Add some Coq realizations.

Makefile.in

@@ -839,10 +839,10 @@ endif
 
 ifeq (@enable_coq_libs@,yes)
 
-COQLIBS_INT_FILES = Int
+COQLIBS_INT_FILES = Abs EuclideanDivision Int MinMax
 COQLIBS_INT = $(addprefix lib/coq/int/, $(COQLIBS_INT_FILES))
 
-COQLIBS_REAL_FILES = Abs MinMax Real Square
+COQLIBS_REAL_FILES = Abs FromInt MinMax Real Square
 COQLIBS_REAL = $(addprefix lib/coq/real/, $(COQLIBS_REAL_FILES))
 
 COQLIBS_FILES = $(COQLIBS_INT) $(COQLIBS_REAL)

lib/coq/int/Abs.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
+Require int.Int.
+
+Notation abs := Zabs (only parsing).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma abs_def : forall (x:Z), ((0%Z <= x)%Z -> ((abs x) = x)) /\
+  ((~ (0%Z <= x)%Z) -> ((abs x) = (-x)%Z)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+split ; intros H.
+now apply Zabs_eq.
+apply Zabs_non_eq.
+apply Znot_gt_le.
+contradict H.
+apply Zlt_le_weak.
+now apply Zgt_lt.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Abs_le : forall (x:Z) (y:Z), ((abs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
+  (x <= y)%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y.
+zify.
+omega.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Abs_pos : forall (x:Z), (0%Z <= (abs x))%Z.
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zabs_pos.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

lib/coq/int/EuclideanDivision.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
+Require int.Int.
+Require int.Abs.
+
+Definition div : Z -> Z -> Z.
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+intros x y.
+case (Z_le_dec 0 (Zmod x y)) ; intros H.
+exact (Zdiv x y).
+exact (Zdiv x y + 1)%Z.
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Definition mod1 : Z -> Z -> Z.
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+intros x y.
+exact (x - y * div x y)%Z.
+Defined.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Div_mod : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (x = ((y * (div x
+  y))%Z + (mod1 x y))%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y Zy.
+unfold mod1, div.
+case Z_le_dec ; intros H ; ring.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Div_bound : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  ((0%Z <= (div x y))%Z /\ ((div x y) <= x)%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y (Hx,Hy).
+unfold div.
+case Z_le_dec ; intros H.
+split.
+apply Z_div_pos with (2 := Hx).
+now apply Zlt_gt.
+destruct (Z_eq_dec y 1) as [H'|H'].
+rewrite H', Zdiv_1_r.
+apply Zle_refl.
+rewrite <- (Zdiv_1_r x) at 2.
+apply Zdiv_le_compat_l with (1 := Hx).
+omega.
+elim H.
+apply Z_mod_lt.
+now apply Zlt_gt.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> ((0%Z <= (mod1 x
+  y))%Z /\ ((mod1 x y) <  (Zabs y))%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y Zy.
+zify.
+assert (H1 := Z_mod_neg x y).
+assert (H2 := Z_mod_lt x y).
+unfold mod1, div.
+case Z_le_dec ; intros H0.
+rewrite Zmult_comm, <- Zmod_eq_full with (1 := Zy).
+omega.
+replace (x - y * (x / y + 1))%Z with (x - x / y * y - y)%Z by ring.
+rewrite <- Zmod_eq_full with (1 := Zy).
+omega.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Mod_1 : forall (x:Z), ((mod1 x 1%Z) = 0%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+unfold mod1, div.
+rewrite Zmod_1_r, Zdiv_1_r, Zmult_1_l.
+apply Zminus_diag.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Div_1 : forall (x:Z), ((div x 1%Z) = x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+unfold div.
+now rewrite Zmod_1_r, Zdiv_1_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

lib/coq/int/MinMax.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
+Require int.Int.
+
+Notation min := Zmin (only parsing).
+
+Notation max := Zmax (only parsing).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Max_is_ge : forall (x:Z) (y:Z), (x <= (max x y))%Z /\ (y <= (max x
+  y))%Z.
+(* YOU MAY EDIT THE PROOF BELOW *)
+split.
+apply Zle_max_l.
+apply Zle_max_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Max_is_some : forall (x:Z) (y:Z), ((max x y) = x) \/ ((max x y) = y).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y.
+unfold Zmax.
+case Zcompare.
+now left.
+now right.
+now left.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Min_is_le : forall (x:Z) (y:Z), ((min x y) <= x)%Z /\ ((min x
+  y) <= y)%Z.
+(* YOU MAY EDIT THE PROOF BELOW *)
+split.
+apply Zle_min_l.
+apply Zle_min_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Min_is_some : forall (x:Z) (y:Z), ((min x y) = x) \/ ((min x y) = y).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y.
+unfold Zmin.
+case Zcompare.
+now left.
+now left.
+now right.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Max_x : forall (x:Z) (y:Z), (y <= x)%Z -> ((max x y) = x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmax_l.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Max_y : forall (x:Z) (y:Z), (x <= y)%Z -> ((max x y) = y).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmax_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Min_x : forall (x:Z) (y:Z), (x <= y)%Z -> ((min x y) = x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmin_l.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Min_y : forall (x:Z) (y:Z), (y <= x)%Z -> ((min x y) = y).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmin_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Max_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((max x y) = (max y x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y _.
+apply Zmax_comm.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Min_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((min x y) = (min y x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y _.
+apply Zmin_comm.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

lib/coq/real/FromInt.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
+(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
+Require int.Int.
+Require real.Real.
+Definition from_int: Z -> R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact IZR.
+Defined.
+(* DO NOT EDIT BELOW *)
+
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Zero : ((from_int 0%Z) = 0%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+split.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma One : ((from_int 1%Z) = 1%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+split.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Add : forall (x:Z) (y:Z),
+  ((from_int (x + y)%Z) = ((from_int x) + (from_int y))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact plus_IZR.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Sub : forall (x:Z) (y:Z),
+  ((from_int (x - y)%Z) = ((from_int x) - (from_int y))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact minus_IZR.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Mul : forall (x:Z) (y:Z),
+  ((from_int (x * y)%Z) = ((from_int x) * (from_int y))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact mult_IZR.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Neg : forall (x:Z), ((from_int (-x)%Z) = (-(from_int x))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact opp_IZR.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+