IEEE-754 encoding of binary floating-point data

Require Import Core Digits Binary.

Section Binary_Bits.

Arguments exist {A} {P}.
Arguments B754_zero {prec} {emax}.
Arguments B754_infinity {prec} {emax}.
Arguments B754_nan {prec} {emax}.
Arguments B754_finite {prec} {emax}.

Variable mw ew : Z.

Definition join_bits (s : bool) m e :=
  (Z.shiftl ((if s then Zpower 2 ew else 0) + e) mw + m)%Z.

Lemma join_bits_range :
  forall s m e,
  (0 <= m < 2^mw)%Z ->
  (0 <= e < 2^ew)%Z ->
  (0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z.
mw, ew:Z
forall (s : bool) (m e : Z),
(0 <= m < 2 ^ mw)%Z ->
(0 <= e < 2 ^ ew)%Z ->
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
Proof.
mw, ew:Z
forall (s : bool) (m e : Z),
(0 <= m < 2 ^ mw)%Z ->
(0 <= e < 2 ^ ew)%Z ->
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
intros s m e Hm He.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
assert (0 <= mw)%Z as Hmw.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
(0 <= mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
  destruct mw as [|mw'|mw'] ; try easy.
mw, ew:Z
s:bool
m, e:Z
mw':positive
Hm:(0 <= m < 2 ^ Z.neg mw')%Z
He:(0 <= e < 2 ^ ew)%Z
(0 <= Z.neg mw')%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
  clear -Hm ; simpl in Hm ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
assert (0 <= ew)%Z as Hew.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
(0 <= ew)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
  destruct ew as [|ew'|ew'] ; try easy.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
ew':positive
He:(0 <= e < 2 ^ Z.neg ew')%Z
Hmw:(0 <= mw)%Z
(0 <= Z.neg ew')%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
  clear -He ; simpl in He ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z
unfold join_bits.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <=
 Z.shiftl ((if s then 2 ^ ew else 0) + e) mw + m <
 2 ^ (mw + ew + 1))%Z
rewrite Z.shiftl_mul_pow2 by easy.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m <
 2 ^ (mw + ew + 1))%Z
split.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m <
 2 ^ (mw + ew + 1))%Z
-
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)%Z apply (Zplus_le_compat 0 _ 0) with (2 := proj1 Hm).
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ((if s then 2 ^ ew else 0) + e) * 2 ^ mw)%Z
  rewrite <- (Zmult_0_l (2^mw)).
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 * 2 ^ mw <=
 ((if s then 2 ^ ew else 0 * 2 ^ mw) + e) * 2 ^ mw)%Z
  apply Zmult_le_compat_r.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= (if s then 2 ^ ew else 0 * 2 ^ mw) + e)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
  case s.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ ew + e)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 0 * 2 ^ mw + e)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
  clear -He ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 0 * 2 ^ mw + e)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
  now rewrite Zmult_0_l.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
  clear -Hm ; omega.
-
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m <
 2 ^ (mw + ew + 1))%Z apply Z.lt_le_trans with (((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw)%Z.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m <
 ((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw <=
 2 ^ (mw + ew + 1))%Z
  rewrite (Zmult_plus_distr_l _ 1).
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m <
 ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + 1 * 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw <=
 2 ^ (mw + ew + 1))%Z
  apply Zplus_lt_compat_l.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(m < 1 * 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw <=
 2 ^ (mw + ew + 1))%Z
  now rewrite Zmult_1_l.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw <=
 2 ^ (mw + ew + 1))%Z
  rewrite <- (Zplus_assoc mw), (Zplus_comm mw), Zpower_plus.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 1) * 2 ^ mw <=
 2 ^ (ew + 1) * 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  apply Zmult_le_compat_r.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((if s then 2 ^ ew else 0) + e + 1 <= 2 ^ (ew + 1))%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  rewrite Zpower_plus by easy.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((if s then 2 ^ ew else 0) + e + 1 <= 2 ^ ew * 2 ^ 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  change (2^1)%Z with 2%Z.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((if s then 2 ^ ew else 0) + e + 1 <= 2 ^ ew * 2)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  case s ; clear -He ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  clear -Hm ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= ew + 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  clear -Hew ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= mw)%Z
  easy.
Qed.

Definition split_bits x :=
  let mm := Zpower 2 mw in
  let em := Zpower 2 ew in
  (Zle_bool (mm * em) x, Zmod x mm, Zmod (Z.div x mm) em)%Z.

Theorem split_join_bits :
  forall s m e,
  (0 <= m < Zpower 2 mw)%Z ->
  (0 <= e < Zpower 2 ew)%Z ->
  split_bits (join_bits s m e) = (s, m, e).
mw, ew:Z
forall (s : bool) (m e : Z),
(0 <= m < 2 ^ mw)%Z ->
(0 <= e < 2 ^ ew)%Z ->
split_bits (join_bits s m e) = (s, m, e)
Proof.
mw, ew:Z
forall (s : bool) (m e : Z),
(0 <= m < 2 ^ mw)%Z ->
(0 <= e < 2 ^ ew)%Z ->
split_bits (join_bits s m e) = (s, m, e)
intros s m e Hm He.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
split_bits (join_bits s m e) = (s, m, e)
assert (0 <= mw)%Z as Hmw.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
(0 <= mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
split_bits (join_bits s m e) = (s, m, e)
  destruct mw as [|mw'|mw'] ; try easy.
mw, ew:Z
s:bool
m, e:Z
mw':positive
Hm:(0 <= m < 2 ^ Z.neg mw')%Z
He:(0 <= e < 2 ^ ew)%Z
(0 <= Z.neg mw')%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
split_bits (join_bits s m e) = (s, m, e)
  clear -Hm ; simpl in Hm ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
split_bits (join_bits s m e) = (s, m, e)
assert (0 <= ew)%Z as Hew.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
(0 <= ew)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
split_bits (join_bits s m e) = (s, m, e)
  destruct ew as [|ew'|ew'] ; try easy.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
ew':positive
He:(0 <= e < 2 ^ Z.neg ew')%Z
Hmw:(0 <= mw)%Z
(0 <= Z.neg ew')%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
split_bits (join_bits s m e) = (s, m, e)
  clear -He ; simpl in He ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
split_bits (join_bits s m e) = (s, m, e)
unfold split_bits, join_bits.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((2 ^ mw * 2 ^ ew <=?
  Z.shiftl ((if s then 2 ^ ew else 0) + e) mw + m)%Z,
((Z.shiftl ((if s then 2 ^ ew else 0) + e) mw + m)
 mod 2 ^ mw)%Z,
(((Z.shiftl ((if s then 2 ^ ew else 0) + e) mw + m) /
  2 ^ mw) mod 2 ^ ew)%Z) = (s, m, e)
rewrite Z.shiftl_mul_pow2 by easy.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((2 ^ mw * 2 ^ ew <=?
  ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)%Z,
((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)
 mod 2 ^ mw)%Z,
(((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m) /
  2 ^ mw) mod 2 ^ ew)%Z) = (s, m, e)
apply f_equal2 ; [apply f_equal2|].
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=?
 ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)%Z = s
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)
 mod 2 ^ mw)%Z = m
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m) /
  2 ^ mw) mod 2 ^ ew)%Z = e
-
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=?
 ((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)%Z = s case s.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=? (2 ^ ew + e) * 2 ^ mw + m)%Z =
true
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=? (0 + e) * 2 ^ mw + m)%Z = false
  +
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=? (2 ^ ew + e) * 2 ^ mw + m)%Z =
true apply Zle_bool_true.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <= (2 ^ ew + e) * 2 ^ mw + m)%Z
    apply Zle_0_minus_le.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= (2 ^ ew + e) * 2 ^ mw + m - 2 ^ mw * 2 ^ ew)%Z
    ring_simplify.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= e * 2 ^ mw + m)%Z
    apply Zplus_le_0_compat.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= e * 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= m)%Z
    apply Zmult_le_0_compat.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= e)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= m)%Z
    apply He.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= m)%Z
    clear -Hm ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= m)%Z
    apply Hm.
  +
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 2 ^ ew <=? (0 + e) * 2 ^ mw + m)%Z = false apply Zle_bool_false.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((0 + e) * 2 ^ mw + m < 2 ^ mw * 2 ^ ew)%Z
    apply Zplus_lt_reg_l with (2^mw * (-e))%Z.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * - e + ((0 + e) * 2 ^ mw + m) <
 2 ^ mw * - e + 2 ^ mw * 2 ^ ew)%Z
    replace (2 ^ mw * - e + ((0 + e) * 2 ^ mw + m))%Z with (m * 1)%Z by ring.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(m * 1 < 2 ^ mw * - e + 2 ^ mw * 2 ^ ew)%Z
    rewrite <- Zmult_plus_distr_r.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(m * 1 < 2 ^ mw * (- e + 2 ^ ew))%Z
    apply Z.lt_le_trans with (2^mw * 1)%Z.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(m * 1 < 2 ^ mw * 1)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 1 <= 2 ^ mw * (- e + 2 ^ ew))%Z
    now apply Zmult_lt_compat_r.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(2 ^ mw * 1 <= 2 ^ mw * (- e + 2 ^ ew))%Z
    apply Zmult_le_compat_l.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(1 <= - e + 2 ^ ew)%Z
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
    clear -He ; omega.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(0 <= 2 ^ mw)%Z
    clear -Hm ; omega.
-
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m)
 mod 2 ^ mw)%Z = m rewrite Zplus_comm.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((m + ((if s then 2 ^ ew else 0) + e) * 2 ^ mw)
 mod 2 ^ mw)%Z = m
  rewrite Z_mod_plus_full.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(m mod 2 ^ mw)%Z = m
  now apply Zmod_small.
-
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((((if s then 2 ^ ew else 0) + e) * 2 ^ mw + m) /
  2 ^ mw) mod 2 ^ ew)%Z = e rewrite Z_div_plus_full_l by (clear -Hm ; omega).
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + m / 2 ^ mw)
 mod 2 ^ ew)%Z = e
  rewrite Zdiv_small with (1 := Hm).
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e + 0) mod 2 ^ ew)%Z = e
  rewrite Zplus_0_r.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(((if s then 2 ^ ew else 0) + e) mod 2 ^ ew)%Z = e
  case s.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((2 ^ ew + e) mod 2 ^ ew)%Z = e
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((0 + e) mod 2 ^ ew)%Z = e
  +
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((2 ^ ew + e) mod 2 ^ ew)%Z = e replace (2^ew + e)%Z with (e + 1 * 2^ew)%Z by ring.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((e + 1 * 2 ^ ew) mod 2 ^ ew)%Z = e
    rewrite Z_mod_plus_full.
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
(e mod 2 ^ ew)%Z = e
    now apply Zmod_small.
  +
mw, ew:Z
s:bool
m, e:Z
Hm:(0 <= m < 2 ^ mw)%Z
He:(0 <= e < 2 ^ ew)%Z
Hmw:(0 <= mw)%Z
Hew:(0 <= ew)%Z
((0 + e) mod 2 ^ ew)%Z = e now apply Zmod_small.
Qed.

Hypothesis Hmw : (0 < mw)%Z.
Hypothesis Hew : (0 < ew)%Z.

Let emax := Zpower 2 (ew - 1).
Let prec := (mw + 1)%Z.
Let emin := (3 - emax - prec)%Z.
Let binary_float := binary_float prec emax.

Let Hprec : (0 < prec)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
(0 < prec)%Z
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
(0 < prec)%Z
unfold prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
(0 < mw + 1)%Z
apply Zle_lt_succ.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
(0 <= mw)%Z
now apply Zlt_le_weak.
Qed.

Let Hm_gt_0 : (0 < 2^mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
(0 < 2 ^ mw)%Z
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
(0 < 2 ^ mw)%Z
apply (Zpower_gt_0 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
(0 <= mw)%Z
now apply Zlt_le_weak.
Qed.

Let He_gt_0 : (0 < 2^ew)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
(0 < 2 ^ ew)%Z
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
(0 < 2 ^ ew)%Z
apply (Zpower_gt_0 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
(0 <= ew)%Z
now apply Zlt_le_weak.
Qed.

Hypothesis Hmax : (prec < emax)%Z.

Theorem join_split_bits :
  forall x,
  (0 <= x < Zpower 2 (mw + ew + 1))%Z ->
  let '(s, m, e) := split_bits x in
  join_bits s m e = x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
let '(s, m, e) := split_bits x in join_bits s m e = x
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
let '(s, m, e) := split_bits x in join_bits s m e = x
intros x Hx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
let '(s, m, e) := split_bits x in join_bits s m e = x
unfold split_bits, join_bits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(Z.shiftl
   ((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
    (x / 2 ^ mw) mod 2 ^ ew) mw + x mod 2 ^ mw)%Z = x
rewrite Z.shiftl_mul_pow2 by now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
  (x / 2 ^ mw) mod 2 ^ ew) * 2 ^ mw + x mod 2 ^ mw)%Z =
x
pattern x at 4 ; rewrite Z_div_mod_eq_full with x (2^mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
  (x / 2 ^ mw) mod 2 ^ ew) * 2 ^ mw + x mod 2 ^ mw)%Z =
(2 ^ mw * (x / 2 ^ mw) + x mod 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply (f_equal (fun v => (v + _)%Z)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
  (x / 2 ^ mw) mod 2 ^ ew) * 2 ^ mw)%Z =
(2 ^ mw * (x / 2 ^ mw))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
rewrite Zmult_comm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw *
 ((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
  (x / 2 ^ mw) mod 2 ^ ew))%Z =
(2 ^ mw * (x / 2 ^ mw))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply f_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
 (x / 2 ^ mw) mod 2 ^ ew)%Z = (x / 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
pattern (x / (2^mw))%Z at 2 ; rewrite Z_div_mod_eq_full with (x / (2^mw))%Z (2^ew)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
((if 2 ^ mw * 2 ^ ew <=? x then 2 ^ ew else 0) +
 (x / 2 ^ mw) mod 2 ^ ew)%Z =
(2 ^ ew * (x / 2 ^ mw / 2 ^ ew) +
 (x / 2 ^ mw) mod 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply (f_equal (fun v => (v + _)%Z)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then (2 ^ ew)%Z else 0%Z) =
(2 ^ ew * (x / 2 ^ mw / 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
replace (x / 2 ^ mw / 2 ^ ew)%Z with (if Zle_bool (2 ^ mw * 2 ^ ew) x then 1 else 0)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then (2 ^ ew)%Z else 0%Z) =
(2 ^ ew * (if 2 ^ mw * 2 ^ ew <=? x then 1 else 0))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z) =
(x / 2 ^ mw / 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
case Zle_bool.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z = (2 ^ ew * 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
0%Z = (2 ^ ew * 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z) =
(x / 2 ^ mw / 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now rewrite Zmult_1_r.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
0%Z = (2 ^ ew * 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z) =
(x / 2 ^ mw / 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now rewrite Zmult_0_r.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z) =
(x / 2 ^ mw / 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
rewrite Zdiv_Zdiv.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z) =
(x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply sym_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(x / (2 ^ mw * 2 ^ ew))%Z =
(if (2 ^ mw * 2 ^ ew <=? x)%Z then 1%Z else 0%Z)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
case Zle_bool_spec ; intros Hs.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 1%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply Zle_antisym.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x / (2 ^ mw * 2 ^ ew) <= 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
cut (x / (2^mw * 2^ew) < 2)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x / (2 ^ mw * 2 ^ ew) < 2)%Z ->
(x / (2 ^ mw * 2 ^ ew) <= 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x / (2 ^ mw * 2 ^ ew) < 2)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x / (2 ^ mw * 2 ^ ew) < 2)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply Zdiv_lt_upper_bound.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(0 < 2 ^ mw * 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x < 2 * (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zmult_lt_0_compat.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x < 2 * (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
rewrite <- Zpower_exp ; try ( apply Z.le_ge ; apply Zlt_le_weak ; assumption ).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x < 2 * 2 ^ (mw + ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
change 2%Z at 1 with (Zpower 2 1).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x < 2 ^ 1 * 2 ^ (mw + ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
rewrite <- Zpower_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(x < 2 ^ (1 + (mw + ew)))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(mw + ew >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now rewrite Zplus_comm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(mw + ew >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(mw + ew >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply Z.le_ge.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(0 <= mw + ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zplus_le_0_compat ; apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 <= x / (2 ^ mw * 2 ^ ew))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply Zdiv_le_lower_bound.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(0 < 2 ^ mw * 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 * (2 ^ mw * 2 ^ ew) <= x)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zmult_lt_0_compat.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(2 ^ mw * 2 ^ ew <= x)%Z
(1 * (2 ^ mw * 2 ^ ew) <= x)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now rewrite Zmult_1_l.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(x / (2 ^ mw * 2 ^ ew))%Z = 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
apply Zdiv_small.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hs:(x < 2 ^ mw * 2 ^ ew)%Z
(0 <= x < 2 ^ mw * 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(0 <= 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ ew)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zgt_not_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(2 ^ mw)%Z <> 0%Z
now apply Zgt_not_eq.
Qed.

Theorem split_bits_inj :
  forall x y,
  (0 <= x < Zpower 2 (mw + ew + 1))%Z ->
  (0 <= y < Zpower 2 (mw + ew + 1))%Z ->
  split_bits x = split_bits y ->
  x = y.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x y : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
(0 <= y < 2 ^ (mw + ew + 1))%Z ->
split_bits x = split_bits y -> x = y
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x y : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
(0 <= y < 2 ^ (mw + ew + 1))%Z ->
split_bits x = split_bits y -> x = y
intros x y Hx Hy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
split_bits x = split_bits y -> x = y
generalize (join_split_bits x Hx) (join_split_bits y Hy).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
(let '(s, m, e) := split_bits x in join_bits s m e = x) ->
(let '(s, m, e) := split_bits y in join_bits s m e = y) ->
split_bits x = split_bits y -> x = y
destruct (split_bits x) as ((sx, mx), ex).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
join_bits sx mx ex = x ->
(let '(s, m, e) := split_bits y in join_bits s m e = y) ->
(sx, mx, ex) = split_bits y -> x = y
destruct (split_bits y) as ((sy, my), ey).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
sy:bool
my, ey:Z
join_bits sx mx ex = x ->
join_bits sy my ey = y ->
(sx, mx, ex) = (sy, my, ey) -> x = y
intros Jx Jy H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
sy:bool
my, ey:Z
Jx:join_bits sx mx ex = x
Jy:join_bits sy my ey = y
H:(sx, mx, ex) = (sy, my, ey)
x = y revert Jx Jy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
sy:bool
my, ey:Z
H:(sx, mx, ex) = (sy, my, ey)
join_bits sx mx ex = x ->
join_bits sy my ey = y -> x = y
inversion_clear H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
sy:bool
my, ey:Z
join_bits sy my ey = x ->
join_bits sy my ey = y -> x = y
intros Jx Jy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x, y:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Hy:(0 <= y < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
sy:bool
my, ey:Z
Jx:join_bits sy my ey = x
Jy:join_bits sy my ey = y
x = y
now rewrite <- Jx.
Qed.

Definition bits_of_binary_float (x : binary_float) :=
  match x with
  | B754_zero sx => join_bits sx 0 0
  | B754_infinity sx => join_bits sx 0 (Zpower 2 ew - 1)
  | B754_nan sx plx _ => join_bits sx (Zpos plx) (Zpower 2 ew - 1)
  | B754_finite sx mx ex _ =>
    let m := (Zpos mx - Zpower 2 mw)%Z in
    if Zle_bool 0 m then
      join_bits sx m (ex - emin + 1)
    else
      join_bits sx (Zpos mx) 0
  end.

Definition split_bits_of_binary_float (x : binary_float) :=
  match x with
  | B754_zero sx => (sx, 0, 0)%Z
  | B754_infinity sx => (sx, 0, Zpower 2 ew - 1)%Z
  | B754_nan sx plx _ => (sx, Zpos plx, Zpower 2 ew - 1)%Z
  | B754_finite sx mx ex _ =>
    let m := (Zpos mx - Zpower 2 mw)%Z in
    if Zle_bool 0 m then
      (sx, m, ex - emin + 1)%Z
    else
      (sx, Zpos mx, 0)%Z
  end.

Theorem split_bits_of_binary_float_correct :
  forall x,
  split_bits (bits_of_binary_float x) = split_bits_of_binary_float x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
split_bits (bits_of_binary_float x) =
split_bits_of_binary_float x
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
split_bits (bits_of_binary_float x) =
split_bits_of_binary_float x
intros [sx|sx|sx plx Hplx|sx mx ex Hx] ;
  try ( simpl ; apply split_join_bits ; split ; try apply Z.le_refl ; try apply Zlt_pred ; trivial ; omega ).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
split_bits
  (bits_of_binary_float (B754_nan sx plx Hplx)) =
split_bits_of_binary_float (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
split_bits (join_bits sx (Z.pos plx) (2 ^ ew - 1)) =
(sx, Z.pos plx, (2 ^ ew - 1)%Z)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx) apply split_join_bits; split; try (zify; omega).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
destruct (digits2_Pnat_correct plx).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < Zpower_nat 2 (S (digits2_Pnat plx)))%Z
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
unfold nan_pl in Hplx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.pos (SpecFloatCopy.digits2_pos plx) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < Zpower_nat 2 (S (digits2_Pnat plx)))%Z
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
rewrite Zpos_digits2_pos, <- Z_of_nat_S_digits2_Pnat in Hplx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < Zpower_nat 2 (S (digits2_Pnat plx)))%Z
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
rewrite Zpower_nat_Z in H0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
eapply Z.lt_le_trans.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(Z.pos plx < ?m)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(?m <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx) apply H0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(2 ^ Z.of_nat (S (digits2_Pnat plx)) <= 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
change 2%Z with (radix_val radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(radix2 ^ Z.of_nat (S (digits2_Pnat plx)) <=
 radix2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx) apply Zpower_le.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) <? prec)%Z =
true
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(Z.of_nat (S (digits2_Pnat plx)) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
rewrite Z.ltb_lt in Hplx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) < prec)%Z
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(Z.of_nat (S (digits2_Pnat plx)) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
unfold prec in *.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - (mw + 1))%Z:Z
binary_float:=Binary.binary_float (mw + 1) emax:Set
Hprec:(0 < mw + 1)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(mw + 1 < emax)%Z
sx:bool
plx:positive
Hplx:(Z.of_nat (S (digits2_Pnat plx)) < mw + 1)%Z
H:(Zpower_nat 2 (digits2_Pnat plx) <= Z.pos plx)%Z
H0:(Z.pos plx < 2 ^ Z.of_nat (S (digits2_Pnat plx)))%Z
(Z.of_nat (S (digits2_Pnat plx)) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx) zify; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (bits_of_binary_float (B754_finite sx mx ex Hx)) =
split_bits_of_binary_float (B754_finite sx mx ex Hx)
(* *)
unfold bits_of_binary_float, split_bits_of_binary_float.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
assert (Hf: (emin <= ex /\ Zdigits radix2 (Zpos mx) <= prec)%Z).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
destruct (andb_prop _ _ Hx) as (Hx', _).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':canonical_mantissa prec emax mx ex = true
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
unfold canonical_mantissa in Hx'.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':Zeq_bool
  (FLT_exp (3 - emax - prec) prec
     (Z.pos (SpecFloatCopy.digits2_pos mx) + ex))
  ex = true
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
rewrite Zpos_digits2_pos in Hx'.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':Zeq_bool
  (FLT_exp (3 - emax - prec) prec
     (Zdigits radix2 (Z.pos mx) + ex)) ex = true
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
generalize (Zeq_bool_eq _ _ Hx').
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':Zeq_bool
  (FLT_exp (3 - emax - prec) prec
     (Zdigits radix2 (Z.pos mx) + ex)) ex = true
FLT_exp (3 - emax - prec) prec
  (Zdigits radix2 (Z.pos mx) + ex) = ex ->
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
unfold FLT_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':Zeq_bool
  (FLT_exp (3 - emax - prec) prec
     (Zdigits radix2 (Z.pos mx) + ex)) ex = true
Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex ->
(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
unfold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hx':Zeq_bool
  (FLT_exp (3 - emax - prec) prec
     (Zdigits radix2 (Z.pos mx) + ex)) ex = true
Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex ->
(3 - emax - prec <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
clear ; zify ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
split_bits
  (if (0 <=? Z.pos mx - 2 ^ mw)%Z
   then
    join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
   else join_bits sx (Z.pos mx) 0) =
(if (0 <=? Z.pos mx - 2 ^ mw)%Z
 then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
 else (sx, Z.pos mx, 0%Z))
case Zle_bool_spec ; intros H ;
  [ apply -> Z.le_0_sub in H | apply -> Z.lt_sub_0 in H ] ;
  apply split_join_bits ; try now split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
(* *)
split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= Z.pos mx - 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
clear -He_gt_0 H ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
cut (Zpos mx < 2 * 2^mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx < 2 * 2 ^ mw)%Z ->
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx < 2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx < 2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
replace (2 * 2^mw)%Z with (2^prec)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Z.pos mx < 2 ^ prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(2 ^ prec)%Z = (2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
apply (Zpower_gt_Zdigits radix2 _ (Zpos mx)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(Zdigits radix2 (Z.pos mx) <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(2 ^ prec)%Z = (2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
apply Hf.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(2 ^ prec)%Z = (2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
unfold prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(2 ^ (mw + 1))%Z = (2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
rewrite Zplus_comm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(2 ^ (1 + mw))%Z = (2 * 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
apply Zpower_exp ; apply Z.le_ge.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
(* *)
split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(0 <= ex - emin + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(ex - emin + 1 < 2 ^ ew)%Z
generalize (proj1 Hf).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(emin <= ex)%Z -> (0 <= ex - emin + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(ex - emin + 1 < 2 ^ ew)%Z
clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
(ex - emin + 1 < 2 ^ ew)%Z
destruct (andb_prop _ _ Hx) as (_, Hx').
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(ex - emin + 1 < 2 ^ ew)%Z
unfold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(ex - (3 - emax - prec) + 1 < 2 ^ ew)%Z
replace (2^ew)%Z with (2 * emax)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(ex - (3 - emax - prec) + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ ew)%Z
generalize (Zle_bool_imp_le _ _ Hx').
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(ex <= emax - prec)%Z ->
(ex - (3 - emax - prec) + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ ew)%Z
clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ ew)%Z
apply sym_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 ^ ew)%Z = (2 * emax)%Z
rewrite (Zsucc_pred ew).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 ^ Z.succ (Z.pred ew))%Z = (2 * emax)%Z
unfold Z.succ.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 ^ (Z.pred ew + 1))%Z = (2 * emax)%Z
rewrite Zplus_comm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(2 ^ (1 + Z.pred ew))%Z = (2 * emax)%Z
apply Zpower_exp ; apply Z.le_ge.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(0 <= 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(0 <= Z.pred ew)%Z
discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Hx:bounded prec emax mx ex = true
Hf:(emin <= ex)%Z /\
(Zdigits radix2 (Z.pos mx) <= prec)%Z
H:(2 ^ mw <= Z.pos mx)%Z
Hx':(ex <=? emax - prec)%Z = true
(0 <= Z.pred ew)%Z
now apply Zlt_0_le_0_pred.
Qed.

Theorem bits_of_binary_float_range:
  forall x, (0 <= bits_of_binary_float x < 2^(mw+ew+1))%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
(0 <= bits_of_binary_float x < 2 ^ (mw + ew + 1))%Z
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
(0 <= bits_of_binary_float x < 2 ^ (mw + ew + 1))%Z
unfold bits_of_binary_float.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
(0 <=
 match x with
 | B754_zero sx => join_bits sx 0 0
 | B754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
 | B754_nan sx plx _ =>
     join_bits sx (Z.pos plx) (2 ^ ew - 1)
 | B754_finite sx mx ex _ =>
     if 0 <=? Z.pos mx - 2 ^ mw
     then
      join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
     else join_bits sx (Z.pos mx) 0
 end < 2 ^ (mw + ew + 1))%Z
intros [sx|sx|sx pl pl_range|sx mx ex H].
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= join_bits sx 0 0 < 2 ^ (mw + ew + 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= join_bits sx 0 (2 ^ ew - 1) < 2 ^ (mw + ew + 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:nan_pl prec pl = true
(0 <= join_bits sx (Z.pos pl) (2 ^ ew - 1) <
 2 ^ (mw + ew + 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
H:bounded prec emax mx ex = true
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 
 2 ^ (mw + ew + 1))%Z
-
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= join_bits sx 0 0 < 2 ^ (mw + ew + 1))%Z apply join_bits_range ; now split.
-
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= join_bits sx 0 (2 ^ ew - 1) < 2 ^ (mw + ew + 1))%Z apply join_bits_range.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= 0 < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  now split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  clear -He_gt_0 ; omega.
-
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:nan_pl prec pl = true
(0 <= join_bits sx (Z.pos pl) (2 ^ ew - 1) <
 2 ^ (mw + ew + 1))%Z apply Z.ltb_lt in pl_range.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= join_bits sx (Z.pos pl) (2 ^ ew - 1) <
 2 ^ (mw + ew + 1))%Z
  apply join_bits_range.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= Z.pos pl < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= Z.pos pl)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(Z.pos pl < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  easy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(Z.pos pl < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  apply (Zpower_gt_Zdigits radix2 _ (Zpos pl)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(Zdigits radix2 (Z.pos pl) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  apply Z.lt_succ_r.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(Zdigits radix2 (Z.pos pl) < Z.succ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  now rewrite <- Zdigits2_Zdigits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
pl:positive
pl_range:(Z.pos (SpecFloatCopy.digits2_pos pl) <
 prec)%Z
(0 <= 2 ^ ew - 1 < 2 ^ ew)%Z
  clear -He_gt_0 ; omega.
-
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
H:bounded prec emax mx ex = true
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z unfold bounded in H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
H:(canonical_mantissa prec emax mx ex &&
 (ex <=? emax - prec)%Z)%bool = true
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z
  apply Bool.andb_true_iff in H ; destruct H as [A B].
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:canonical_mantissa prec emax mx ex = true
B:(ex <=? emax - prec)%Z = true
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z
  apply Z.leb_le in B.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:canonical_mantissa prec emax mx ex = true
B:(ex <= emax - prec)%Z
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z
  unfold canonical_mantissa, FLT_exp in A.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Zeq_bool
  (Z.max
     (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
      prec) (3 - emax - prec)) ex = true
B:(ex <= emax - prec)%Z
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z apply Zeq_bool_eq in A.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
(0 <=
 (if 0 <=? Z.pos mx - 2 ^ mw
  then
   join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
  else join_bits sx (Z.pos mx) 0) < 2 ^ (mw + ew + 1))%Z
  case Zle_bool_spec ; intros H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <=
 join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1) <
 2 ^ (mw + ew + 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(Z.pos mx - 2 ^ mw < 0)%Z
(0 <= join_bits sx (Z.pos mx) 0 < 2 ^ (mw + ew + 1))%Z
  +
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <=
 join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1) <
 2 ^ (mw + ew + 1))%Z apply join_bits_range.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z
    *
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= Z.pos mx - 2 ^ mw < 2 ^ mw)%Z split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= Z.pos mx - 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
      clear -H ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
      rewrite Zpos_digits2_pos in A.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
      cut (Zpos mx < 2 ^ prec)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z ->
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z
      unfold prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ (mw + 1))%Z ->
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z
      rewrite Zpower_plus by (clear -Hmw ; omega).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ mw * 2 ^ 1)%Z ->
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z
      change (2^1)%Z with 2%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ mw * 2)%Z ->
(Z.pos mx - 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z
      clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Z.pos mx < 2 ^ prec)%Z
      apply (Zpower_gt_Zdigits radix2 _ (Zpos mx)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(Zdigits radix2 (Z.pos mx) <= prec)%Z
      clear -A ; zify ; omega.
    *
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= ex - emin + 1 < 2 ^ ew)%Z split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(0 <= ex - emin + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1 < 2 ^ ew)%Z
      unfold emin ; clear -A ; zify ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1 < 2 ^ ew)%Z
      replace ew with ((ew - 1) + 1)%Z by ring.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1 < 2 ^ (ew - 1 + 1))%Z
      rewrite Zpower_plus by (clear - Hew ; omega).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1 < 2 ^ (ew - 1) * 2 ^ 1)%Z
      unfold emin, emax in *.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - 2 ^ (ew - 1) - prec)%Z:Z
binary_float:=Binary.binary_float prec (2 ^ (ew - 1)):Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < 2 ^ (ew - 1))%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - 2 ^ (ew - 1) - prec) = ex
B:(ex <= 2 ^ (ew - 1) - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - (3 - 2 ^ (ew - 1) - prec) + 1 <
 2 ^ (ew - 1) * 2 ^ 1)%Z
      change (2^1)%Z with 2%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - 2 ^ (ew - 1) - prec)%Z:Z
binary_float:=Binary.binary_float prec (2 ^ (ew - 1)):Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < 2 ^ (ew - 1))%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - 2 ^ (ew - 1) - prec) = ex
B:(ex <= 2 ^ (ew - 1) - prec)%Z
H:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - (3 - 2 ^ (ew - 1) - prec) + 1 < 2 ^ (ew - 1) * 2)%Z
      clear -B ; omega.
  +
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(Z.pos mx - 2 ^ mw < 0)%Z
(0 <= join_bits sx (Z.pos mx) 0 < 2 ^ (mw + ew + 1))%Z apply -> Z.lt_sub_0 in H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
A:Z.max
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex -
   prec) (3 - emax - prec) = ex
B:(ex <= emax - prec)%Z
H:(Z.pos mx < 2 ^ mw)%Z
(0 <= join_bits sx (Z.pos mx) 0 < 2 ^ (mw + ew + 1))%Z
    apply join_bits_range ; now split.
Qed.

Definition binary_float_of_bits_aux x :=
  let '(sx, mx, ex) := split_bits x in
  if Zeq_bool ex 0 then
    match mx with
    | Z0 => F754_zero sx
    | Zpos px => F754_finite sx px emin
    | Zneg _ => F754_nan false xH (* dummy *)
    end
  else if Zeq_bool ex (Zpower 2 ew - 1) then
    match mx with
      | Z0 => F754_infinity sx
      | Zpos plx => F754_nan sx plx
      | Zneg _ => F754_nan false xH (* dummy *)
    end
  else
    match (mx + Zpower 2 mw)%Z with
    | Zpos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false xH (* dummy *)
    end.

Lemma binary_float_of_bits_aux_correct :
  forall x,
  valid_binary prec emax (binary_float_of_bits_aux x) = true.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
valid_binary prec emax (binary_float_of_bits_aux x) =
true
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
valid_binary prec emax (binary_float_of_bits_aux x) =
true
intros x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
valid_binary prec emax (binary_float_of_bits_aux x) =
true
unfold binary_float_of_bits_aux, split_bits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true
assert (Hnan: nan_pl prec 1 = true).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
nan_pl prec 1 = true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true
  apply Z.ltb_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
(Z.pos (SpecFloatCopy.digits2_pos 1) < prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true
  simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
(1 < prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true unfold prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
(1 < mw + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true
  clear -Hmw ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) 0
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
    then
     match (x mod 2 ^ mw)%Z with
     | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
     | Z.pos plx =>
         F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (x mod 2 ^ mw + 2 ^ mw)%Z with
     | Z.pos px =>
         F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
           ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
     | _ => F754_nan false 1
     end) = true
case Zeq_bool_spec ; intros He1.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
valid_binary prec emax
  match (x mod 2 ^ mw)%Z with
  | 0%Z => F754_zero (2 ^ mw * 2 ^ ew <=? x)%Z
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin
  | Z.neg _ => F754_nan false 1
  end = true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
case_eq (x mod 2^mw)%Z ; try easy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
forall p : positive,
(x mod 2 ^ mw)%Z = Z.pos p ->
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z p emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
(* subnormal *)
intros px Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
assert (Zdigits radix2 (Zpos px) <= mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
(Zdigits radix2 (Z.pos px) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply Zdigits_le_Zpower.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
(Z.abs (Z.pos px) < radix2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
(Z.pos px < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
rewrite <- Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
eapply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
(2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
now apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px emin) =
true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply bounded_canonical_lt_emax ; try assumption.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
canonical radix2 (FLT_exp (3 - emax - prec) prec)
  {| Fnum := Z.pos px; Fexp := emin |}
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
unfold canonical, cexp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Fexp {| Fnum := Z.pos px; Fexp := emin |} =
FLT_exp (3 - emax - prec) prec
  (mag radix2
     (F2R {| Fnum := Z.pos px; Fexp := emin |}))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
fold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Fexp {| Fnum := Z.pos px; Fexp := emin |} =
FLT_exp emin prec
  (mag radix2
     (F2R {| Fnum := Z.pos px; Fexp := emin |}))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
rewrite mag_F2R_Zdigits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Fexp {| Fnum := Z.pos px; Fexp := emin |} =
FLT_exp emin prec (Zdigits radix2 (Z.pos px) + emin)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Z.pos px <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Fexp {| Fnum := Z.pos px; Fexp := emin |} =
FLT_exp emin prec (Zdigits radix2 (Z.pos px) + emin)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
unfold Fexp, FLT_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
emin =
Z.max (Zdigits radix2 (Z.pos px) + emin - prec) emin
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply sym_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
Z.max (Zdigits radix2 (Z.pos px) + emin - prec) emin =
emin
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply Zmax_right.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(Zdigits radix2 (Z.pos px) + emin - prec <= emin)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
clear -H Hprec.
mw, ew:Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
Hprec:(0 < prec)%Z
px:positive
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(Zdigits radix2 (Z.pos px) + emin - prec <= emin)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
unfold prec ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
(F2R {| Fnum := Z.pos px; Fexp := emin |} <
 bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply Rnot_le_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
~
(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
intros H0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
refine (_ (mag_le radix2 _ _ _ H0)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(mag radix2 (bpow radix2 emax) <=
 mag radix2 (F2R {| Fnum := Z.pos px; Fexp := emin |}))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
rewrite mag_bpow.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(emax + 1 <=
 mag radix2 (F2R {| Fnum := Z.pos px; Fexp := emin |}))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
rewrite mag_F2R_Zdigits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(emax + 1 <= Zdigits radix2 (Z.pos px) + emin)%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
Z.pos px <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(emax + 1 <= Zdigits radix2 (Z.pos px) + emin)%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
unfold emin, prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(emax + 1 <=
 Zdigits radix2 (Z.pos px) + (3 - emax - (mw + 1)))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply Zlt_not_le.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(Zdigits radix2 (Z.pos px) + (3 - emax - (mw + 1)) <
 emax + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
cut (0 < emax)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < emax)%Z ->
(Zdigits radix2 (Z.pos px) + (3 - emax - (mw + 1)) <
 emax + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true clear -H Hew ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply (Zpower_gt_0 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 <= ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
clear -Hew ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z = 0%Z
px:positive
Hm:(x mod 2 ^ mw)%Z = Z.pos px
H:(Zdigits radix2 (Z.pos px) <= mw)%Z
H0:(bpow radix2 emax <=
 F2R {| Fnum := Z.pos px; Fexp := emin |})%R
(0 < bpow radix2 emax)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
apply bpow_gt_0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
valid_binary prec emax
  (if Zeq_bool ((x / 2 ^ mw) mod 2 ^ ew) (2 ^ ew - 1)
   then
    match (x mod 2 ^ mw)%Z with
    | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
    | Z.pos plx =>
        F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (x mod 2 ^ mw + 2 ^ mw)%Z with
    | Z.pos px =>
        F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
          ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
    | _ => F754_nan false 1
    end) = true
case Zeq_bool_spec ; intros He2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw)%Z with
  | 0%Z => F754_infinity (2 ^ mw * 2 ^ ew <=? x)%Z
  | Z.pos plx =>
      F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx
  | Z.neg _ => F754_nan false 1
  end = true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
case_eq (x mod 2 ^ mw)%Z; try easy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
forall p : positive,
(x mod 2 ^ mw)%Z = Z.pos p ->
valid_binary prec emax
  (F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z p) = true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
(* nan *)
intros plx Eqplx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
valid_binary prec emax
  (F754_nan (2 ^ mw * 2 ^ ew <=? x)%Z plx) = true
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true apply Z.ltb_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(Z.pos (SpecFloatCopy.digits2_pos plx) < prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
rewrite Zpos_digits2_pos.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(Zdigits radix2 (Z.pos plx) < prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
assert (forall a b, a <= b -> a < b+1)%Z by (intros; omega).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
H:forall a b : Z, (a <= b)%Z -> (a < b + 1)%Z
(Zdigits radix2 (Z.pos plx) < prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true apply H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
H:forall a b : Z, (a <= b)%Z -> (a < b + 1)%Z
(Zdigits radix2 (Z.pos plx) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true clear H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(Zdigits radix2 (Z.pos plx) <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
apply Zdigits_le_Zpower.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(Z.abs (Z.pos plx) < radix2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(Z.pos plx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
rewrite <- Eqplx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(x mod 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true edestruct Z_mod_lt; eauto.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
change 2%Z with (radix_val radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(radix2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
apply Z.lt_gt, Zpower_gt_0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z = (2 ^ ew - 1)%Z
plx:positive
Eqplx:(x mod 2 ^ mw)%Z = Z.pos plx
(0 <= mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
valid_binary prec emax
  match (x mod 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
        ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)
  | _ => F754_nan false 1
  end = true
case_eq (x mod 2^mw + 2^mw)%Z ; try easy.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
forall p : positive,
(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos p ->
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z p
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
(* normal *)
intros px Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
assert (prec = Zdigits radix2 (Zpos px)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
prec = Zdigits radix2 (Z.pos px)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
(* . *)
rewrite Zdigits_mag.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
prec = mag radix2 (IZR (Z.pos px))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
Z.pos px <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
prec = mag radix2 (IZR (Z.pos px))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply sym_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
mag radix2 (IZR (Z.pos px)) = prec
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply mag_unique.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(bpow radix2 (prec - 1) <= Rabs (IZR (Z.pos px)) <
 bpow radix2 prec)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite <- abs_IZR.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(bpow radix2 (prec - 1) <= IZR (Z.abs (Z.pos px)) <
 bpow radix2 prec)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
unfold Z.abs.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(bpow radix2 (prec - 1) <= IZR (Z.pos px) <
 bpow radix2 prec)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
replace (prec - 1)%Z with mw by ( unfold prec ; ring ).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(bpow radix2 mw <= IZR (Z.pos px) < bpow radix2 prec)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite <- IZR_Zpower with (1 := Zlt_le_weak _ _ Hmw).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (radix2 ^ mw) <= IZR (Z.pos px) <
 bpow radix2 prec)%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite <- IZR_Zpower.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (radix2 ^ mw) <= IZR (Z.pos px) <
 IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(0 <= prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true 2: now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (radix2 ^ mw) <= IZR (Z.pos px) <
 IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite <- Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (radix2 ^ mw) <= IZR (x mod 2 ^ mw + 2 ^ mw) <
 IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
split.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (radix2 ^ mw) <= IZR (x mod 2 ^ mw + 2 ^ mw))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply IZR_le.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(radix2 ^ mw <= x mod 2 ^ mw + 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
change (radix2^mw)%Z with (0 + 2^mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(0 + 2 ^ mw <= x mod 2 ^ mw + 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply Zplus_le_compat_r.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(0 <= x mod 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
eapply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
now apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(IZR (x mod 2 ^ mw + 2 ^ mw) < IZR (radix2 ^ prec))%R
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply IZR_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
unfold prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ (mw + 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite Zpower_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ mw * radix2 ^ 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(mw >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true 2: now apply Z.le_ge ; apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ mw * radix2 ^ 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ mw * radix2 ^ 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
rewrite <- Zplus_diag_eq_mult_2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw + 2 ^ mw < radix2 ^ mw + radix2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
apply Zplus_lt_compat_r.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(x mod 2 ^ mw < radix2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
eapply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
(radix2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
now apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
valid_binary prec emax
  (F754_finite (2 ^ mw * 2 ^ ew <=? x)%Z px
     ((x / 2 ^ mw) mod 2 ^ ew + emin - 1)) = true
(* . *)
apply bounded_canonical_lt_emax ; try assumption.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
canonical radix2 (FLT_exp (3 - emax - prec) prec)
  {|
  Fnum := Z.pos px;
  Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |}
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
unfold canonical, cexp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
Fexp
  {|
  Fnum := Z.pos px;
  Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} =
FLT_exp (3 - emax - prec) prec
  (mag radix2
     (F2R
        {|
        Fnum := Z.pos px;
        Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |}))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
rewrite mag_F2R_Zdigits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
Fexp
  {|
  Fnum := Z.pos px;
  Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} =
FLT_exp (3 - emax - prec) prec
  (Zdigits radix2 (Z.pos px) +
   ((x / 2 ^ mw) mod 2 ^ ew + emin - 1))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
Z.pos px <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
Fexp
  {|
  Fnum := Z.pos px;
  Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} =
FLT_exp (3 - emax - prec) prec
  (Zdigits radix2 (Z.pos px) +
   ((x / 2 ^ mw) mod 2 ^ ew + emin - 1))
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
unfold Fexp, FLT_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
((x / 2 ^ mw) mod 2 ^ ew + emin - 1)%Z =
Z.max
  (Zdigits radix2 (Z.pos px) +
   ((x / 2 ^ mw) mod 2 ^ ew + emin - 1) - prec)
  (3 - emax - prec)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
rewrite <- H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
((x / 2 ^ mw) mod 2 ^ ew + emin - 1)%Z =
Z.max
  (prec + ((x / 2 ^ mw) mod 2 ^ ew + emin - 1) - prec)
  (3 - emax - prec)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
set (ex := ((x / 2^mw) mod 2^ew)%Z).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex + emin - 1)%Z =
Z.max (prec + (ex + emin - 1) - prec)
  (3 - emax - prec)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
replace (prec + (ex + emin - 1) - prec)%Z with (ex + emin - 1)%Z by ring.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex + emin - 1)%Z =
Z.max (ex + emin - 1) (3 - emax - prec)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
apply sym_eq.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
Z.max (ex + emin - 1) (3 - emax - prec) =
(ex + emin - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
apply Zmax_left.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex + emin - 1 >= 3 - emax - prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
revert He1.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z ->
(ex + emin - 1 >= 3 - emax - prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
fold ex.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
ex <> 0%Z -> (ex + emin - 1 >= 3 - emax - prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
cut (0 <= ex)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ex)%Z ->
ex <> 0%Z -> (ex + emin - 1 >= 3 - emax - prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ex)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
unfold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ex)%Z ->
ex <> 0%Z ->
(ex + (3 - emax - prec) - 1 >= 3 - emax - prec)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ex)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
clear ; intros H1 H2 ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ex)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
eapply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 ^ ew > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
apply (Zpower_gt_0 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
(F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |} <
 bpow radix2 emax)%R
apply Rnot_le_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
~
(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
intros H0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
False
refine (_ (mag_le radix2 _ _ _ H0)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(mag radix2 (bpow radix2 emax) <=
 mag radix2
   (F2R
      {|
      Fnum := Z.pos px;
      Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |}))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
rewrite mag_bpow.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(emax + 1 <=
 mag radix2
   (F2R
      {|
      Fnum := Z.pos px;
      Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |}))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
rewrite mag_F2R_Zdigits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(emax + 1 <=
 Zdigits radix2 (Z.pos px) +
 ((x / 2 ^ mw) mod 2 ^ ew + emin - 1))%Z -> False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
Z.pos px <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R 2: discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(emax + 1 <=
 Zdigits radix2 (Z.pos px) +
 ((x / 2 ^ mw) mod 2 ^ ew + emin - 1))%Z -> False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
rewrite <- H.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(emax + 1 <=
 prec + ((x / 2 ^ mw) mod 2 ^ ew + emin - 1))%Z ->
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply Zlt_not_le.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(prec + ((x / 2 ^ mw) mod 2 ^ ew + emin - 1) <
 emax + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
unfold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(prec +
 ((x / 2 ^ mw) mod 2 ^ ew + (3 - emax - prec) - 1) <
 emax + 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply Zplus_lt_reg_r with (emax - 1)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(prec +
 ((x / 2 ^ mw) mod 2 ^ ew + (3 - emax - prec) - 1) +
 (emax - 1) < emax + 1 + (emax - 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
ring_simplify.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
((x / 2 ^ mw) mod 2 ^ ew + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
revert He2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z ->
((x / 2 ^ mw) mod 2 ^ ew + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
set (ex := ((x / 2^mw) mod 2^ew)%Z).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
ex <> (2 ^ ew - 1)%Z -> (ex + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
cut (ex < 2^ew)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z ->
ex <> (2 ^ ew - 1)%Z -> (ex + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
replace (2^ew)%Z with (2 * emax)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 * emax)%Z ->
ex <> (2 * emax - 1)%Z -> (ex + 1 < 2 * emax)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 * emax)%Z = (2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
clear ; intros H1 H2 ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 * emax)%Z = (2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
replace ew with (1 + (ew - 1))%Z by ring.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 * emax)%Z = (2 ^ (1 + (ew - 1)))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
rewrite Zpower_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 * emax)%Z = (2 ^ 1 * 2 ^ (ew - 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply refl_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
clear -Hew ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(ex < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
eapply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(2 ^ ew > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply (Zpower_gt_0 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
ex:=((x / 2 ^ mw) mod 2 ^ ew)%Z:Z
(0 <= ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
now apply Zlt_le_weak.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hnan:nan_pl prec 1 = true
He1:((x / 2 ^ mw) mod 2 ^ ew)%Z <> 0%Z
He2:((x / 2 ^ mw) mod 2 ^ ew)%Z <> (2 ^ ew - 1)%Z
px:positive
Hm:(x mod 2 ^ mw + 2 ^ mw)%Z = Z.pos px
H:prec = Zdigits radix2 (Z.pos px)
H0:(bpow radix2 emax <=
 F2R
   {|
   Fnum := Z.pos px;
   Fexp := (x / 2 ^ mw) mod 2 ^ ew + emin - 1 |})%R
(0 < bpow radix2 emax)%R
apply bpow_gt_0.
Qed.

Definition binary_float_of_bits x :=
  FF2B prec emax _ (binary_float_of_bits_aux_correct x).

Theorem binary_float_of_bits_of_binary_float :
  forall x,
  binary_float_of_bits (bits_of_binary_float x) = x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
binary_float_of_bits (bits_of_binary_float x) = x
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : binary_float,
binary_float_of_bits (bits_of_binary_float x) = x
intros x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
binary_float_of_bits (bits_of_binary_float x) = x
apply B2FF_inj.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
B2FF prec emax
  (binary_float_of_bits (bits_of_binary_float x)) =
B2FF prec emax x
unfold binary_float_of_bits.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
B2FF prec emax
  (FF2B prec emax
     (binary_float_of_bits_aux
        (bits_of_binary_float x))
     (binary_float_of_bits_aux_correct
        (bits_of_binary_float x))) = B2FF prec emax x
rewrite B2FF_FF2B.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
binary_float_of_bits_aux (bits_of_binary_float x) =
B2FF prec emax x
unfold binary_float_of_bits_aux.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
(let
 '(sx, mx, ex) := split_bits (bits_of_binary_float x)
  in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx
   | Z.pos px => F754_finite sx px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax x
rewrite split_bits_of_binary_float_correct.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:binary_float
(let
 '(sx, mx, ex) := split_bits_of_binary_float x in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx
   | Z.pos px => F754_finite sx px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax x
destruct x as [sx|sx|sx plx Hplx|sx mx ex Bx].
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_zero sx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_zero sx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_infinity sx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_infinity sx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
apply refl_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_infinity sx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_infinity sx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
(* *)
simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(if Zeq_bool (2 ^ ew - 1) 0
 then F754_zero sx
 else
  if Zeq_bool (2 ^ ew - 1) (2 ^ ew - 1)
  then F754_infinity sx
  else
   match (2 ^ mw)%Z with
   | Z.pos px =>
       F754_finite sx px (2 ^ ew - 1 + emin - 1)
   | _ => F754_nan false 1
   end) = F754_infinity sx
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zeq_bool_false.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(if Zeq_bool (2 ^ ew - 1) (2 ^ ew - 1)
 then F754_infinity sx
 else
  match (2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite sx px (2 ^ ew - 1 + emin - 1)
  | _ => F754_nan false 1
  end) = F754_infinity sx
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
now rewrite Zeq_bool_true.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
cut (1 < 2^ew)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(1 < 2 ^ ew)%Z -> (2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(1 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx) clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
(1 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
now apply (Zpower_gt_1 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(let
 '(sx0, mx, ex) :=
  split_bits_of_binary_float (B754_nan sx plx Hplx) in
  if Zeq_bool ex 0
  then
   match mx with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx0
    | Z.pos plx0 => F754_nan sx0 plx0
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_nan sx plx Hplx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
(* *)
simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(if Zeq_bool (2 ^ ew - 1) 0
 then F754_finite sx plx emin
 else
  if Zeq_bool (2 ^ ew - 1) (2 ^ ew - 1)
  then F754_nan sx plx
  else
   match
     match (2 ^ mw)%Z with
     | 0%Z => Z.pos plx
     | Z.pos y' => Z.pos (plx + y')
     | Z.neg y' => Z.pos_sub plx y'
     end
   with
   | Z.pos px =>
       F754_finite sx px (2 ^ ew - 1 + emin - 1)
   | _ => F754_nan false 1
   end) = F754_nan sx plx
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zeq_bool_false.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(if Zeq_bool (2 ^ ew - 1) (2 ^ ew - 1)
 then F754_nan sx plx
 else
  match
    match (2 ^ mw)%Z with
    | 0%Z => Z.pos plx
    | Z.pos y' => Z.pos (plx + y')
    | Z.neg y' => Z.pos_sub plx y'
    end
  with
  | Z.pos px =>
      F754_finite sx px (2 ^ ew - 1 + emin - 1)
  | _ => F754_nan false 1
  end) = F754_nan sx plx
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zeq_bool_true; auto.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
cut (1 < 2^ew)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(1 < 2 ^ ew)%Z -> (2 ^ ew - 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(1 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx) clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
plx:positive
Hplx:nan_pl prec plx = true
(1 < 2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
now apply (Zpower_gt_1 radix2).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  split_bits_of_binary_float (B754_finite sx mx ex Bx)
  in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
(* *)
unfold split_bits_of_binary_float.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
(let
 '(sx0, mx0, ex0) :=
  if (0 <=? Z.pos mx - 2 ^ mw)%Z
  then (sx, (Z.pos mx - 2 ^ mw)%Z, (ex - emin + 1)%Z)
  else (sx, Z.pos mx, 0%Z) in
  if Zeq_bool ex0 0
  then
   match mx0 with
   | 0%Z => F754_zero sx0
   | Z.pos px => F754_finite sx0 px emin
   | Z.neg _ => F754_nan false 1
   end
  else
   if Zeq_bool ex0 (2 ^ ew - 1)
   then
    match mx0 with
    | 0%Z => F754_infinity sx0
    | Z.pos plx => F754_nan sx0 plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx0 + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx0 px (ex0 + emin - 1)
    | _ => F754_nan false 1
    end) = B2FF prec emax (B754_finite sx mx ex Bx)
case Zle_bool_spec ; intros Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(if Zeq_bool (ex - emin + 1) 0
 then
  match (Z.pos mx - 2 ^ mw)%Z with
  | 0%Z => F754_zero sx
  | Z.pos px => F754_finite sx px emin
  | Z.neg _ => F754_nan false 1
  end
 else
  if Zeq_bool (ex - emin + 1) (2 ^ ew - 1)
  then
   match (Z.pos mx - 2 ^ mw)%Z with
   | 0%Z => F754_infinity sx
   | Z.pos plx => F754_nan sx plx
   | Z.neg _ => F754_nan false 1
   end
  else
   match (Z.pos mx - 2 ^ mw + 2 ^ mw)%Z with
   | Z.pos px =>
       F754_finite sx px (ex - emin + 1 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
(* . *)
rewrite Zeq_bool_false.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(if Zeq_bool (ex - emin + 1) (2 ^ ew - 1)
 then
  match (Z.pos mx - 2 ^ mw)%Z with
  | 0%Z => F754_infinity sx
  | Z.pos plx => F754_nan sx plx
  | Z.neg _ => F754_nan false 1
  end
 else
  match (Z.pos mx - 2 ^ mw + 2 ^ mw)%Z with
  | Z.pos px =>
      F754_finite sx px (ex - emin + 1 + emin - 1)
  | _ => F754_nan false 1
  end) = B2FF prec emax (B754_finite sx mx ex Bx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zeq_bool_false.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
match (Z.pos mx - 2 ^ mw + 2 ^ mw)%Z with
| Z.pos px =>
    F754_finite sx px (ex - emin + 1 + emin - 1)
| _ => F754_nan false 1
end = B2FF prec emax (B754_finite sx mx ex Bx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> (2 ^ ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
now ring_simplify (Zpos mx - 2 ^ mw + 2 ^ mw)%Z (ex - emin + 1 + emin - 1)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> (2 ^ ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
destruct (andb_prop _ _ Bx) as (_, H1).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ex - emin + 1)%Z <> (2 ^ ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
generalize (Zle_bool_imp_le _ _ H1).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ex <= emax - prec)%Z ->
(ex - emin + 1)%Z <> (2 ^ ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
unfold emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ex <= emax - prec)%Z ->
(ex - (3 - emax - prec) + 1)%Z <> (2 ^ ew - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
replace (2^ew)%Z with (2 * emax)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ex <= emax - prec)%Z ->
(ex - (3 - emax - prec) + 1)%Z <> (2 * emax - 1)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
clear ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ ew)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
replace ew with (1 + (ew - 1))%Z by ring.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ (1 + (ew - 1)))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zpower_exp.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(2 * emax)%Z = (2 ^ 1 * 2 ^ (ew - 1))%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
apply refl_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
discriminate.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:(ex <=? emax - prec)%Z = true
(ew - 1 >= 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
clear -Hew ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
destruct (andb_prop _ _ Bx) as (H1, _).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
generalize (Zeq_bool_eq _ _ H1).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
FLT_exp (3 - emax - prec) prec
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex) = ex ->
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
rewrite Zpos_digits2_pos.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
FLT_exp (3 - emax - prec) prec
  (Zdigits radix2 (Z.pos mx) + ex) = ex ->
(ex - emin + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
unfold FLT_exp, emin.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
Z.max (Zdigits radix2 (Z.pos mx) + ex - prec)
  (3 - emax - prec) = ex ->
(ex - (3 - emax - prec) + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
generalize (Zdigits radix2 (Zpos mx)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(0 <= Z.pos mx - 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
forall z : Z,
Z.max (z + ex - prec) (3 - emax - prec) = ex ->
(ex - (3 - emax - prec) + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
clear.
mw, ew:Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
ex:Z
forall z : Z,
Z.max (z + ex - prec) (3 - emax - prec) = ex ->
(ex - (3 - emax - prec) + 1)%Z <> 0%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
intros ; zify ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
(if Zeq_bool 0 0
 then F754_finite sx mx emin
 else
  if Zeq_bool 0 (2 ^ ew - 1)
  then F754_nan sx mx
  else
   match (Z.pos mx + 2 ^ mw)%Z with
   | Z.pos px => F754_finite sx px (0 + emin - 1)
   | _ => F754_nan false 1
   end) = B2FF prec emax (B754_finite sx mx ex Bx)
(* . *)
rewrite Zeq_bool_true.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
F754_finite sx mx emin =
B2FF prec emax (B754_finite sx mx ex Bx)
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
0%Z = 0%Z 2: apply refl_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
F754_finite sx mx emin =
B2FF prec emax (B754_finite sx mx ex Bx)
simpl.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
F754_finite sx mx emin = F754_finite sx mx ex
apply f_equal.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
emin = ex
destruct (andb_prop _ _ Bx) as (H1, _).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
H1:canonical_mantissa prec emax mx ex = true
emin = ex
generalize (Zeq_bool_eq _ _ H1).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
H1:canonical_mantissa prec emax mx ex = true
FLT_exp (3 - emax - prec) prec
  (Z.pos (SpecFloatCopy.digits2_pos mx) + ex) = ex ->
emin = ex
rewrite Zpos_digits2_pos.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
H1:canonical_mantissa prec emax mx ex = true
FLT_exp (3 - emax - prec) prec
  (Zdigits radix2 (Z.pos mx) + ex) = ex -> emin = ex
unfold FLT_exp, emin, prec.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx - 2 ^ mw < 0)%Z
H1:canonical_mantissa prec emax mx ex = true
Z.max (Zdigits radix2 (Z.pos mx) + ex - (mw + 1))
  (3 - emax - (mw + 1)) = ex ->
(3 - emax - (mw + 1))%Z = ex
apply -> Z.lt_sub_0 in Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx < 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
Z.max (Zdigits radix2 (Z.pos mx) + ex - (mw + 1))
  (3 - emax - (mw + 1)) = ex ->
(3 - emax - (mw + 1))%Z = ex
generalize (Zdigits_le_Zpower radix2 _ (Zpos mx) Hm).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx < 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
(Zdigits radix2 (Z.pos mx) <= mw)%Z ->
Z.max (Zdigits radix2 (Z.pos mx) + ex - (mw + 1))
  (3 - emax - (mw + 1)) = ex ->
(3 - emax - (mw + 1))%Z = ex
generalize (Zdigits radix2 (Zpos mx)).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
sx:bool
mx:positive
ex:Z
Bx:bounded prec emax mx ex = true
Hm:(Z.pos mx < 2 ^ mw)%Z
H1:canonical_mantissa prec emax mx ex = true
forall z : Z,
(z <= mw)%Z ->
Z.max (z + ex - (mw + 1)) (3 - emax - (mw + 1)) = ex ->
(3 - emax - (mw + 1))%Z = ex
clear.
mw, ew:Z
emax:=(2 ^ (ew - 1))%Z:Z
ex:Z
forall z : Z,
(z <= mw)%Z ->
Z.max (z + ex - (mw + 1)) (3 - emax - (mw + 1)) = ex ->
(3 - emax - (mw + 1))%Z = ex
intros ; zify ; omega.
Qed.

Theorem bits_of_binary_float_of_bits :
  forall x,
  (0 <= x < 2^(mw+ew+1))%Z ->
  bits_of_binary_float (binary_float_of_bits x) = x.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
bits_of_binary_float (binary_float_of_bits x) = x
Proof.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
forall x : Z,
(0 <= x < 2 ^ (mw + ew + 1))%Z ->
bits_of_binary_float (binary_float_of_bits x) = x
intros x Hx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
bits_of_binary_float (binary_float_of_bits x) = x
unfold binary_float_of_bits, bits_of_binary_float.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
match
  FF2B prec emax (binary_float_of_bits_aux x)
    (binary_float_of_bits_aux_correct x)
with
| B754_zero sx => join_bits sx 0 0
| B754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| B754_nan sx plx _ =>
    join_bits sx (Z.pos plx) (2 ^ ew - 1)
| B754_finite sx mx ex _ =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
set (Cx := binary_float_of_bits_aux_correct x).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Cx:=binary_float_of_bits_aux_correct x:valid_binary prec emax
  (binary_float_of_bits_aux x) = true
match
  FF2B prec emax (binary_float_of_bits_aux x) Cx
with
| B754_zero sx => join_bits sx 0 0
| B754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| B754_nan sx plx _ =>
    join_bits sx (Z.pos plx) (2 ^ ew - 1)
| B754_finite sx mx ex _ =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
clearbody Cx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Cx:valid_binary prec emax
  (binary_float_of_bits_aux x) = true
match
  FF2B prec emax (binary_float_of_bits_aux x) Cx
with
| B754_zero sx => join_bits sx 0 0
| B754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| B754_nan sx plx _ =>
    join_bits sx (Z.pos plx) (2 ^ ew - 1)
| B754_finite sx mx ex _ =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
rewrite match_FF2B.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
Cx:valid_binary prec emax
  (binary_float_of_bits_aux x) = true
match binary_float_of_bits_aux x with
| F754_zero sx => join_bits sx 0 0
| F754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx mx ex =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
revert Cx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
valid_binary prec emax (binary_float_of_bits_aux x) =
true ->
match binary_float_of_bits_aux x with
| F754_zero sx => join_bits sx 0 0
| F754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx mx ex =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
generalize (join_split_bits x Hx).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(let '(s, m, e) := split_bits x in join_bits s m e = x) ->
valid_binary prec emax (binary_float_of_bits_aux x) =
true ->
match binary_float_of_bits_aux x with
| F754_zero sx => join_bits sx 0 0
| F754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx mx ex =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
unfold binary_float_of_bits_aux.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
(let '(s, m, e) := split_bits x in join_bits s m e = x) ->
valid_binary prec emax
  (let
   '(sx, mx, ex) := split_bits x in
    if Zeq_bool ex 0
    then
     match mx with
     | 0%Z => F754_zero sx
     | Z.pos px => F754_finite sx px emin
     | Z.neg _ => F754_nan false 1
     end
    else
     if Zeq_bool ex (2 ^ ew - 1)
     then
      match mx with
      | 0%Z => F754_infinity sx
      | Z.pos plx => F754_nan sx plx
      | Z.neg _ => F754_nan false 1
      end
     else
      match (mx + 2 ^ mw)%Z with
      | Z.pos px => F754_finite sx px (ex + emin - 1)
      | _ => F754_nan false 1
      end) = true ->
match
  (let
   '(sx, mx, ex) := split_bits x in
    if Zeq_bool ex 0
    then
     match mx with
     | 0%Z => F754_zero sx
     | Z.pos px => F754_finite sx px emin
     | Z.neg _ => F754_nan false 1
     end
    else
     if Zeq_bool ex (2 ^ ew - 1)
     then
      match mx with
      | 0%Z => F754_infinity sx
      | Z.pos plx => F754_nan sx plx
      | Z.neg _ => F754_nan false 1
      end
     else
      match (mx + 2 ^ mw)%Z with
      | Z.pos px => F754_finite sx px (ex + emin - 1)
      | _ => F754_nan false 1
      end)
with
| F754_zero sx => join_bits sx 0 0
| F754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx mx ex =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
case_eq (split_bits x).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
forall (p : bool * Z) (z : Z),
split_bits x = (p, z) ->
(let '(s, m) := p in join_bits s m z = x) ->
valid_binary prec emax
  (let
   '(sx, mx) := p in
    if Zeq_bool z 0
    then
     match mx with
     | 0%Z => F754_zero sx
     | Z.pos px => F754_finite sx px emin
     | Z.neg _ => F754_nan false 1
     end
    else
     if Zeq_bool z (2 ^ ew - 1)
     then
      match mx with
      | 0%Z => F754_infinity sx
      | Z.pos plx => F754_nan sx plx
      | Z.neg _ => F754_nan false 1
      end
     else
      match (mx + 2 ^ mw)%Z with
      | Z.pos px => F754_finite sx px (z + emin - 1)
      | _ => F754_nan false 1
      end) = true ->
match
  (let
   '(sx, mx) := p in
    if Zeq_bool z 0
    then
     match mx with
     | 0%Z => F754_zero sx
     | Z.pos px => F754_finite sx px emin
     | Z.neg _ => F754_nan false 1
     end
    else
     if Zeq_bool z (2 ^ ew - 1)
     then
      match mx with
      | 0%Z => F754_infinity sx
      | Z.pos plx => F754_nan sx plx
      | Z.neg _ => F754_nan false 1
      end
     else
      match (mx + 2 ^ mw)%Z with
      | Z.pos px => F754_finite sx px (z + emin - 1)
      | _ => F754_nan false 1
      end)
with
| F754_zero sx => join_bits sx 0 0
| F754_infinity sx => join_bits sx 0 (2 ^ ew - 1)
| F754_nan b p0 => join_bits b (Z.pos p0) (2 ^ ew - 1)
| F754_finite sx mx ex =>
    if (0 <=? Z.pos mx - 2 ^ mw)%Z
    then
     join_bits sx (Z.pos mx - 2 ^ mw) (ex - emin + 1)
    else join_bits sx (Z.pos mx) 0
end = x
intros (sx, mx) ex Sx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end) = true ->
match
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
assert (Bm: (0 <= mx < 2^mw)%Z).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
(0 <= mx < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end) = true ->
match
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
inversion_clear Sx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
(0 <= x mod 2 ^ mw < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end) = true ->
match
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
apply Z_mod_lt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
(2 ^ mw > 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end) = true ->
match
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now apply Z.lt_gt.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end) = true ->
match
  (if Zeq_bool ex 0
   then
    match mx with
    | 0%Z => F754_zero sx
    | Z.pos px => F754_finite sx px emin
    | Z.neg _ => F754_nan false 1
    end
   else
    if Zeq_bool ex (2 ^ ew - 1)
    then
     match mx with
     | 0%Z => F754_infinity sx
     | Z.pos plx => F754_nan sx plx
     | Z.neg _ => F754_nan false 1
     end
    else
     match (mx + 2 ^ mw)%Z with
     | Z.pos px => F754_finite sx px (ex + emin - 1)
     | _ => F754_nan false 1
     end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
case Zeq_bool_spec ; intros He1.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match mx with
  | 0%Z => F754_zero sx
  | Z.pos px => F754_finite sx px emin
  | Z.neg _ => F754_nan false 1
  end = true ->
match
  match mx with
  | 0%Z => F754_zero sx
  | Z.pos px => F754_finite sx px emin
  | Z.neg _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
(* subnormal *)
case_eq mx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
mx = 0%Z ->
join_bits sx 0 ex = x ->
valid_binary prec emax (F754_zero sx) = true ->
join_bits sx 0 0 = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.pos p ->
join_bits sx (Z.pos p) ex = x ->
valid_binary prec emax (F754_finite sx p emin) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw) (emin - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
intros Hm Jx _.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
Hm:mx = 0%Z
Jx:join_bits sx 0 ex = x
join_bits sx 0 0 = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.pos p ->
join_bits sx (Z.pos p) ex = x ->
valid_binary prec emax (F754_finite sx p emin) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw) (emin - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite He1 in Jx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.pos p ->
join_bits sx (Z.pos p) ex = x ->
valid_binary prec emax (F754_finite sx p emin) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw) (emin - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
intros px Hm Jx _.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.pos px
Jx:join_bits sx (Z.pos px) ex = x
(if (0 <=? Z.pos px - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos px - 2 ^ mw) (emin - emin + 1)
 else join_bits sx (Z.pos px) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
rewrite Zle_bool_false.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.pos px
Jx:join_bits sx (Z.pos px) ex = x
join_bits sx (Z.pos px) 0 = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.pos px
Jx:join_bits sx (Z.pos px) ex = x
(Z.pos px - 2 ^ mw < 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite <- He1.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.pos px
Jx:join_bits sx (Z.pos px) ex = x
(Z.pos px - 2 ^ mw < 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
apply <- Z.lt_sub_0.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.pos px
Jx:join_bits sx (Z.pos px) ex = x
(Z.pos px < 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite <- Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
forall p : positive,
mx = Z.neg p ->
join_bits sx (Z.neg p) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
intros px Hm _ _.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.neg px
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
apply False_ind.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.neg px
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
apply Zle_not_lt with (1 := proj1 Bm).
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex = 0%Z
px:positive
Hm:mx = Z.neg px
(mx < 0)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end) = true ->
match
  (if Zeq_bool ex (2 ^ ew - 1)
   then
    match mx with
    | 0%Z => F754_infinity sx
    | Z.pos plx => F754_nan sx plx
    | Z.neg _ => F754_nan false 1
    end
   else
    match (mx + 2 ^ mw)%Z with
    | Z.pos px => F754_finite sx px (ex + emin - 1)
    | _ => F754_nan false 1
    end)
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
case Zeq_bool_spec ; intros He2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match mx with
  | 0%Z => F754_infinity sx
  | Z.pos plx => F754_nan sx plx
  | Z.neg _ => F754_nan false 1
  end = true ->
match
  match mx with
  | 0%Z => F754_infinity sx
  | Z.pos plx => F754_nan sx plx
  | Z.neg _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
(* infinity/nan *)
case_eq mx; intros Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:mx = 0%Z
join_bits sx 0 ex = x ->
valid_binary prec emax (F754_infinity sx) = true ->
join_bits sx 0 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
mx = Z.pos Hm ->
join_bits sx (Z.pos Hm) ex = x ->
valid_binary prec emax (F754_nan sx Hm) = true ->
join_bits sx (Z.pos Hm) (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
mx = Z.neg Hm ->
join_bits sx (Z.neg Hm) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite He2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
mx = Z.pos Hm ->
join_bits sx (Z.pos Hm) ex = x ->
valid_binary prec emax (F754_nan sx Hm) = true ->
join_bits sx (Z.pos Hm) (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
mx = Z.neg Hm ->
join_bits sx (Z.neg Hm) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
now rewrite He2.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
mx = Z.neg Hm ->
join_bits sx (Z.neg Hm) ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
intros.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex = (2 ^ ew - 1)%Z
Hm:positive
H:mx = Z.neg Hm
H0:join_bits sx (Z.neg Hm) ex = x
Cx:valid_binary prec emax (F754_nan false 1) = true
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x zify; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
join_bits sx mx ex = x ->
valid_binary prec emax
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end = true ->
match
  match (mx + 2 ^ mw)%Z with
  | Z.pos px => F754_finite sx px (ex + emin - 1)
  | _ => F754_nan false 1
  end
with
| F754_zero sx0 => join_bits sx0 0 0
| F754_infinity sx0 => join_bits sx0 0 (2 ^ ew - 1)
| F754_nan b p => join_bits b (Z.pos p) (2 ^ ew - 1)
| F754_finite sx0 mx0 ex0 =>
    if (0 <=? Z.pos mx0 - 2 ^ mw)%Z
    then
     join_bits sx0 (Z.pos mx0 - 2 ^ mw)
       (ex0 - emin + 1)
    else join_bits sx0 (Z.pos mx0) 0
end = x
(* normal *)
case_eq (mx + 2 ^ mw)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
(mx + 2 ^ mw)%Z = 0%Z ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.pos p ->
join_bits sx mx ex = x ->
valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
intros Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
Hm:(mx + 2 ^ mw)%Z = 0%Z
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.pos p ->
join_bits sx mx ex = x ->
valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
apply False_ind.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
Hm:(mx + 2 ^ mw)%Z = 0%Z
False
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.pos p ->
join_bits sx mx ex = x ->
valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
clear -Bm Hm ; omega.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.pos p ->
join_bits sx mx ex = x ->
valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true ->
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
intros p Hm Jx Cx.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.pos p
Jx:join_bits sx mx ex = x
Cx:valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true
(if (0 <=? Z.pos p - 2 ^ mw)%Z
 then
  join_bits sx (Z.pos p - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (Z.pos p) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
rewrite <- Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.pos p
Jx:join_bits sx mx ex = x
Cx:valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true
(if (0 <=? mx + 2 ^ mw - 2 ^ mw)%Z
 then
  join_bits sx (mx + 2 ^ mw - 2 ^ mw)
    (ex + emin - 1 - emin + 1)
 else join_bits sx (mx + 2 ^ mw) 0) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
rewrite Zle_bool_true.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.pos p
Jx:join_bits sx mx ex = x
Cx:valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true
join_bits sx (mx + 2 ^ mw - 2 ^ mw)
  (ex + emin - 1 - emin + 1) = x
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.pos p
Jx:join_bits sx mx ex = x
Cx:valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true
(0 <= mx + 2 ^ mw - 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
now ring_simplify (mx + 2^mw - 2^mw)%Z (ex + emin - 1 - emin + 1)%Z.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.pos p
Jx:join_bits sx mx ex = x
Cx:valid_binary prec emax
  (F754_finite sx p (ex + emin - 1)) = true
(0 <= mx + 2 ^ mw - 2 ^ mw)%Z
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
now ring_simplify.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
forall p : positive,
(mx + 2 ^ mw)%Z = Z.neg p ->
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
intros p Hm.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.neg p
join_bits sx mx ex = x ->
valid_binary prec emax (F754_nan false 1) = true ->
join_bits false 1 (2 ^ ew - 1) = x
apply False_ind.
mw, ew:Z
Hmw:(0 < mw)%Z
Hew:(0 < ew)%Z
emax:=(2 ^ (ew - 1))%Z:Z
prec:=(mw + 1)%Z:Z
emin:=(3 - emax - prec)%Z:Z
binary_float:=Binary.binary_float prec emax:Set
Hprec:(0 < prec)%Z
Hm_gt_0:(0 < 2 ^ mw)%Z
He_gt_0:(0 < 2 ^ ew)%Z
Hmax:(prec < emax)%Z
x:Z
Hx:(0 <= x < 2 ^ (mw + ew + 1))%Z
sx:bool
mx, ex:Z
Sx:split_bits x = (sx, mx, ex)
Bm:(0 <= mx < 2 ^ mw)%Z
He1:ex <> 0%Z
He2:ex <> (2 ^ ew - 1)%Z
p:positive
Hm:(mx + 2 ^ mw)%Z = Z.neg p
False
clear -Bm Hm ; zify ; omega.
Qed.

End Binary_Bits.