Digits.v

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This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/

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From Coq Require Import ZArith Zquot.

Require Import Zaux.
Require Import SpecFloatCompat.

Notation digits2_pos := digits2_pos (only parsing).
Notation Zdigits2 := Zdigits2 (only parsing).

Number of bits (radix 2) of a positive integer.

It serves as an upper bound on the number of digits to ensure termination.

Fixpoint digits2_Pnat (n : positive) : nat :=
  match n with
  | xH => O
  | xO p => S (digits2_Pnat p)
  | xI p => S (digits2_Pnat p)
  end.

Theorem digits2_Pnat_correct :
  forall n,
  let d := digits2_Pnat n in
  (Zpower_nat 2 d <= Zpos n < Zpower_nat 2 (S d))%Z.forall n : positive,
let d := digits2_Pnat n in
(Zpower_nat 2 d <= Z.pos n < Zpower_nat 2 (S d))%Z
Proof.forall n : positive,
let d := digits2_Pnat n in
(Zpower_nat 2 d <= Z.pos n < Zpower_nat 2 (S d))%Z
intros n d.n:positive
d:=digits2_Pnat n:nat
(Zpower_nat 2 d <= Z.pos n < Zpower_nat 2 (S d))%Z unfold d.n:positive
d:=digits2_Pnat n:nat
(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n <
 Zpower_nat 2 (S (digits2_Pnat n)))%Z clear.n:positive
(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n <
 Zpower_nat 2 (S (digits2_Pnat n)))%Z
assert (Hp: forall m, (Zpower_nat 2 (S m) = 2 * Zpower_nat 2 m)%Z) by easy.n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n <
 Zpower_nat 2 (S (digits2_Pnat n)))%Z
induction n ; simpl digits2_Pnat.n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(Zpower_nat 2 (S (digits2_Pnat n)) <= Z.pos n~1 <
 Zpower_nat 2 (S (S (digits2_Pnat n))))%Z
n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(Zpower_nat 2 (S (digits2_Pnat n)) <= Z.pos n~0 <
 Zpower_nat 2 (S (S (digits2_Pnat n))))%Z
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 0 <= 1 < Zpower_nat 2 1)%Z
rewrite Zpos_xI, 2!Hp.n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(2 * Zpower_nat 2 (digits2_Pnat n) <=
 2 * Z.pos n + 1 <
 2 * Zpower_nat 2 (S (digits2_Pnat n)))%Z
n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(Zpower_nat 2 (S (digits2_Pnat n)) <= Z.pos n~0 <
 Zpower_nat 2 (S (S (digits2_Pnat n))))%Z
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 0 <= 1 < Zpower_nat 2 1)%Z
omega.n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(Zpower_nat 2 (S (digits2_Pnat n)) <= Z.pos n~0 <
 Zpower_nat 2 (S (S (digits2_Pnat n))))%Z
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 0 <= 1 < Zpower_nat 2 1)%Z
rewrite (Zpos_xO n), 2!Hp.n:positive
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
IHn:(Zpower_nat 2 (digits2_Pnat n) <= Z.pos n < Zpower_nat 2 (S (digits2_Pnat n)))%Z
(2 * Zpower_nat 2 (digits2_Pnat n) <= 2 * Z.pos n <
 2 * Zpower_nat 2 (S (digits2_Pnat n)))%Z
Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 0 <= 1 < Zpower_nat 2 1)%Z
omega.Hp:forall m : nat,
Zpower_nat 2 (S m) = (2 * Zpower_nat 2 m)%Z
(Zpower_nat 2 0 <= 1 < Zpower_nat 2 1)%Z
now split.
Qed.

Section Fcore_digits.

Variable beta : radix.

Definition Zdigit n k := Z.rem (Z.quot n (Zpower beta k)) beta.

Theorem Zdigit_lt :
  forall n k,
  (k < 0)%Z ->
  Zdigit n k = Z0.beta:radix
forall n k : Z, (k < 0)%Z -> Zdigit n k = 0%Z
Proof.beta:radix
forall n k : Z, (k < 0)%Z -> Zdigit n k = 0%Z
intros n [|k|k] Hk ; try easy.beta:radix
n:Z
k:positive
Hk:(Z.neg k < 0)%Z
Zdigit n (Z.neg k) = 0%Z
now case n.
Qed.

Theorem Zdigit_0 :
  forall k, Zdigit 0 k = Z0.beta:radix
forall k : Z, Zdigit 0 k = 0%Z
Proof.beta:radix
forall k : Z, Zdigit 0 k = 0%Z
intros k.beta:radix
k:Z
Zdigit 0 k = 0%Z
unfold Zdigit.beta:radix
k:Z
Z.rem (0 ÷ beta ^ k) beta = 0%Z
rewrite Zquot_0_l.beta:radix
k:Z
Z.rem 0 beta = 0%Z
apply Zrem_0_l.
Qed.

Theorem Zdigit_opp :
  forall n k,
  Zdigit (-n) k = Z.opp (Zdigit n k).beta:radix
forall n k : Z, Zdigit (- n) k = (- Zdigit n k)%Z
Proof.beta:radix
forall n k : Z, Zdigit (- n) k = (- Zdigit n k)%Z
intros n k.beta:radix
n, k:Z
Zdigit (- n) k = (- Zdigit n k)%Z
unfold Zdigit.beta:radix
n, k:Z
Z.rem (- n ÷ beta ^ k) beta =
(- Z.rem (n ÷ beta ^ k) beta)%Z
rewrite Zquot_opp_l.beta:radix
n, k:Z
Z.rem (- (n ÷ beta ^ k)) beta =
(- Z.rem (n ÷ beta ^ k) beta)%Z
apply Zrem_opp_l.
Qed.

Theorem Zdigit_ge_Zpower_pos :
  forall e n,
  (0 <= n < Zpower beta e)%Z ->
  forall k, (e <= k)%Z -> Zdigit n k = Z0.beta:radix
forall e n : Z,
(0 <= n < beta ^ e)%Z ->
forall k : Z, (e <= k)%Z -> Zdigit n k = 0%Z
Proof.beta:radix
forall e n : Z,
(0 <= n < beta ^ e)%Z ->
forall k : Z, (e <= k)%Z -> Zdigit n k = 0%Z
intros e n Hn k Hk.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
Zdigit n k = 0%Z
unfold Zdigit.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
Z.rem (n ÷ beta ^ k) beta = 0%Z
rewrite Z.quot_small.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
Z.rem 0 beta = 0%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= n < beta ^ k)%Z
apply Zrem_0_l.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= n < beta ^ k)%Z
split.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= n)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(n < beta ^ k)%Z
apply Hn.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(n < beta ^ k)%Z
apply Z.lt_le_trans with (1 := proj2 Hn).beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(beta ^ e <= beta ^ k)%Z
replace k with (e + (k - e))%Z by ring.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(beta ^ e <= beta ^ (e + (k - e)))%Z
rewrite Zpower_plus.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(beta ^ e <= beta ^ e * beta ^ (k - e))%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
rewrite <- (Zmult_1_r (beta ^ e)) at 1.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(beta ^ e * 1 <= beta ^ e * beta ^ (k - e))%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
apply Zmult_le_compat_l.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(1 <= beta ^ (k - e))%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
apply (Zlt_le_succ 0).beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 < beta ^ (k - e))%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
apply Zpower_gt_0.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
now apply Zle_minus_le_0.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
apply Zlt_le_weak.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 < beta ^ e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
now apply Z.le_lt_trans with n.beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
generalize (Z.le_lt_trans _ _ _ (proj1 Hn) (proj2 Hn)).beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 < beta ^ e)%Z -> (0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
clear.beta:radix
e:Z
(0 < beta ^ e)%Z -> (0 <= e)%Z
beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
now destruct e as [|e|e].beta:radix
e, n:Z
Hn:(0 <= n < beta ^ e)%Z
k:Z
Hk:(e <= k)%Z
(0 <= k - e)%Z
now apply Zle_minus_le_0.
Qed.

Theorem Zdigit_ge_Zpower :
  forall e n,
  (Z.abs n < Zpower beta e)%Z ->
  forall k, (e <= k)%Z -> Zdigit n k = Z0.beta:radix
forall e n : Z,
(Z.abs n < beta ^ e)%Z ->
forall k : Z, (e <= k)%Z -> Zdigit n k = 0%Z
Proof.beta:radix
forall e n : Z,
(Z.abs n < beta ^ e)%Z ->
forall k : Z, (e <= k)%Z -> Zdigit n k = 0%Z
intros e [|n|n] Hn k.beta:radix
e:Z
Hn:(Z.abs 0 < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit 0 k = 0%Z
beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.pos n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.pos n) k = 0%Z
beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.neg n) k = 0%Z
easy.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.pos n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.pos n) k = 0%Z
beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.neg n) k = 0%Z
apply Zdigit_ge_Zpower_pos.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.pos n) < beta ^ e)%Z
k:Z
(0 <= Z.pos n < beta ^ e)%Z
beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.neg n) k = 0%Z
now split.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
(e <= k)%Z -> Zdigit (Z.neg n) k = 0%Z
intros He.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
Zdigit (Z.neg n) k = 0%Z
change (Zneg n) with (Z.opp (Zpos n)).beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
Zdigit (- Z.pos n) k = 0%Z
rewrite Zdigit_opp.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
(- Zdigit (Z.pos n) k)%Z = 0%Z
rewrite Zdigit_ge_Zpower_pos with (2 := He).beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
(- 0)%Z = 0%Z
beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
(0 <= Z.pos n < beta ^ e)%Z
apply Z.opp_0.beta:radix
e:Z
n:positive
Hn:(Z.abs (Z.neg n) < beta ^ e)%Z
k:Z
He:(e <= k)%Z
(0 <= Z.pos n < beta ^ e)%Z
now split.
Qed.

Theorem Zdigit_not_0_pos :
  forall e n, (0 <= e)%Z ->
  (Zpower beta e <= n < Zpower beta (e + 1))%Z ->
  Zdigit n e <> Z0.beta:radix
forall e n : Z,
(0 <= e)%Z ->
(beta ^ e <= n < beta ^ (e + 1))%Z ->
Zdigit n e <> 0%Z
Proof.beta:radix
forall e n : Z,
(0 <= e)%Z ->
(beta ^ e <= n < beta ^ (e + 1))%Z ->
Zdigit n e <> 0%Z
intros e n He (Hn1,Hn2).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
Zdigit n e <> 0%Z
unfold Zdigit.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
Z.rem (n ÷ beta ^ e) beta <> 0%Z
rewrite <- ZOdiv_mod_mult.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(Z.rem n (beta ^ e * beta) ÷ beta ^ e)%Z <> 0%Z
rewrite Z.rem_small.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(n ÷ beta ^ e)%Z <> 0%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
intros H.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
False
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
apply Zle_not_lt with (1 := Hn1).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(n < beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
rewrite (Z.quot_rem' n (beta ^ e)).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(beta ^ e * (n ÷ beta ^ e) + Z.rem n (beta ^ e) <
 beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
rewrite H, Zmult_0_r, Zplus_0_l.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(Z.rem n (beta ^ e) < beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
apply Zrem_lt_pos_pos.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(0 <= n)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(0 < beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
apply Z.le_trans with (2 := Hn1).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(0 < beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
apply Zpower_ge_0.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
H:(n ÷ beta ^ e)%Z = 0%Z
(0 < beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
now apply Zpower_gt_0.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n < beta ^ e * beta)%Z
split.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= n)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(n < beta ^ e * beta)%Z
apply Z.le_trans with (2 := Hn1).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(0 <= beta ^ e)%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(n < beta ^ e * beta)%Z
apply Zpower_ge_0.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(n < beta ^ e * beta)%Z
replace (beta ^ e * beta)%Z with (beta ^ (e + 1))%Z.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(n < beta ^ (e + 1))%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(beta ^ (e + 1))%Z = (beta ^ e * beta)%Z
exact Hn2.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(beta ^ (e + 1))%Z = (beta ^ e * beta)%Z
rewrite <- (Zmult_1_r beta) at 3.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn1:(beta ^ e <= n)%Z
Hn2:(n < beta ^ (e + 1))%Z
(beta ^ (e + 1))%Z = (beta ^ e * (beta * 1))%Z
now apply (Zpower_plus beta e 1).
Qed.

Theorem Zdigit_not_0 :
  forall e n, (0 <= e)%Z ->
  (Zpower beta e <= Z.abs n < Zpower beta (e + 1))%Z ->
  Zdigit n e <> Z0.beta:radix
forall e n : Z,
(0 <= e)%Z ->
(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z ->
Zdigit n e <> 0%Z
Proof.beta:radix
forall e n : Z,
(0 <= e)%Z ->
(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z ->
Zdigit n e <> 0%Z
intros e n He Hn.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Zdigit n e <> 0%Z
destruct (Zle_or_lt 0 n) as [Hn'|Hn'].beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Hn':(0 <= n)%Z
Zdigit n e <> 0%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
Zdigit n e <> 0%Z
rewrite (Z.abs_eq _ Hn') in Hn.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= n < beta ^ (e + 1))%Z
Hn':(0 <= n)%Z
Zdigit n e <> 0%Z
beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
Zdigit n e <> 0%Z
now apply Zdigit_not_0_pos.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
Zdigit n e <> 0%Z
intros H.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= Z.abs n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
H:Zdigit n e = 0%Z
False
rewrite (Zabs_non_eq n) in Hn by now apply Zlt_le_weak.beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= - n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
H:Zdigit n e = 0%Z
False
apply (Zdigit_not_0_pos _ _ He Hn).beta:radix
e, n:Z
He:(0 <= e)%Z
Hn:(beta ^ e <= - n < beta ^ (e + 1))%Z
Hn':(n < 0)%Z
H:Zdigit n e = 0%Z
Zdigit (- n) e = 0%Z
now rewrite Zdigit_opp, H.
Qed.

Theorem Zdigit_mul_pow :
  forall n k k', (0 <= k')%Z ->
  Zdigit (n * Zpower beta k') k = Zdigit n (k - k').beta:radix
forall n k k' : Z,
(0 <= k')%Z ->
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
Proof.beta:radix
forall n k k' : Z,
(0 <= k')%Z ->
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
intros n k k' Hk'.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
destruct (Zle_or_lt k' k) as [H|H].beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k' <= k)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
revert k H.beta:radix
n, k':Z
Hk':(0 <= k')%Z
forall k : Z,
(k' <= k)%Z ->
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
pattern k' ; apply Zlt_0_ind with (2 := Hk').beta:radix
n, k':Z
Hk':(0 <= k')%Z
forall x : Z,
(forall y : Z,
 (0 <= y < x)%Z ->
 forall k : Z,
 (y <= k)%Z ->
 Zdigit (n * beta ^ y) k = Zdigit n (k - y)) ->
(0 <= x)%Z ->
forall k : Z,
(x <= k)%Z ->
Zdigit (n * beta ^ x) k = Zdigit n (k - x)
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
clear k' Hk'.beta:radix
n:Z
forall x : Z,
(forall y : Z,
 (0 <= y < x)%Z ->
 forall k : Z,
 (y <= k)%Z ->
 Zdigit (n * beta ^ y) k = Zdigit n (k - y)) ->
(0 <= x)%Z ->
forall k : Z,
(x <= k)%Z ->
Zdigit (n * beta ^ x) k = Zdigit n (k - x)
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
intros k' IHk' Hk' k H.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
unfold Zdigit.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
Z.rem (n * beta ^ k' ÷ beta ^ k) beta =
Z.rem (n ÷ beta ^ (k - k')) beta
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply (f_equal (fun x => Z.rem x beta)).beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(n * beta ^ k' ÷ beta ^ k)%Z = (n ÷ beta ^ (k - k'))%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
pattern k at 1 ; replace k with (k - k' + k')%Z by ring.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(n * beta ^ k' ÷ beta ^ (k - k' + k'))%Z =
(n ÷ beta ^ (k - k'))%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zpower_plus with (2 := Hk').beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(n * beta ^ k' ÷ (beta ^ (k - k') * beta ^ k'))%Z =
(n ÷ beta ^ (k - k'))%Z
beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(0 <= k - k')%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply Zquot_mult_cancel_r.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(beta ^ k')%Z <> 0%Z
beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(0 <= k - k')%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply Zgt_not_eq.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(0 < beta ^ k')%Z
beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(0 <= k - k')%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
now apply Zpower_gt_0.beta:radix
n, k':Z
IHk':forall y : Z,
(0 <= y < k')%Z ->
forall k0 : Z,
(y <= k0)%Z ->
Zdigit (n * beta ^ y) k0 = Zdigit n (k0 - y)
Hk':(0 <= k')%Z
k:Z
H:(k' <= k)%Z
(0 <= k - k')%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
now apply Zle_minus_le_0.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
destruct (Zle_or_lt 0 k) as [H0|H0].beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite (Zdigit_lt n) by omega.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Zdigit (n * beta ^ k') k = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
unfold Zdigit.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ k' ÷ beta ^ k) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
replace k' with (k' - k + k)%Z by ring.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k + k) ÷ beta ^ k) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zpower_plus with (2 := H0).beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * (beta ^ (k' - k) * beta ^ k) ÷ beta ^ k)
  beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zmult_assoc, Z_quot_mult.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k)) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
replace (k' - k)%Z with (k' - k - 1 + 1)%Z by ring.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k - 1 + 1)) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zpower_exp by omega.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * (beta ^ (k' - k - 1) * beta ^ 1)) beta =
0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zmult_assoc.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k - 1) * beta ^ 1) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
change (Zpower beta 1) with (beta * 1)%Z.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k - 1) * (beta * 1)) beta =
0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zmult_1_r.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
Z.rem (n * beta ^ (k' - k - 1) * beta) beta = 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply Z_rem_mult.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply Zgt_not_eq.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 < beta ^ k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
now apply Zpower_gt_0.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(0 <= k' - k)%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
apply Zle_minus_le_0.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(0 <= k)%Z
(k <= k')%Z
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
now apply Zlt_le_weak.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit (n * beta ^ k') k = Zdigit n (k - k')
rewrite Zdigit_lt with (1 := H0).beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
0%Z = Zdigit n (k - k')
apply sym_eq.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
Zdigit n (k - k') = 0%Z
apply Zdigit_lt.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
H:(k < k')%Z
H0:(k < 0)%Z
(k - k' < 0)%Z
omega.
Qed.

Theorem Zdigit_div_pow :
  forall n k k', (0 <= k)%Z -> (0 <= k')%Z ->
  Zdigit (Z.quot n (Zpower beta k')) k = Zdigit n (k + k').beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
(0 <= k')%Z ->
Zdigit (n ÷ beta ^ k') k = Zdigit n (k + k')
Proof.beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
(0 <= k')%Z ->
Zdigit (n ÷ beta ^ k') k = Zdigit n (k + k')
intros n k k' Hk Hk'.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
Zdigit (n ÷ beta ^ k') k = Zdigit n (k + k')
unfold Zdigit.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
Z.rem (n ÷ beta ^ k' ÷ beta ^ k) beta =
Z.rem (n ÷ beta ^ (k + k')) beta
rewrite Zquot_Zquot.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
Z.rem (n ÷ (beta ^ k' * beta ^ k)) beta =
Z.rem (n ÷ beta ^ (k + k')) beta
rewrite Zplus_comm.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
Z.rem (n ÷ (beta ^ k' * beta ^ k)) beta =
Z.rem (n ÷ beta ^ (k' + k)) beta
now rewrite Zpower_plus.
Qed.

Theorem Zdigit_mod_pow :
  forall n k k', (k < k')%Z ->
  Zdigit (Z.rem n (Zpower beta k')) k = Zdigit n k.beta:radix
forall n k k' : Z,
(k < k')%Z ->
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
Proof.beta:radix
forall n k k' : Z,
(k < k')%Z ->
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
intros n k k' Hk.beta:radix
n, k, k':Z
Hk:(k < k')%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
destruct (Zle_or_lt 0 k) as [H|H].beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
unfold Zdigit.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem (Z.rem n (beta ^ k') ÷ beta ^ k) beta =
Z.rem (n ÷ beta ^ k) beta
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
rewrite <- 2!ZOdiv_mod_mult.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
(Z.rem (Z.rem n (beta ^ k')) (beta ^ k * beta)
 ÷ beta ^ k)%Z =
(Z.rem n (beta ^ k * beta) ÷ beta ^ k)%Z
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
apply (f_equal (fun x => Z.quot x (beta ^ k))).beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem (Z.rem n (beta ^ k')) (beta ^ k * beta) =
Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
replace k' with (k + 1 + (k' - (k + 1)))%Z by ring.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem (Z.rem n (beta ^ (k + 1 + (k' - (k + 1)))))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
rewrite Zpower_exp by omega.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem
  (Z.rem n (beta ^ (k + 1) * beta ^ (k' - (k + 1))))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
rewrite Zmult_comm.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem
  (Z.rem n (beta ^ (k' - (k + 1)) * beta ^ (k + 1)))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
rewrite Zpower_plus by easy.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem
  (Z.rem n
     (beta ^ (k' - (k + 1)) * (beta ^ k * beta ^ 1)))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
change (Zpower beta 1) with (beta * 1)%Z.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem
  (Z.rem n
     (beta ^ (k' - (k + 1)) * (beta ^ k * (beta * 1))))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
rewrite Zmult_1_r.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(0 <= k)%Z
Z.rem
  (Z.rem n (beta ^ (k' - (k + 1)) * (beta ^ k * beta)))
  (beta ^ k * beta) = Z.rem n (beta ^ k * beta)
beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
apply ZOmod_mod_mult.beta:radix
n, k, k':Z
Hk:(k < k')%Z
H:(k < 0)%Z
Zdigit (Z.rem n (beta ^ k')) k = Zdigit n k
now rewrite 2!Zdigit_lt.
Qed.

Theorem Zdigit_mod_pow_out :
  forall n k k', (0 <= k' <= k)%Z ->
  Zdigit (Z.rem n (Zpower beta k')) k = Z0.beta:radix
forall n k k' : Z,
(0 <= k' <= k)%Z ->
Zdigit (Z.rem n (beta ^ k')) k = 0%Z
Proof.beta:radix
forall n k k' : Z,
(0 <= k' <= k)%Z ->
Zdigit (Z.rem n (beta ^ k')) k = 0%Z
intros n k k' Hk.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
Zdigit (Z.rem n (beta ^ k')) k = 0%Z
unfold Zdigit.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
Z.rem (Z.rem n (beta ^ k') ÷ beta ^ k) beta = 0%Z
rewrite ZOdiv_small_abs.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
Z.rem 0 beta = 0%Z
beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(Z.abs (Z.rem n (beta ^ k')) < beta ^ k)%Z
apply Zrem_0_l.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(Z.abs (Z.rem n (beta ^ k')) < beta ^ k)%Z
apply Z.lt_le_trans with (Zpower beta k').beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(Z.abs (Z.rem n (beta ^ k')) < beta ^ k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k' <= beta ^ k)%Z
rewrite <- (Z.abs_eq (beta ^ k')) at 2 by apply Zpower_ge_0.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(Z.abs (Z.rem n (beta ^ k')) < Z.abs (beta ^ k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k' <= beta ^ k)%Z
apply Zrem_lt.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k')%Z <> 0%Z
beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k' <= beta ^ k)%Z
apply Zgt_not_eq.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(0 < beta ^ k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k' <= beta ^ k)%Z
now apply Zpower_gt_0.beta:radix
n, k, k':Z
Hk:(0 <= k' <= k)%Z
(beta ^ k' <= beta ^ k)%Z
now apply Zpower_le.
Qed.

Fixpoint Zsum_digit f k :=
  match k with
  | O => Z0
  | S k => (Zsum_digit f k + f (Z_of_nat k) * Zpower beta (Z_of_nat k))%Z
  end.

Theorem Zsum_digit_digit :
  forall n k,
  Zsum_digit (Zdigit n) k = Z.rem n (Zpower beta (Z_of_nat k)).beta:radix
forall (n : Z) (k : nat),
Zsum_digit (Zdigit n) k = Z.rem n (beta ^ Z.of_nat k)
Proof.beta:radix
forall (n : Z) (k : nat),
Zsum_digit (Zdigit n) k = Z.rem n (beta ^ Z.of_nat k)
intros n.beta:radix
n:Z
forall k : nat,
Zsum_digit (Zdigit n) k = Z.rem n (beta ^ Z.of_nat k)
induction k.beta:radix
n:Z
Zsum_digit (Zdigit n) 0 = Z.rem n (beta ^ Z.of_nat 0)
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
Zsum_digit (Zdigit n) (S k) =
Z.rem n (beta ^ Z.of_nat (S k))
apply sym_eq.beta:radix
n:Z
Z.rem n (beta ^ Z.of_nat 0) = Zsum_digit (Zdigit n) 0
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
Zsum_digit (Zdigit n) (S k) =
Z.rem n (beta ^ Z.of_nat (S k))
apply Z.rem_1_r.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
Zsum_digit (Zdigit n) (S k) =
Z.rem n (beta ^ Z.of_nat (S k))
simpl Zsum_digit.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Zsum_digit (Zdigit n) k +
 Zdigit n (Z.of_nat k) * beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
rewrite IHk.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem n (beta ^ Z.of_nat k) +
 Zdigit n (Z.of_nat k) * beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
unfold Zdigit.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem n (beta ^ Z.of_nat k) +
 Z.rem (n ÷ beta ^ Z.of_nat k) beta *
 beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
rewrite <- ZOdiv_mod_mult.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem n (beta ^ Z.of_nat k) +
 Z.rem n (beta ^ Z.of_nat k * beta)
 ÷ beta ^ Z.of_nat k * beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
rewrite <- (ZOmod_mod_mult n beta).beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem (Z.rem n (beta * beta ^ Z.of_nat k))
   (beta ^ Z.of_nat k) +
 Z.rem n (beta ^ Z.of_nat k * beta)
 ÷ beta ^ Z.of_nat k * beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
rewrite Zmult_comm.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem (Z.rem n (beta ^ Z.of_nat k * beta))
   (beta ^ Z.of_nat k) +
 Z.rem n (beta ^ Z.of_nat k * beta)
 ÷ beta ^ Z.of_nat k * beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
replace (beta ^ Z_of_nat k * beta)%Z with (Zpower beta (Z_of_nat (S k))).beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(Z.rem (Z.rem n (beta ^ Z.of_nat (S k)))
   (beta ^ Z.of_nat k) +
 Z.rem n (beta ^ Z.of_nat (S k)) ÷ beta ^ Z.of_nat k *
 beta ^ Z.of_nat k)%Z =
Z.rem n (beta ^ Z.of_nat (S k))
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.of_nat (S k))%Z =
(beta ^ Z.of_nat k * beta)%Z
rewrite Zplus_comm, Zmult_comm.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.of_nat k *
 (Z.rem n (beta ^ Z.of_nat (S k)) ÷ beta ^ Z.of_nat k) +
 Z.rem (Z.rem n (beta ^ Z.of_nat (S k)))
   (beta ^ Z.of_nat k))%Z =
Z.rem n (beta ^ Z.of_nat (S k))
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.of_nat (S k))%Z =
(beta ^ Z.of_nat k * beta)%Z
apply sym_eq.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
Z.rem n (beta ^ Z.of_nat (S k)) =
(beta ^ Z.of_nat k *
 (Z.rem n (beta ^ Z.of_nat (S k)) ÷ beta ^ Z.of_nat k) +
 Z.rem (Z.rem n (beta ^ Z.of_nat (S k)))
   (beta ^ Z.of_nat k))%Z
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.of_nat (S k))%Z =
(beta ^ Z.of_nat k * beta)%Z
apply Z.quot_rem'.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.of_nat (S k))%Z =
(beta ^ Z.of_nat k * beta)%Z
rewrite inj_S.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.succ (Z.of_nat k))%Z =
(beta ^ Z.of_nat k * beta)%Z
rewrite <- (Zmult_1_r beta) at 3.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(beta ^ Z.succ (Z.of_nat k))%Z =
(beta ^ Z.of_nat k * (beta * 1))%Z
apply Zpower_plus.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(0 <= Z.of_nat k)%Z
beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(0 <= 1)%Z
apply Zle_0_nat.beta:radix
n:Z
k:nat
IHk:Zsum_digit (Zdigit n) k =
Z.rem n (beta ^ Z.of_nat k)
(0 <= 1)%Z
easy.
Qed.

Theorem Zdigit_ext :
  forall n1 n2,
  (forall k, (0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k) ->
  n1 = n2.beta:radix
forall n1 n2 : Z,
(forall k : Z, (0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k) ->
n1 = n2
Proof.beta:radix
forall n1 n2 : Z,
(forall k : Z, (0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k) ->
n1 = n2
intros n1 n2 H.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n1 = n2
rewrite <- (ZOmod_small_abs n1 (Zpower beta (Z.max (Z.abs n1) (Z.abs n2)))).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.rem n1 (beta ^ Z.max (Z.abs n1) (Z.abs n2)) = n2
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
rewrite <- (ZOmod_small_abs n2 (Zpower beta (Z.max (Z.abs n1) (Z.abs n2)))) at 2.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.rem n1 (beta ^ Z.max (Z.abs n1) (Z.abs n2)) =
Z.rem n2 (beta ^ Z.max (Z.abs n1) (Z.abs n2))
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
replace (Z.max (Z.abs n1) (Z.abs n2)) with (Z_of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2)))).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.rem n1
  (beta
   ^ Z.of_nat
       (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2)))) =
Z.rem n2
  (beta
   ^ Z.of_nat
       (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))))
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
rewrite <- 2!Zsum_digit_digit.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Zsum_digit (Zdigit n1)
  (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Zsum_digit (Zdigit n2)
  (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2)))
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
induction (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Zsum_digit (Zdigit n1) 0 = Zsum_digit (Zdigit n2) 0
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
Zsum_digit (Zdigit n1) (S n) =
Zsum_digit (Zdigit n2) (S n)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
easy.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
Zsum_digit (Zdigit n1) (S n) =
Zsum_digit (Zdigit n2) (S n)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
simpl.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
(Zsum_digit (Zdigit n1) n +
 Zdigit n1 (Z.of_nat n) * beta ^ Z.of_nat n)%Z =
(Zsum_digit (Zdigit n2) n +
 Zdigit n2 (Z.of_nat n) * beta ^ Z.of_nat n)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
rewrite H, IHn.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
(Zsum_digit (Zdigit n2) n +
 Zdigit n2 (Z.of_nat n) * beta ^ Z.of_nat n)%Z =
(Zsum_digit (Zdigit n2) n +
 Zdigit n2 (Z.of_nat n) * beta ^ Z.of_nat n)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
(0 <= Z.of_nat n)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply refl_equal.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
n:nat
IHn:Zsum_digit (Zdigit n1) n =
Zsum_digit (Zdigit n2) n
(0 <= Z.of_nat n)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zle_0_nat.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.of_nat (Z.abs_nat (Z.max (Z.abs n1) (Z.abs n2))) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
rewrite inj_Zabs_nat.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
Z.abs (Z.max (Z.abs n1) (Z.abs n2)) =
Z.max (Z.abs n1) (Z.abs n2)
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.abs_eq.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(0 <= Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.le_trans with (Z.abs n1).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(0 <= Z.abs n1)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 <= Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zabs_pos.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 <= Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.le_max_l.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.lt_le_trans with (Zpower beta (Z.abs n2)).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 < beta ^ Z.abs n2)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(beta ^ Z.abs n2 <= beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zpower_gt_id.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(beta ^ Z.abs n2 <= beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zpower_le.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n2 <= Z.max (Z.abs n1) (Z.abs n2))%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.le_max_r.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.lt_le_trans with (Zpower beta (Z.abs n1)).beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 < beta ^ Z.abs n1)%Z
beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(beta ^ Z.abs n1 <= beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zpower_gt_id.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(beta ^ Z.abs n1 <= beta ^ Z.max (Z.abs n1) (Z.abs n2))%Z
apply Zpower_le.beta:radix
n1, n2:Z
H:forall k : Z,
(0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k
(Z.abs n1 <= Z.max (Z.abs n1) (Z.abs n2))%Z
apply Z.le_max_l.
Qed.

Theorem ZOmod_plus_pow_digit :
  forall u v n, (0 <= u * v)%Z ->
  (forall k, (0 <= k < n)%Z -> Zdigit u k = Z0 \/ Zdigit v k = Z0) ->
  Z.rem (u + v) (Zpower beta n) = (Z.rem u (Zpower beta n) + Z.rem v (Zpower beta n))%Z.beta:radix
forall u v n : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k < n)%Z ->
 Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
Proof.beta:radix
forall u v n : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k < n)%Z ->
 Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
intros u v n Huv Hd.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
destruct (Zle_or_lt 0 n) as [Hn|Hn].beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite Zplus_rem with (1 := Huv).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
Z.rem (Z.rem u (beta ^ n) + Z.rem v (beta ^ n))
  (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply ZOmod_small_abs.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.abs (Z.rem u (beta ^ n) + Z.rem v (beta ^ n)) <
 beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
generalize (Z.le_refl n).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(n <= n)%Z ->
(Z.abs (Z.rem u (beta ^ n) + Z.rem v (beta ^ n)) <
 beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
pattern n at -2 ; rewrite <- Z.abs_eq with (1 := Hn).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.abs n <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.abs n) +
    Z.rem v (beta ^ Z.abs n)) < beta ^ Z.abs n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite <- (inj_Zabs_nat n).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.of_nat (Z.abs_nat n) <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat (Z.abs_nat n)) +
    Z.rem v (beta ^ Z.of_nat (Z.abs_nat n))) <
 beta ^ Z.of_nat (Z.abs_nat n))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
induction (Z.abs_nat n) as [|p IHp].beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.of_nat 0 <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat 0) +
    Z.rem v (beta ^ Z.of_nat 0)) < beta ^ Z.of_nat 0)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
(Z.of_nat (S p) <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat (S p)) +
    Z.rem v (beta ^ Z.of_nat (S p))) <
 beta ^ Z.of_nat (S p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now rewrite 2!Z.rem_1_r.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
(Z.of_nat (S p) <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat (S p)) +
    Z.rem v (beta ^ Z.of_nat (S p))) <
 beta ^ Z.of_nat (S p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite <- 2!Zsum_digit_digit.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
(Z.of_nat (S p) <= n)%Z ->
(Z.abs
   (Zsum_digit (Zdigit u) (S p) +
    Zsum_digit (Zdigit v) (S p)) <
 beta ^ Z.of_nat (S p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
simpl Zsum_digit.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
(Z.of_nat (S p) <= n)%Z ->
(Z.abs
   (Zsum_digit (Zdigit u) p +
    Zdigit u (Z.of_nat p) * beta ^ Z.of_nat p +
    (Zsum_digit (Zdigit v) p +
     Zdigit v (Z.of_nat p) * beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat (S p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite inj_S.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
(Z.succ (Z.of_nat p) <= n)%Z ->
(Z.abs
   (Zsum_digit (Zdigit u) p +
    Zdigit u (Z.of_nat p) * beta ^ Z.of_nat p +
    (Zsum_digit (Zdigit v) p +
     Zdigit v (Z.of_nat p) * beta ^ Z.of_nat p)) <
 beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
intros Hn'.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Zsum_digit (Zdigit u) p +
    Zdigit u (Z.of_nat p) * beta ^ Z.of_nat p +
    (Zsum_digit (Zdigit v) p +
     Zdigit v (Z.of_nat p) * beta ^ Z.of_nat p)) <
 beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
replace (Zsum_digit (Zdigit u) p + Zdigit u (Z_of_nat p) * beta ^ Z_of_nat p +
  (Zsum_digit (Zdigit v) p + Zdigit v (Z_of_nat p) * beta ^ Z_of_nat p))%Z with
  (Zsum_digit (Zdigit u) p + Zsum_digit (Zdigit v) p +
  (Zdigit u (Z_of_nat p) + Zdigit v (Z_of_nat p)) * beta ^ Z_of_nat p)%Z by ring.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Zsum_digit (Zdigit u) p + Zsum_digit (Zdigit v) p +
    (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) < beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply (Z.le_lt_trans _ _ _ (Z.abs_triangle _ _)).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Zsum_digit (Zdigit u) p + Zsum_digit (Zdigit v) p) +
 Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) < beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
replace (beta ^ Z.succ (Z_of_nat p))%Z with (beta ^ Z_of_nat p + (beta - 1) * beta ^ Z_of_nat p)%Z.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Zsum_digit (Zdigit u) p + Zsum_digit (Zdigit v) p) +
 Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) <
 beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zplus_lt_le_compat.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Zsum_digit (Zdigit u) p + Zsum_digit (Zdigit v) p) <
 beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) <=
 (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite 2!Zsum_digit_digit.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) < beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) <=
 (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply IHp.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.of_nat p <= n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) <=
 (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now apply Zle_succ_le.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs
   ((Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
    beta ^ Z.of_nat p) <=
 (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite Zabs_Zmult.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
 Z.abs (beta ^ Z.of_nat p) <=
 (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite (Z.abs_eq (beta ^ Z_of_nat p)) by apply Zpower_ge_0.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) *
 beta ^ Z.of_nat p <= (beta - 1) * beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zmult_le_compat_r.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <=
 beta - 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= beta ^ Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z 2: apply Zpower_ge_0.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <=
 beta - 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zlt_succ_le.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
assert (forall u v, Z.abs (Zdigit u v) < Z.succ (beta -  1))%Z.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
clear ; intros n k.beta:radix
n, k:Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
assert (0 < beta)%Z.beta:radix
n, k:Z
(0 < beta)%Z
beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Z.lt_le_trans with 2%Z.beta:radix
n, k:Z
(0 < 2)%Z
beta:radix
n, k:Z
(2 <= beta)%Z
beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply refl_equal.beta:radix
n, k:Z
(2 <= beta)%Z
beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zle_bool_imp_le.beta:radix
n, k:Z
(2 <=? beta)%Z = true
beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply beta.beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
replace (Z.succ (beta - 1)) with (Z.abs beta).beta:radix
n, k:Z
H:(0 < beta)%Z
(Z.abs (Zdigit n k) < Z.abs beta)%Z
beta:radix
n, k:Z
H:(0 < beta)%Z
Z.abs beta = Z.succ (beta - 1)
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zrem_lt.beta:radix
n, k:Z
H:(0 < beta)%Z
beta <> 0%Z
beta:radix
n, k:Z
H:(0 < beta)%Z
Z.abs beta = Z.succ (beta - 1)
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now apply Zgt_not_eq.beta:radix
n, k:Z
H:(0 < beta)%Z
Z.abs beta = Z.succ (beta - 1)
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite Z.abs_eq.beta:radix
n, k:Z
H:(0 < beta)%Z
beta = Z.succ (beta - 1)
beta:radix
n, k:Z
H:(0 < beta)%Z
(0 <= beta)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zsucc_pred.beta:radix
n, k:Z
H:(0 < beta)%Z
(0 <= beta)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now apply Zlt_le_weak.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
assert (0 <= Z_of_nat p < n)%Z.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(0 <= Z.of_nat p < n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
split.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(0 <= Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.of_nat p < n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zle_0_nat.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(Z.of_nat p < n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Z.gt_lt.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
(n > Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now apply Zle_succ_gt.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
(Z.abs (Zdigit u (Z.of_nat p) + Zdigit v (Z.of_nat p)) <
 Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
destruct (Hd (Z_of_nat p) H0) as [K|K] ; rewrite K.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
K:Zdigit u (Z.of_nat p) = 0%Z
(Z.abs (0 + Zdigit v (Z.of_nat p)) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
K:Zdigit v (Z.of_nat p) = 0%Z
(Z.abs (Zdigit u (Z.of_nat p) + 0) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply H.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
K:Zdigit v (Z.of_nat p) = 0%Z
(Z.abs (Zdigit u (Z.of_nat p) + 0) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite Zplus_0_r.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
H:forall u0 v0 : Z,
(Z.abs (Zdigit u0 v0) < Z.succ (beta - 1))%Z
H0:(0 <= Z.of_nat p < n)%Z
K:Zdigit v (Z.of_nat p) = 0%Z
(Z.abs (Zdigit u (Z.of_nat p)) < Z.succ (beta - 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply H.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ Z.succ (Z.of_nat p))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
unfold Z.succ.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p + (beta - 1) * beta ^ Z.of_nat p)%Z =
(beta ^ (Z.of_nat p + 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
ring_simplify.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p * beta)%Z =
(beta ^ (Z.of_nat p + 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
rewrite Zpower_plus.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p * beta)%Z =
(beta ^ Z.of_nat p * beta ^ 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
change (beta ^1)%Z with (beta * 1)%Z.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(beta ^ Z.of_nat p * beta)%Z =
(beta ^ Z.of_nat p * (beta * 1))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
now rewrite Zmult_1_r.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= Z.of_nat p)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
apply Zle_0_nat.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
p:nat
IHp:(Z.of_nat p <= n)%Z ->
(Z.abs
   (Z.rem u (beta ^ Z.of_nat p) +
    Z.rem v (beta ^ Z.of_nat p)) <
 beta ^ Z.of_nat p)%Z
Hn':(Z.succ (Z.of_nat p) <= n)%Z
(0 <= 1)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
easy.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
Z.rem (u + v) (beta ^ n) =
(Z.rem u (beta ^ n) + Z.rem v (beta ^ n))%Z
destruct n as [|n|n] ; try easy.beta:radix
u, v:Z
n:positive
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < Z.neg n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(Z.neg n < 0)%Z
Z.rem (u + v) (beta ^ Z.neg n) =
(Z.rem u (beta ^ Z.neg n) + Z.rem v (beta ^ Z.neg n))%Z
now rewrite 3!Zrem_0_r.
Qed.

Theorem ZOdiv_plus_pow_digit :
  forall u v n, (0 <= u * v)%Z ->
  (forall k, (0 <= k < n)%Z -> Zdigit u k = Z0 \/ Zdigit v k = Z0) ->
  Z.quot (u + v) (Zpower beta n) = (Z.quot u (Zpower beta n) + Z.quot v (Zpower beta n))%Z.beta:radix
forall u v n : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k < n)%Z ->
 Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
((u + v) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n)%Z
Proof.beta:radix
forall u v n : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k < n)%Z ->
 Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
((u + v) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n)%Z
intros u v n Huv Hd.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
((u + v) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n)%Z
rewrite <- (Zplus_0_r (Z.quot u (Zpower beta n) + Z.quot v (Zpower beta n))).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
((u + v) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n + 0)%Z
rewrite ZOdiv_plus with (1 := Huv).beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
(u ÷ beta ^ n + v ÷ beta ^ n +
 (Z.rem u (beta ^ n) + Z.rem v (beta ^ n)) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n + 0)%Z
rewrite <- ZOmod_plus_pow_digit by assumption.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
(u ÷ beta ^ n + v ÷ beta ^ n +
 Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z =
(u ÷ beta ^ n + v ÷ beta ^ n + 0)%Z
apply f_equal.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
destruct (Zle_or_lt 0 n) as [Hn|Hn].beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
apply ZOdiv_small_abs.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.abs (Z.rem (u + v) (beta ^ n)) < beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
rewrite <- Z.abs_eq.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(Z.abs (Z.rem (u + v) (beta ^ n)) < Z.abs (beta ^ n))%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(0 <= beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
apply Zrem_lt.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(beta ^ n)%Z <> 0%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(0 <= beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
apply Zgt_not_eq.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(0 < beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(0 <= beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
now apply Zpower_gt_0.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(0 <= n)%Z
(0 <= beta ^ n)%Z
beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
apply Zpower_ge_0.beta:radix
u, v, n:Z
Huv:(0 <= u * v)%Z
Hd:forall k : Z,
(0 <= k < n)%Z ->
Zdigit u k = 0%Z \/ Zdigit v k = 0%Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
clear -Hn.beta:radix
u, v, n:Z
Hn:(n < 0)%Z
(Z.rem (u + v) (beta ^ n) ÷ beta ^ n)%Z = 0%Z
destruct n as [|n|n] ; try easy.beta:radix
u, v:Z
n:positive
Hn:(Z.neg n < 0)%Z
(Z.rem (u + v) (beta ^ Z.neg n) ÷ beta ^ Z.neg n)%Z =
0%Z
apply Zquot_0_r.
Qed.

Theorem Zdigit_plus :
  forall u v, (0 <= u * v)%Z ->
  (forall k, (0 <= k)%Z -> Zdigit u k = Z0 \/ Zdigit v k = Z0) ->
  forall k,
  Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z.beta:radix
forall u v : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k)%Z -> Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
forall k : Z,
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
Proof.beta:radix
forall u v : Z,
(0 <= u * v)%Z ->
(forall k : Z,
 (0 <= k)%Z -> Zdigit u k = 0%Z \/ Zdigit v k = 0%Z) ->
forall k : Z,
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
intros u v Huv Hd k.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
destruct (Zle_or_lt 0 k) as [Hk|Hk].beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
unfold Zdigit.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Z.rem ((u + v) ÷ beta ^ k) beta =
(Z.rem (u ÷ beta ^ k) beta + Z.rem (v ÷ beta ^ k) beta)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
rewrite ZOdiv_plus_pow_digit with (1 := Huv).beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Z.rem (u ÷ beta ^ k + v ÷ beta ^ k) beta =
(Z.rem (u ÷ beta ^ k) beta + Z.rem (v ÷ beta ^ k) beta)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
rewrite <- (Zmult_1_r beta) at 3 5 7.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Z.rem (u ÷ beta ^ k + v ÷ beta ^ k) (beta * 1) =
(Z.rem (u ÷ beta ^ k) (beta * 1) +
 Z.rem (v ÷ beta ^ k) (beta * 1))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
change (beta * 1)%Z with (beta ^1)%Z.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Z.rem (u ÷ beta ^ k + v ÷ beta ^ k) (beta ^ 1) =
(Z.rem (u ÷ beta ^ k) (beta ^ 1) +
 Z.rem (v ÷ beta ^ k) (beta ^ 1))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply ZOmod_plus_pow_digit.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u ÷ beta ^ k * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zsame_sign_trans_weak with v.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
v = 0%Z -> (v ÷ beta ^ k)%Z = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u ÷ beta ^ k * v)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
intros Zv ; rewrite Zv.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Zv:v = 0%Z
(0 ÷ beta ^ k)%Z = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u ÷ beta ^ k * v)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zquot_0_l.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u ÷ beta ^ k * v)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
rewrite Zmult_comm.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (u ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zsame_sign_trans_weak with u.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
u = 0%Z -> (u ÷ beta ^ k)%Z = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * u)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u * (u ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
intros Zu ; rewrite Zu.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
Zu:u = 0%Z
(0 ÷ beta ^ k)%Z = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * u)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u * (u ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zquot_0_l.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * u)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u * (u ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
now rewrite Zmult_comm.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= u * (u ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zsame_sign_odiv.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= beta ^ k)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zpower_ge_0.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= v * (v ÷ beta ^ k))%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zsame_sign_odiv.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
(0 <= beta ^ k)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Zpower_ge_0.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < 1)%Z ->
Zdigit (u ÷ beta ^ k) k0 = 0%Z \/
Zdigit (v ÷ beta ^ k) k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
intros k' (Hk1,Hk2).beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
k':Z
Hk1:(0 <= k')%Z
Hk2:(k' < 1)%Z
Zdigit (u ÷ beta ^ k) k' = 0%Z \/
Zdigit (v ÷ beta ^ k) k' = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
rewrite 2!Zdigit_div_pow by assumption.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
k':Z
Hk1:(0 <= k')%Z
Hk2:(k' < 1)%Z
Zdigit u (k' + k) = 0%Z \/ Zdigit v (k' + k) = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
apply Hd.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
k':Z
Hk1:(0 <= k')%Z
Hk2:(k' < 1)%Z
(0 <= k' + k)%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
now apply Zplus_le_0_compat.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0 < k)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
intros k' (Hk1,Hk2).beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(0 <= k)%Z
k':Z
Hk1:(0 <= k')%Z
Hk2:(k' < k)%Z
Zdigit u k' = 0%Z \/ Zdigit v k' = 0%Z
beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
now apply Hd.beta:radix
u, v:Z
Huv:(0 <= u * v)%Z
Hd:forall k0 : Z,
(0 <= k0)%Z ->
Zdigit u k0 = 0%Z \/ Zdigit v k0 = 0%Z
k:Z
Hk:(k < 0)%Z
Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z
now rewrite 3!Zdigit_lt.
Qed.

Left and right shifts

Definition Zscale n k :=
  if Zle_bool 0 k then (n * Zpower beta k)%Z else Z.quot n (Zpower beta (-k)).

Theorem Zdigit_scale :
  forall n k k', (0 <= k')%Z ->
  Zdigit (Zscale n k) k' = Zdigit n (k' - k).beta:radix
forall n k k' : Z,
(0 <= k')%Z ->
Zdigit (Zscale n k) k' = Zdigit n (k' - k)
Proof.beta:radix
forall n k k' : Z,
(0 <= k')%Z ->
Zdigit (Zscale n k) k' = Zdigit n (k' - k)
intros n k k' Hk'.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Zdigit (Zscale n k) k' = Zdigit n (k' - k)
unfold Zscale.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Zdigit
  (if (0 <=? k)%Z
   then (n * beta ^ k)%Z
   else (n ÷ beta ^ (- k))%Z) k' = Zdigit n (k' - k)
case Zle_bool_spec ; intros Hk.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Hk:(0 <= k)%Z
Zdigit (n * beta ^ k) k' = Zdigit n (k' - k)
beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Hk:(k < 0)%Z
Zdigit (n ÷ beta ^ (- k)) k' = Zdigit n (k' - k)
now apply Zdigit_mul_pow.beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Hk:(k < 0)%Z
Zdigit (n ÷ beta ^ (- k)) k' = Zdigit n (k' - k)
apply Zdigit_div_pow with (1 := Hk').beta:radix
n, k, k':Z
Hk':(0 <= k')%Z
Hk:(k < 0)%Z
(0 <= - k)%Z
omega.
Qed.

Theorem Zscale_0 :
  forall k,
  Zscale 0 k = Z0.beta:radix
forall k : Z, Zscale 0 k = 0%Z
Proof.beta:radix
forall k : Z, Zscale 0 k = 0%Z
intros k.beta:radix
k:Z
Zscale 0 k = 0%Z
unfold Zscale.beta:radix
k:Z
(if (0 <=? k)%Z
 then (0 * beta ^ k)%Z
 else (0 ÷ beta ^ (- k))%Z) = 0%Z
case Zle_bool.beta:radix
k:Z
(0 * beta ^ k)%Z = 0%Z
beta:radix
k:Z
(0 ÷ beta ^ (- k))%Z = 0%Z
apply Zmult_0_l.beta:radix
k:Z
(0 ÷ beta ^ (- k))%Z = 0%Z
apply Zquot_0_l.
Qed.

Theorem Zsame_sign_scale :
  forall n k,
  (0 <= n * Zscale n k)%Z.beta:radix
forall n k : Z, (0 <= n * Zscale n k)%Z
Proof.beta:radix
forall n k : Z, (0 <= n * Zscale n k)%Z
intros n k.beta:radix
n, k:Z
(0 <= n * Zscale n k)%Z
unfold Zscale.beta:radix
n, k:Z
(0 <=
 n *
 (if 0 <=? k then n * beta ^ k else n ÷ beta ^ (- k)))%Z
case Zle_bool_spec ; intros Hk.beta:radix
n, k:Z
Hk:(0 <= k)%Z
(0 <= n * (n * beta ^ k))%Z
beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= n * (n ÷ beta ^ (- k)))%Z
rewrite Zmult_assoc.beta:radix
n, k:Z
Hk:(0 <= k)%Z
(0 <= n * n * beta ^ k)%Z
beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= n * (n ÷ beta ^ (- k)))%Z
apply Zmult_le_0_compat.beta:radix
n, k:Z
Hk:(0 <= k)%Z
(0 <= n * n)%Z
beta:radix
n, k:Z
Hk:(0 <= k)%Z
(0 <= beta ^ k)%Z
beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= n * (n ÷ beta ^ (- k)))%Z
apply Zsame_sign_imp ; apply Zlt_le_weak.beta:radix
n, k:Z
Hk:(0 <= k)%Z
(0 <= beta ^ k)%Z
beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= n * (n ÷ beta ^ (- k)))%Z
apply Zpower_ge_0.beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= n * (n ÷ beta ^ (- k)))%Z
apply Zsame_sign_odiv.beta:radix
n, k:Z
Hk:(k < 0)%Z
(0 <= beta ^ (- k))%Z
apply Zpower_ge_0.
Qed.

Theorem Zscale_mul_pow :
  forall n k k', (0 <= k)%Z ->
  Zscale (n * Zpower beta k) k' = Zscale n (k + k').beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
Zscale (n * beta ^ k) k' = Zscale n (k + k')
Proof.beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
Zscale (n * beta ^ k) k' = Zscale n (k + k')
intros n k k' Hk.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Zscale (n * beta ^ k) k' = Zscale n (k + k')
unfold Zscale.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
(if (0 <=? k')%Z
 then (n * beta ^ k * beta ^ k')%Z
 else (n * beta ^ k ÷ beta ^ (- k'))%Z) =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
case Zle_bool_spec ; intros Hk'.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(n * beta ^ k * beta ^ k')%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
rewrite Zle_bool_true.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(n * beta ^ k * beta ^ k')%Z = (n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(0 <= k + k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
rewrite <- Zmult_assoc.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(n * (beta ^ k * beta ^ k'))%Z =
(n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(0 <= k + k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
apply f_equal.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(beta ^ k * beta ^ k')%Z = (beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(0 <= k + k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
now rewrite Zpower_plus.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(0 <= k')%Z
(0 <= k + k')%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
now apply Zplus_le_0_compat.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(if (0 <=? k + k')%Z
 then (n * beta ^ (k + k'))%Z
 else (n ÷ beta ^ (- (k + k')))%Z)
case Zle_bool_spec ; intros Hk''.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
pattern k at 1 ; replace k with (k + k' + -k')%Z by ring.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
(n * beta ^ (k + k' + - k') ÷ beta ^ (- k'))%Z =
(n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
assert (0 <= -k')%Z by omega.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(n * beta ^ (k + k' + - k') ÷ beta ^ (- k'))%Z =
(n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
rewrite Zpower_plus by easy.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(n * (beta ^ (k + k') * beta ^ (- k')) ÷ beta ^ (- k'))%Z =
(n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
rewrite Zmult_assoc, Z_quot_mult.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(n * beta ^ (k + k'))%Z = (n * beta ^ (k + k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(beta ^ (- k'))%Z <> 0%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
apply refl_equal.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(beta ^ (- k'))%Z <> 0%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
apply Zgt_not_eq.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(0 <= k + k')%Z
H:(0 <= - k')%Z
(0 < beta ^ (- k'))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
now apply Zpower_gt_0.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- k'))%Z =
(n ÷ beta ^ (- (k + k')))%Z
replace (-k')%Z with (-(k+k') + k)%Z by ring.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ beta ^ (- (k + k') + k))%Z =
(n ÷ beta ^ (- (k + k')))%Z
rewrite Zpower_plus with (2 := Hk).beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(n * beta ^ k ÷ (beta ^ (- (k + k')) * beta ^ k))%Z =
(n ÷ beta ^ (- (k + k')))%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(0 <= - (k + k'))%Z
apply Zquot_mult_cancel_r.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(beta ^ k)%Z <> 0%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(0 <= - (k + k'))%Z
apply Zgt_not_eq.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(0 < beta ^ k)%Z
beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(0 <= - (k + k'))%Z
now apply Zpower_gt_0.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Hk':(k' < 0)%Z
Hk'':(k + k' < 0)%Z
(0 <= - (k + k'))%Z
omega.
Qed.

Theorem Zscale_scale :
  forall n k k', (0 <= k)%Z ->
  Zscale (Zscale n k) k' = Zscale n (k + k').beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
Zscale (Zscale n k) k' = Zscale n (k + k')
Proof.beta:radix
forall n k k' : Z,
(0 <= k)%Z ->
Zscale (Zscale n k) k' = Zscale n (k + k')
intros n k k' Hk.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Zscale (Zscale n k) k' = Zscale n (k + k')
unfold Zscale at 2.beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Zscale
  (if (0 <=? k)%Z
   then (n * beta ^ k)%Z
   else (n ÷ beta ^ (- k))%Z) k' = Zscale n (k + k')
rewrite Zle_bool_true with (1 := Hk).beta:radix
n, k, k':Z
Hk:(0 <= k)%Z
Zscale (n * beta ^ k) k' = Zscale n (k + k')
now apply Zscale_mul_pow.
Qed.

Slice of an integer

Definition Zslice n k1 k2 :=
  if Zle_bool 0 k2 then Z.rem (Zscale n (-k1)) (Zpower beta k2) else Z0.

Theorem Zdigit_slice :
  forall n k1 k2 k, (0 <= k < k2)%Z ->
  Zdigit (Zslice n k1 k2) k = Zdigit n (k1 + k).beta:radix
forall n k1 k2 k : Z,
(0 <= k < k2)%Z ->
Zdigit (Zslice n k1 k2) k = Zdigit n (k1 + k)
Proof.beta:radix
forall n k1 k2 k : Z,
(0 <= k < k2)%Z ->
Zdigit (Zslice n k1 k2) k = Zdigit n (k1 + k)
intros n k1 k2 k Hk.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit (Zslice n k1 k2) k = Zdigit n (k1 + k)
unfold Zslice.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit
  (if (0 <=? k2)%Z
   then Z.rem (Zscale n (- k1)) (beta ^ k2)
   else 0%Z) k = Zdigit n (k1 + k)
rewrite Zle_bool_true.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit (Z.rem (Zscale n (- k1)) (beta ^ k2)) k =
Zdigit n (k1 + k)
beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
(0 <= k2)%Z
rewrite Zdigit_mod_pow by apply Hk.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit (Zscale n (- k1)) k = Zdigit n (k1 + k)
beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
(0 <= k2)%Z
rewrite Zdigit_scale by apply Hk.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit n (k - - k1) = Zdigit n (k1 + k)
beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
(0 <= k2)%Z
unfold Zminus.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
Zdigit n (k + - - k1) = Zdigit n (k1 + k)
beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
(0 <= k2)%Z
now rewrite Z.opp_involutive, Zplus_comm.beta:radix
n, k1, k2, k:Z
Hk:(0 <= k < k2)%Z
(0 <= k2)%Z
omega.
Qed.

Theorem Zdigit_slice_out :
  forall n k1 k2 k, (k2 <= k)%Z ->
  Zdigit (Zslice n k1 k2) k = Z0.beta:radix
forall n k1 k2 k : Z,
(k2 <= k)%Z -> Zdigit (Zslice n k1 k2) k = 0%Z
Proof.beta:radix
forall n k1 k2 k : Z,
(k2 <= k)%Z -> Zdigit (Zslice n k1 k2) k = 0%Z
intros n k1 k2 k Hk.beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Zdigit (Zslice n k1 k2) k = 0%Z
unfold Zslice.beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Zdigit
  (if (0 <=? k2)%Z
   then Z.rem (Zscale n (- k1)) (beta ^ k2)
   else 0%Z) k = 0%Z
case Zle_bool_spec ; intros Hk2.beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Hk2:(0 <= k2)%Z
Zdigit (Z.rem (Zscale n (- k1)) (beta ^ k2)) k = 0%Z
beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Hk2:(k2 < 0)%Z
Zdigit 0 k = 0%Z
apply Zdigit_mod_pow_out.beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Hk2:(0 <= k2)%Z
(0 <= k2 <= k)%Z
beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Hk2:(k2 < 0)%Z
Zdigit 0 k = 0%Z
now split.beta:radix
n, k1, k2, k:Z
Hk:(k2 <= k)%Z
Hk2:(k2 < 0)%Z
Zdigit 0 k = 0%Z
apply Zdigit_0.
Qed.

Theorem Zslice_0 :
  forall k k',
  Zslice 0 k k' = Z0.beta:radix
forall k k' : Z, Zslice 0 k k' = 0%Z
Proof.beta:radix
forall k k' : Z, Zslice 0 k k' = 0%Z
intros k k'.beta:radix
k, k':Z
Zslice 0 k k' = 0%Z
unfold Zslice.beta:radix
k, k':Z
(if (0 <=? k')%Z
 then Z.rem (Zscale 0 (- k)) (beta ^ k')
 else 0%Z) = 0%Z
case Zle_bool.beta:radix
k, k':Z
Z.rem (Zscale 0 (- k)) (beta ^ k') = 0%Z
beta:radix
k, k':Z
0%Z = 0%Z
rewrite Zscale_0.beta:radix
k, k':Z
Z.rem 0 (beta ^ k') = 0%Z
beta:radix
k, k':Z
0%Z = 0%Z
apply Zrem_0_l.beta:radix
k, k':Z
0%Z = 0%Z
apply refl_equal.
Qed.

Theorem Zsame_sign_slice :
  forall n k k',
  (0 <= n * Zslice n k k')%Z.beta:radix
forall n k k' : Z, (0 <= n * Zslice n k k')%Z
Proof.beta:radix
forall n k k' : Z, (0 <= n * Zslice n k k')%Z
intros n k k'.beta:radix
n, k, k':Z
(0 <= n * Zslice n k k')%Z
unfold Zslice.beta:radix
n, k, k':Z
(0 <=
 n *
 (if 0 <=? k'
  then Z.rem (Zscale n (- k)) (beta ^ k')
  else 0))%Z
case Zle_bool.beta:radix
n, k, k':Z
(0 <= n * Z.rem (Zscale n (- k)) (beta ^ k'))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
apply Zsame_sign_trans_weak with (Zscale n (-k)).beta:radix
n, k, k':Z
Zscale n (- k) = 0%Z ->
Z.rem (Zscale n (- k)) (beta ^ k') = 0%Z
beta:radix
n, k, k':Z
(0 <= n * Zscale n (- k))%Z
beta:radix
n, k, k':Z
(0 <=
 Zscale n (- k) * Z.rem (Zscale n (- k)) (beta ^ k'))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
intros H ; rewrite H.beta:radix
n, k, k':Z
H:Zscale n (- k) = 0%Z
Z.rem 0 (beta ^ k') = 0%Z
beta:radix
n, k, k':Z
(0 <= n * Zscale n (- k))%Z
beta:radix
n, k, k':Z
(0 <=
 Zscale n (- k) * Z.rem (Zscale n (- k)) (beta ^ k'))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
apply Zrem_0_l.beta:radix
n, k, k':Z
(0 <= n * Zscale n (- k))%Z
beta:radix
n, k, k':Z
(0 <=
 Zscale n (- k) * Z.rem (Zscale n (- k)) (beta ^ k'))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
apply Zsame_sign_scale.beta:radix
n, k, k':Z
(0 <=
 Zscale n (- k) * Z.rem (Zscale n (- k)) (beta ^ k'))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
rewrite Zmult_comm.beta:radix
n, k, k':Z
(0 <=
 Z.rem (Zscale n (- k)) (beta ^ k') * Zscale n (- k))%Z
beta:radix
n, k, k':Z
(0 <= n * 0)%Z
apply Zrem_sgn2.beta:radix
n, k, k':Z
(0 <= n * 0)%Z
now rewrite Zmult_0_r.
Qed.

Theorem Zslice_slice :
  forall n k1 k2 k1' k2', (0 <= k1' <= k2)%Z ->
  Zslice (Zslice n k1 k2) k1' k2' = Zslice n (k1 + k1') (Z.min (k2 - k1') k2').beta:radix
forall n k1 k2 k1' k2' : Z,
(0 <= k1' <= k2)%Z ->
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
Proof.beta:radix
forall n k1 k2 k1' k2' : Z,
(0 <= k1' <= k2)%Z ->
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
intros n k1 k2 k1' k2' Hk1'.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
destruct (Zle_or_lt 0 k2') as [Hk2'|Hk2'].beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
apply Zdigit_ext.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
forall k : Z,
(0 <= k)%Z ->
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
intros k Hk.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
destruct (Zle_or_lt (Z.min (k2 - k1') k2') k) as [Hk'|Hk'].beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
rewrite (Zdigit_slice_out n (k1 + k1')) with (1 := Hk').beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k = 0%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
destruct (Zle_or_lt k2' k) as [Hk''|Hk''].beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Hk'':(k2' <= k)%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k = 0%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Hk'':(k < k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k = 0%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
now apply Zdigit_slice_out.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Hk'':(k < k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k = 0%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
rewrite Zdigit_slice by now split.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Hk'':(k < k2')%Z
Zdigit (Zslice n k1 k2) (k1' + k) = 0%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
apply Zdigit_slice_out.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(Z.min (k2 - k1') k2' <= k)%Z
Hk'':(k < k2')%Z
(k2 <= k1' + k)%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
zify ; omega.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice (Zslice n k1 k2) k1' k2') k =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
rewrite Zdigit_slice by (zify ; omega).beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice n k1 k2) (k1' + k) =
Zdigit (Zslice n (k1 + k1') (Z.min (k2 - k1') k2')) k
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
rewrite (Zdigit_slice n (k1 + k1')) by now split.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit (Zslice n k1 k2) (k1' + k) =
Zdigit n (k1 + k1' + k)
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
rewrite Zdigit_slice.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
Zdigit n (k1 + (k1' + k)) = Zdigit n (k1 + k1' + k)
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
(0 <= k1' + k < k2)%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
now rewrite Zplus_assoc.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(0 <= k2')%Z
k:Z
Hk:(0 <= k)%Z
Hk':(k < Z.min (k2 - k1') k2')%Z
(0 <= k1' + k < k2)%Z
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
zify ; omega.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
Zslice (Zslice n k1 k2) k1' k2' =
Zslice n (k1 + k1') (Z.min (k2 - k1') k2')
unfold Zslice.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
(if (0 <=? k2')%Z
 then
  Z.rem
    (Zscale
       (if (0 <=? k2)%Z
        then Z.rem (Zscale n (- k1)) (beta ^ k2)
        else 0%Z) (- k1')) (beta ^ k2')
 else 0%Z) =
(if (0 <=? Z.min (k2 - k1') k2')%Z
 then
  Z.rem (Zscale n (- (k1 + k1')))
    (beta ^ Z.min (k2 - k1') k2')
 else 0%Z)
rewrite Z.min_r.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
(if (0 <=? k2')%Z
 then
  Z.rem
    (Zscale
       (if (0 <=? k2)%Z
        then Z.rem (Zscale n (- k1)) (beta ^ k2)
        else 0%Z) (- k1')) (beta ^ k2')
 else 0%Z) =
(if (0 <=? k2')%Z
 then Z.rem (Zscale n (- (k1 + k1'))) (beta ^ k2')
 else 0%Z)
beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
(k2' <= k2 - k1')%Z
now rewrite Zle_bool_false.beta:radix
n, k1, k2, k1', k2':Z
Hk1':(0 <= k1' <= k2)%Z
Hk2':(k2' < 0)%Z
(k2' <= k2 - k1')%Z
omega.
Qed.

Theorem Zslice_mul_pow :
  forall n k k1 k2, (0 <= k)%Z ->
  Zslice (n * Zpower beta k) k1 k2 = Zslice n (k1 - k) k2.beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
Zslice (n * beta ^ k) k1 k2 = Zslice n (k1 - k) k2
Proof.beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
Zslice (n * beta ^ k) k1 k2 = Zslice n (k1 - k) k2
intros n k k1 k2 Hk.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Zslice (n * beta ^ k) k1 k2 = Zslice n (k1 - k) k2
unfold Zslice.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
(if (0 <=? k2)%Z
 then Z.rem (Zscale (n * beta ^ k) (- k1)) (beta ^ k2)
 else 0%Z) =
(if (0 <=? k2)%Z
 then Z.rem (Zscale n (- (k1 - k))) (beta ^ k2)
 else 0%Z)
case Zle_bool_spec ; intros Hk2.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk2:(0 <= k2)%Z
Z.rem (Zscale (n * beta ^ k) (- k1)) (beta ^ k2) =
Z.rem (Zscale n (- (k1 - k))) (beta ^ k2)
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk2:(k2 < 0)%Z
0%Z = 0%Z
2: apply refl_equal.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk2:(0 <= k2)%Z
Z.rem (Zscale (n * beta ^ k) (- k1)) (beta ^ k2) =
Z.rem (Zscale n (- (k1 - k))) (beta ^ k2)
rewrite Zscale_mul_pow with (1 := Hk).beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk2:(0 <= k2)%Z
Z.rem (Zscale n (k + - k1)) (beta ^ k2) =
Z.rem (Zscale n (- (k1 - k))) (beta ^ k2)
now replace (- (k1 - k))%Z with (k + -k1)%Z by ring.
Qed.

Theorem Zslice_div_pow :
  forall n k k1 k2, (0 <= k)%Z -> (0 <= k1)%Z ->
  Zslice (Z.quot n (Zpower beta k)) k1 k2 = Zslice n (k1 + k) k2.beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
(0 <= k1)%Z ->
Zslice (n ÷ beta ^ k) k1 k2 = Zslice n (k1 + k) k2
Proof.beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
(0 <= k1)%Z ->
Zslice (n ÷ beta ^ k) k1 k2 = Zslice n (k1 + k) k2
intros n k k1 k2 Hk Hk1.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Zslice (n ÷ beta ^ k) k1 k2 = Zslice n (k1 + k) k2
unfold Zslice.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
(if (0 <=? k2)%Z
 then Z.rem (Zscale (n ÷ beta ^ k) (- k1)) (beta ^ k2)
 else 0%Z) =
(if (0 <=? k2)%Z
 then Z.rem (Zscale n (- (k1 + k))) (beta ^ k2)
 else 0%Z)
case Zle_bool_spec ; intros Hk2.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Z.rem (Zscale (n ÷ beta ^ k) (- k1)) (beta ^ k2) =
Z.rem (Zscale n (- (k1 + k))) (beta ^ k2)
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(k2 < 0)%Z
0%Z = 0%Z
2: apply refl_equal.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Z.rem (Zscale (n ÷ beta ^ k) (- k1)) (beta ^ k2) =
Z.rem (Zscale n (- (k1 + k))) (beta ^ k2)
apply (f_equal (fun x => Z.rem x (beta ^ k2))).beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Zscale (n ÷ beta ^ k) (- k1) = Zscale n (- (k1 + k))
unfold Zscale.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
(if (0 <=? - k1)%Z
 then (n ÷ beta ^ k * beta ^ (- k1))%Z
 else (n ÷ beta ^ k ÷ beta ^ (- - k1))%Z) =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
case Zle_bool_spec ; intros Hk1'.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
(n ÷ beta ^ k * beta ^ (- k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
replace k1 with Z0 by omega.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(if (0 <=? - (0 + k))%Z
 then (n * beta ^ (- (0 + k)))%Z
 else (n ÷ beta ^ (- - (0 + k)))%Z)
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
case Zle_bool_spec ; intros Hk'.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(0 <= - (0 + k))%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n * beta ^ (- (0 + k)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(- (0 + k) < 0)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n ÷ beta ^ (- - (0 + k)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
replace k with Z0 by omega.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(0 <= - (0 + k))%Z
(n ÷ beta ^ 0 * beta ^ (- 0))%Z =
(n * beta ^ (- (0 + 0)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(- (0 + k) < 0)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n ÷ beta ^ (- - (0 + k)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
simpl.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(0 <= - (0 + k))%Z
(n ÷ 1 * 1)%Z = (n * 1)%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(- (0 + k) < 0)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n ÷ beta ^ (- - (0 + k)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
now rewrite Z.quot_1_r.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(- (0 + k) < 0)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n ÷ beta ^ (- - (0 + k)))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
rewrite Z.opp_involutive.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(0 <= - k1)%Z
Hk':(- (0 + k) < 0)%Z
(n ÷ beta ^ k * beta ^ (- 0))%Z =
(n ÷ beta ^ (0 + k))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
apply Zmult_1_r.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(if (0 <=? - (k1 + k))%Z
 then (n * beta ^ (- (k1 + k)))%Z
 else (n ÷ beta ^ (- - (k1 + k)))%Z)
rewrite Zle_bool_false by omega.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ (- - k1))%Z =
(n ÷ beta ^ (- - (k1 + k)))%Z
rewrite 2!Z.opp_involutive, Zplus_comm.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ k1)%Z = (n ÷ beta ^ (k + k1))%Z
rewrite Zpower_plus by assumption.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Hk1:(0 <= k1)%Z
Hk2:(0 <= k2)%Z
Hk1':(- k1 < 0)%Z
(n ÷ beta ^ k ÷ beta ^ k1)%Z =
(n ÷ (beta ^ k * beta ^ k1))%Z
apply Zquot_Zquot.
Qed.

Theorem Zslice_scale :
  forall n k k1 k2, (0 <= k1)%Z ->
  Zslice (Zscale n k) k1 k2 = Zslice n (k1 - k) k2.beta:radix
forall n k k1 k2 : Z,
(0 <= k1)%Z ->
Zslice (Zscale n k) k1 k2 = Zslice n (k1 - k) k2
Proof.beta:radix
forall n k k1 k2 : Z,
(0 <= k1)%Z ->
Zslice (Zscale n k) k1 k2 = Zslice n (k1 - k) k2
intros n k k1 k2 Hk1.beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Zslice (Zscale n k) k1 k2 = Zslice n (k1 - k) k2
unfold Zscale.beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Zslice
  (if (0 <=? k)%Z
   then (n * beta ^ k)%Z
   else (n ÷ beta ^ (- k))%Z) k1 k2 =
Zslice n (k1 - k) k2
case Zle_bool_spec; intros Hk.beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Hk:(0 <= k)%Z
Zslice (n * beta ^ k) k1 k2 = Zslice n (k1 - k) k2
beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Hk:(k < 0)%Z
Zslice (n ÷ beta ^ (- k)) k1 k2 = Zslice n (k1 - k) k2
now apply Zslice_mul_pow.beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Hk:(k < 0)%Z
Zslice (n ÷ beta ^ (- k)) k1 k2 = Zslice n (k1 - k) k2
apply Zslice_div_pow with (2 := Hk1).beta:radix
n, k, k1, k2:Z
Hk1:(0 <= k1)%Z
Hk:(k < 0)%Z
(0 <= - k)%Z
omega.
Qed.

Theorem Zslice_div_pow_scale :
  forall n k k1 k2, (0 <= k)%Z ->
  Zslice (Z.quot n (Zpower beta k)) k1 k2 = Zscale (Zslice n k (k1 + k2)) (-k1).beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
Zslice (n ÷ beta ^ k) k1 k2 =
Zscale (Zslice n k (k1 + k2)) (- k1)
Proof.beta:radix
forall n k k1 k2 : Z,
(0 <= k)%Z ->
Zslice (n ÷ beta ^ k) k1 k2 =
Zscale (Zslice n k (k1 + k2)) (- k1)
intros n k k1 k2 Hk.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
Zslice (n ÷ beta ^ k) k1 k2 =
Zscale (Zslice n k (k1 + k2)) (- k1)
apply Zdigit_ext.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k0 =
Zdigit (Zscale (Zslice n k (k1 + k2)) (- k1)) k0
intros k' Hk'.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zscale (Zslice n k (k1 + k2)) (- k1)) k'
rewrite Zdigit_scale with (1 := Hk').beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k' - - k1)
unfold Zminus.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k' + - - k1)
rewrite (Zplus_comm k'), Z.opp_involutive.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
destruct (Zle_or_lt k2 k') as [Hk2|Hk2].beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k2 <= k')%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
rewrite Zdigit_slice_out with (1 := Hk2).beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k2 <= k')%Z
0%Z = Zdigit (Zslice n k (k1 + k2)) (k1 + k')
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
apply sym_eq.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k2 <= k')%Z
Zdigit (Zslice n k (k1 + k2)) (k1 + k') = 0%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
apply Zdigit_slice_out.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k2 <= k')%Z
(k1 + k2 <= k1 + k')%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
now apply Zplus_le_compat_l.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (Zslice (n ÷ beta ^ k) k1 k2) k' =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
rewrite Zdigit_slice by now split.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
destruct (Zle_or_lt 0 (k1 + k')) as [Hk1'|Hk1'].beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(0 <= k1 + k')%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(k1 + k' < 0)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
rewrite Zdigit_slice by omega.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(0 <= k1 + k')%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit n (k + (k1 + k'))
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(k1 + k' < 0)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
rewrite Zdigit_div_pow by assumption.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(0 <= k1 + k')%Z
Zdigit n (k1 + k' + k) = Zdigit n (k + (k1 + k'))
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(k1 + k' < 0)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
apply f_equal.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(0 <= k1 + k')%Z
(k1 + k' + k)%Z = (k + (k1 + k'))%Z
beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(k1 + k' < 0)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
ring.beta:radix
n, k, k1, k2:Z
Hk:(0 <= k)%Z
k':Z
Hk':(0 <= k')%Z
Hk2:(k' < k2)%Z
Hk1':(k1 + k' < 0)%Z
Zdigit (n ÷ beta ^ k) (k1 + k') =
Zdigit (Zslice n k (k1 + k2)) (k1 + k')
now rewrite 2!Zdigit_lt.
Qed.

Theorem Zplus_slice :
  forall n k l1 l2, (0 <= l1)%Z -> (0 <= l2)%Z ->
  (Zslice n k l1 + Zscale (Zslice n (k + l1) l2) l1)%Z = Zslice n k (l1 + l2).beta:radix
forall n k l1 l2 : Z,
(0 <= l1)%Z ->
(0 <= l2)%Z ->
(Zslice n k l1 + Zscale (Zslice n (k + l1) l2) l1)%Z =
Zslice n k (l1 + l2)
Proof.beta:radix
forall n k l1 l2 : Z,
(0 <= l1)%Z ->
(0 <= l2)%Z ->
(Zslice n k l1 + Zscale (Zslice n (k + l1) l2) l1)%Z =
Zslice n k (l1 + l2)
intros n k1 l1 l2 Hl1 Hl2.beta:radix
n, k1, l1, l2:Z
Hl1:(0 <= l1)%Z
Hl2:(0 <= l2)%Z
(Zslice n k1 l1 + Zscale (Zslice n (k1 + l1) l2) l1)%Z =
Zslice n k1 (l1 + l2)
clear Hl1.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
(Zslice n k1 l1 + Zscale (Zslice n (k1 + l1) l2) l1)%Z =
Zslice n k1 (l1 + l2)
apply Zdigit_ext.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
forall k : Z,
(0 <= k)%Z ->
Zdigit
  (Zslice n k1 l1 + Zscale (Zslice n (k1 + l1) l2) l1)
  k = Zdigit (Zslice n k1 (l1 + l2)) k
intros k Hk.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Zdigit
  (Zslice n k1 l1 + Zscale (Zslice n (k1 + l1) l2) l1)
  k = Zdigit (Zslice n k1 (l1 + l2)) k
rewrite Zdigit_plus.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k)%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_scale with (1 := Hk).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
destruct (Zle_or_lt (l1 + l2) k) as [Hk2|Hk2].beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(l1 + l2 <= k)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_slice_out with (1 := Hk2).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(l1 + l2 <= k)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
now rewrite 2!Zdigit_slice_out by omega.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit (Zslice n k1 (l1 + l2)) k
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_slice with (1 := conj Hk Hk2).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
destruct (Zle_or_lt l1 k) as [Hk1|Hk1].beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(l1 <= k)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_slice_out with (1 := Hk1).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(l1 <= k)%Z
(0 + Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_slice by omega.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(l1 <= k)%Z
(0 + Zdigit n (k1 + l1 + (k - l1)))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
simpl ; apply f_equal.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(l1 <= k)%Z
(k1 + l1 + (k - l1))%Z = (k1 + k)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
ring.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit (Zslice n k1 l1) k +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zdigit_slice with (1 := conj Hk Hk1).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit n (k1 + k) +
 Zdigit (Zslice n (k1 + l1) l2) (k - l1))%Z =
Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite (Zdigit_lt _ (k - l1)) by omega.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk2:(k < l1 + l2)%Z
Hk1:(k < l1)%Z
(Zdigit n (k1 + k) + 0)%Z = Zdigit n (k1 + k)
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zplus_0_r.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n k1 l1 * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zmult_comm.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zscale (Zslice n (k1 + l1) l2) l1 * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zsame_sign_trans_weak with n.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
n = 0%Z -> Zslice n k1 l1 = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= Zscale (Zslice n (k1 + l1) l2) l1 * n)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
intros H ; rewrite H.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
H:n = 0%Z
Zslice 0 k1 l1 = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= Zscale (Zslice n (k1 + l1) l2) l1 * n)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zslice_0.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= Zscale (Zslice n (k1 + l1) l2) l1 * n)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
rewrite Zmult_comm.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zsame_sign_trans_weak with (Zslice n (k1 + l1) l2).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Zslice n (k1 + l1) l2 = 0%Z ->
Zscale (Zslice n (k1 + l1) l2) l1 = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n (k1 + l1) l2)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n (k1 + l1) l2 *
 Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
intros H ; rewrite H.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
H:Zslice n (k1 + l1) l2 = 0%Z
Zscale 0 l1 = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n (k1 + l1) l2)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n (k1 + l1) l2 *
 Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zscale_0.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n (k1 + l1) l2)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n (k1 + l1) l2 *
 Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zsame_sign_slice.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <=
 Zslice n (k1 + l1) l2 *
 Zscale (Zslice n (k1 + l1) l2) l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zsame_sign_scale.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
(0 <= n * Zslice n k1 l1)%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
apply Zsame_sign_slice.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
forall k0 : Z,
(0 <= k0)%Z ->
Zdigit (Zslice n k1 l1) k0 = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k0 = 0%Z
clear k Hk ; intros k Hk.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zscale (Zslice n (k1 + l1) l2) l1) k = 0%Z
rewrite Zdigit_scale with (1 := Hk).beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
destruct (Zle_or_lt l1 k) as [Hk1|Hk1].beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(l1 <= k)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(k < l1)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
left.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(l1 <= k)%Z
Zdigit (Zslice n k1 l1) k = 0%Z
beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(k < l1)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
now apply Zdigit_slice_out.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(k < l1)%Z
Zdigit (Zslice n k1 l1) k = 0%Z \/
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
right.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(k < l1)%Z
Zdigit (Zslice n (k1 + l1) l2) (k - l1) = 0%Z
apply Zdigit_lt.beta:radix
n, k1, l1, l2:Z
Hl2:(0 <= l2)%Z
k:Z
Hk:(0 <= k)%Z
Hk1:(k < l1)%Z
(k - l1 < 0)%Z
omega.
Qed.

Section digits_aux.

Variable p : Z.

Fixpoint Zdigits_aux (nb pow : Z) (n : nat) { struct n } : Z :=
  match n with
  | O => nb
  | S n => if Zlt_bool p pow then nb else Zdigits_aux (nb + 1) (Zmult beta pow) n
  end.

End digits_aux.

Number of digits of an integer

Definition Zdigits n :=
  match n with
  | Z0 => Z0
  | Zneg p => Zdigits_aux (Zpos p) 1 beta (digits2_Pnat p)
  | Zpos p => Zdigits_aux n 1 beta (digits2_Pnat p)
  end.

Theorem Zdigits_correct :
  forall n,
  (Zpower beta (Zdigits n - 1) <= Z.abs n < Zpower beta (Zdigits n))%Z.beta:radix
forall n : Z,
(beta ^ (Zdigits n - 1) <= Z.abs n < beta ^ Zdigits n)%Z
Proof.beta:radix
forall n : Z,
(beta ^ (Zdigits n - 1) <= Z.abs n < beta ^ Zdigits n)%Z
cut (forall p, Zpower beta (Zdigits (Zpos p) - 1) <= Zpos p < Zpower beta (Zdigits (Zpos p)))%Z.beta:radix
(forall p : positive,
 (beta ^ (Zdigits (Z.pos p) - 1) <= Z.pos p <
  beta ^ Zdigits (Z.pos p))%Z) ->
forall n : Z,
(beta ^ (Zdigits n - 1) <= Z.abs n < beta ^ Zdigits n)%Z
beta:radix
forall p : positive,
(beta ^ (Zdigits (Z.pos p) - 1) <= 
 Z.pos p < beta ^ Zdigits (Z.pos p))%Z
intros H [|n|n] ; try exact (H n).beta:radix
H:forall p : positive,
(beta ^ (Zdigits (Z.pos p) - 1) <= 
 Z.pos p < beta ^ Zdigits (Z.pos p))%Z
(beta ^ (Zdigits 0 - 1) <= Z.abs 0 < beta ^ Zdigits 0)%Z
beta:radix
forall p : positive,
(beta ^ (Zdigits (Z.pos p) - 1) <= 
 Z.pos p < beta ^ Zdigits (Z.pos p))%Z
now split.beta:radix
forall p : positive,
(beta ^ (Zdigits (Z.pos p) - 1) <= Z.pos p <
 beta ^ Zdigits (Z.pos p))%Z
intros n.beta:radix
n:positive
(beta ^ (Zdigits (Z.pos n) - 1) <= Z.pos n <
 beta ^ Zdigits (Z.pos n))%Z
simpl.beta:radix
n:positive
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
(* *)
assert (U: (Zpos n < Zpower beta (Z_of_nat (S (digits2_Pnat n))))%Z).beta:radix
n:positive
(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply Z.lt_le_trans with (1 := proj2 (digits2_Pnat_correct n)).beta:radix
n:positive
(Zpower_nat 2 (S (digits2_Pnat n)) <=
 beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite Zpower_Zpower_nat.beta:radix
n:positive
(Zpower_nat 2 (S (digits2_Pnat n)) <=
 Zpower_nat beta
   (Z.abs_nat (Z.of_nat (S (digits2_Pnat n)))))%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite Zabs_nat_Z_of_nat.beta:radix
n:positive
(Zpower_nat 2 (S (digits2_Pnat n)) <=
 Zpower_nat beta (S (digits2_Pnat n)))%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
induction (S (digits2_Pnat n)).beta:radix
n:positive
(Zpower_nat 2 0 <= Zpower_nat beta 0)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(Zpower_nat 2 (S n0) <= Zpower_nat beta (S n0))%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
easy.beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(Zpower_nat 2 (S n0) <= Zpower_nat beta (S n0))%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite 2!(Zpower_nat_S).beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(2 * Zpower_nat 2 n0 <= beta * Zpower_nat beta n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply Zmult_le_compat with (2 := IHn0).beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(2 <= beta)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= 2)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Zpower_nat 2 n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply Zle_bool_imp_le.beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(2 <=? beta)%Z = true
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= 2)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Zpower_nat 2 n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply beta.beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= 2)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Zpower_nat 2 n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
easy.beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Zpower_nat 2 n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite <- (Zabs_nat_Z_of_nat n0).beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Zpower_nat 2 (Z.abs_nat (Z.of_nat n0)))%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite <- Zpower_Zpower_nat.beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= 2 ^ Z.of_nat n0)%Z
beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Z.of_nat n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply (Zpower_ge_0 (Build_radix 2 (refl_equal true))).beta:radix
n:positive
n0:nat
IHn0:(Zpower_nat 2 n0 <= Zpower_nat beta n0)%Z
(0 <= Z.of_nat n0)%Z
beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply Zle_0_nat.beta:radix
n:positive
(0 <= Z.of_nat (S (digits2_Pnat n)))%Z
beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
apply Zle_0_nat.beta:radix
n:positive
U:(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
(* *)
revert U.beta:radix
n:positive
(Z.pos n < beta ^ Z.of_nat (S (digits2_Pnat n)))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
rewrite inj_S.beta:radix
n:positive
(Z.pos n < beta ^ Z.succ (Z.of_nat (digits2_Pnat n)))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
unfold Z.succ.beta:radix
n:positive
(Z.pos n < beta ^ (Z.of_nat (digits2_Pnat n) + 1))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) 1 beta (digits2_Pnat n))%Z
generalize (digits2_Pnat n).beta:radix
n:positive
forall n0 : nat,
(Z.pos n < beta ^ (Z.of_nat n0 + 1))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) 1 beta n0 - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 beta n0)%Z
intros u U.beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
(beta ^ (Zdigits_aux (Z.pos n) 1 beta u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 beta u)%Z
pattern (radix_val beta) at 2 4 ; replace (radix_val beta) with (Zpower beta 1) by apply Zmult_1_r.beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
assert (V: (Zpower beta (1 - 1) <= Zpos n)%Z).beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
(beta ^ (1 - 1) <= Z.pos n)%Z
beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
V:(beta ^ (1 - 1) <= Z.pos n)%Z
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
now apply (Zlt_le_succ 0).beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
V:(beta ^ (1 - 1) <= Z.pos n)%Z
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
generalize (conj V U).beta:radix
n:positive
u:nat
U:(Z.pos n < beta ^ (Z.of_nat u + 1))%Z
V:(beta ^ (1 - 1) <= Z.pos n)%Z
(beta ^ (1 - 1) <= Z.pos n < beta ^ (Z.of_nat u + 1))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
clear.beta:radix
n:positive
u:nat
(beta ^ (1 - 1) <= Z.pos n < beta ^ (Z.of_nat u + 1))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
generalize (Z.le_refl 1).beta:radix
n:positive
u:nat
(1 <= 1)%Z ->
(beta ^ (1 - 1) <= Z.pos n < beta ^ (Z.of_nat u + 1))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) 1 (beta ^ 1) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) 1 (beta ^ 1) u)%Z
generalize 1%Z at 2 3 5 6 7 9 10.beta:radix
n:positive
u:nat
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n < beta ^ (Z.of_nat u + z))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
(* *)
induction u.beta:radix
n:positive
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n < beta ^ (Z.of_nat 0 + z))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) z (beta ^ z) 0 - 1) <=
 Z.pos n < beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) 0)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat (S u) + z))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) z (beta ^ z) (S u) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) (S u))%Z
easy.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat (S u) + z))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) z (beta ^ z) (S u) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) (S u))%Z
rewrite inj_S; unfold Z.succ.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + z))%Z ->
(beta ^ (Zdigits_aux (Z.pos n) z (beta ^ z) (S u) - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) (S u))%Z
simpl Zdigits_aux.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + z))%Z ->
(beta
 ^ ((if Z.pos n <? beta ^ z
     then z
     else
      Zdigits_aux (Z.pos n) (z + 1) (beta * beta ^ z)
        u) - 1) <= Z.pos n <
 beta
 ^ (if Z.pos n <? beta ^ z
    then z
    else
     Zdigits_aux (Z.pos n) (z + 1) (beta * beta ^ z) u))%Z
intros v Hv U.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
(beta
 ^ ((if Z.pos n <? beta ^ v
     then v
     else
      Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v)
        u) - 1) <= Z.pos n <
 beta
 ^ (if Z.pos n <? beta ^ v
    then v
    else
     Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v) u))%Z
case Zlt_bool_spec ; intros K.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(Z.pos n < beta ^ v)%Z
(beta ^ (v - 1) <= Z.pos n < beta ^ v)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta
 ^ (Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v) u -
    1) <= Z.pos n <
 beta
 ^ Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v) u)%Z
now split.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta
 ^ (Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v) u -
    1) <= Z.pos n <
 beta
 ^ Zdigits_aux (Z.pos n) (v + 1) (beta * beta ^ v) u)%Z
pattern (radix_val beta) at 2 5 ; replace (radix_val beta) with (Zpower beta 1) by apply Zmult_1_r.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta
 ^ (Zdigits_aux (Z.pos n) (v + 1)
      (beta ^ 1 * beta ^ v) u - 1) <= Z.pos n <
 beta
 ^ Zdigits_aux (Z.pos n) (v + 1) (beta ^ 1 * beta ^ v)
     u)%Z
rewrite <- Zpower_plus.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta
 ^ (Zdigits_aux (Z.pos n) (v + 1) (beta ^ (1 + v)) u -
    1) <= Z.pos n <
 beta
 ^ Zdigits_aux (Z.pos n) (v + 1) (beta ^ (1 + v)) u)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
rewrite Zplus_comm.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta
 ^ (Zdigits_aux (Z.pos n) (1 + v) (beta ^ (1 + v)) u -
    1) <= Z.pos n <
 beta
 ^ Zdigits_aux (Z.pos n) (1 + v) (beta ^ (1 + v)) u)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
apply IHu.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(1 <= 1 + v)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta ^ (1 + v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + (1 + v)))%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
clear -Hv ; omega.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta ^ (1 + v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + (1 + v)))%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
split.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(beta ^ (1 + v - 1) <= Z.pos n)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(Z.pos n < beta ^ (Z.of_nat u + (1 + v)))%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
now ring_simplify (1 + v - 1)%Z.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(Z.pos n < beta ^ (Z.of_nat u + (1 + v)))%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
now rewrite Zplus_assoc.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= 1)%Z
beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
easy.beta:radix
n:positive
u:nat
IHu:forall z : Z,
(1 <= z)%Z ->
(beta ^ (z - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + z))%Z ->
(beta
 ^ (Zdigits_aux (Z.pos n) z (beta ^ z) u - 1) <=
 Z.pos n <
 beta ^ Zdigits_aux (Z.pos n) z (beta ^ z) u)%Z
v:Z
Hv:(1 <= v)%Z
U:(beta ^ (v - 1) <= Z.pos n <
 beta ^ (Z.of_nat u + 1 + v))%Z
K:(beta ^ v <= Z.pos n)%Z
(0 <= v)%Z
apply Zle_succ_le with (1 := Hv).
Qed.

Theorem Zdigits_unique :
  forall n d,
  (Zpower beta (d - 1) <= Z.abs n < Zpower beta d)%Z ->
  Zdigits n = d.beta:radix
forall n d : Z,
(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z ->
Zdigits n = d
Proof.beta:radix
forall n d : Z,
(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z ->
Zdigits n = d
intros n d Hd.beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Zdigits n = d
assert (Hd' := Zdigits_correct n).beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
Zdigits n = d
apply Zle_antisym.beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(Zdigits n <= d)%Z
beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(d <= Zdigits n)%Z
apply (Zpower_lt_Zpower beta).beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(beta ^ (Zdigits n - 1) < beta ^ d)%Z
beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(d <= Zdigits n)%Z
now apply Z.le_lt_trans with (Z.abs n).beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(d <= Zdigits n)%Z
apply (Zpower_lt_Zpower beta).beta:radix
n, d:Z
Hd:(beta ^ (d - 1) <= Z.abs n < beta ^ d)%Z
Hd':(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ Zdigits n)%Z
(beta ^ (d - 1) < beta ^ Zdigits n)%Z
now apply Z.le_lt_trans with (Z.abs n).
Qed.

Theorem Zdigits_abs :
  forall n, Zdigits (Z.abs n) = Zdigits n.beta:radix
forall n : Z, Zdigits (Z.abs n) = Zdigits n
Proof.beta:radix
forall n : Z, Zdigits (Z.abs n) = Zdigits n
now intros [|n|n].
Qed.

Theorem Zdigits_gt_0 :
  forall n, n <> Z0 -> (0 < Zdigits n)%Z.beta:radix
forall n : Z, n <> 0%Z -> (0 < Zdigits n)%Z
Proof.beta:radix
forall n : Z, n <> 0%Z -> (0 < Zdigits n)%Z
intros n Zn.beta:radix
n:Z
Zn:n <> 0%Z
(0 < Zdigits n)%Z
rewrite <- (Zdigits_abs n).beta:radix
n:Z
Zn:n <> 0%Z
(0 < Zdigits (Z.abs n))%Z
assert (Hn: (0 < Z.abs n)%Z).beta:radix
n:Z
Zn:n <> 0%Z
(0 < Z.abs n)%Z
beta:radix
n:Z
Zn:n <> 0%Z
Hn:(0 < Z.abs n)%Z
(0 < Zdigits (Z.abs n))%Z
destruct n ; [|easy|easy].beta:radix
Zn:0%Z <> 0%Z
(0 < Z.abs 0)%Z
beta:radix
n:Z
Zn:n <> 0%Z
Hn:(0 < Z.abs n)%Z
(0 < Zdigits (Z.abs n))%Z
now elim Zn.beta:radix
n:Z
Zn:n <> 0%Z
Hn:(0 < Z.abs n)%Z
(0 < Zdigits (Z.abs n))%Z
destruct (Z.abs n) as [|p|p] ; try easy ; clear.beta:radix
p:positive
(0 < Zdigits (Z.pos p))%Z
simpl.beta:radix
p:positive
(0 < Zdigits_aux (Z.pos p) 1 beta (digits2_Pnat p))%Z
generalize 1%Z (radix_val beta) (refl_equal Lt : (0 < 1)%Z).beta:radix
p:positive
forall z z0 : Z,
(0 < z)%Z ->
(0 < Zdigits_aux (Z.pos p) z z0 (digits2_Pnat p))%Z
induction (digits2_Pnat p).beta:radix
p:positive
forall z z0 : Z,
(0 < z)%Z -> (0 < Zdigits_aux (Z.pos p) z z0 0)%Z
beta:radix
p:positive
n:nat
IHn:forall z z0 : Z,
(0 < z)%Z ->
(0 < Zdigits_aux (Z.pos p) z z0 n)%Z
forall z z0 : Z,
(0 < z)%Z -> (0 < Zdigits_aux (Z.pos p) z z0 (S n))%Z
easy.beta:radix
p:positive
n:nat
IHn:forall z z0 : Z,
(0 < z)%Z ->
(0 < Zdigits_aux (Z.pos p) z z0 n)%Z
forall z z0 : Z,
(0 < z)%Z -> (0 < Zdigits_aux (Z.pos p) z z0 (S n))%Z
simpl.beta:radix
p:positive
n:nat
IHn:forall z z0 : Z,
(0 < z)%Z ->
(0 < Zdigits_aux (Z.pos p) z z0 n)%Z
forall z z0 : Z,
(0 < z)%Z ->
(0 <
 (if Z.pos p <? z0
  then z
  else Zdigits_aux (Z.pos p) (z + 1) (beta * z0) n))%Z
intros.beta:radix
p:positive
n:nat
IHn:forall z1 z2 : Z,
(0 < z1)%Z ->
(0 < Zdigits_aux (Z.pos p) z1 z2 n)%Z
z, z0:Z
H:(0 < z)%Z
(0 <
 (if Z.pos p <? z0
  then z
  else Zdigits_aux (Z.pos p) (z + 1) (beta * z0) n))%Z
case Zlt_bool.beta:radix
p:positive
n:nat
IHn:forall z1 z2 : Z,
(0 < z1)%Z ->
(0 < Zdigits_aux (Z.pos p) z1 z2 n)%Z
z, z0:Z
H:(0 < z)%Z
(0 < z)%Z
beta:radix
p:positive
n:nat
IHn:forall z1 z2 : Z,
(0 < z1)%Z ->
(0 < Zdigits_aux (Z.pos p) z1 z2 n)%Z
z, z0:Z
H:(0 < z)%Z
(0 < Zdigits_aux (Z.pos p) (z + 1) (beta * z0) n)%Z
exact H.beta:radix
p:positive
n:nat
IHn:forall z1 z2 : Z,
(0 < z1)%Z ->
(0 < Zdigits_aux (Z.pos p) z1 z2 n)%Z
z, z0:Z
H:(0 < z)%Z
(0 < Zdigits_aux (Z.pos p) (z + 1) (beta * z0) n)%Z
apply IHn.beta:radix
p:positive
n:nat
IHn:forall z1 z2 : Z,
(0 < z1)%Z ->
(0 < Zdigits_aux (Z.pos p) z1 z2 n)%Z
z, z0:Z
H:(0 < z)%Z
(0 < z + 1)%Z
now apply Zlt_lt_succ.
Qed.

Theorem Zdigits_ge_0 :
  forall n, (0 <= Zdigits n)%Z.beta:radix
forall n : Z, (0 <= Zdigits n)%Z
Proof.beta:radix
forall n : Z, (0 <= Zdigits n)%Z
intros n.beta:radix
n:Z
(0 <= Zdigits n)%Z
destruct (Z.eq_dec n 0) as [H|H].beta:radix
n:Z
H:n = 0%Z
(0 <= Zdigits n)%Z
beta:radix
n:Z
H:n <> 0%Z
(0 <= Zdigits n)%Z
now rewrite H.beta:radix
n:Z
H:n <> 0%Z
(0 <= Zdigits n)%Z
apply Zlt_le_weak.beta:radix
n:Z
H:n <> 0%Z
(0 < Zdigits n)%Z
now apply Zdigits_gt_0.
Qed.

Theorem Zdigit_out :
  forall n k, (Zdigits n <= k)%Z ->
  Zdigit n k = Z0.beta:radix
forall n k : Z, (Zdigits n <= k)%Z -> Zdigit n k = 0%Z
Proof.beta:radix
forall n k : Z, (Zdigits n <= k)%Z -> Zdigit n k = 0%Z
intros n k Hk.beta:radix
n, k:Z
Hk:(Zdigits n <= k)%Z
Zdigit n k = 0%Z
apply Zdigit_ge_Zpower with (2 := Hk).beta:radix
n, k:Z
Hk:(Zdigits n <= k)%Z
(Z.abs n < beta ^ Zdigits n)%Z
apply Zdigits_correct.
Qed.

Theorem Zdigit_digits :
  forall n, n <> Z0 ->
  Zdigit n (Zdigits n - 1) <> Z0.beta:radix
forall n : Z,
n <> 0%Z -> Zdigit n (Zdigits n - 1) <> 0%Z
Proof.beta:radix
forall n : Z,
n <> 0%Z -> Zdigit n (Zdigits n - 1) <> 0%Z
intros n Zn.beta:radix
n:Z
Zn:n <> 0%Z
Zdigit n (Zdigits n - 1) <> 0%Z
apply Zdigit_not_0.beta:radix
n:Z
Zn:n <> 0%Z
(0 <= Zdigits n - 1)%Z
beta:radix
n:Z
Zn:n <> 0%Z
(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ (Zdigits n - 1 + 1))%Z
apply Zlt_0_le_0_pred.beta:radix
n:Z
Zn:n <> 0%Z
(0 < Zdigits n)%Z
beta:radix
n:Z
Zn:n <> 0%Z
(beta ^ (Zdigits n - 1) <= 
 Z.abs n < beta ^ (Zdigits n - 1 + 1))%Z
now apply Zdigits_gt_0.beta:radix
n:Z
Zn:n <> 0%Z
(beta ^ (Zdigits n - 1) <= Z.abs n <
 beta ^ (Zdigits n - 1 + 1))%Z
ring_simplify (Zdigits n - 1 + 1)%Z.beta:radix
n:Z
Zn:n <> 0%Z
(beta ^ (Zdigits n - 1) <= Z.abs n < beta ^ Zdigits n)%Z
apply Zdigits_correct.
Qed.

Theorem Zdigits_slice :
  forall n k l, (0 <= l)%Z ->
  (Zdigits (Zslice n k l) <= l)%Z.beta:radix
forall n k l : Z,
(0 <= l)%Z -> (Zdigits (Zslice n k l) <= l)%Z
Proof.beta:radix
forall n k l : Z,
(0 <= l)%Z -> (Zdigits (Zslice n k l) <= l)%Z
intros n k l Hl.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
(Zdigits (Zslice n k l) <= l)%Z
unfold Zslice.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
(Zdigits
   (if 0 <=? l
    then Z.rem (Zscale n (- k)) (beta ^ l)
    else 0) <= l)%Z
rewrite Zle_bool_true with (1 := Hl).beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
(Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) <= l)%Z
destruct (Zdigits_correct (Z.rem (Zscale n (- k)) (Zpower beta l))) as (H1,H2).beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) <= l)%Z
apply Zpower_lt_Zpower with beta.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) - 1) <
 beta ^ l)%Z
apply Z.le_lt_trans with (1 := H1).beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) < beta ^ l)%Z
rewrite <- (Z.abs_eq (beta ^ l)) at 2 by apply Zpower_ge_0.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 Z.abs (beta ^ l))%Z
apply Zrem_lt.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(beta ^ l)%Z <> 0%Z
apply Zgt_not_eq.beta:radix
n, k, l:Z
Hl:(0 <= l)%Z
H1:(beta
 ^ (Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)) -
    1) <=
 Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
H2:(Z.abs (Z.rem (Zscale n (- k)) (beta ^ l)) <
 beta
 ^ Zdigits (Z.rem (Zscale n (- k)) (beta ^ l)))%Z
(0 < beta ^ l)%Z
now apply Zpower_gt_0.
Qed.

Theorem Zdigits_mult_Zpower :
  forall m e,
  m <> Z0 -> (0 <= e)%Z ->
  Zdigits (m * Zpower beta e) = (Zdigits m + e)%Z.beta:radix
forall m e : Z,
m <> 0%Z ->
(0 <= e)%Z ->
Zdigits (m * beta ^ e) = (Zdigits m + e)%Z
Proof.beta:radix
forall m e : Z,
m <> 0%Z ->
(0 <= e)%Z ->
Zdigits (m * beta ^ e) = (Zdigits m + e)%Z
intros m e Hm He.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
Zdigits (m * beta ^ e) = (Zdigits m + e)%Z
assert (H := Zdigits_correct m).beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
Zdigits (m * beta ^ e) = (Zdigits m + e)%Z
apply Zdigits_unique.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m + e - 1) <= Z.abs (m * beta ^ e) <
 beta ^ (Zdigits m + e))%Z
rewrite Z.abs_mul, Z.abs_pow, (Z.abs_eq beta).beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m + e - 1) <= Z.abs m * beta ^ e <
 beta ^ (Zdigits m + e))%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= beta)%Z
2: now apply Zlt_le_weak, radix_gt_0.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m + e - 1) <= Z.abs m * beta ^ e <
 beta ^ (Zdigits m + e))%Z
split.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m + e - 1) <= Z.abs m * beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
replace (Zdigits m + e - 1)%Z with (Zdigits m - 1 + e)%Z by ring.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m - 1 + e) <= Z.abs m * beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
rewrite Zpower_plus with (2 := He).beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m - 1) * beta ^ e <=
 Z.abs m * beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
apply Zmult_le_compat_r.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m - 1) <= Z.abs m)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
apply H.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
apply Zpower_ge_0.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
now apply Zlt_0_le_0_pred, Zdigits_gt_0.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ (Zdigits m + e))%Z
rewrite Zpower_plus with (2 := He).beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m * beta ^ e < beta ^ Zdigits m * beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m)%Z
apply Zmult_lt_compat_r.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 < beta ^ e)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m < beta ^ Zdigits m)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m)%Z
now apply Zpower_gt_0.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(Z.abs m < beta ^ Zdigits m)%Z
beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m)%Z
apply H.beta:radix
m, e:Z
Hm:m <> 0%Z
He:(0 <= e)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(0 <= Zdigits m)%Z
now apply Zlt_le_weak, Zdigits_gt_0.
Qed.

Theorem Zdigits_Zpower :
  forall e,
  (0 <= e)%Z ->
  Zdigits (Zpower beta e) = (e + 1)%Z.beta:radix
forall e : Z,
(0 <= e)%Z -> Zdigits (beta ^ e) = (e + 1)%Z
Proof.beta:radix
forall e : Z,
(0 <= e)%Z -> Zdigits (beta ^ e) = (e + 1)%Z
intros e He.beta:radix
e:Z
He:(0 <= e)%Z
Zdigits (beta ^ e) = (e + 1)%Z
rewrite <- (Zmult_1_l (Zpower beta e)).beta:radix
e:Z
He:(0 <= e)%Z
Zdigits (1 * beta ^ e) = (e + 1)%Z
rewrite Zdigits_mult_Zpower ; try easy.beta:radix
e:Z
He:(0 <= e)%Z
(Zdigits 1 + e)%Z = (e + 1)%Z
apply Zplus_comm.
Qed.

Theorem Zdigits_le :
  forall x y,
  (0 <= x)%Z -> (x <= y)%Z ->
  (Zdigits x <= Zdigits y)%Z.beta:radix
forall x y : Z,
(0 <= x)%Z -> (x <= y)%Z -> (Zdigits x <= Zdigits y)%Z
Proof.beta:radix
forall x y : Z,
(0 <= x)%Z -> (x <= y)%Z -> (Zdigits x <= Zdigits y)%Z
intros x y Zx Hxy.beta:radix
x, y:Z
Zx:(0 <= x)%Z
Hxy:(x <= y)%Z
(Zdigits x <= Zdigits y)%Z
assert (Hx := Zdigits_correct x).beta:radix
x, y:Z
Zx:(0 <= x)%Z
Hxy:(x <= y)%Z
Hx:(beta ^ (Zdigits x - 1) <= 
 Z.abs x < beta ^ Zdigits x)%Z
(Zdigits x <= Zdigits y)%Z
assert (Hy := Zdigits_correct y).beta:radix
x, y:Z
Zx:(0 <= x)%Z
Hxy:(x <= y)%Z
Hx:(beta ^ (Zdigits x - 1) <= 
 Z.abs x < beta ^ Zdigits x)%Z
Hy:(beta ^ (Zdigits y - 1) <= 
 Z.abs y < beta ^ Zdigits y)%Z
(Zdigits x <= Zdigits y)%Z
apply (Zpower_lt_Zpower beta).beta:radix
x, y:Z
Zx:(0 <= x)%Z
Hxy:(x <= y)%Z
Hx:(beta ^ (Zdigits x - 1) <= 
 Z.abs x < beta ^ Zdigits x)%Z
Hy:(beta ^ (Zdigits y - 1) <= 
 Z.abs y < beta ^ Zdigits y)%Z
(beta ^ (Zdigits x - 1) < beta ^ Zdigits y)%Z
zify ; omega.
Qed.

Theorem lt_Zdigits :
  forall x y,
  (0 <= y)%Z ->
  (Zdigits x < Zdigits y)%Z ->
  (x < y)%Z.beta:radix
forall x y : Z,
(0 <= y)%Z -> (Zdigits x < Zdigits y)%Z -> (x < y)%Z
Proof.beta:radix
forall x y : Z,
(0 <= y)%Z -> (Zdigits x < Zdigits y)%Z -> (x < y)%Z
intros x y Hy.beta:radix
x, y:Z
Hy:(0 <= y)%Z
(Zdigits x < Zdigits y)%Z -> (x < y)%Z
cut (y <= x -> Zdigits y <= Zdigits x)%Z.beta:radix
x, y:Z
Hy:(0 <= y)%Z
((y <= x)%Z -> (Zdigits y <= Zdigits x)%Z) ->
(Zdigits x < Zdigits y)%Z -> (x < y)%Z
beta:radix
x, y:Z
Hy:(0 <= y)%Z
(y <= x)%Z -> (Zdigits y <= Zdigits x)%Z omega.beta:radix
x, y:Z
Hy:(0 <= y)%Z
(y <= x)%Z -> (Zdigits y <= Zdigits x)%Z
now apply Zdigits_le.
Qed.

Theorem Zpower_le_Zdigits :
  forall e x,
  (e < Zdigits x)%Z ->
  (Zpower beta e <= Z.abs x)%Z.beta:radix
forall e x : Z,
(e < Zdigits x)%Z -> (beta ^ e <= Z.abs x)%Z
Proof.beta:radix
forall e x : Z,
(e < Zdigits x)%Z -> (beta ^ e <= Z.abs x)%Z
intros e x Hex.beta:radix
e, x:Z
Hex:(e < Zdigits x)%Z
(beta ^ e <= Z.abs x)%Z
destruct (Zdigits_correct x) as [H1 H2].beta:radix
e, x:Z
Hex:(e < Zdigits x)%Z
H1:(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
H2:(Z.abs x < beta ^ Zdigits x)%Z
(beta ^ e <= Z.abs x)%Z
apply Z.le_trans with (2 := H1).beta:radix
e, x:Z
Hex:(e < Zdigits x)%Z
H1:(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
H2:(Z.abs x < beta ^ Zdigits x)%Z
(beta ^ e <= beta ^ (Zdigits x - 1))%Z
apply Zpower_le.beta:radix
e, x:Z
Hex:(e < Zdigits x)%Z
H1:(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
H2:(Z.abs x < beta ^ Zdigits x)%Z
(e <= Zdigits x - 1)%Z
clear -Hex ; omega.
Qed.

Theorem Zdigits_le_Zpower :
  forall e x,
  (Z.abs x < Zpower beta e)%Z ->
  (Zdigits x <= e)%Z.beta:radix
forall e x : Z,
(Z.abs x < beta ^ e)%Z -> (Zdigits x <= e)%Z
Proof.beta:radix
forall e x : Z,
(Z.abs x < beta ^ e)%Z -> (Zdigits x <= e)%Z
intros e x.beta:radix
e, x:Z
(Z.abs x < beta ^ e)%Z -> (Zdigits x <= e)%Z
generalize (Zpower_le_Zdigits e x).beta:radix
e, x:Z
((e < Zdigits x)%Z -> (beta ^ e <= Z.abs x)%Z) ->
(Z.abs x < beta ^ e)%Z -> (Zdigits x <= e)%Z
omega.
Qed.

Theorem Zpower_gt_Zdigits :
  forall e x,
  (Zdigits x <= e)%Z ->
  (Z.abs x < Zpower beta e)%Z.beta:radix
forall e x : Z,
(Zdigits x <= e)%Z -> (Z.abs x < beta ^ e)%Z
Proof.beta:radix
forall e x : Z,
(Zdigits x <= e)%Z -> (Z.abs x < beta ^ e)%Z
intros e x Hex.beta:radix
e, x:Z
Hex:(Zdigits x <= e)%Z
(Z.abs x < beta ^ e)%Z
destruct (Zdigits_correct x) as [H1 H2].beta:radix
e, x:Z
Hex:(Zdigits x <= e)%Z
H1:(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
H2:(Z.abs x < beta ^ Zdigits x)%Z
(Z.abs x < beta ^ e)%Z
apply Z.lt_le_trans with (1 := H2).beta:radix
e, x:Z
Hex:(Zdigits x <= e)%Z
H1:(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
H2:(Z.abs x < beta ^ Zdigits x)%Z
(beta ^ Zdigits x <= beta ^ e)%Z
now apply Zpower_le.
Qed.

Theorem Zdigits_gt_Zpower :
  forall e x,
  (Zpower beta e <= Z.abs x)%Z ->
  (e < Zdigits x)%Z.beta:radix
forall e x : Z,
(beta ^ e <= Z.abs x)%Z -> (e < Zdigits x)%Z
Proof.beta:radix
forall e x : Z,
(beta ^ e <= Z.abs x)%Z -> (e < Zdigits x)%Z
intros e x Hex.beta:radix
e, x:Z
Hex:(beta ^ e <= Z.abs x)%Z
(e < Zdigits x)%Z
generalize (Zpower_gt_Zdigits e x).beta:radix
e, x:Z
Hex:(beta ^ e <= Z.abs x)%Z
((Zdigits x <= e)%Z -> (Z.abs x < beta ^ e)%Z) ->
(e < Zdigits x)%Z
omega.
Qed.

Number of digits of a product.

This strong version is needed for proofs of division and square root algorithms, since they involve operation remainders.

Theorem Zdigits_mult_strong :
  forall x y,
  (0 <= x)%Z -> (0 <= y)%Z ->
  (Zdigits (x + y + x * y) <= Zdigits x + Zdigits y)%Z.beta:radix
forall x y : Z,
(0 <= x)%Z ->
(0 <= y)%Z ->
(Zdigits (x + y + x * y) <= Zdigits x + Zdigits y)%Z
Proof.beta:radix
forall x y : Z,
(0 <= x)%Z ->
(0 <= y)%Z ->
(Zdigits (x + y + x * y) <= Zdigits x + Zdigits y)%Z
intros x y Hx Hy.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(Zdigits (x + y + x * y) <= Zdigits x + Zdigits y)%Z
apply Zdigits_le_Zpower.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(Z.abs (x + y + x * y) <
 beta ^ (Zdigits x + Zdigits y))%Z
rewrite Z.abs_eq.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(x + y + x * y < beta ^ (Zdigits x + Zdigits y))%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Z.lt_le_trans with ((x + 1) * (y + 1))%Z.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(x + y + x * y < (x + 1) * (y + 1))%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
((x + 1) * (y + 1) <= beta ^ (Zdigits x + Zdigits y))%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
ring_simplify.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(x * y + x + y < x * y + x + y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
((x + 1) * (y + 1) <= beta ^ (Zdigits x + Zdigits y))%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zle_lt_succ, Z.le_refl.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
((x + 1) * (y + 1) <= beta ^ (Zdigits x + Zdigits y))%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
rewrite Zpower_plus by apply Zdigits_ge_0.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
((x + 1) * (y + 1) <=
 beta ^ Zdigits x * beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zmult_le_compat.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(x + 1 <= beta ^ Zdigits x)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(y + 1 <= beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zlt_le_succ.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(x < beta ^ Zdigits x)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(y + 1 <= beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
rewrite <- (Z.abs_eq x) at 1 by easy.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(Z.abs x < beta ^ Zdigits x)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(y + 1 <= beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zdigits_correct.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(y + 1 <= beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zlt_le_succ.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(y < beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
rewrite <- (Z.abs_eq y) at 1 by easy.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(Z.abs y < beta ^ Zdigits y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
apply Zdigits_correct.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
clear -Hx ; omega.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= y + 1)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
clear -Hy ; omega.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x + y + x * y)%Z
change Z0 with (0 + 0 + 0)%Z.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 + 0 + 0 <= x + y + x * y)%Z
apply Zplus_le_compat.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 + 0 <= x + y)%Z
beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x * y)%Z
now apply Zplus_le_compat.beta:radix
x, y:Z
Hx:(0 <= x)%Z
Hy:(0 <= y)%Z
(0 <= x * y)%Z
now apply Zmult_le_0_compat.
Qed.

Theorem Zdigits_mult :
  forall x y,
  (Zdigits (x * y) <= Zdigits x + Zdigits y)%Z.beta:radix
forall x y : Z,
(Zdigits (x * y) <= Zdigits x + Zdigits y)%Z
Proof.beta:radix
forall x y : Z,
(Zdigits (x * y) <= Zdigits x + Zdigits y)%Z
intros x y.beta:radix
x, y:Z
(Zdigits (x * y) <= Zdigits x + Zdigits y)%Z
rewrite <- Zdigits_abs.beta:radix
x, y:Z
(Zdigits (Z.abs (x * y)) <= Zdigits x + Zdigits y)%Z
rewrite <- (Zdigits_abs x).beta:radix
x, y:Z
(Zdigits (Z.abs (x * y)) <=
 Zdigits (Z.abs x) + Zdigits y)%Z
rewrite <- (Zdigits_abs y).beta:radix
x, y:Z
(Zdigits (Z.abs (x * y)) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
apply Z.le_trans with (Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y)).beta:radix
x, y:Z
(Zdigits (Z.abs (x * y)) <=
 Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y))%Z
beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
apply Zdigits_le.beta:radix
x, y:Z
(0 <= Z.abs (x * y))%Z
beta:radix
x, y:Z
(Z.abs (x * y) <=
 Z.abs x + Z.abs y + Z.abs x * Z.abs y)%Z
beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
apply Zabs_pos.beta:radix
x, y:Z
(Z.abs (x * y) <=
 Z.abs x + Z.abs y + Z.abs x * Z.abs y)%Z
beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
rewrite Zabs_Zmult.beta:radix
x, y:Z
(Z.abs x * Z.abs y <=
 Z.abs x + Z.abs y + Z.abs x * Z.abs y)%Z
beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
generalize (Zabs_pos x) (Zabs_pos y).beta:radix
x, y:Z
(0 <= Z.abs x)%Z ->
(0 <= Z.abs y)%Z ->
(Z.abs x * Z.abs y <=
 Z.abs x + Z.abs y + Z.abs x * Z.abs y)%Z
beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
omega.beta:radix
x, y:Z
(Zdigits (Z.abs x + Z.abs y + Z.abs x * Z.abs y) <=
 Zdigits (Z.abs x) + Zdigits (Z.abs y))%Z
apply Zdigits_mult_strong ; apply Zabs_pos.
Qed.

Theorem Zdigits_mult_ge :
  forall x y,
  (x <> 0)%Z -> (y <> 0)%Z ->
  (Zdigits x + Zdigits y - 1 <= Zdigits (x * y))%Z.beta:radix
forall x y : Z,
x <> 0%Z ->
y <> 0%Z ->
(Zdigits x + Zdigits y - 1 <= Zdigits (x * y))%Z
Proof.beta:radix
forall x y : Z,
x <> 0%Z ->
y <> 0%Z ->
(Zdigits x + Zdigits y - 1 <= Zdigits (x * y))%Z
intros x y Zx Zy.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x + Zdigits y - 1 <= Zdigits (x * y))%Z
cut ((Zdigits x - 1) + (Zdigits y - 1) < Zdigits (x * y))%Z.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 + (Zdigits y - 1) < Zdigits (x * y))%Z ->
(Zdigits x + Zdigits y - 1 <= Zdigits (x * y))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 + (Zdigits y - 1) < Zdigits (x * y))%Z omega.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 + (Zdigits y - 1) < Zdigits (x * y))%Z
apply Zdigits_gt_Zpower.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits x - 1 + (Zdigits y - 1)) <=
 Z.abs (x * y))%Z
rewrite Zabs_Zmult.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits x - 1 + (Zdigits y - 1)) <=
 Z.abs x * Z.abs y)%Z
rewrite Zpower_exp.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits x - 1) * beta ^ (Zdigits y - 1) <=
 Z.abs x * Z.abs y)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zmult_le_compat.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits x - 1) <= Z.abs x)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits y - 1) <= Z.abs y)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits x - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zpower_le_Zdigits.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 < Zdigits x)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits y - 1) <= Z.abs y)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits x - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zlt_pred.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(beta ^ (Zdigits y - 1) <= Z.abs y)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits x - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zpower_le_Zdigits.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 < Zdigits y)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits x - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zlt_pred.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits x - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zpower_ge_0.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(0 <= beta ^ (Zdigits y - 1))%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
apply Zpower_ge_0.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
generalize (Zdigits_gt_0 x).beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(x <> 0%Z -> (0 < Zdigits x)%Z) ->
(Zdigits x - 1 >= 0)%Z
beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z omega.beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(Zdigits y - 1 >= 0)%Z
generalize (Zdigits_gt_0 y).beta:radix
x, y:Z
Zx:x <> 0%Z
Zy:y <> 0%Z
(y <> 0%Z -> (0 < Zdigits y)%Z) ->
(Zdigits y - 1 >= 0)%Z omega.
Qed.

Theorem Zdigits_div_Zpower :
  forall m e,
  (0 <= m)%Z ->
  (0 <= e <= Zdigits m)%Z ->
  Zdigits (m / Zpower beta e) = (Zdigits m - e)%Z.beta:radix
forall m e : Z,
(0 <= m)%Z ->
(0 <= e <= Zdigits m)%Z ->
Zdigits (m / beta ^ e) = (Zdigits m - e)%Z
Proof.beta:radix
forall m e : Z,
(0 <= m)%Z ->
(0 <= e <= Zdigits m)%Z ->
Zdigits (m / beta ^ e) = (Zdigits m - e)%Z
intros m e Hm He.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
Zdigits (m / beta ^ e) = (Zdigits m - e)%Z
assert (H := Zdigits_correct m).beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
Zdigits (m / beta ^ e) = (Zdigits m - e)%Z
apply Zdigits_unique.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= Z.abs (m / beta ^ e) <
 beta ^ (Zdigits m - e))%Z
destruct (Zle_lt_or_eq _ _ (proj2 He)) as [He'|He'].beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= Z.abs (m / beta ^ e) <
 beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  rewrite Z.abs_eq in H by easy.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= Z.abs (m / beta ^ e) <
 beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  destruct H as [H1 H2].beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= Z.abs (m / beta ^ e) <
 beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  rewrite Z.abs_eq.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= m / beta ^ e <
 beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  split.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - e - 1) <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  replace (Zdigits m - e - 1)%Z with (Zdigits m - 1 - e)%Z by ring.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - 1 - e) <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  rewrite Z.pow_sub_r.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - 1) / beta ^ e <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
beta <> 0%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= e <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  2: apply Zgt_not_eq, radix_gt_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - 1) / beta ^ e <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= e <= Zdigits m - 1)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  2: clear -He He' ; omega.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ (Zdigits m - 1) / beta ^ e <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply Z_div_le with (2 := H1).beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ e > 0)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now apply Z.lt_gt, Zpower_gt_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e < beta ^ (Zdigits m - e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply Zmult_lt_reg_r with (Zpower beta e).beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 < beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e * beta ^ e <
 beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now apply Zpower_gt_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e * beta ^ e <
 beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply Z.le_lt_trans with m.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m / beta ^ e * beta ^ e <= m)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m < beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  rewrite Zmult_comm.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ e * (m / beta ^ e) <= m)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m < beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply Z_mult_div_ge.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ e > 0)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m < beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now apply Z.lt_gt, Zpower_gt_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m < beta ^ (Zdigits m - e) * beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  rewrite <- Zpower_plus.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(m < beta ^ (Zdigits m - e + e))%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= Zdigits m - e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now replace (Zdigits m - e + e)%Z with (Zdigits m) by ring.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= Zdigits m - e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now apply Zle_minus_le_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply He.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(0 <= m / beta ^ e)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  apply Z_div_pos with (2 := Hm).beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H1:(beta ^ (Zdigits m - 1) <= m)%Z
H2:(m < beta ^ Zdigits m)%Z
He':(e < Zdigits m)%Z
(beta ^ e > 0)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= 
 Z.abs (m / beta ^ e) < beta ^ (Zdigits m - e))%Z
  now apply Z.lt_gt, Zpower_gt_0.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - e - 1) <= Z.abs (m / beta ^ e) <
 beta ^ (Zdigits m - e))%Z
rewrite He'.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (Zdigits m - Zdigits m - 1) <=
 Z.abs (m / beta ^ Zdigits m) <
 beta ^ (Zdigits m - Zdigits m))%Z
rewrite (Zeq_minus _ (Zdigits m)) by reflexivity.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(beta ^ (0 - 1) <= Z.abs (m / beta ^ Zdigits m) <
 beta ^ 0)%Z
simpl.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(0 <= Z.abs (m / beta ^ Zdigits m) < 1)%Z
rewrite Zdiv_small.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(0 <= Z.abs 0 < 1)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(0 <= m < beta ^ Zdigits m)%Z
easy.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(0 <= m < beta ^ Zdigits m)%Z
split.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(0 <= m)%Z
beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(m < beta ^ Zdigits m)%Z
exact Hm.beta:radix
m, e:Z
Hm:(0 <= m)%Z
He:(0 <= e <= Zdigits m)%Z
H:(beta ^ (Zdigits m - 1) <= 
 Z.abs m < beta ^ Zdigits m)%Z
He':e = Zdigits m
(m < beta ^ Zdigits m)%Z
now rewrite <- (Z.abs_eq m) at 1.
Qed.

End Fcore_digits.

Specialized version for computing the number of bits of an integer

Section Zdigits2.

Theorem Z_of_nat_S_digits2_Pnat :
  forall m : positive,
  Z_of_nat (S (digits2_Pnat m)) = Zdigits radix2 (Zpos m).forall m : positive,
Z.of_nat (S (digits2_Pnat m)) =
Zdigits radix2 (Z.pos m)
Proof.forall m : positive,
Z.of_nat (S (digits2_Pnat m)) =
Zdigits radix2 (Z.pos m)
intros m.m:positive
Z.of_nat (S (digits2_Pnat m)) =
Zdigits radix2 (Z.pos m)
apply eq_sym, Zdigits_unique.m:positive
(radix2 ^ (Z.of_nat (S (digits2_Pnat m)) - 1) <=
 Z.abs (Z.pos m) <
 radix2 ^ Z.of_nat (S (digits2_Pnat m)))%Z
rewrite <- Zpower_nat_Z.m:positive
(radix2 ^ (Z.of_nat (S (digits2_Pnat m)) - 1) <=
 Z.abs (Z.pos m) <
 Zpower_nat radix2 (S (digits2_Pnat m)))%Z
rewrite Nat2Z.inj_succ.m:positive
(radix2 ^ (Z.succ (Z.of_nat (digits2_Pnat m)) - 1) <=
 Z.abs (Z.pos m) <
 Zpower_nat radix2 (S (digits2_Pnat m)))%Z
change (_ - 1)%Z with (Z.pred (Z.succ (Z.of_nat (digits2_Pnat m)))).m:positive
(radix2 ^ Z.pred (Z.succ (Z.of_nat (digits2_Pnat m))) <=
 Z.abs (Z.pos m) <
 Zpower_nat radix2 (S (digits2_Pnat m)))%Z
rewrite <- Zpred_succ.m:positive
(radix2 ^ Z.of_nat (digits2_Pnat m) <=
 Z.abs (Z.pos m) <
 Zpower_nat radix2 (S (digits2_Pnat m)))%Z
rewrite <- Zpower_nat_Z.m:positive
(Zpower_nat radix2 (digits2_Pnat m) <=
 Z.abs (Z.pos m) <
 Zpower_nat radix2 (S (digits2_Pnat m)))%Z
apply digits2_Pnat_correct.
Qed.

Theorem Zpos_digits2_pos :
  forall m : positive,
  Zpos (digits2_pos m) = Zdigits radix2 (Zpos m).forall m : positive,
Z.pos (SpecFloatCopy.digits2_pos m) =
Zdigits radix2 (Z.pos m)
Proof.forall m : positive,
Z.pos (SpecFloatCopy.digits2_pos m) =
Zdigits radix2 (Z.pos m)
intros m.m:positive
Z.pos (SpecFloatCopy.digits2_pos m) =
Zdigits radix2 (Z.pos m)
rewrite <- Z_of_nat_S_digits2_Pnat.m:positive
Z.pos (SpecFloatCopy.digits2_pos m) =
Z.of_nat (S (digits2_Pnat m))
unfold Z.of_nat.m:positive
Z.pos (SpecFloatCopy.digits2_pos m) =
Z.pos (Pos.of_succ_nat (digits2_Pnat m))
apply f_equal.m:positive
SpecFloatCopy.digits2_pos m =
Pos.of_succ_nat (digits2_Pnat m)
induction m ; simpl ; try easy ;
  apply f_equal, IHm.
Qed.

Lemma Zdigits2_Zdigits :
  forall n, Zdigits2 n = Zdigits radix2 n.forall n : Z,
SpecFloatCopy.Zdigits2 n = Zdigits radix2 n
Proof.forall n : Z,
SpecFloatCopy.Zdigits2 n = Zdigits radix2 n
intros [|p|p] ; try easy ;
  apply Zpos_digits2_pos.
Qed.

End Zdigits2.