Update some Coq realizations with the new script format.

Note: they are stable with respect to the following command.
why3 --realize -D drivers/coq-realize.drv -T toto.Titi -o lib/coq/toto/

lib/coq/real/Abs.v

@@ -3,38 +3,26 @@
 Require Import ZArith.
 Require Import Rbase.
 Require Import Rbasic_fun.
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*)
-(*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*)
 Require real.Real.
+
+(* Why3 goal *)
 Definition abs: R -> R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rabs.
 Defined.
-(* DO NOT EDIT BELOW *)
-
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma abs_def : forall (x:R), ((0%R <= x)%R -> ((abs x) = x)) /\
   ((~ (0%R <= x)%R) -> ((abs x) = (-x)%R)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 split ; intros H.
 apply Rabs_right.
 now apply Rle_ge.
 apply Rabs_left.
 now apply Rnot_le_lt.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Abs_le : forall (x:R) (y:R), ((abs x) <= y)%R <-> (((-y)%R <= x)%R /\
   (x <= y)%R).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 unfold abs, Rabs.
 case Rcase_abs ; intros H ; (split ; [intros H0;split | intros (H0,H1)]).
@@ -57,16 +45,30 @@ now apply Ropp_le_contravar.
 exact H0.
 exact H1.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Abs_pos : forall (x:R), (0%R <= (abs x))%R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rabs_pos.
 Qed.
-(* DO NOT EDIT BELOW *)
+
+(* Why3 goal *)
+Lemma Abs_sum : forall (x:R) (y:R),
+  ((abs (x + y)%R) <= ((abs x) + (abs y))%R)%R.
+exact Rabs_triang.
+Qed.
+
+(* Why3 goal *)
+Lemma Abs_prod : forall (x:R) (y:R),
+  ((abs (x * y)%R) = ((abs x) * (abs y))%R).
+exact Rabs_mult.
+Qed.
+
+(* Why3 goal *)
+Lemma triangular_inequality : forall (x:R) (y:R) (z:R),
+  ((abs (x - z)%R) <= ((abs (x - y)%R) + (abs (y - z)%R))%R)%R.
+intros x y z.
+replace (x - z)%R with ((x - y) + (y - z))%R by ring.
+apply Rabs_triang.
+Qed.
 
 

lib/coq/real/FromInt.v

@@ -2,78 +2,45 @@
 (* 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.
+
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
+(* Why3 goal *)
 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 *)
 
 

lib/coq/real/Real.v

@@ -2,264 +2,165 @@
 (* 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".*)
 
+(* Why3 goal *)
 Definition infix_ls: R -> R -> Prop.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rlt.
 Defined.
-(* DO NOT EDIT BELOW *)
-
 
+(* Why3 assumption *)
 Definition infix_lseq(x:R) (y:R): Prop := (infix_ls x y) \/ (x = y).
 
+(* Why3 goal *)
 Definition infix_pl: R -> R -> R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rplus.
 Defined.
-(* DO NOT EDIT BELOW *)
-
 
+(* Why3 goal *)
 Definition prefix_mn: R -> R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Ropp.
 Defined.
-(* DO NOT EDIT BELOW *)
-
 
+(* Why3 goal *)
 Definition infix_as: R -> R -> R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rmult.
 Defined.
-(* DO NOT EDIT BELOW *)
-
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Unit_def : forall (x:R), ((infix_pl x 0%R) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rplus_0_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Assoc : forall (x:R) (y:R) (z:R), ((infix_pl (infix_pl x y)
   z) = (infix_pl x (infix_pl y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rplus_assoc.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Inv_def : forall (x:R), ((infix_pl x (prefix_mn x)) = 0%R).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rplus_opp_r.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Comm : forall (x:R) (y:R), ((infix_pl x y) = (infix_pl y x)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rplus_comm.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Assoc1 : forall (x:R) (y:R) (z:R), ((infix_as (infix_as x y)
   z) = (infix_as x (infix_as y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rmult_assoc.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Mul_distr : forall (x:R) (y:R) (z:R), ((infix_as x (infix_pl y
   z)) = (infix_pl (infix_as x y) (infix_as x z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rmult_plus_distr_l.
 Qed.
-(* DO NOT EDIT BELOW *)
 
+(* Why3 assumption *)
 Definition infix_mn(x:R) (y:R): R := (infix_pl x (prefix_mn y)).
 
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma Comm1 : forall (x:R) (y:R), ((infix_as x y) = (infix_as y x)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rmult_comm.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Unitary : forall (x:R), ((infix_as 1%R x) = x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rmult_1_l.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma NonTrivialRing : ~ (0%R = 1%R).
-(* YOU MAY EDIT THE PROOF BELOW *)
 apply not_eq_sym.
 exact R1_neq_R0.
 Qed.
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Definition inv: R -> R.
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rinv.
 Defined.
-(* DO NOT EDIT BELOW *)
-
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Inverse : forall (x:R), (~ (x = 0%R)) -> ((infix_as x (inv x)) = 1%R).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rinv_r.
 Qed.
-(* DO NOT EDIT BELOW *)
 
+(* Why3 assumption *)
 Definition infix_sl(x:R) (y:R): R := (infix_as x (inv y)).
 
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
-
+(* Why3 goal *)
 Lemma assoc_mul_div : forall (x:R) (y:R) (z:R), (~ (z = 0%R)) ->
   ((infix_sl (infix_as x y) z) = (infix_as x (infix_sl y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z _.
 apply Rmult_assoc.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma assoc_div_mul : forall (x:R) (y:R) (z:R), ((~ (y = 0%R)) /\
   ~ (z = 0%R)) -> ((infix_sl (infix_sl x y) z) = (infix_sl x (infix_as y
   z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z (Zy, Zz).
 unfold infix_sl, infix_as, inv.
 rewrite Rmult_assoc.
 now rewrite Rinv_mult_distr.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma assoc_div_div : forall (x:R) (y:R) (z:R), ((~ (y = 0%R)) /\
   ~ (z = 0%R)) -> ((infix_sl x (infix_sl y z)) = (infix_sl (infix_as x z)
   y)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z (Zy, Zz).
 unfold infix_sl, infix_as, inv.
 field.
 now split.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Refl : forall (x:R), (infix_lseq x x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rle_refl.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Trans : forall (x:R) (y:R) (z:R), (infix_lseq x y) -> ((infix_lseq y
   z) -> (infix_lseq x z)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rle_trans.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Antisymm : forall (x:R) (y:R), (infix_lseq x y) -> ((infix_lseq y x) ->
   (x = y)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 exact Rle_antisym.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma Total : forall (x:R) (y:R), (infix_lseq x y) \/ (infix_lseq y x).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y.
 destruct (Rle_or_lt x y) as [H|H].
 now left.
 right.
 now apply Rlt_le.
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
 
-(* DO NOT EDIT BELOW *)
+(* Why3 goal *)
+Lemma ZeroLessOne : (infix_lseq 0%R 1%R).
+exact Rle_0_1.
+Qed.
 
+(* Why3 goal *)
 Lemma CompatOrderAdd : forall (x:R) (y:R) (z:R), (infix_lseq x y) ->
   (infix_lseq (infix_pl x z) (infix_pl y z)).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z.
 exact (Rplus_le_compat_r z x y).
 Qed.
-(* DO NOT EDIT BELOW *)
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
 
+(* Why3 goal *)
 Lemma CompatOrderMult : forall (x:R) (y:R) (z:R), (infix_lseq x y) ->
   ((infix_lseq 0%R z) -> (infix_lseq (infix_as x z) (infix_as y z))).
-(* YOU MAY EDIT THE PROOF BELOW *)
 intros x y z H Zz.
 now apply Rmult_le_compat_r.
 Qed.
-(* DO NOT EDIT BELOW *)