Start a Coq realization for int.why.

Makefile.in

@@ -839,10 +839,13 @@ endif
 
 ifeq (@enable_coq_libs@,yes)
 
+COQLIBS_INT_FILES = Int
+COQLIBS_INT = $(addprefix lib/coq/int/, $(COQLIBS_INT_FILES))
+
 COQLIBS_REAL_FILES = Abs MinMax Real Square
 COQLIBS_REAL = $(addprefix lib/coq/real/, $(COQLIBS_REAL_FILES))
 
-COQLIBS_FILES = $(COQLIBS_REAL)
+COQLIBS_FILES = $(COQLIBS_INT) $(COQLIBS_REAL)
 
 COQV  = $(addsuffix .v,  $(COQLIBS_FILES))
 COQVO = $(addsuffix .vo, $(COQLIBS_FILES))
@@ -854,7 +857,9 @@ all: $(COQVO)
 
 install_no_local::
 	mkdir -p $(LIBDIR)/why3/coq
+	mkdir -p $(LIBDIR)/why3/coq/int
 	mkdir -p $(LIBDIR)/why3/coq/real
+	cp $(addsuffix .vo, $(COQLIBS_INT)) $(LIBDIR)/why3/coq/int/
 	cp $(addsuffix .vo, $(COQLIBS_REAL)) $(LIBDIR)/why3/coq/real/
 
 install_local: $(COQVO)

lib/coq/int/Int.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".*)
+
+Notation infix_ls := Zlt (only parsing).
+
+Definition infix_lseq(x:Z) (y:Z): Prop := (infix_ls x y) \/ (x = y).
+
+Lemma infix_lseq_Zle :
+  forall x y, infix_lseq x y <-> Zle x y.
+Proof.
+intros x y.
+apply iff_Symmetric.
+apply Zle_lt_or_eq_iff.
+Qed.
+
+Notation infix_pl := Zplus (only parsing).
+
+Notation prefix_mn := Zopp (only parsing).
+
+Notation infix_as := Zmult (only parsing).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Unit_def : forall (x:Z), ((infix_pl x 0%Z) = x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zplus_0_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Assoc : forall (x:Z) (y:Z) (z:Z), ((infix_pl (infix_pl x y)
+  z) = (infix_pl x (infix_pl y z))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y z.
+apply sym_eq.
+apply Zplus_assoc.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Inv_def : forall (x:Z), ((infix_pl x (prefix_mn x)) = 0%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zplus_opp_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Comm : forall (x:Z) (y:Z), ((infix_pl x y) = (infix_pl y x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zplus_comm.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Assoc1 : forall (x:Z) (y:Z) (z:Z), ((infix_as (infix_as x y)
+  z) = (infix_as x (infix_as y z))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y z.
+apply sym_eq.
+apply Zmult_assoc.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Mul_distr : forall (x:Z) (y:Z) (z:Z), ((infix_as x (infix_pl y
+  z)) = (infix_pl (infix_as x y) (infix_as x z))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmult_plus_distr_r.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition infix_mn(x:Z) (y:Z): Z := (infix_pl x (prefix_mn y)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Comm1 : forall (x:Z) (y:Z), ((infix_as x y) = (infix_as y x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmult_comm.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Unitary : forall (x:Z), ((infix_as 1%Z x) = x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact Zmult_1_l.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma NonTrivialRing : ~ (0%Z = 1%Z).
+(* YOU MAY EDIT THE PROOF BELOW *)
+discriminate.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Refl : forall (x:Z), (infix_lseq x x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+apply infix_lseq_Zle.
+apply Zle_refl.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Trans : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) -> ((infix_lseq y
+  z) -> (infix_lseq x z)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y z H1 H2.
+apply infix_lseq_Zle.
+apply Zle_trans with y ;
+  now apply infix_lseq_Zle.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Antisymm : forall (x:Z) (y:Z), (infix_lseq x y) -> ((infix_lseq y x) ->
+  (x = y)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y H1 H2.
+apply Zle_antisym ;
+  now apply infix_lseq_Zle.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Total : forall (x:Z) (y:Z), (infix_lseq x y) \/ (infix_lseq y x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y.
+destruct (Zle_or_lt x y) as [H|H].
+left.
+now apply infix_lseq_Zle.
+right.
+apply infix_lseq_Zle.
+now apply Zlt_le_weak.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma CompatOrderAdd : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) ->
+  (infix_lseq (infix_pl x z) (infix_pl y z)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y z H.
+apply infix_lseq_Zle.
+apply Zplus_le_compat_r.
+now apply infix_lseq_Zle.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma CompatOrderMult : forall (x:Z) (y:Z) (z:Z), (infix_lseq x y) ->
+  ((infix_lseq 0%Z z) -> (infix_lseq (infix_as x z) (infix_as y z))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x y z H1 H2.
+apply infix_lseq_Zle.
+apply Zmult_le_compat_r ;
+  now apply infix_lseq_Zle.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+