1 x)) <= / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:R(Rabs (x - (rd + / 2 * ulp beta fexp1 x)) <= / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:R(Rabs (x - (rd + / 2 * u1)) <= / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:R(Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) x) -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xFx:generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xFx:generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xnow apply (generic_inclusion_mag beta fexp1); [omega|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xFx:generic_format beta fexp1 xgeneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)x < midp fexp1 x - / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)x < round beta fexp1 Zfloor x + / 2 * ulp beta fexp1 x - / 2 * ulp beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)x < rd + / 2 * u1 - / 2 * u2now rewrite <- Rmult_minus_distr_l.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:x - rd < / 2 * (u1 - u2)x - rd < / 2 * u1 - / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:x - rd <= / 2 * (u1 + u2)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdH1:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdround beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdrd + u1 - x < / 2 * (u1 - u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdH1:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xunfold midp'; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:/ 2 * (u1 + u2) < x - rdH1:round beta fexp1 Zceil x - x < / 2 * (ulp beta fexp1 x - ulp beta fexp2 x)midp' fexp1 x + / 2 * ulp beta fexp2 x < xapply Cmid, Rabs_le; split; lra. } Qed. Section Double_round_sqrt. Definition round_round_sqrt_hyp fexp1 fexp2 := (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex))%Z) /\ (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex - 1))%Z) /\ (forall ex, (fexp1 (2 * ex) < 2 * ex)%Z -> (fexp2 ex + ex <= 2 * fexp1 ex - 2)%Z).beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2f1:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCmid:Rabs (x - (rd + / 2 * u1)) <= / 2 * u2 -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = rd + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZH:/ 2 * (u1 - u2) <= x - rdH0:x - rd <= / 2 * (u1 + u2)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixforall x : R, 0 < x -> mag x = (2 * mag (sqrt x) - 1)%Z \/ mag x = (2 * mag (sqrt x))%Zbeta:radixforall x : R, 0 < x -> mag x = (2 * mag (sqrt x) - 1)%Z \/ mag x = (2 * mag (sqrt x))%Zbeta:radixx:RPx:0 < xmag x = (2 * mag (sqrt x) - 1)%Z \/ mag x = (2 * mag (sqrt x))%Zbeta:radixx:RPx:0 < xmag x = (2 * Z.div2 (mag x + 1) - 1)%Z \/ mag x = (2 * Z.div2 (mag x + 1))%Zdestruct Z.odd ; intros ; omega. Qed.beta:radixx:RPx:0 < x(mag x + 1)%Z = (2 * Z.div2 (mag x + 1) + (if Z.odd (mag x + 1) then 1 else 0))%Z -> mag x = (2 * Z.div2 (mag x + 1) - 1)%Z \/ mag x = (2 * Z.div2 (mag x + 1))%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_sqrt_hyp fexp1 fexp2 -> forall x : R, 0 < x -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z -> generic_format beta fexp1 x -> / 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_sqrt_hyp fexp1 fexp2 -> forall x : R, 0 < x -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z -> generic_format beta fexp1 x -> / 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 x/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 x(2 <= beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Z/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 x(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xbeta_val:Zbeta_prop:(2 <=? beta_val)%Z = trueHf2:(fexp2 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} (sqrt x)) <= fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} (sqrt x)) - 1)%ZFx:generic_format {| radix_val := beta_val; radix_prop := beta_prop |} fexp1 x(2 <= {| radix_val := beta_val; radix_prop := beta_prop |})%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Z/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):R/ 2 * bpow (cexp beta fexp2 (sqrt x)) < Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Fa:generic_format beta fexp1 aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 (round beta fexp1 Zfloor (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_rnd Zfloorexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_exp fexp1now apply valid_rnd_DN.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_rnd Zfloorbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Fa:generic_format beta fexp1 aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 x -> generic_format beta fexp1 a -> Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))x = IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) -> a = IZR (Ztrunc (a * bpow (- fexp1 (mag a)))) * bpow (fexp1 (mag a)) -> Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zx = IZR mx * bpow (fexp1 (mag x)) -> a = IZR (Ztrunc (a * bpow (- fexp1 (mag a)))) * bpow (fexp1 (mag a)) -> Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):Zx = IZR mx * bpow (fexp1 (mag x)) -> a = IZR ma * bpow (fexp1 (mag a)) -> Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))round beta fexp1 Zfloor 0 <= abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_rnd Zfloorbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= sqrt xexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_exp fexp1now apply valid_rnd_DN.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_rnd Zfloorapply sqrt_pos.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= a0 < / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Falseapply Rmult_lt_0_compat; [lra|apply bpow_gt_0].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= a0 < / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u10 < / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Falseapply Rmult_lt_0_compat; [lra|apply bpow_gt_0].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u10 < / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < / 2 * (u1 - u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2/ 2 * 0 < / 2 * (u1 - u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < u1 - u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2u2 < u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2bpow (fexp2 (mag (sqrt x))) < bpow (fexp1 (mag (sqrt x)))omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2(fexp2 (mag (sqrt x)) < fexp1 (mag (sqrt x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < b0 < b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Falsenow unfold b'; rewrite Rmult_plus_distr_l; apply Rplus_lt_0_compat.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < b0 < b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x <= a + b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x <= a + b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x - / 2 * u1 - a <= / 2 * u2now apply Rabs_le_inv.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x - (a + / 2 * u1) <= / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'a + b <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'a + b <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- / 2 * u2 <= - a - / 2 * u1 + sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- / 2 * u2 <= sqrt x - (a + / 2 * u1)now apply Rabs_le_inv in H; destruct H.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- (/ 2 * u2) <= sqrt x - (a + / 2 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (mag x) < mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (mag x) < mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Za * a + u1 * a - u2 * a + b * b <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Za * a + u1 * a - u2 * a + b * b <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(a + b) * (a + b) <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(a + b) * (a + b) <= sqrt x * sqrt xnow apply Rmult_le_compat.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZH':0 <= a + b(a + b) * (a + b) <= sqrt x * sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= a * a + u1 * a + u2 * a + b' * b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= a * a + u1 * a + u2 * a + b' * b'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= (a + b') * (a + b')beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xsqrt x * sqrt x <= (a + b') * (a + b')now apply Rmult_le_compat.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xH':0 <= sqrt xsqrt x * sqrt x <= (a + b') * (a + b')beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 < 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= xnow rewrite Ropp_0; do 3 rewrite Rplus_0_l.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 + 0 + - 0 + b * b <= x -> b * b <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx * bpow (fexp1 (mag x)) <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) <= 0 * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR (Zfloor (x * bpow (- fexp1 (mag x)))) <= 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Zfloor (x * bpow (- fexp1 (mag x))) = 0%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 <= x * bpow (- fexp1 (mag x)) < IZR (0 + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x * bpow (- fexp1 (mag x)) < 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) < 1 * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (2 * fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag (sqrt x)) + fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0sqrt x * sqrt x < bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0sqrt x < bpow (fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b' < bpow (fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0/ 2 * (u1 + u2) < u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0/ 2 * (u1 + u2) < / 2 * (u1 + u1)unfold u2, u1, ulp, cexp; apply bpow_lt; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0u2 < u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < amag a = mag (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < amag a = mag (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aValid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < a0 < round beta fexp1 Zfloor (sqrt x)exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aValid_exp fexp1now fold a.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < a0 < round beta fexp1 Zfloor (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 < - (u2 * a) + b * bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 < - (u2 * a) + b * bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * a < b ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * a < u2 ^ 2 * (/ 2) ^ 2 - 2 * u2 * (/ 2) ^ 2 * u1 + (/ 2) ^ 2 * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)(2 * u2 * a + u2 * u1) / 2 < (2 * u2 ^ 2 + 2 * u1 ^ 2) / 8beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < (2 * u2 ^ 2 + 2 * u1 ^ 2) / 8beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < u2 * bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < u2 * bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * u1 < bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < bpow (mag a)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < a + bpow (fexp1 (mag a))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + bpow (fexp1 (mag a)) <= bpow (mag a)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < a + bpow (fexp1 (mag a))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ 2 * bpow (fexp1 (mag a)) < bpow (fexp1 (mag a))apply Rmult_lt_compat_r; [apply bpow_gt_0|lra].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ 2 * bpow (fexp1 (mag a)) < 1 * bpow (fexp1 (mag a))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + bpow (fexp1 (mag a)) <= bpow (mag a)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + bpow (fexp1 (mag a)) <= a + ulp beta fexp1 abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + ulp beta fexp1 a <= bpow (mag a)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + ulp beta fexp1 a <= bpow (mag a)apply Rabs_lt_inv, bpow_mag_gt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a < bpow (mag a)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-2) * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-2) * u1 ^ 2 <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-2) * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-2) * (u1 * u1)now apply Hexp.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 2)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-2) * u1 ^ 2 <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= bpow (-2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-2) <= / 4beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u1 ^ 2 <= u2 ^ 2 + u1 ^ 2apply bpow_ge_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= bpow (-2)apply pow2_ge_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-2) <= / 4beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ IZR (beta * beta) <= / 4change 4%Z with (2 * 2)%Z; apply IZR_le, Zmult_le_compat; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)4 <= IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u1 ^ 2 <= u2 ^ 2 + u1 ^ 2apply pow2_ge_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u2 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (a + u1) * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (a + bpow (fexp1 (mag (sqrt x)))) * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (IZR ma + 1) * bpow (fexp1 (mag (sqrt x))) * (IZR ma * bpow (fexp1 (mag (sqrt x))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= IZR ma * ((IZR ma + 1) * bpow (fexp1 (mag (sqrt x)))) * bpow (fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= IZR ma * (IZR ma + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx * bpow (-2 * fexp1 (mag (sqrt x))) <= IZR ma * (IZR ma + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x))) <= IZR ma * (IZR ma + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * IZR (beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) <= IZR ma * (IZR ma + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR (mx * beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) <= IZR (ma * (ma + 1))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR (mx * beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) < IZR (Z.succ (ma * (ma + 1)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * IZR (beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) < IZR ma * (IZR ma + 1) + 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x))) < IZR ma * (IZR ma + 1) + 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x)) < (IZR ma * (IZR ma + 1) + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * bpow (fexp1 (mag (sqrt x)) + fexp1 (mag (sqrt x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * (bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < a * a + u1 * a + u1 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba * a + u1 * a + u2 * a + b' * b' < a * a + u1 * a + u1 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * a + b' * b' < u1 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * a + u2 * / 2 * u1 < u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < bpow (mag (sqrt x)) * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < bpow (mag (sqrt x)) * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + / 2 * u1 < bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + u1 <= bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + bpow (cexp beta fexp1 (sqrt x)) <= bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + bpow (cexp beta fexp1 a) <= bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + ulp beta fexp1 a <= bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b0 < abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bgeneric_format beta fexp1 abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba < bpow (mag (sqrt x))exact Pa.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b0 < anow apply round_DN_pt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bgeneric_format beta fexp1 abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba < bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bsqrt x < bpow (mag (sqrt x))now apply round_DN_pt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bsqrt x < bpow (mag (sqrt x))apply bpow_mag_gt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bRabs (sqrt x) < bpow (mag (sqrt x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= / 2 * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ 2 * u1 ^ 2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= / 2 * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-2) * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-2) * u1 ^ 2 <= / 2 * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-2) * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-2) * (u1 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * bpow (fexp2 (mag (sqrt x))) <= bpow (-2) * (bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x) + fexp2 (mag (sqrt x))) <= bpow (2 * fexp1 (mag (sqrt x)) - 2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(mag (sqrt x) + fexp2 (mag (sqrt x)) <= 2 * fexp1 (mag (sqrt x)) - 2)%Znow apply Hexp.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 2)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-2) * u1 ^ 2 <= / 2 * u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-2) <= / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ IZR (beta * (beta * 1)) <= / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ IZR (beta * beta) <= / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b2 <= IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(2 <= beta * beta)%Zapply Zmult_le_compat; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(1 * 2 <= beta * beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ 2 * u1 ^ 2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 ^ 2 < u1 ^ 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * u2 < u1 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bH':0 <= u2u2 * u2 < u1 * u1unfold u1, u2, ulp, cexp; apply bpow_lt; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bH':0 <= u2u2 < u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * a - u2 * a + b * bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a - u2 * a + b * b <= a * a + u1 * abeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * a - u2 * a + b * bbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a + 0 < a * a + u1 * a - u2 * a + b * bnow apply Rplus_lt_compat_l.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a + 0 < a * a + u1 * a + (- (u2 * a) + b * b)now apply Rle_trans with x. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHbeta:(2 <= beta)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a - u2 * a + b * b <= a * a + u1 * abeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_sqrt_hyp fexp1 fexp2 -> forall x : R, generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_sqrt_hyp fexp1 fexp2 -> forall x : R, generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)now rewrite round_0; [|apply valid_rnd_N].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) 0) = round beta fexp1 (Znearest choice1) 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(mag x <= mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1x <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x * sqrt x <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x * sqrt x <= sqrt x * 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 10 <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= 1apply sqrt_pos.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 10 <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= 1now apply sqrt_le_1_alt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= sqrt 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(2 * fexp1 1 <= fexp1 (2 * 1 - 1))%Z -> (fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(2 * fexp1 1 <= fexp1 1)%Z -> (fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHexp10:(2 * fexp1 1 <= fexp1 1)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z(1 <= mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zbpow (1 - 1) <= Rabs (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zbpow 0 <= Rabs (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= Rabs (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zsqrt 1 <= sqrt xnow apply Rlt_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (mag x) < mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (mag x) < mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 2)%Z -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z0 < sqrt xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp1 (mag (sqrt x)) <= mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZRabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x) -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp1exact Vfexp2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp2now apply sqrt_lt_R0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z0 < sqrt xomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zomega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp1 (mag (sqrt x)) <= mag (sqrt x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZRabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x) -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)apply (round_round_sqrt_aux fexp1 fexp2 Vfexp1 Vfexp2 Hexp x Px Hf2 Fx). Qed. Section Double_round_sqrt_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHmid:Rabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x)/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)unfold round_round_sqrt_hyp; split; [|split]; intro ex; omega. Qed.beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 2 <= prec')%Z -> forall x : R, FLX_format beta prec x -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (2 * prec + 2 <= prec')%Z -> forall x : R, FLX_format beta prec x -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xround_round_sqrt_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xgeneric_format beta (FLX_exp prec) xnow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec')now apply FLX_round_round_sqrt_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xround_round_sqrt_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX. Qed. End Double_round_sqrt_FLX. Section Double_round_sqrt_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLX_format beta prec xgeneric_format beta (FLX_exp prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin <= 0)%Z -> (emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%Z -> (2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin <= 0)%Z -> (emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%Z -> (2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * ex - 1 - prec >= emin)%Z /\ Z.max (2 * ex - 1 - prec) emin = (2 * ex - 1 - prec)%Z \/ (2 * ex - 1 - prec < emin)%Z /\ Z.max (2 * ex - 1 - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 2)%Zomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin <= 0)%Z -> (emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%Z -> (2 * prec + 2 <= prec')%Z -> forall x : R, FLT_format beta emin prec x -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (emin <= 0)%Z -> (emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%Z -> (2 * prec + 2 <= prec')%Z -> forall x : R, FLT_format beta emin prec x -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_sqrt_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xgeneric_format beta (FLT_exp emin prec) xnow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin' prec')now apply FLT_round_round_sqrt_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_sqrt_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT. Qed. End Double_round_sqrt_FLT. Section Double_round_sqrt_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 2)%Z \/ (2 * emin' <= emin - 4 * prec - 2)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FLT_format beta emin prec xgeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 2 <= prec')%Z -> round_round_sqrt_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zround_round_sqrt_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - 1 - prec <? emin then emin + prec - 1 else 2 * ex - 1 - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (2 * ex - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec))%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (2 * ex - 1 - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - 1 - prec <? emin then emin + prec - 1 else 2 * ex - 1 - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:Z((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(emin <= 2 * ex - prec)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zdestruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ex - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(emin <= 2 * ex - prec)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 2)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%ZFalseomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zex:ZH:(emin + prec - 1 < 2 * ex)%ZH':(2 * ex - prec < emin)%ZFalsebeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 2 <= prec')%Z -> forall x : R, FTZ_format beta emin prec x -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 2 <= prec')%Z -> forall x : R, FTZ_format beta emin prec x -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_sqrt_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xgeneric_format beta (FTZ_exp emin prec) xnow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_sqrt_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_sqrt_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ. Qed. End Double_round_sqrt_FTZ. Section Double_round_sqrt_radix_ge_4. Definition round_round_sqrt_radix_ge_4_hyp fexp1 fexp2 := (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex))%Z) /\ (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex - 1))%Z) /\ (forall ex, (fexp1 (2 * ex) < 2 * ex)%Z -> (fexp2 ex + ex <= 2 * fexp1 ex - 1)%Z).beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 2 <= prec')%Zx:RFx:FTZ_format beta emin prec xgeneric_format beta (FTZ_exp emin prec) xbeta:radix(4 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_sqrt_radix_ge_4_hyp fexp1 fexp2 -> forall x : R, 0 < x -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z -> generic_format beta fexp1 x -> / 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radix(4 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> round_round_sqrt_radix_ge_4_hyp fexp1 fexp2 -> forall x : R, 0 < x -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z -> generic_format beta fexp1 x -> / 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 x/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):R/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):R/ 2 * bpow (cexp beta fexp2 (sqrt x)) < Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x))))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Fa:generic_format beta fexp1 aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 (round beta fexp1 Zfloor (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_exp fexp1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_rnd Zfloorexact Vfexp1.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_exp fexp1now apply valid_rnd_DN.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Valid_rnd Zfloorbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZFx:generic_format beta fexp1 xa:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))Fa:generic_format beta fexp1 aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))generic_format beta fexp1 x -> generic_format beta fexp1 a -> Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))x = IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) -> a = IZR (Ztrunc (a * bpow (- fexp1 (mag a)))) * bpow (fexp1 (mag a)) -> Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zx = IZR mx * bpow (fexp1 (mag x)) -> a = IZR (Ztrunc (a * bpow (- fexp1 (mag a)))) * bpow (fexp1 (mag a)) -> Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):Zx = IZR mx * bpow (fexp1 (mag x)) -> a = IZR ma * bpow (fexp1 (mag a)) -> Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))round beta fexp1 Zfloor 0 <= abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_exp fexp1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_rnd Zfloorbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= sqrt xexact Vfexp1.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_exp fexp1now apply valid_rnd_DN.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Valid_rnd Zfloorapply sqrt_pos.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))0 <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= a0 < / 2 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Falseapply Rmult_lt_0_compat; [lra|apply bpow_gt_0].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= a0 < / 2 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u10 < / 2 * u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Falseapply Rmult_lt_0_compat; [lra|apply bpow_gt_0].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u10 < / 2 * u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < / 2 * (u1 - u2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2/ 2 * 0 < / 2 * (u1 - u2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u20 < u1 - u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2u2 < u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2bpow (fexp2 (mag (sqrt x))) < bpow (fexp1 (mag (sqrt x)))omega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2(fexp2 (mag (sqrt x)) < fexp1 (mag (sqrt x)))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < b0 < b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Falsenow unfold b'; rewrite Rmult_plus_distr_l; apply Rplus_lt_0_compat.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < b0 < b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x <= a + b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x <= a + b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x - / 2 * u1 - a <= / 2 * u2now apply Rabs_le_inv.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'sqrt x - (a + / 2 * u1) <= / 2 * u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'a + b <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'a + b <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- / 2 * u2 <= - a - / 2 * u1 + sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- / 2 * u2 <= sqrt x - (a + / 2 * u1)now apply Rabs_le_inv in H; destruct H.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'- (/ 2 * u2) <= sqrt x - (a + / 2 * u1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Za * a + u1 * a - u2 * a + b * b <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Za * a + u1 * a - u2 * a + b * b <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(a + b) * (a + b) <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(a + b) * (a + b) <= sqrt x * sqrt xnow apply Rmult_le_compat.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZH':0 <= a + b(a + b) * (a + b) <= sqrt x * sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= a * a + u1 * a + u2 * a + b' * b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= a * a + u1 * a + u2 * a + b' * b'beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xx <= (a + b') * (a + b')beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xsqrt x * sqrt x <= (a + b') * (a + b')now apply Rmult_le_compat.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xH':0 <= sqrt xsqrt x * sqrt x <= (a + b') * (a + b')beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 < 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b * b <= xnow rewrite Ropp_0; do 3 rewrite Rplus_0_l.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 + 0 + - 0 + b * b <= x -> b * b <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx * bpow (fexp1 (mag x)) <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) <= 0 * bpow (- fexp1 (mag x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR mx <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0IZR (Zfloor (x * bpow (- fexp1 (mag x)))) <= 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0Zfloor (x * bpow (- fexp1 (mag x))) = 0%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 00 <= x * bpow (- fexp1 (mag x)) < IZR (0 + 1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x * bpow (- fexp1 (mag x)) < 1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x * bpow (- fexp1 (mag x)) * bpow (fexp1 (mag x)) < 1 * bpow (fexp1 (mag x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (2 * fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag (sqrt x)) + fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0x < bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0sqrt x * sqrt x < bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0sqrt x < bpow (fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0b' < bpow (fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0/ 2 * (u1 + u2) < u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0/ 2 * (u1 + u2) < / 2 * (u1 + u1)unfold u2, u1, ulp, cexp; apply bpow_lt; omega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Za:a = 0u2 < u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < amag a = mag (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < amag a = mag (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aValid_exp fexp1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < a0 < round beta fexp1 Zfloor (sqrt x)exact Vfexp1.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aValid_exp fexp1now fold a.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < a0 < round beta fexp1 Zfloor (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Falsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 < - (u2 * a) + b * bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 < - (u2 * a) + b * bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * a < b ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * a < u2 ^ 2 * (/ 2) ^ 2 - 2 * u2 * (/ 2) ^ 2 * u1 + (/ 2) ^ 2 * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)(2 * u2 * a + u2 * u1) / 2 < (2 * u2 ^ 2 + 2 * u1 ^ 2) / 8beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < (2 * u2 ^ 2 + 2 * u1 ^ 2) / 8beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < u2 * bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * (a + / 2 * u1) < u2 * bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * u1 < bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < bpow (mag a)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < a + ulp beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + ulp beta fexp1 a <= bpow (mag a)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + / 2 * bpow (fexp1 (mag a)) < a + ulp beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ 2 * bpow (fexp1 (mag a)) < ulp beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ 2 * bpow (fexp1 (mag a)) < 1 * ulp beta fexp1 aapply Rmult_lt_compat_r; [apply bpow_gt_0|lra].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ 2 * bpow (fexp1 (mag a)) < 1 * bpow (cexp beta fexp1 a)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a + ulp beta fexp1 a <= bpow (mag a)apply Rabs_lt_inv, bpow_mag_gt.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)a < bpow (mag a)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-1) * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-1) * u1 ^ 2 <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-1) * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u2 * bpow (mag (sqrt x)) <= bpow (-1) * (u1 * u1)now apply Hexp.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-1) * u1 ^ 2 <= / 4 * (u2 ^ 2 + u1 ^ 2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= bpow (-1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-1) <= / 4beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u1 ^ 2 <= u2 ^ 2 + u1 ^ 2apply bpow_ge_0.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= bpow (-1)apply pow2_ge_0.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)bpow (-1) <= / 4beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)/ IZR beta <= / 4now apply IZR_le.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)4 <= IZR betabeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)u1 ^ 2 <= u2 ^ 2 + u1 ^ 2apply pow2_ge_0.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)0 <= u2 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= a * a + u1 * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (a + u1) * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (a + bpow (fexp1 (mag (sqrt x)))) * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= (IZR ma + 1) * bpow (fexp1 (mag (sqrt x))) * (IZR ma * bpow (fexp1 (mag (sqrt x))))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= IZR ma * ((IZR ma + 1) * bpow (fexp1 (mag (sqrt x)))) * bpow (fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx <= IZR ma * (IZR ma + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx * bpow (-2 * fexp1 (mag (sqrt x))) <= IZR ma * (IZR ma + 1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x))) <= IZR ma * (IZR ma + 1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * IZR (beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) <= IZR ma * (IZR ma + 1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR (mx * beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) <= IZR (ma * (ma + 1))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR (mx * beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) < IZR (Z.succ (ma * (ma + 1)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * IZR (beta ^ (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x)))) < IZR ma * (IZR ma + 1) + 1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x) - 2 * fexp1 (mag (sqrt x))) < IZR ma * (IZR ma + 1) + 1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bIZR mx * bpow (fexp1 (mag x)) < (IZR ma * (IZR ma + 1) + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * bpow (2 * fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * bpow (fexp1 (mag (sqrt x)) + fexp1 (mag (sqrt x)))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < (IZR ma * (IZR ma + 1) + 1) * (bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x))))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bx < a * a + u1 * a + u1 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba * a + u1 * a + u2 * a + b' * b' < a * a + u1 * a + u1 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * a + b' * b' < u1 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * a + u2 * / 2 * u1 < u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < bpow (mag (sqrt x)) * u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(a + / 2 * u1) * u2 < bpow (mag (sqrt x)) * u2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + / 2 * u1 < bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + u1 <= bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + bpow (cexp beta fexp1 (sqrt x)) <= bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + bpow (cexp beta fexp1 a) <= bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba + ulp beta fexp1 a <= bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b0 < abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bgeneric_format beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba < bpow (mag (sqrt x))exact Pa.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b0 < anow apply round_DN_pt.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bgeneric_format beta fexp1 abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba < bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bsqrt x < bpow (mag (sqrt x))now apply round_DN_pt.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * ba <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bsqrt x < bpow (mag (sqrt x))apply bpow_mag_gt.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bRabs (sqrt x) < bpow (mag (sqrt x))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= / 2 * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ 2 * u1 ^ 2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= / 2 * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-1) * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-1) * u1 ^ 2 <= / 2 * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-1) * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * u2 <= bpow (-1) * (u1 * u1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x)) * bpow (fexp2 (mag (sqrt x))) <= bpow (-1) * (bpow (fexp1 (mag (sqrt x))) * bpow (fexp1 (mag (sqrt x))))beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (mag (sqrt x) + fexp2 (mag (sqrt x))) <= bpow (2 * fexp1 (mag (sqrt x)) - 1)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(mag (sqrt x) + fexp2 (mag (sqrt x)) <= 2 * fexp1 (mag (sqrt x)) - 1)%Znow apply Hexp.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-1) * u1 ^ 2 <= / 2 * u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bbpow (-1) <= / 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ IZR (beta * 1) <= / 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ IZR beta <= / 2apply IZR_le; omega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b2 <= IZR betabeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * b/ 2 * u1 ^ 2 <= u2 * / 2 * u1 - b' ^ 2 + u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 ^ 2 < u1 ^ 2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bu2 * u2 < u1 * u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bH':0 <= u2u2 * u2 < u1 * u1unfold u1, u2, ulp, cexp; apply bpow_lt; omega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag (sqrt x)))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bH':0 <= u2u2 < u1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aFalsebeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * a - u2 * a + b * bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a - u2 * a + b * b <= a * a + u1 * abeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a < a * a + u1 * a - u2 * a + b * bbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a + 0 < a * a + u1 * a - u2 * a + b * bnow apply Rplus_lt_compat_l.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a + 0 < a * a + u1 * a + (- (u2 * a) + b * b)now apply Rle_trans with x. Qed.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2Hexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RPx:0 < xHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Za:=round beta fexp1 Zfloor (sqrt x):Ru1:=bpow (fexp1 (mag (sqrt x))):Ru2:=bpow (fexp2 (mag (sqrt x))):Rb:=/ 2 * (u1 - u2):Rb':=/ 2 * (u1 + u2):RH:Rabs (sqrt x - (round beta fexp1 Zfloor (sqrt x) + / 2 * bpow (cexp beta fexp1 (sqrt x)))) <= / 2 * bpow (cexp beta fexp2 (sqrt x))mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zma:=Ztrunc (a * bpow (- fexp1 (mag a))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fa:a = IZR ma * bpow (fexp1 (mag a))Nna:0 <= aPhu1:0 < / 2 * u1Phu2:0 < / 2 * u2Pb:0 < bPb':0 < b'Hr:sqrt x <= a + b'Hl:a + b <= sqrt xHf1:(2 * fexp1 (mag (sqrt x)) <= fexp1 (mag x))%ZHlx:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%ZHsl:a * a + u1 * a - u2 * a + b * b <= xHsr:x <= a * a + u1 * a + u2 * a + b' * b'Nza:a <> 0Pa:0 < aHla:mag a = mag (sqrt x)Hl':0 < - (u2 * a) + b * bHr':x <= a * a + u1 * aa * a + u1 * a - u2 * a + b * b <= a * a + u1 * abeta:radix(4 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_sqrt_radix_ge_4_hyp fexp1 fexp2 -> forall x : R, generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radix(4 <= beta)%Z -> forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, round_round_sqrt_radix_ge_4_hyp fexp1 fexp2 -> forall x : R, generic_format beta fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0sqrt x = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0sqrt x = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Zx:x = 0sqrt x = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0sqrt x = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Zx:x = 0sqrt x = 0exact sqrt_0.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Zx:x = 0sqrt 0 = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0sqrt x = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0match Rcase_abs x with | left _ => 0 | right a => Rsqrt {| nonneg := x; cond_nonneg := Rge_le x 0 a |} end = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0r:x < 00 = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0r:x >= 0Rsqrt {| nonneg := x; cond_nonneg := Rge_le x 0 r |} = 0reflexivity.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0r:x < 00 = 0casetype False; lra.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Nzx:x <> 0r:x >= 0Rsqrt {| nonneg := x; cond_nonneg := Rge_le x 0 r |} = 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) 0) = round beta fexp1 (Znearest choice1) 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0round beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0Valid_rnd (Znearest choice2)reflexivity.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0round beta fexp1 (Znearest choice1) 0 = round beta fexp1 (Znearest choice1) 0now apply valid_rnd_N.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xNpx:x <= 0Hs:sqrt x = 0Valid_rnd (Znearest choice2)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1(mag x <= mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1x <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x * sqrt x <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x * sqrt x <= sqrt x * 1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 10 <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= 1apply sqrt_pos.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 10 <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= 1now apply sqrt_le_1_alt.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:x <= 1sqrt x <= sqrt 1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(2 * fexp1 1 <= fexp1 (2 * 1 - 1))%Z -> (fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < x(2 * fexp1 1 <= fexp1 1)%Z -> (fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHexp10:(2 * fexp1 1 <= fexp1 1)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z(1 <= mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zbpow (1 - 1) <= Rabs (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zbpow 0 <= Rabs (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= Rabs (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Zsqrt 1 <= sqrt xnow apply Rlt_le.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHx:1 < xHf0:(fexp1 1 < 1)%Z1 <= xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x) - 1)%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Znow apply mag_generic_gt; [|apply Rgt_not_eq|].beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHlx:mag x = (2 * mag (sqrt x))%Z(fexp1 (mag x) < mag x)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zomega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZH:(fexp1 (2 * mag (sqrt x)) < 2 * mag (sqrt x))%Z(fexp2 (mag (sqrt x)) + mag (sqrt x) <= 2 * fexp1 (mag (sqrt x)) - 1)%Z -> (fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (sqrt x)) = round beta fexp1 (Znearest choice1) (sqrt x)beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp1beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp2beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z0 < sqrt xbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp1 (mag (sqrt x)) <= mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZRabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x) -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)exact Vfexp1.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp1exact Vfexp2.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZValid_exp fexp2now apply sqrt_lt_R0.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z0 < sqrt xomega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Zomega.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z(fexp1 (mag (sqrt x)) <= mag (sqrt x))%Zbeta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZRabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x) -> round_round_eq fexp1 fexp2 choice1 choice2 (sqrt x)apply (round_round_sqrt_radix_ge_4_aux Hbeta fexp1 fexp2 Vfexp1 Vfexp2 Hexp x Px Hf2 Fx). Qed. Section Double_round_sqrt_radix_ge_4_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixHbeta:(4 <= beta)%Zfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_sqrt_radix_ge_4_hyp fexp1 fexp2x:RFx:generic_format beta fexp1 xPx:0 < xHfx:(fexp1 (mag x) < mag x)%ZHfsx:(fexp1 (mag (sqrt x)) < mag (sqrt x))%ZHf2:(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%ZHmid:Rabs (sqrt x - midp fexp1 (sqrt x)) <= / 2 * ulp beta fexp2 (sqrt x)/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x))beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)unfold round_round_sqrt_radix_ge_4_hyp; split; [|split]; intro ex; omega. Qed.beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x : R, FLX_format beta prec x -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * prec + 1 <= prec')%Z -> forall x : R, FLX_format beta prec x -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (sqrt x)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec x(4 <= beta)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xround_round_sqrt_radix_ge_4_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xgeneric_format beta (FLX_exp prec) xexact Hbeta.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec x(4 <= beta)%Znow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xValid_exp (FLX_exp prec')now apply FLX_round_round_sqrt_radix_ge_4_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xround_round_sqrt_radix_ge_4_hyp (FLX_exp prec) (FLX_exp prec')now apply generic_format_FLX. Qed. End Double_round_sqrt_radix_ge_4_FLX. Section Double_round_sqrt_radix_ge_4_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLX_format beta prec xgeneric_format beta (FLX_exp prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin <= 0)%Z -> (emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin <= 0)%Z -> (emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * ex - 1 - prec >= emin)%Z /\ Z.max (2 * ex - 1 - prec) emin = (2 * ex - 1 - prec)%Z \/ (2 * ex - 1 - prec < emin)%Z /\ Z.max (2 * ex - 1 - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (2 * Z.max (ex - prec) emin <= Z.max (2 * ex - 1 - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 1)%Zomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (2 * ex - prec >= emin)%Z /\ Z.max (2 * ex - prec) emin = (2 * ex - prec)%Z \/ (2 * ex - prec < emin)%Z /\ Z.max (2 * ex - prec) emin = emin -> (Z.max (2 * ex - prec) emin < 2 * ex)%Z -> (Z.max (ex - prec') emin' + ex <= 2 * Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin <= 0)%Z -> (emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%Z -> (2 * prec + 1 <= prec')%Z -> forall x : R, FLT_format beta emin prec x -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (emin <= 0)%Z -> (emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%Z -> (2 * prec + 1 <= prec')%Z -> forall x : R, FLT_format beta emin prec x -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec x(4 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_sqrt_radix_ge_4_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xgeneric_format beta (FLT_exp emin prec) xexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec x(4 <= beta)%Znow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xValid_exp (FLT_exp emin' prec')now apply FLT_round_round_sqrt_radix_ge_4_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xround_round_sqrt_radix_ge_4_hyp (FLT_exp emin prec) (FLT_exp emin' prec')now apply generic_format_FLT. Qed. End Double_round_sqrt_radix_ge_4_FLT. Section Double_round_sqrt_radix_ge_4_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(emin <= 0)%ZHeminprec:(emin' <= emin - prec - 1)%Z \/ (2 * emin' <= emin - 4 * prec)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FLT_format beta emin prec xgeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 1 <= prec')%Z -> round_round_sqrt_radix_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zround_round_sqrt_radix_ge_4_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - 1 - prec <? emin then emin + prec - 1 else 2 * ex - 1 - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (2 * ex - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec))%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (2 * ex - 1 - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z(2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) <= (if 2 * ex - 1 - prec <? emin then emin + prec - 1 else 2 * ex - 1 - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:Z((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z -> ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(emin <= 2 * ex - prec)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zdestruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ex - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(emin <= 2 * ex - prec)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') + ex <= 2 * (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:((if 2 * ex - prec <? emin then emin + prec - 1 else 2 * ex - prec) < 2 * ex)%ZH':(2 * ex - prec < emin)%ZFalseomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':(0 < prec')%ZHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zex:ZH:(emin + prec - 1 < 2 * ex)%ZH':(2 * ex - prec < emin)%ZFalsebeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x : R, FTZ_format beta emin prec x -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(4 <= beta)%Z -> forall choice1 choice2 : Z -> bool, (2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x : R, FTZ_format beta emin prec x -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec x(4 <= beta)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_sqrt_radix_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xgeneric_format beta (FTZ_exp emin prec) xexact Hbeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec x(4 <= beta)%Znow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xValid_exp (FTZ_exp emin' prec')now apply FTZ_round_round_sqrt_radix_ge_4_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xround_round_sqrt_radix_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')now apply generic_format_FTZ. Qed. End Double_round_sqrt_radix_ge_4_FTZ. End Double_round_sqrt_radix_ge_4. End Double_round_sqrt. Section Double_round_div.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hbeta:(4 <= beta)%Zchoice1, choice2:Z -> boolHemin:(2 * (emin' + prec') <= emin + prec <= 1)%ZHprec:(2 * prec + 1 <= prec')%Zx:RFx:FTZ_format beta emin prec xgeneric_format beta (FTZ_exp emin prec) xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> forall x : R, 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> forall x : R, 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zx = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zx = midp fexp1 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zx = round beta fexp1 Zfloor x + / 2 * ulp beta fexp1 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Rx = rd + / 2 * ulp beta fexp1 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:Rx = rd + / 2 * u -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RH:- rd + x = - rd + (rd + / 2 * u)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:R- rd + x = / 2 * u -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:Rx - rd = / 2 * u -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * uround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * uround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * u(2 <= beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * uHbeta:(2 <= beta)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * u(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln, beta_val:Zbeta_prop:(2 <=? beta_val)%Z = trueEbeta:{| radix_val := beta_val; radix_prop := beta_prop |} = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) <= fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) - 1)%ZHf1:(fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) <= Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x)%Zrd:=round {| radix_val := beta_val; radix_prop := beta_prop |} fexp1 Zfloor x:Ru:=ulp {| radix_val := beta_val; radix_prop := beta_prop |} fexp1 x:RXmid:x - rd = / 2 * u(2 <= {| radix_val := beta_val; radix_prop := beta_prop |})%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x - rd = / 2 * uHbeta:(2 <= beta)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zgeneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betageneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaF2R f = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaHf:F2R f = xgeneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaF2R f = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x))) * bpow (cexp beta fexp2 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float beta(IZR (Zfloor (scaled_mantissa beta fexp2 rd)) + IZR (n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)))) * bpow (cexp beta fexp2 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR (n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x))) * bpow (cexp beta fexp2 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * IZR (beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x))) * bpow (cexp beta fexp2 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * bpow (fexp1 (mag x) - 1 - fexp2 (mag x)) * bpow (cexp beta fexp2 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * bpow (fexp1 (mag x) - 1) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * (bpow (fexp1 (mag x)) * bpow (- (1))) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * (bpow (-1) * bpow (fexp1 (mag x))) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * bpow (-1) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * / IZR (beta * 1) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * / IZR beta * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * / IZR (2 * n) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * / (2 * IZR n) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + IZR n * (/ 2 * / IZR n) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + / 2 * (IZR n * / IZR n) * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + / 2 * bpow (fexp1 (mag x)) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) + / 2 * bpow (cexp beta fexp1 x) = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * ulp beta fexp2 x + / 2 * ulp beta fexp1 x = xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * ulp beta fexp2 x + / 2 * u = rd + / 2 * ubeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * ulp beta fexp2 x = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaZrd:rd = 0IZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0IZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaZrd:rd = 0IZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaZrd:rd = 0IZR (Zfloor (scaled_mantissa beta fexp2 0)) * bpow (cexp beta fexp2 x) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaZrd:rd = 0IZR (Zfloor 0) * bpow (cexp beta fexp2 x) = 0now rewrite Rmult_0_l.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaZrd:rd = 00 * bpow (cexp beta fexp2 x) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0IZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 00 <= rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 00 <= rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Valid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0generic_format beta fexp1 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 00 <= xexact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Valid_exp fexp1apply generic_format_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0generic_format beta fexp1 0now apply Rlt_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 00 <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdmag rd = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdmag rd = mag xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rd(mag rd <= mag x)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rd(mag x <= mag rd)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rd(mag rd <= mag x)%Znow apply round_DN_pt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdrd <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rd(mag x <= mag rd)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdValid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdValid_rnd Zfloorbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdround beta fexp1 Zfloor x <> 0exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdValid_exp fexp1now apply valid_rnd_DN.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdValid_rnd Zfloorexact Nzrd.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdround beta fexp1 Zfloor x <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (scaled_mantissa beta fexp2 rd)) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (rd * bpow (- cexp beta fexp2 rd))) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (round beta fexp1 Zfloor x * bpow (- cexp beta fexp2 rd))) * bpow (cexp beta fexp2 x) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) * bpow (- fexp2 (mag rd)))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x) - fexp2 (mag rd)))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x) - fexp2 (mag x)))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (IZR (Zfloor (x * bpow (- fexp1 (mag x)))) * IZR (beta ^ (fexp1 (mag x) - fexp2 (mag x))))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) <= IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) < IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x)) + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x)))) * IZR (beta ^ (fexp1 (mag x) - fexp2 (mag x))) * bpow (fexp2 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x) - fexp2 (mag x)) * bpow (fexp2 (mag x)) = rdnow unfold rd.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = rdbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) <= IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) < IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x)) + 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) < IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x)) + 1)simpl; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaNzrd:rd <> 0Nnrd:0 <= rdPrd:0 < rdLrd:mag rd = mag xIZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) < IZR (Zfloor (x * bpow (- fexp1 (mag x))) * beta ^ (fexp1 (mag x) - fexp2 (mag x))) + 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaHf:F2R f = xgeneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaHf:F2R f = xx <> 0 -> (cexp beta fexp2 x <= Fexp f)%Zapply Z.le_refl. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> booln:ZEbeta:beta = (2 * n)%Zx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%ZHf1:(fexp1 (mag x) <= mag x)%Zrd:=round beta fexp1 Zfloor x:Ru:=ulp beta fexp1 x:RXmid:x = rd + / 2 * uHbeta:(2 <= beta)%Zf:={| Fnum := Zfloor (scaled_mantissa beta fexp2 rd) + n * beta ^ (fexp1 (mag x) - 1 - fexp2 (mag x)); Fexp := cexp beta fexp2 x |}:float betaHf:F2R f = x(cexp beta fexp2 x <= Fexp f)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (mag x <= fexp1 (mag x) - 2)%Z -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (mag x <= fexp1 (mag x) - 2)%Z -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%Zbpow (mag x - 1) <= x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%Zbpow (mag x - 1) <= x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:(Build_mag_prop beta x ex Hex <= fexp1 (Build_mag_prop beta x ex Hex) - 2)%Zbpow (ex - 1) <= x < bpow exbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:(Build_mag_prop beta x ex Hex <= fexp1 (Build_mag_prop beta x ex Hex) - 2)%Zbpow (ex - 1) <= Rabs x < bpow exnow apply Rgt_not_eq.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:(Build_mag_prop beta x ex Hex <= fexp1 (Build_mag_prop beta x ex Hex) - 2)%Zx <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:Rround beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%Zround beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(mag x < fexp2 (mag x))%Zround beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%Zround beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)bpow (mag x - 1) <= x''beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)0 < round beta fexp2 (Znearest choice2) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)0 <= round beta fexp2 (Znearest choice2) xnow apply round_le; [|apply valid_rnd_N|apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:x'' < bpow (mag x)round beta fexp2 (Znearest choice2) 0 <= round beta fexp2 (Znearest choice2) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''x'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''x'' = bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''x'' <= bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''x'' <= round beta fexp2 (Znearest choice2) (bpow (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''generic_format beta fexp2 (bpow (mag x))now apply round_le; [|apply valid_rnd_N|apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''x'' <= round beta fexp2 (Znearest choice2) (bpow (mag x))now apply generic_format_bpow'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''generic_format beta fexp2 (bpow (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)round beta fexp1 (Znearest choice1) (bpow (mag x)) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)IZR (Znearest choice1 (bpow (mag x) * bpow (- fexp1 (mag (bpow (mag x)))))) * bpow (fexp1 (mag (bpow (mag x)))) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)IZR (Znearest choice1 (bpow (mag x) * bpow (- fexp1 (mag x + 1)%Z))) * bpow (fexp1 (mag x + 1)%Z) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)IZR (Znearest choice1 (bpow (mag x) * bpow (- fexp1 (mag x + 1)%Z))) * bpow (fexp1 (mag x + 1)%Z) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)IZR (Znearest choice1 (bpow (mag x) * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)IZR (Znearest choice1 (bpow (mag x) * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) = 0 * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)IZR (Znearest choice1 (bpow (mag x - fexp1 (mag x)))) = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Znearest choice1 (bpow (mag x - fexp1 (mag x))) = 0%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Rabs (bpow (mag x - fexp1 (mag x)) - 0) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Rabs (bpow (mag x - fexp1 (mag x))) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)bpow (mag x - fexp1 (mag x)) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)bpow (-2) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)/ IZR (beta * beta) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)(2 <= beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z/ IZR (beta * beta) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)(2 <= beta)%Znow apply Zle_bool_imp_le.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xbeta_val:Zbeta_prop:(2 <=? beta_val)%Z = trueHf1:(Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x <= fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) - 2)%ZHlx:Raux.bpow {| radix_val := beta_val; radix_prop := beta_prop |} (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x - 1) <= x < Raux.bpow {| radix_val := beta_val; radix_prop := beta_prop |} (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x)x'':=round {| radix_val := beta_val; radix_prop := beta_prop |} fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) <= Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x)%ZH0:Raux.bpow {| radix_val := beta_val; radix_prop := beta_prop |} (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x) <= x''Hx'':x'' = Raux.bpow {| radix_val := beta_val; radix_prop := beta_prop |} (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x)Hf11:fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x + 1)%Z = fexp1 (Raux.mag {| radix_val := beta_val; radix_prop := beta_prop |} x)(2 <= beta_val)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z/ IZR (beta * beta) < / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z0 < 2 * IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z2 < IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z0 < 2 * IZR (beta * beta)rewrite mult_IZR; apply Rmult_lt_0_compat; apply IZR_lt; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z0 < IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z2 < IZR (beta * beta)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z(2 < beta * beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z(beta < beta * beta)%Zapply Zmult_lt_compat_l; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(fexp2 (mag x) <= mag x)%ZH0:bpow (mag x) <= x''Hx'':x'' = bpow (mag x)Hf11:fexp1 (mag x + 1)%Z = fexp1 (mag x)Hbeta:(2 <= beta)%Z(beta * 1 < beta * beta)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(mag x < fexp2 (mag x))%Zround beta fexp1 (Znearest choice1) x'' = 0now apply (round_N_small_pos beta _ _ _ (mag x)). Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:(mag x <= fexp1 (mag x) - 2)%ZHlx:bpow (mag x - 1) <= x < bpow (mag x)x'':=round beta fexp2 (Znearest choice2) x:RNzx'':x'' <> 0H:(mag x < fexp2 (mag x))%Zx'' = 0beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp1 (mag x) = (mag x + 1)%Z -> x < bpow (mag x) - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> fexp1 (mag x) = (mag x + 1)%Z -> x < bpow (mag x) - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx < bpow (mag x) - / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx < bpow (mag x) - / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) x) = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Rx < bpow (mag x) - / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Rx < bpow (mag x) - / 2 * ulp beta fexp2 x -> round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:Rx < bpow (mag x) - / 2 * u2 -> round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2bpow (mag x - 1) <= x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2bpow (mag x - 1) <= x < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:fexp1 (Build_mag_prop beta x ex Hex) = (Build_mag_prop beta x ex Hex + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (Build_mag_prop beta x ex Hex) - / 2 * u2bpow (ex - 1) <= x < bpow exbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:fexp1 (Build_mag_prop beta x ex Hex) = (Build_mag_prop beta x ex Hex + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (Build_mag_prop beta x ex Hex) - / 2 * u2bpow (ex - 1) <= Rabs x < bpow exnow apply Rgt_not_eq.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:fexp1 (Build_mag_prop beta x ex Hex) = (Build_mag_prop beta x ex Hex + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (Build_mag_prop beta x ex Hex) - / 2 * u2x <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)round beta fexp1 (Znearest choice1) x'' = round beta fexp1 (Znearest choice1) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0round beta fexp1 (Znearest choice1) x'' = 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= x'' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= x''beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0x'' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= x''beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 00 < round beta fexp2 (Znearest choice2) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= xnow apply valid_rnd_N.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0Valid_rnd (Znearest choice2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 00 < round beta fexp2 (Znearest choice2) xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 00 <= x''now apply round_le; [|apply valid_rnd_N|apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0round beta fexp2 (Znearest choice2) 0 <= x''beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0bpow (mag x - 1) <= Rabs xnow apply Rgt_not_eq.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xex:ZHex:x <> 0 -> bpow (ex - 1) <= Rabs x < bpow exHf1:fexp1 (Build_mag_prop beta x ex Hex) = (Build_mag_prop beta x ex Hex + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (Build_mag_prop beta x ex Hex) - / 2 * u2Hlx:bpow (Build_mag_prop beta x ex Hex - 1) <= x < bpow (Build_mag_prop beta x ex Hex)Nzx'':x'' <> 0x <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0x'' < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0x + (x'' - x) < bpow (mag x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0x + (x'' - x) < bpow (mag x) - / 2 * u2 + / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0x'' - x <= / 2 * u2now apply error_le_half_ulp. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf1:fexp1 (mag x) = (mag x + 1)%Zx'':=round beta fexp2 (Znearest choice2) x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RHx:x < bpow (mag x) - / 2 * u2Hlx:bpow (mag x - 1) <= x < bpow (mag x)Nzx'':x'' <> 0Rabs (x'' - x) <= / 2 * u2beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * ulp beta fexp2 x <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * ulp beta fexp2 x <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool) (x : R), 0 < x -> (fexp2 (mag x) <= fexp1 (mag x) - 1)%Z -> (fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * ulp beta fexp2 x <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * ulp beta fexp2 x <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Z(fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * ulp beta fexp2 x <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * ulp beta fexp2 x <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:R(fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * ulp beta fexp2 x <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * ulp beta fexp2 x <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:R(fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * ulp beta fexp2 x <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * ulp beta fexp2 x <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * ulp beta fexp2 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:R(fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> ((fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 x) -> round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHlt:(mag x < fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHlt:(mag x < fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_really_zero.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHlt:(mag x < fexp1 (mag x) - 1)%ZH:(mag x <= fexp1 (mag x) - 2)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%ZH:fexp1 (mag x) = (mag x + 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%ZH:fexp1 (mag x) = (mag x + 1)%ZHlt':x < bpow (mag x) - / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%ZH:fexp1 (mag x) = (mag x + 1)%ZHge':bpow (mag x) - / 2 * u2 <= xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_zero.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%ZH:fexp1 (mag x) = (mag x + 1)%ZHlt':x < bpow (mag x) - / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xnow apply Cz.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHeq:mag x = (fexp1 (mag x) - 1)%ZH:fexp1 (mag x) = (mag x + 1)%ZHge':bpow (mag x) - / 2 * u2 <= xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHeq':x = midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xHlt'':x < midp fexp1 x - / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xHle'':midp fexp1 x - / 2 * u2 <= xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply round_round_lt_mid_further_place; [| | |omega| |].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xHlt'':x < midp fexp1 x - / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xnow apply Clt; [|split].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHlt':x < midp fexp1 xHle'':midp fexp1 x - / 2 * u2 <= xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply Ceq.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHeq':x = midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHlt'':x <= midp fexp1 x + / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply Cgt; [|split].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHlt'':x <= midp fexp1 x + / 2 * u2round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xNfx:~ generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xFx:generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xnow apply (generic_inclusion_mag beta fexp1); [omega|].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xFx:generic_format beta fexp1 xgeneric_format beta fexp2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xNfx:~ generic_format beta fexp1 xround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = x' + u1round_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = x' + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zround_round_eq fexp1 fexp2 choice1 choice2 xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xHle'':midp fexp1 x + / 2 * u2 < xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = x' + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zmidp' fexp1 x + / 2 * ulp beta fexp2 x < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = x' + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx' + / 2 * ulp beta fexp1 x + / 2 * u2 < x -> round beta fexp1 Zceil x - / 2 * ulp beta fexp1 x + / 2 * ulp beta fexp2 x < xlra. } Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolx:RPx:0 < xHf2:(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx':=round beta fexp1 Zfloor x:Ru1:=ulp beta fexp1 x:Ru2:=ulp beta fexp2 x:RCz:fexp1 (mag x) = (mag x + 1)%Z -> bpow (mag x) - / 2 * u2 <= x -> round_round_eq fexp1 fexp2 choice1 choice2 xClt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x - / 2 * u2 <= x < midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCeq:(fexp1 (mag x) <= mag x)%Z -> x = midp fexp1 x -> round_round_eq fexp1 fexp2 choice1 choice2 xCgt:(fexp1 (mag x) <= mag x)%Z -> midp fexp1 x < x <= midp fexp1 x + / 2 * u2 -> round_round_eq fexp1 fexp2 choice1 choice2 xHgt:(mag x > fexp1 (mag x) - 1)%ZH:(fexp1 (mag x) <= mag x)%ZHgt':x > midp fexp1 xNfx:~ generic_format beta fexp1 xHceil:round beta fexp1 Zceil x = x' + u1Hf2':(fexp2 (mag x) <= fexp1 (mag x) - 1)%Zx' + / 2 * u1 + / 2 * u2 < x -> x' + u1 - / 2 * u1 + / 2 * u2 < xbeta:radixforall x y : R, 0 < x -> 0 < y -> mag (x / y) = (mag x - mag y)%Z \/ mag (x / y) = (mag x - mag y + 1)%Zbeta:radixforall x y : R, 0 < x -> 0 < y -> mag (x / y) = (mag x - mag y)%Z \/ mag (x / y) = (mag x - mag y + 1)%Zbeta:radixx, y:RPx:0 < xPy:0 < ymag (x / y) = (mag x - mag y)%Z \/ mag (x / y) = (mag x - mag y + 1)%Zomega. Qed. Definition round_round_div_hyp fexp1 fexp2 := (forall ex, (fexp2 ex <= fexp1 ex - 1)%Z) /\ (forall ex ey, (fexp1 ex < ex)%Z -> (fexp1 ey < ey)%Z -> (fexp1 (ex - ey) <= ex - ey + 1)%Z -> (fexp2 (ex - ey) <= fexp1 ex - ey)%Z) /\ (forall ex ey, (fexp1 ex < ex)%Z -> (fexp1 ey < ey)%Z -> (fexp1 (ex - ey + 1) <= ex - ey + 1 + 1)%Z -> (fexp2 (ex - ey + 1) <= fexp1 ex - ey)%Z) /\ (forall ex ey, (fexp1 ex < ex)%Z -> (fexp1 ey < ey)%Z -> (fexp1 (ex - ey) <= ex - ey)%Z -> (fexp2 (ex - ey) <= fexp1 (ex - ey) + fexp1 ey - ey)%Z) /\ (forall ex ey, (fexp1 ex < ex)%Z -> (fexp1 ey < ey)%Z -> (fexp1 (ex - ey) = ex - ey + 1)%Z -> (fexp2 (ex - ey) <= ex - ey - ey + fexp1 ey)%Z).beta:radixx, y:RPx:0 < xPy:0 < y(mag x - mag y <= mag (x / y) <= mag x - mag y + 1)%Z -> mag (x / y) = (mag x - mag y)%Z \/ mag (x / y) = (mag x - mag y + 1)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> fexp1 (mag (x / y)) = (mag (x / y) + 1)%Z -> ~ bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> fexp1 (mag (x / y)) = (mag (x / y) + 1)%Z -> ~ bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%Z~ bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%Z~ bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):R~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):R~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rgeneric_format beta fexp1 x -> generic_format beta fexp1 y -> ~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rx = IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR my * bpow (fexp1 (mag y)) -> ~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ p - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))x / y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))/ y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pp - / 2 * u2 < p - / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < px / y < p - / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < px / y * y < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < px * (/ y * y) < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < px < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx <= p * y - p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zp * y - p * bpow (fexp1 (mag y)) < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx <= p * y - p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) <= p * (IZR my * bpow (fexp1 (mag y))) - p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) <= p * IZR my * bpow (fexp1 (mag y)) - p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) <= IZR my * p * bpow (fexp1 (mag y)) - p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) <= IZR my * bpow (mag (x / y) + fexp1 (mag y)) - bpow (mag (x / y) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) * bpow (- mag (x / y) - fexp1 (mag y)) <= (IZR my * bpow (mag (x / y) + fexp1 (mag y)) - bpow (mag (x / y) + fexp1 (mag y))) * bpow (- mag (x / y) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) * bpow (- mag (x / y) - fexp1 (mag y)) <= IZR my * bpow (mag (x / y) + fexp1 (mag y)) * bpow (- mag (x / y) - fexp1 (mag y)) - bpow (mag (x / y) + fexp1 (mag y)) * bpow (- mag (x / y) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) <= IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * IZR (beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y))) <= IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR (mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y))) <= IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR (mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y))) <= IZR (my - 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) <= my - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) + 1 <= my)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) < my)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR (mx * beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y))) < IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * IZR (beta ^ (fexp1 (mag x) - mag (x / y) - fexp1 (mag y))) < IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) < IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x) - mag (x / y) - fexp1 (mag y)) * bpow (fexp1 (mag y) + mag (x / y)) < IZR my * bpow (fexp1 (mag y) + mag (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZIZR mx * bpow (fexp1 (mag x)) < IZR my * bpow (fexp1 (mag y) + mag (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < IZR my * bpow (fexp1 (mag y) + mag (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < IZR my * (bpow (fexp1 (mag y)) * bpow (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < y * bpow (mag (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx < y * pfield_simplify; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zx * / y < y * p * / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zp * y - p * bpow (fexp1 (mag y)) < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zp * y - p * bpow (fexp1 (mag y)) < p * y - / 2 * u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z- (p * bpow (fexp1 (mag y))) < - (/ 2 * u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * u2 * y < p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * u2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zu2 * bpow (mag y) <= p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * u2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * u2 * y < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * (u2 * y) < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 * (u2 * y) < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z0 <= / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z0 <= u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 < 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z0 <= / 2now apply Rmult_le_pos; [apply bpow_ge_0|apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z0 <= u2 * ylra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z/ 2 < 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zy < bpow (mag y)apply bpow_mag_gt. }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZRabs y < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Zu2 * bpow (mag y) <= p * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) + mag y <= mag (x / y) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= - mag y + mag (x / y) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= mag (x / y) - mag y + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x - mag y + 1) <= mag x - mag y + 1 - mag y + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x + 1 - mag y) <= mag x + 1 - mag y - mag y + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Zfexp1 (mag x + 1 - mag y)%Z = (mag x + 1 - mag y + 1)%Znow assert (fexp1 (mag x + 1) <= mag x)%Z; [apply valid_exp|omega].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Zfexp1 (mag x + 1 - mag y)%Z = (mag x + 1 - mag y + 1)%Zassumption.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Zfexp1 (mag x + 1 - mag y)%Z = (mag x + 1 - mag y + 1)%Znow rewrite <- Hxy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(0 <= fexp1 (mag x) - mag (x / y) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Zfexp1 (mag x - mag y + 1)%Z = (mag x - mag y + 1 + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx <= p * y - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zp * y - bpow (fexp1 (mag x)) < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx <= p * y - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx * bpow (fexp1 (mag x)) <= p * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) <= (p * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) <= p * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag x)) - bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= p * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= IZR my * p * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= IZR my * bpow (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= IZR my * IZR (beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= IZR (my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx <= IZR (my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)) - 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(mx <= my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(mx + 1 <= my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(mx < my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx < IZR (my * beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx < IZR my * IZR (beta ^ (mag (x / y) + fexp1 (mag y) - fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx < IZR my * bpow (mag (x / y) + fexp1 (mag y) - fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZIZR mx * bpow (fexp1 (mag x)) < IZR my * bpow (mag (x / y) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < IZR my * bpow (mag (x / y) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < IZR my * (bpow (fexp1 (mag y)) * bpow (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < y * bpow (mag (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx < y * pfield_simplify; lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zx * / y < y * p * / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zp * y - bpow (fexp1 (mag x)) < (p - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zp * y - bpow (fexp1 (mag x)) < p * y - / 2 * u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z- bpow (fexp1 (mag x)) < - (/ 2 * u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * u2 * y < bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * u2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * u2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * u2 * y < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * (u2 * y) < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 * (u2 * y) < 1 * (u2 * bpow (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z0 <= / 2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z0 <= u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 < 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)lra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z0 <= / 2now apply Rmult_le_pos; [apply bpow_ge_0|apply Rlt_le].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z0 <= u2 * ylra.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z/ 2 < 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zy < bpow (mag y)apply bpow_mag_gt. }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%ZRabs y < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) + mag y <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= - mag y + fexp1 (mag x))%Zdestruct (mag_div_disj x y Px Py) as [Hxy|Hxy]; rewrite Hxy; apply Hexp; try assumption; rewrite <- Hxy; rewrite Hf1; apply Z.le_refl. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zp:=bpow (mag (x / y)):Ru2:=bpow (fexp2 (mag (x / y))):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hl:p - / 2 * bpow (cexp beta fexp2 (x / y)) <= x / yHr:x / y < pHe:(fexp1 (mag x) - mag (x / y) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= fexp1 (mag x) + - mag y)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> (fexp1 (mag (x / y)) <= mag (x / y))%Z -> ~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> (fexp1 (mag (x / y)) <= mag (x / y))%Z -> ~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%Z~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%Z~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zx / y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z/ y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y) -> ~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y) -> ~ midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)/ 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)/ 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)round beta fexp1 Zfloor (x / y) + / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= round beta fexp1 Zfloor (x / y) + (x / y - round beta fexp1 Zfloor (x / y))apply H'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)round beta fexp1 Zfloor (x / y) + (x / y - round beta fexp1 Zfloor (x / y)) < round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y)apply H'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0H:~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)H':midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)x / y < round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):R~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):R~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - round beta fexp1 Zfloor (x / y) < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):R~ / 2 * (ulp beta fexp1 (x / y) - ulp beta fexp2 (x / y)) <= x / y - x' < / 2 * ulp beta fexp1 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):R~ / 2 * (bpow (cexp beta fexp1 (x / y)) - bpow (cexp beta fexp2 (x / y))) <= x / y - x' < / 2 * bpow (cexp beta fexp1 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rgeneric_format beta fexp1 x -> generic_format beta fexp1 y -> ~ / 2 * (bpow (cexp beta fexp1 (x / y)) - bpow (cexp beta fexp2 (x / y))) <= x / y - x' < / 2 * bpow (cexp beta fexp1 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rx = IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ / 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ / 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR my * bpow (fexp1 (mag y)) -> ~ / 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ / 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))/ 2 * (u1 - u2) < / 2 * (u1 - u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x / y - x' < / 2 * (u1 - u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x / y < x' + / 2 * u1 - / 2 * u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x / y * y < (x' + / 2 * u1 - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x * (/ y * y) < (x' + / 2 * u1 - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x < (x' + / 2 * u1 - / 2 * u2) * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))x < x' * y + / 2 * u1 * y - / 2 * u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))2 * x < 2 * (x' * y + / 2 * u1 * y - / 2 * u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))2 * x < 2 * (x' * y) + 2 * (/ 2 * u1 * y) - 2 * (/ 2 * u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))2 * x < 2 * x' * y + 2 * / 2 * u1 * y - 2 * / 2 * u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))2 * x < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x <= 2 * x' * y + u1 * y - bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x' * y + u1 * y - bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x <= 2 * x' * y + u1 * y - bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR mx * bpow (fexp1 (mag x))) <= 2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) <= (2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag (x / y)) + fexp1 (mag y))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * x' * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) + u1 * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) - bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * x' * IZR my * bpow (- fexp1 (mag (x / y))) + u1 * IZR my * bpow (- fexp1 (mag (x / y))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * IZR my * x' * bpow (- fexp1 (mag (x / y))) + u1 * IZR my * bpow (- fexp1 (mag (x / y))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * IZR my * x' * bpow (- fexp1 (mag (x / y))) + IZR my * u1 * bpow (- fexp1 (mag (x / y))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)))) * bpow (- fexp1 (mag (x / y))) + IZR my * bpow (fexp1 (mag (x / y))) * bpow (- fexp1 (mag (x / y))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * IZR (beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) <= 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my - 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) + 1 <= 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) < 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) < IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) + my)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) < IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * IZR (beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))) < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR mybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * IZR mx * bpow (fexp1 (mag x)) < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR mx * bpow (fexp1 (mag x))) < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + IZR my) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * (x' * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * IZR my * (bpow (fexp1 (mag y)) * x') + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * (IZR my * (bpow (fexp1 (mag y)) * x')) + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * (IZR my * bpow (fexp1 (mag y)) * x') + IZR my * bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * (y * x') + y * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x < 2 * (y * x') + y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * (2 * x) < / 2 * (2 * (y * x') + y * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * (2 * x) < / 2 * (2 * (y * x')) + / 2 * (y * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * 2 * x < / 2 * 2 * y * x' + / 2 * y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zx < y * x' + / 2 * y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zx - y * x' < y * / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * (x - y * x') < / y * (y * / 2 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * x - / y * (y * x') < / y * (y * / 2 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * x - / y * y * x' < / y * y * / 2 * u1now rewrite Rmult_comm.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * x - x' < / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * x' * y + u1 * y - bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z- bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) < - (u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * bpow (mag y) <= bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy < bpow (mag y)apply bpow_gt_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy < bpow (mag y)apply bpow_mag_gt. }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZRabs y < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * bpow (mag y) <= bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) + mag y <= fexp1 (mag (x / y)) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= - mag y + fexp1 (mag (x / y)) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= fexp1 (mag (x / y)) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x - mag y + 1) <= fexp1 (mag x - mag y + 1) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x + 1 - mag y) <= fexp1 (mag x + 1 - mag y) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Znow assert (fexp1 (mag x + 1) <= mag x)%Z; [apply valid_exp|omega].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Zassumption.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Znow rewrite <- Hxy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x - mag y + 1) <= mag x - mag y + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x <= 2 * x' * y + u1 * y - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * y + u1 * y - bpow (fexp1 (mag x)) < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x <= 2 * x' * y + u1 * y - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR mx * bpow (fexp1 (mag x))) <= 2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x)) <= (2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) - bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x)) <= 2 * x' * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag x)) + u1 * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag x)) - bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * x' * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) + u1 * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * IZR my * x' * bpow (fexp1 (mag y) - fexp1 (mag x)) + u1 * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * IZR my * x' * bpow (fexp1 (mag y) - fexp1 (mag x)) + IZR my * u1 * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)))) * bpow (fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx <= 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR my * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * mx) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR (my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * mx) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) - 1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * mx) <= IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) - 1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * mx <= 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * mx + 1 <= 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * mx < 2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * mx) < IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * mx) < IZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR (my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR my * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR mx * bpow (fexp1 (mag x)) < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR mx * bpow (fexp1 (mag x))) < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < (2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) * bpow (fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * (x' * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * IZR my * (bpow (fexp1 (mag y)) * x') + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * (IZR my * (bpow (fexp1 (mag y)) * x')) + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * (IZR my * bpow (fexp1 (mag y)) * x') + IZR my * bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * (y * x') + y * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x < 2 * (y * x') + y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * (2 * x) < / 2 * (2 * (y * x') + y * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * (2 * x) < / 2 * (2 * (y * x')) + / 2 * (y * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * 2 * x < / 2 * 2 * y * x' + / 2 * y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zx < y * x' + / 2 * y * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zx - y * x' < y * / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * (x - y * x') < / y * (y * / 2 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * x - / y * (y * x') < / y * (y * / 2 * u1)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * x - / y * y * x' < / y * y * / 2 * u1now rewrite Rmult_comm.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * x - x' < / 2 * u1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * y + u1 * y - bpow (fexp1 (mag x)) < 2 * x' * y + u1 * y - u2 * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z- bpow (fexp1 (mag x)) < - (u2 * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy < bpow (mag y)apply bpow_gt_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy < bpow (mag y)apply bpow_mag_gt. }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZRabs y < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) + mag y <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= - mag y + fexp1 (mag x))%Zdestruct (mag_div_disj x y Px Py) as [Hxy|Hxy]; rewrite Hxy; apply Hexp; try assumption; rewrite <- Hxy; omega. Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZS:x / y <> 0u1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rmx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * (bpow (fexp1 (mag (x / y))) - bpow (fexp2 (mag (x / y)))) <= x / y - x' < / 2 * bpow (fexp1 (mag (x / y)))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= fexp1 (mag x) + - mag y)%Zbeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> (fexp1 (mag (x / y)) <= mag (x / y))%Z -> ~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> (Z -> bool) -> (Z -> bool) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> (fexp1 (mag (x / y)) <= mag (x / y))%Z -> ~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%Z~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%Z~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y)) -> ~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y)) -> ~ midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)/ 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)/ 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y) < round beta fexp1 Zfloor (x / y) + (x / y - round beta fexp1 Zfloor (x / y))apply H'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y) < x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)round beta fexp1 Zfloor (x / y) + (x / y - round beta fexp1 Zfloor (x / y)) <= round beta fexp1 Zfloor (x / y) + / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))apply H'.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%ZH:~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))H':midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)x / y <= round beta fexp1 Zfloor (x / y) + / 2 * ulp beta fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Z~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):R~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):R~ / 2 * ulp beta fexp1 (x / y) < x / y - round beta fexp1 Zfloor (x / y) <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):R~ / 2 * ulp beta fexp1 (x / y) < x / y - x' <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):Rx / y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0~ / 2 * ulp beta fexp1 (x / y) < x / y - x' <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):R/ y <> 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0~ / 2 * ulp beta fexp1 (x / y) < x / y - x' <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0~ / 2 * ulp beta fexp1 (x / y) < x / y - x' <= / 2 * (ulp beta fexp1 (x / y) + ulp beta fexp2 (x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0~ / 2 * bpow (cexp beta fexp1 (x / y)) < x / y - x' <= / 2 * (bpow (cexp beta fexp1 (x / y)) + bpow (cexp beta fexp2 (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0generic_format beta fexp1 x -> generic_format beta fexp1 y -> ~ / 2 * bpow (cexp beta fexp1 (x / y)) < x / y - x' <= / 2 * (bpow (cexp beta fexp1 (x / y)) + bpow (cexp beta fexp2 (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0x = IZR (Ztrunc (x * bpow (- fexp1 (mag x)))) * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ / 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR (Ztrunc (y * bpow (- fexp1 (mag y)))) * bpow (fexp1 (mag y)) -> ~ / 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):Zx = IZR mx * bpow (fexp1 (mag x)) -> y = IZR my * bpow (fexp1 (mag y)) -> ~ / 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))~ / 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))/ 2 * (u1 + u2) < / 2 * (u1 + u2)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))/ 2 * (u1 + u2) < x / y - x'beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))/ 2 * u1 + / 2 * u2 + x' < x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))(/ 2 * u1 + / 2 * u2 + x') * y < x / y * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))(/ 2 * u1 + / 2 * u2 + x') * y < x * (/ y * y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))(/ 2 * u1 + / 2 * u2 + x') * y < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))/ 2 * u1 * y + / 2 * u2 * y + x' * y < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))2 * (/ 2 * u1 * y + / 2 * u2 * y + x' * y) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))2 * (/ 2 * u1 * y) + 2 * (/ 2 * u2 * y) + 2 * (x' * y) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))2 * / 2 * u1 * y + 2 * / 2 * u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))u1 * y + u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu1 * y + u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + u2 * y + 2 * x' * y < u1 * y + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * y <= 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + u2 * y + 2 * x' * y < u1 * y + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * bpow (mag y) <= bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy < bpow (mag y)apply bpow_gt_0.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy < bpow (mag y)apply bpow_mag_gt. }beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZRabs y < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu2 * bpow (mag y) <= bpow (fexp1 (mag (x / y)) + fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) + mag y <= fexp1 (mag (x / y)) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= - mag y + fexp1 (mag (x / y)) + fexp1 (mag y))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(fexp2 (mag (x / y)) <= fexp1 (mag (x / y)) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x - mag y + 1) <= fexp1 (mag x - mag y + 1) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp2 (mag x + 1 - mag y) <= fexp1 (mag x + 1 - mag y) + fexp1 (mag y) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Znow assert (fexp1 (mag x + 1) <= mag x)%Z; [apply valid_exp|omega].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1) < mag x + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Zassumption.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag y) < mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x + 1 - mag y) <= mag x + 1 - mag y)%Znow rewrite <- Hxy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZHxy:mag (x / y) = (mag x - mag y + 1)%Z(fexp1 (mag x - mag y + 1) <= mag x - mag y + 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * y <= 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * y + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * y <= 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * (IZR my * bpow (fexp1 (mag y))) + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * y <= 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * (IZR my * bpow (fexp1 (mag y))) + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * (IZR my * bpow (fexp1 (mag y))) <= 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(u1 * (IZR my * bpow (fexp1 (mag y))) + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + 2 * x' * (IZR my * bpow (fexp1 (mag y)))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) + bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) + 2 * x' * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y)) <= 2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zu1 * IZR my * bpow (- fexp1 (mag (x / y))) + 1 + 2 * x' * IZR my * bpow (- fexp1 (mag (x / y))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my * u1 * bpow (- fexp1 (mag (x / y))) + 1 + 2 * x' * IZR my * bpow (- fexp1 (mag (x / y))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * x' * IZR my * bpow (- fexp1 (mag (x / y))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (x' * IZR my) * bpow (- fexp1 (mag (x / y))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (IZR my * x') * bpow (- fexp1 (mag (x / y))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (IZR my * x' * bpow (- fexp1 (mag (x / y)))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)))) * bpow (- fexp1 (mag (x / y)))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) <= 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + 2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) <= 2 * IZR mx * IZR (beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR my + 1 + IZR (2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) <= IZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (my + 1 + 2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) <= IZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(my + 1 + 2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) <= 2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + my + 1 <= 2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + my < 2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) + my) < IZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%ZIZR (2 * (my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) + IZR my < IZR (2 * mx * beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) + IZR my < 2 * IZR mx * IZR (beta ^ (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) + IZR my < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) + IZR my) * bpow (fexp1 (mag y)) < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) * bpow (fexp1 (mag y)) + IZR my * bpow (fexp1 (mag y)) < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * IZR my) * bpow (fexp1 (mag y)) + IZR my * bpow (fexp1 (mag y)) < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * (IZR my * bpow (fexp1 (mag y)))) + IZR my * bpow (fexp1 (mag y)) < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y)) * bpow (fexp1 (mag y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * IZR mx * bpow (fexp1 (mag x) - fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * IZR mx * (bpow (fexp1 (mag x)) * bpow (- fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * (IZR mx * (bpow (fexp1 (mag x)) * bpow (- fexp1 (mag (x / y)))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * (IZR mx * bpow (fexp1 (mag x)) * bpow (- fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y < 2 * (x * bpow (- fexp1 (mag (x / y))))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z(2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) + y) * bpow (fexp1 (mag (x / y))) < 2 * (x * bpow (- fexp1 (mag (x / y)))) * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) * bpow (fexp1 (mag (x / y))) + y * bpow (fexp1 (mag (x / y))) < 2 * (x * bpow (- fexp1 (mag (x / y)))) * bpow (fexp1 (mag (x / y)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * y) * bpow (fexp1 (mag (x / y))) + y * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (y * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y)))))) * bpow (fexp1 (mag (x / y))) + y * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (y * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y))))) + y * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (y * x') + y * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z2 * (y * x') + y * u1 < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * (2 * (y * x') + y * u1) < / 2 * (2 * x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * (2 * (y * x')) + / 2 * (y * u1) < / 2 * (2 * x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * 2 * y * x' + / 2 * y * u1 < / 2 * 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy * x' + / 2 * y * u1 < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Zy * / 2 * u1 < - y * x' + xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * (y * / 2 * u1) < / y * (- y * x' + x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * (y * / 2 * u1) < / y * (- y * x') + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * y * / 2 * u1 < / y * - y * x' + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * y * / 2 * u1 < - (/ y * y) * x' + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ y * y * / 2 * u1 < - (/ y * y * x') + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * u1 < - x' + / y * xnow rewrite (Rplus_comm (- x')).beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(0 <= fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y))%Z/ 2 * u1 < - x' + x * / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu1 * y + u2 * y + 2 * x' * y < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu1 * y + u2 * y + 2 * x' * y < 2 * x' * y + u1 * y + bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * y + u1 * y + bpow (fexp1 (mag x)) <= 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu1 * y + u2 * y + 2 * x' * y < 2 * x' * y + u1 * y + bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * y < u2 * bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z0 < u2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy < bpow (mag y)now apply Rabs_lt_inv; apply bpow_mag_gt.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy < bpow (mag y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zu2 * bpow (mag y) <= bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) + mag y <= fexp1 (mag x))%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= - mag y + fexp1 (mag x))%Zdestruct (mag_div_disj x y Px Py) as [Hxy|Hxy]; rewrite Hxy; apply Hexp; try assumption; rewrite <- Hxy; omega.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(fexp2 (mag (x / y)) <= fexp1 (mag x) + - mag y)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * y + u1 * y + bpow (fexp1 (mag x)) <= 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * y + u1 * y + bpow (fexp1 (mag x)) <= 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) + bpow (fexp1 (mag x)) <= 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * x' * (IZR my * bpow (fexp1 (mag y))) + u1 * (IZR my * bpow (fexp1 (mag y))) + bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x)) <= 2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag x)) + u1 * (IZR my * bpow (fexp1 (mag y))) * bpow (- fexp1 (mag x)) + bpow (fexp1 (mag x)) * bpow (- fexp1 (mag x)) <= 2 * (IZR mx * bpow (fexp1 (mag x))) * bpow (- fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * x' * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) + u1 * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * x' * bpow (fexp1 (mag y) - fexp1 (mag x)) + u1 * IZR my * bpow (fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * x' * bpow (fexp1 (mag y) - fexp1 (mag x)) + IZR my * u1 * bpow (fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)))) * bpow (fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR my * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + 1 <= 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR (my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + 1 <= IZR (2 * mx)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + 1) <= IZR (2 * mx)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + 1 <= 2 * mx)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) < 2 * mx)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) < IZR (2 * mx)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%ZIZR (2 * my * Zfloor (x / y * bpow (- fexp1 (mag (x / y)))) * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR (my * beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) < IZR (2 * mx)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) + IZR my * IZR (beta ^ (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) < 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) < 2 * IZR mxbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x)) < 2 * IZR mx * bpow (fexp1 (mag x))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x)) < 2 * (IZR mx * bpow (fexp1 (mag x)))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z(2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x))) * bpow (fexp1 (mag x)) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) * bpow (fexp1 (mag x)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y) - fexp1 (mag x)) * bpow (fexp1 (mag x)) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) + IZR my * bpow (fexp1 (mag (x / y)) + fexp1 (mag y)) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y)))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * (IZR (Zfloor (x / y * bpow (- fexp1 (mag (x / y))))) * bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * (x' * bpow (fexp1 (mag y))) + IZR my * (bpow (fexp1 (mag (x / y))) * bpow (fexp1 (mag y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * IZR my * (bpow (fexp1 (mag y)) * x') + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y)))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR my * (bpow (fexp1 (mag y)) * x')) + IZR my * (bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y)))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (IZR my * bpow (fexp1 (mag y)) * x') + IZR my * bpow (fexp1 (mag y)) * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (y * x') + y * bpow (fexp1 (mag (x / y))) < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z2 * (y * x') + y * u1 < 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * (2 * (y * x') + y * u1) < / 2 * (2 * x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * (2 * (y * x')) + / 2 * (y * u1) < / 2 * (2 * x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * 2 * y * x' + / 2 * y * u1 < / 2 * 2 * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy * x' + / 2 * y * u1 < xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Zy * / 2 * u1 < - y * x' + xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * (y * / 2 * u1) < / y * (- y * x' + x)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * (y * / 2 * u1) < / y * (- y * x') + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * y * / 2 * u1 < / y * - y * x' + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * y * / 2 * u1 < - (/ y * y) * x' + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ y * y * / 2 * u1 < - (/ y * y * x') + / y * xbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * u1 < - x' + / y * xnow rewrite (Rplus_comm (- x')). Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHfx:(fexp1 (mag x) < mag x)%ZHfy:(fexp1 (mag y) < mag y)%Zu1:=bpow (fexp1 (mag (x / y))):Ru2:=bpow (fexp2 (mag (x / y))):Rx':=round beta fexp1 Zfloor (x / y):RS:x / y <> 0mx:=Ztrunc (x * bpow (- fexp1 (mag x))):Zmy:=Ztrunc (y * bpow (- fexp1 (mag y))):ZFx:x = IZR mx * bpow (fexp1 (mag x))Fy:y = IZR my * bpow (fexp1 (mag y))Hlr:/ 2 * bpow (fexp1 (mag (x / y))) < x / y - x' <= / 2 * (bpow (fexp1 (mag (x / y))) + bpow (fexp2 (mag (x / y))))He:(fexp1 (mag x) - fexp1 (mag (x / y)) - fexp1 (mag y) < 0)%Z/ 2 * u1 < - x' + x * / ybeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 y0 < x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yround_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 y0 < x / ynow apply Rinv_0_lt_compat.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 y0 < / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yround_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yValid_exp fexp1beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yValid_exp fexp2beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y0 < x / ybeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp2 (mag (x / y)) <= fexp1 (mag (x / y)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yfexp1 (mag (x / y)) = (mag (x / y) + 1)%Z -> bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> x / y = midp fexp1 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)exact Vfexp1.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yValid_exp fexp1exact Vfexp2.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yValid_exp fexp2exact Pxy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y0 < x / yapply Hexp.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp2 (mag (x / y)) <= fexp1 (mag (x / y)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yfexp1 (mag (x / y)) = (mag (x / y) + 1)%Z -> bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHlxy:bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / yround_round_eq fexp1 fexp2 choice1 choice2 (x / y)now apply (round_round_div_aux0 fexp1 fexp2 _ _ choice1 choice2 Hexp x y).beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:fexp1 (mag (x / y)) = (mag (x / y) + 1)%ZHlxy:bpow (mag (x / y)) - / 2 * ulp beta fexp2 (x / y) <= x / yFalsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHlxy:midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)round_round_eq fexp1 fexp2 choice1 choice2 (x / y)now apply (round_round_div_aux1 fexp1 fexp2 _ _ choice1 choice2 Hexp x y).beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHlxy:midp fexp1 (x / y) - / 2 * ulp beta fexp2 (x / y) <= x / y < midp fexp1 (x / y)Falsebeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> x / y = midp fexp1 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yH:(fexp1 (mag (x / y)) <= mag (x / y))%Zx / y = midp fexp1 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)apply Hexp.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yH:(fexp1 (mag (x / y)) <= mag (x / y))%Z(fexp2 (mag (x / y)) <= fexp1 (mag (x / y)) - 1)%Zbeta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / y(fexp1 (mag (x / y)) <= mag (x / y))%Z -> midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y) -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHlxy:midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)round_round_eq fexp1 fexp2 choice1 choice2 (x / y)now apply (round_round_div_aux2 fexp1 fexp2 _ _ choice1 choice2 Hexp x y). Qed.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RPx:0 < xPy:0 < yFx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPxy:0 < x / yHf1:(fexp1 (mag (x / y)) <= mag (x / y))%ZHlxy:midp fexp1 (x / y) < x / y <= midp fexp1 (x / y) + / 2 * ulp beta fexp2 (x / y)Falsebeta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, y <> 0 -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixforall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> round_round_div_hyp fexp1 fexp2 -> forall x y : R, y <> 0 -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> round_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround_round_eq fexp1 fexp2 choice1 choice2 (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- - x / y)) = round beta fexp1 (Znearest choice1) (- - x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- - x / - - y)) = round beta fexp1 (Znearest choice1) (- - x / - - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (- x / - - y))) = round beta fexp1 (Znearest choice1) (- (- x / - - y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (- x * - / - y))) = round beta fexp1 (Znearest choice1) (- (- x * - / - y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- - (- x * / - y))) = round beta fexp1 (Znearest choice1) (- - (- x * / - y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x * / - y)) = round beta fexp1 (Znearest choice1) (- x * / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:- 0 < - xNy:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:- 0 < - xNy:- 0 < - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:0 < - xNy:0 < - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 yNx:0 < - xNy:0 < - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)now apply round_round_div_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 (- y)Nx:0 < - xNy:0 < - yround beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- x / - y)) = round beta fexp1 (Znearest choice1) (- x / - y)now casetype False; apply Nzy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- - x / y)) = round beta fexp1 (Znearest choice1) (- - x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (- x / y))) = round beta fexp1 (Znearest choice1) (- (- x / y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x / y)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:x < 0Py:y > 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x / y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:- 0 < - xPy:y > 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x / y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yNx:0 < - xPy:y > 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x / y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x / y)now apply round_round_div_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 (- x)Fy:generic_format beta fexp1 yNx:0 < - xPy:y > 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (- x / y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (- x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (0 / y)) = round beta fexp1 (Znearest choice1) (0 / y)now rewrite round_0; [|apply valid_rnd_N].beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yZx:x = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) 0) = round beta fexp1 (Znearest choice1) 0beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / - - y)) = round beta fexp1 (Znearest choice1) (x / - - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x * - / - y)) = round beta fexp1 (Znearest choice1) (x * - / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (- (x * / - y))) = round beta fexp1 (Znearest choice1) (- (x * / - y))beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0- round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (x * / - y)) = - round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (x * / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:y < 0round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (x * / - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (x * / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:- 0 < - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (x * / - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (x * / - y)beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Ny:0 < - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (x * / - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (x * / - y)now apply round_round_div_aux.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 (- y)Px:x > 0Ny:0 < - yround beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (round beta fexp2 (Znearest (fun t : Z => negb (choice2 (- (t + 1))%Z))) (x * / - y)) = round beta fexp1 (Znearest (fun t : Z => negb (choice1 (- (t + 1))%Z))) (x * / - y)now casetype False; apply Nzy.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Zy:y = 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)now apply round_round_div_aux. Qed. Section Double_round_div_FLX. Variable prec : Z. Variable prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixfexp1, fexp2:Z -> ZVfexp1:Valid_exp fexp1Vfexp2:Valid_exp fexp2choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHexp:round_round_div_hyp fexp1 fexp2x, y:RNzy:y <> 0Fx:generic_format beta fexp1 xFy:generic_format beta fexp1 yPx:x > 0Py:y > 0round beta fexp1 (Znearest choice1) (round beta fexp2 (Znearest choice2) (x / y)) = round beta fexp1 (Znearest choice1) (x / y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> round_round_div_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(2 * prec <= prec')%Z -> round_round_div_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zround_round_div_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zround_round_div_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => (e - prec)%Z) (fun e : Z => (e - prec')%Z)beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Z(forall ex : Z, (ex - prec' <= ex - prec - 1)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec <= ex - ey + 1)%Z -> (ex - ey - prec' <= ex - prec - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey + 1 - prec <= ex - ey + 1 + 1)%Z -> (ex - ey + 1 - prec' <= ex - prec - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec <= ex - ey)%Z -> (ex - ey - prec' <= ex - ey - prec + (ey - prec) - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec)%Z = (ex - ey + 1)%Z -> (ex - ey - prec' <= ex - ey - ey + (ey - prec))%Z)split; [|split; [|split]]; intros ex ey; omega. Qed.beta:radixprec, prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hprec:(2 * prec <= prec')%Z(forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec <= ex - ey + 1)%Z -> (ex - ey - prec' <= ex - prec - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey + 1 - prec <= ex - ey + 1 + 1)%Z -> (ex - ey + 1 - prec' <= ex - prec - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec <= ex - ey)%Z -> (ex - ey - prec' <= ex - ey - prec + (ey - prec) - ey)%Z) /\ (forall ex ey : Z, (ex - prec < ex)%Z -> (ey - prec < ey)%Z -> (ex - ey - prec)%Z = (ex - ey + 1)%Z -> (ex - ey - prec' <= ex - ey - ey + (ey - prec))%Z)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x / y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FLX_format beta prec x -> FLX_format beta prec y -> round_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x / y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yround_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x / y)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yexists n : Z, beta = (2 * n)%Zbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yround_round_div_hyp (FLX_exp prec) (FLX_exp prec')beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yy <> 0beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xbeta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ynow apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec)now apply FLX_exp_valid.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yValid_exp (FLX_exp prec')exact Ebeta.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yexists n : Z, beta = (2 * n)%Znow apply FLX_round_round_div_hyp.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yround_round_div_hyp (FLX_exp prec) (FLX_exp prec')exact Nzy.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec yy <> 0now apply generic_format_FLX.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) xnow apply generic_format_FLX. Qed. End Double_round_div_FLX. Section Double_round_div_FLT. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixprec, prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLX_format beta prec xFy:FLX_format beta prec ygeneric_format beta (FLX_exp prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin - prec - 2)%Z -> (2 * prec <= prec')%Z -> round_round_div_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' <= emin - prec - 2)%Z -> (2 * prec <= prec')%Z -> round_round_div_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => Z.max (e - prec) emin) (fun e : Z => Z.max (e - prec') emin')beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Z(forall ex : Z, (Z.max (ex - prec') emin' <= Z.max (ex - prec) emin - 1)%Z) /\ (forall ex ey : Z, (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Z) /\ (forall ex ey : Z, (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Z) /\ (forall ex ey : Z, (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Z) /\ (forall ex ey : Z, (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex:Z(Z.max (ex - prec') emin' <= Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex:Z(Z.max (ex - prec') emin' <= Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex:Z(ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (Z.max (ex - prec') emin' <= Z.max (ex - prec) emin - 1)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (ex - prec' >= emin')%Z /\ Z.max (ex - prec') emin' = (ex - prec')%Z \/ (ex - prec' < emin')%Z /\ Z.max (ex - prec') emin' = emin' -> (Z.max (ex - prec') emin' <= Z.max (ex - prec) emin - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec' >= emin')%Z /\ Z.max (ex - ey - prec') emin' = (ex - ey - prec')%Z \/ (ex - ey - prec' < emin')%Z /\ Z.max (ex - ey - prec') emin' = emin' -> (ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey + 1 - prec >= emin)%Z /\ Z.max (ex - ey + 1 - prec) emin = (ex - ey + 1 - prec)%Z \/ (ex - ey + 1 - prec < emin)%Z /\ Z.max (ex - ey + 1 - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey + 1 - prec' >= emin')%Z /\ Z.max (ex - ey + 1 - prec') emin' = (ex - ey + 1 - prec')%Z \/ (ex - ey + 1 - prec' < emin')%Z /\ Z.max (ex - ey + 1 - prec') emin' = emin' -> (ex - ey + 1 - prec >= emin)%Z /\ Z.max (ex - ey + 1 - prec) emin = (ex - ey + 1 - prec)%Z \/ (ex - ey + 1 - prec < emin)%Z /\ Z.max (ex - ey + 1 - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey + 1 - prec) emin <= ex - ey + 1 + 1)%Z -> (Z.max (ex - ey + 1 - prec') emin' <= Z.max (ex - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zomega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec' >= emin')%Z /\ Z.max (ex - ey - prec') emin' = (ex - ey - prec')%Z \/ (ex - ey - prec' < emin')%Z /\ Z.max (ex - ey - prec') emin' = emin' -> (ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> (Z.max (ex - ey - prec) emin <= ex - ey)%Z -> (Z.max (ex - ey - prec') emin' <= Z.max (ex - ey - prec) emin + Z.max (ey - prec) emin - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zomega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zex, ey:Z(ex - ey - prec' >= emin')%Z /\ Z.max (ex - ey - prec') emin' = (ex - ey - prec')%Z \/ (ex - ey - prec' < emin')%Z /\ Z.max (ex - ey - prec') emin' = emin' -> (ex - ey - prec >= emin)%Z /\ Z.max (ex - ey - prec) emin = (ex - ey - prec)%Z \/ (ex - ey - prec < emin)%Z /\ Z.max (ex - ey - prec) emin = emin -> (ey - prec >= emin)%Z /\ Z.max (ey - prec) emin = (ey - prec)%Z \/ (ey - prec < emin)%Z /\ Z.max (ey - prec) emin = emin -> (ex - prec >= emin)%Z /\ Z.max (ex - prec) emin = (ex - prec)%Z \/ (ex - prec < emin)%Z /\ Z.max (ex - prec) emin = emin -> (Z.max (ex - prec) emin < ex)%Z -> (Z.max (ey - prec) emin < ey)%Z -> Z.max (ex - ey - prec) emin = (ex - ey + 1)%Z -> (Z.max (ex - ey - prec') emin' <= ex - ey - ey + Z.max (ey - prec) emin)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (emin' <= emin - prec - 2)%Z -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (emin' <= emin - prec - 2)%Z -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FLT_format beta emin prec x -> FLT_format beta emin prec y -> round_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_eq (FLT_exp emin prec) (FLT_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yexists n : Z, beta = (2 * n)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_div_hyp (FLT_exp emin prec) (FLT_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yy <> 0beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ynow apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin prec)now apply FLT_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yValid_exp (FLT_exp emin' prec')exact Ebeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yexists n : Z, beta = (2 * n)%Znow apply FLT_round_round_div_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yround_round_div_hyp (FLT_exp emin prec) (FLT_exp emin' prec')exact Nzy.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec yy <> 0now apply generic_format_FLT.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) xnow apply generic_format_FLT. Qed. End Double_round_div_FLT. Section Double_round_div_FTZ. Variable emin prec : Z. Variable emin' prec' : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Context { prec_gt_0_' : Prec_gt_0 prec' }.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' <= emin - prec - 2)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FLT_format beta emin prec xFy:FLT_format beta emin prec ygeneric_format beta (FLT_exp emin prec) ybeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin - 1)%Z -> (2 * prec <= prec')%Z -> round_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'(emin' + prec' <= emin - 1)%Z -> (2 * prec <= prec')%Z -> round_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zround_round_div_hyp (fun e : Z => if (e - prec <? emin)%Z then (emin + prec - 1)%Z else (e - prec)%Z) (fun e : Z => if (e - prec' <? emin')%Z then (emin' + prec' - 1)%Z else (e - prec')%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Z(forall ex : Z, ((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Z) /\ (forall ex ey : Z, ((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Z) /\ (forall ex ey : Z, ((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey + 1 - prec <? emin then emin + prec - 1 else ex - ey + 1 - prec) <= ex - ey + 1 + 1)%Z -> ((if ex - ey + 1 - prec' <? emin' then emin' + prec' - 1 else ex - ey + 1 - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Z) /\ (forall ex ey : Z, ((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) + (if ey - prec <? emin then emin + prec - 1 else ey - prec) - ey)%Z) /\ (forall ex ey : Z, ((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> (if (ex - ey - prec <? emin)%Z then (emin + prec - 1)%Z else (ex - ey - prec)%Z) = (ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= ex - ey - ey + (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Z)beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex:Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey + 1 - prec <? emin then emin + prec - 1 else ex - ey + 1 - prec) <= ex - ey + 1 + 1)%Z -> ((if ex - ey + 1 - prec' <? emin' then emin' + prec' - 1 else ex - ey + 1 - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) + (if ey - prec <? emin then emin + prec - 1 else ey - prec) - ey)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> (if (ex - ey - prec <? emin)%Z then (emin + prec - 1)%Z else (ex - ey - prec)%Z) = (ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= ex - ey - ey + (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zdestruct (Z.ltb_spec (ex - prec') emin'); destruct (Z.ltb_spec (ex - prec) emin); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex:Z((if ex - prec' <? emin' then emin' + prec' - 1 else ex - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - 1)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec') emin'); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ey - prec) emin); destruct (Z.ltb_spec (ex - ey + 1 - prec) emin); destruct (Z.ltb_spec (ex - ey + 1 - prec') emin'); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey + 1 - prec <? emin then emin + prec - 1 else ex - ey + 1 - prec) <= ex - ey + 1 + 1)%Z -> ((if ex - ey + 1 - prec' <? emin' then emin' + prec' - 1 else ex - ey + 1 - prec') <= (if ex - prec <? emin then emin + prec - 1 else ex - prec) - ey)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec') emin'); omega.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> ((if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) <= ex - ey)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= (if ex - ey - prec <? emin then emin + prec - 1 else ex - ey - prec) + (if ey - prec <? emin then emin + prec - 1 else ey - prec) - ey)%Zdestruct (Z.ltb_spec (ex - prec) emin); destruct (Z.ltb_spec (ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec) emin); destruct (Z.ltb_spec (ex - ey - prec') emin'); omega. Qed.beta:radixemin, prec, emin', prec':Zprec_gt_0_:(0 < prec)%Zprec_gt_0_':Prec_gt_0 prec'Hemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zex, ey:Z((if ex - prec <? emin then emin + prec - 1 else ex - prec) < ex)%Z -> ((if ey - prec <? emin then emin + prec - 1 else ey - prec) < ey)%Z -> (if (ex - ey - prec <? emin)%Z then (emin + prec - 1)%Z else (ex - ey - prec)%Z) = (ex - ey + 1)%Z -> ((if ex - ey - prec' <? emin' then emin' + prec' - 1 else ex - ey - prec') <= ex - ey - ey + (if ey - prec <? emin then emin + prec - 1 else ey - prec))%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (emin' + prec' <= emin - 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'forall choice1 choice2 : Z -> bool, (exists n : Z, beta = (2 * n)%Z) -> (emin' + prec' <= emin - 1)%Z -> (2 * prec <= prec')%Z -> forall x y : R, y <> 0 -> FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> round_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x / y)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yexists n : Z, beta = (2 * n)%Zbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yy <> 0beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xbeta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) ynow apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin prec)now apply FTZ_exp_valid.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yValid_exp (FTZ_exp emin' prec')exact Ebeta.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yexists n : Z, beta = (2 * n)%Znow apply FTZ_round_round_div_hyp.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yround_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec')exact Nzy.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec yy <> 0now apply generic_format_FTZ.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) xnow apply generic_format_FTZ. Qed. End Double_round_div_FTZ. End Double_round_div. End Double_round.beta:radixemin, prec, emin', prec':Zprec_gt_0_:Prec_gt_0 precprec_gt_0_':Prec_gt_0 prec'choice1, choice2:Z -> boolEbeta:exists n : Z, beta = (2 * n)%ZHemin:(emin' + prec' <= emin - 1)%ZHprec:(2 * prec <= prec')%Zx, y:RNzy:y <> 0Fx:FTZ_format beta emin prec xFy:FTZ_format beta emin prec ygeneric_format beta (FTZ_exp emin prec) y