mergesort_list: completed proof

.gitignore

@@ -154,7 +154,6 @@ why3.conf
 /examples/programs/vacid_0_red_black_trees/
 /examples/programs/vacid_0_red_black_trees_harness/
 /examples/programs/next_digit_sum/
-/examples/programs/mergesort_list/
 /examples/programs/algo63/
 /examples/programs/algo64/
 /examples/programs/algo65/

examples/programs/mergesort_list.mlw

@@ -9,14 +9,14 @@ module M
   use import list.Append
   use import list.Permut
 
-  lemma Permut_append:
-    forall l1 l2 k1 k2 : list 'a.
-    permut l1 k1 -> permut l2 k2 -> permut (l1 ++ l2) (k1 ++ k2)
-
   lemma Permut_cons_append:
     forall x : 'a, l1 l2 : list 'a.
     permut (Cons x l1 ++ l2) (l1 ++ Cons x l2)
 
+  lemma Permut_append:
+    forall l1 l2 k1 k2 : list 'a.
+    permut l1 k1 -> permut l2 k2 -> permut (l1 ++ l2) (k1 ++ k2)
+
   let split l0 =
     {  length l0 >= 2 }
     let rec split_aux (l1 : list 'a) l2 l variant { length l } =

examples/programs/mergesort_list/mergesort_list_WP_M_Permut_append_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Definition unit  := unit.
+
+Parameter mark : Type.
+
+Parameter at1: forall (a:Type), a -> mark  -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a  -> a.
+
+Implicit Arguments old.
+
+Inductive list (a:Type) :=
+  | Nil : list a
+  | Cons : a -> (list a) -> list a.
+Set Contextual Implicit.
+Implicit Arguments Nil.
+Unset Contextual Implicit.
+Implicit Arguments Cons.
+
+Set Implicit Arguments.
+Fixpoint length (a:Type)(l:(list a)) {struct l}: Z :=
+  match l with
+  | Nil  => 0%Z
+  | Cons _ r => (1%Z + (length r))%Z
+  end.
+Unset Implicit Arguments.
+
+Axiom Length_nonnegative : forall (a:Type), forall (l:(list a)),
+  (0%Z <= (length l))%Z.
+
+Axiom Length_nil : forall (a:Type), forall (l:(list a)),
+  ((length l) = 0%Z) <-> (l = (Nil:(list a))).
+
+Inductive sorted : (list Z) -> Prop :=
+  | Sorted_Nil : (sorted (Nil:(list Z)))
+  | Sorted_One : forall (x:Z), (sorted (Cons x (Nil:(list Z))))
+  | Sorted_Two : forall (x:Z) (y:Z) (l:(list Z)), (x <= y)%Z ->
+      ((sorted (Cons y l)) -> (sorted (Cons x (Cons y l)))).
+
+Set Implicit Arguments.
+Fixpoint mem (a:Type)(x:a) (l:(list a)) {struct l}: Prop :=
+  match l with
+  | Nil  => False
+  | Cons y r => (x = y) \/ (mem x r)
+  end.
+Unset Implicit Arguments.
+
+Axiom Sorted_mem : forall (x:Z) (l:(list Z)), ((forall (y:Z), (mem y l) ->
+  (x <= y)%Z) /\ (sorted l)) <-> (sorted (Cons x l)).
+
+Set Implicit Arguments.
+Fixpoint infix_plpl (a:Type)(l1:(list a)) (l2:(list a)) {struct l1}: (list
+  a) :=
+  match l1 with
+  | Nil  => l2
+  | Cons x1 r1 => (Cons x1 (infix_plpl r1 l2))
+  end.
+Unset Implicit Arguments.
+
+Axiom Append_assoc : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), ((infix_plpl l1 (infix_plpl l2
+  l3)) = (infix_plpl (infix_plpl l1 l2) l3)).
+
+Axiom Append_l_nil : forall (a:Type), forall (l:(list a)), ((infix_plpl l
+  (Nil:(list a))) = l).
+
+Axiom Append_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  ((length (infix_plpl l1 l2)) = ((length l1) + (length l2))%Z).
+
+Axiom mem_append : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (mem x (infix_plpl l1 l2)) <-> ((mem x l1) \/ (mem x l2)).
+
+Axiom mem_decomp : forall (a:Type), forall (x:a) (l:(list a)), (mem x l) ->
+  exists l1:(list a), exists l2:(list a), (l = (infix_plpl l1 (Cons x l2))).
+
+Parameter num_occ: forall (a:Type), a -> (list a)  -> Z.
+
+Implicit Arguments num_occ.
+
+Axiom num_occ_def : forall (a:Type), forall (x:a) (l:(list a)),
+  match l with
+  | Nil  => ((num_occ x l) = 0%Z)
+  | Cons y r => ((x = y) -> ((num_occ x l) = (1%Z + (num_occ x r))%Z)) /\
+      ((~ (x = y)) -> ((num_occ x l) = (0%Z + (num_occ x r))%Z))
+  end.
+
+Axiom Mem_Num_Occ : forall (a:Type), forall (x:a) (l:(list a)), (mem x l) <->
+  (0%Z <  (num_occ x l))%Z.
+
+Definition permut (a:Type)(l1:(list a)) (l2:(list a)): Prop := forall (x:a),
+  ((num_occ x l1) = (num_occ x l2)).
+Implicit Arguments permut.
+
+Axiom Permut_refl : forall (a:Type), forall (l:(list a)), (permut l l).
+
+Axiom Permut_sym : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> (permut l2 l1).
+
+Axiom Permut_trans : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), (permut l1 l2) -> ((permut l2 l3) -> (permut l1 l3)).
+
+Axiom Permut_cons : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list
+  a)), (permut l1 l2) -> (permut (Cons x l1) (Cons x l2)).
+
+Axiom Permut_swap : forall (a:Type), forall (x:a) (y:a) (l:(list a)),
+  (permut (Cons x (Cons y l)) (Cons y (Cons x l))).
+
+Axiom Permut_mem : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((mem x l1) -> (mem x l2)).
+
+Axiom Permut_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((length l1) = (length l2)).
+
+Axiom Permut_cons_append : forall (a:Type), forall (x:a) (l1:(list a))
+  (l2:(list a)), (permut (infix_plpl (Cons x l1) l2) (infix_plpl l1 (Cons x
+  l2))).
+
+Theorem Permut_append : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (k1:(list a)) (k2:(list a)), (permut l1 k1) -> ((permut l2 k2) ->
+  (permut (infix_plpl l1 l2) (infix_plpl k1 k2))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intro a.
+(*** moved to list.why
+assert (mem_decomp: forall (x:a) (l:list a), mem x l -> 
+   exists l1: list a, exists l2 : list a, l = infix_plpl l1 (Cons x l2)).
+induction l; simpl; intros.
+elim H.
+destruct H.
+subst a0. exists Nil. exists l; auto.
+destruct (IHl H) as (e1, (e2, h12)).
+exists (Cons a0 e1); exists e2.
+simpl; apply f_equal; auto.
+***)
+induction l1; simpl.
+intros l2 k1 k2 hk1.
+assert (k1 = Nil).
+assert (length (@Nil a) = length k1)%Z.
+apply Permut_length; auto.
+simpl in H.
+destruct k1; auto.
+absurd (0%Z = length (Cons a0 k1)); auto.
+unfold length; fold length.
+generalize (Length_nonnegative _ k1).
+omega.
+subst k1; simpl; auto.
+
+intros.
+assert (mem a0 k1).
+apply Permut_mem with (Cons a0 l1); auto.
+red; auto.
+destruct (mem_decomp _ a0 k1 H1) as (e1, (e2, heq)).
+subst k1.
+
+apply Permut_trans with (Cons a0 (infix_plpl (infix_plpl e1 e2) k2)); auto.
+apply Permut_cons.
+apply IHl1; auto.
+
+assert (permut (Cons a0 l1) (Cons a0 (infix_plpl e1 e2))).
+apply Permut_trans with (infix_plpl e1 (Cons a0 e2)); auto.
+apply Permut_sym.
+apply Permut_cons_append.
+
+red in H2; simpl in H2. 
+red; intros.
+generalize (H2 x); clear H2; intro.
+generalize (num_occ_def _ x (Cons a0 l1)).
+generalize (num_occ_def _ x (Cons a0 (infix_plpl e1 e2))).
+assert (h: (x = a0 \/ x <> a0)%Z) by admit (* excluded middle *).
+destruct h; intuition.
+do 2 rewrite <- Append_assoc.
+simpl.
+apply Permut_cons_append.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/mergesort_list/mergesort_list_WP_M_Permut_cons_append_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Definition unit  := unit.
+
+Parameter mark : Type.
+
+Parameter at1: forall (a:Type), a -> mark  -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a  -> a.
+
+Implicit Arguments old.
+
+Inductive list (a:Type) :=
+  | Nil : list a
+  | Cons : a -> (list a) -> list a.
+Set Contextual Implicit.
+Implicit Arguments Nil.
+Unset Contextual Implicit.
+Implicit Arguments Cons.
+
+Set Implicit Arguments.
+Fixpoint length (a:Type)(l:(list a)) {struct l}: Z :=
+  match l with
+  | Nil  => 0%Z
+  | Cons _ r => (1%Z + (length r))%Z
+  end.
+Unset Implicit Arguments.
+
+Axiom Length_nonnegative : forall (a:Type), forall (l:(list a)),
+  (0%Z <= (length l))%Z.
+
+Axiom Length_nil : forall (a:Type), forall (l:(list a)),
+  ((length l) = 0%Z) <-> (l = (Nil:(list a))).
+
+Inductive sorted : (list Z) -> Prop :=
+  | Sorted_Nil : (sorted (Nil:(list Z)))
+  | Sorted_One : forall (x:Z), (sorted (Cons x (Nil:(list Z))))
+  | Sorted_Two : forall (x:Z) (y:Z) (l:(list Z)), (x <= y)%Z ->
+      ((sorted (Cons y l)) -> (sorted (Cons x (Cons y l)))).
+
+Set Implicit Arguments.
+Fixpoint mem (a:Type)(x:a) (l:(list a)) {struct l}: Prop :=
+  match l with
+  | Nil  => False
+  | Cons y r => (x = y) \/ (mem x r)
+  end.
+Unset Implicit Arguments.
+
+Axiom Sorted_mem : forall (x:Z) (l:(list Z)), ((forall (y:Z), (mem y l) ->
+  (x <= y)%Z) /\ (sorted l)) <-> (sorted (Cons x l)).
+
+Set Implicit Arguments.
+Fixpoint infix_plpl (a:Type)(l1:(list a)) (l2:(list a)) {struct l1}: (list
+  a) :=
+  match l1 with
+  | Nil  => l2
+  | Cons x1 r1 => (Cons x1 (infix_plpl r1 l2))
+  end.
+Unset Implicit Arguments.
+
+Axiom Append_assoc : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), ((infix_plpl l1 (infix_plpl l2
+  l3)) = (infix_plpl (infix_plpl l1 l2) l3)).
+
+Axiom Append_l_nil : forall (a:Type), forall (l:(list a)), ((infix_plpl l
+  (Nil:(list a))) = l).
+
+Axiom Append_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  ((length (infix_plpl l1 l2)) = ((length l1) + (length l2))%Z).
+
+Axiom mem_append : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (mem x (infix_plpl l1 l2)) <-> ((mem x l1) \/ (mem x l2)).
+
+Parameter num_occ: forall (a:Type), a -> (list a)  -> Z.
+
+Implicit Arguments num_occ.
+
+Axiom num_occ_def : forall (a:Type), forall (x:a) (l:(list a)),
+  match l with
+  | Nil  => ((num_occ x l) = 0%Z)
+  | Cons y r => ((x = y) -> ((num_occ x l) = (1%Z + (num_occ x r))%Z)) /\
+      ((~ (x = y)) -> ((num_occ x l) = (0%Z + (num_occ x r))%Z))
+  end.
+
+Axiom Mem_Num_Occ : forall (a:Type), forall (x:a) (l:(list a)), (mem x l) <->
+  (0%Z <  (num_occ x l))%Z.
+
+Definition permut (a:Type)(l1:(list a)) (l2:(list a)): Prop := forall (x:a),
+  ((num_occ x l1) = (num_occ x l2)).
+Implicit Arguments permut.
+
+Axiom Permut_refl : forall (a:Type), forall (l:(list a)), (permut l l).
+
+Axiom Permut_sym : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> (permut l2 l1).
+
+Axiom Permut_trans : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), (permut l1 l2) -> ((permut l2 l3) -> (permut l1 l3)).
+
+Axiom Permut_cons : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list
+  a)), (permut l1 l2) -> (permut (Cons x l1) (Cons x l2)).
+
+Axiom Permut_swap : forall (a:Type), forall (x:a) (y:a) (l:(list a)),
+  (permut (Cons x (Cons y l)) (Cons y (Cons x l))).
+
+Axiom Permut_mem : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((mem x l1) -> (mem x l2)).
+
+Axiom Permut_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((length l1) = (length l2)).
+
+Theorem Permut_cons_append : forall (a:Type), forall (x:a) (l1:(list a))
+  (l2:(list a)), (permut (infix_plpl (Cons x l1) l2) (infix_plpl l1 (Cons x
+  l2))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+induction l1; simpl; intros.
+apply Permut_cons. apply Permut_refl.
+apply Permut_trans with (Cons a0 (Cons x (infix_plpl l1 l2))); auto.
+apply Permut_swap.
+apply Permut_cons.
+simpl in IHl1.
+auto.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/mergesort_list/mergesort_list_WP_M_WP_parameter_mergesort_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Definition unit  := unit.
+
+Parameter mark : Type.
+
+Parameter at1: forall (a:Type), a -> mark  -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a  -> a.
+
+Implicit Arguments old.
+
+Inductive list (a:Type) :=
+  | Nil : list a
+  | Cons : a -> (list a) -> list a.
+Set Contextual Implicit.
+Implicit Arguments Nil.
+Unset Contextual Implicit.
+Implicit Arguments Cons.
+
+Set Implicit Arguments.
+Fixpoint length (a:Type)(l:(list a)) {struct l}: Z :=
+  match l with
+  | Nil  => 0%Z
+  | Cons _ r => (1%Z + (length r))%Z
+  end.
+Unset Implicit Arguments.
+
+Axiom Length_nonnegative : forall (a:Type), forall (l:(list a)),
+  (0%Z <= (length l))%Z.
+
+Axiom Length_nil : forall (a:Type), forall (l:(list a)),
+  ((length l) = 0%Z) <-> (l = (Nil:(list a))).
+
+Inductive sorted : (list Z) -> Prop :=
+  | Sorted_Nil : (sorted (Nil:(list Z)))
+  | Sorted_One : forall (x:Z), (sorted (Cons x (Nil:(list Z))))
+  | Sorted_Two : forall (x:Z) (y:Z) (l:(list Z)), (x <= y)%Z ->
+      ((sorted (Cons y l)) -> (sorted (Cons x (Cons y l)))).
+
+Set Implicit Arguments.
+Fixpoint mem (a:Type)(x:a) (l:(list a)) {struct l}: Prop :=
+  match l with
+  | Nil  => False
+  | Cons y r => (x = y) \/ (mem x r)
+  end.
+Unset Implicit Arguments.
+
+Axiom Sorted_mem : forall (x:Z) (l:(list Z)), ((forall (y:Z), (mem y l) ->
+  (x <= y)%Z) /\ (sorted l)) <-> (sorted (Cons x l)).
+
+Set Implicit Arguments.
+Fixpoint infix_plpl (a:Type)(l1:(list a)) (l2:(list a)) {struct l1}: (list
+  a) :=
+  match l1 with
+  | Nil  => l2
+  | Cons x1 r1 => (Cons x1 (infix_plpl r1 l2))
+  end.
+Unset Implicit Arguments.
+
+Axiom Append_assoc : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), ((infix_plpl l1 (infix_plpl l2
+  l3)) = (infix_plpl (infix_plpl l1 l2) l3)).
+
+Axiom Append_l_nil : forall (a:Type), forall (l:(list a)), ((infix_plpl l
+  (Nil:(list a))) = l).
+
+Axiom Append_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  ((length (infix_plpl l1 l2)) = ((length l1) + (length l2))%Z).
+
+Axiom mem_append : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (mem x (infix_plpl l1 l2)) <-> ((mem x l1) \/ (mem x l2)).
+
+Parameter num_occ: forall (a:Type), a -> (list a)  -> Z.
+
+Implicit Arguments num_occ.
+
+Axiom num_occ_def : forall (a:Type), forall (x:a) (l:(list a)),
+  match l with
+  | Nil  => ((num_occ x l) = 0%Z)
+  | Cons y r => ((x = y) -> ((num_occ x l) = (1%Z + (num_occ x r))%Z)) /\
+      ((~ (x = y)) -> ((num_occ x l) = (0%Z + (num_occ x r))%Z))
+  end.
+
+Axiom Mem_Num_Occ : forall (a:Type), forall (x:a) (l:(list a)), (mem x l) <->
+  (0%Z <  (num_occ x l))%Z.
+
+Definition permut (a:Type)(l1:(list a)) (l2:(list a)): Prop := forall (x:a),
+  ((num_occ x l1) = (num_occ x l2)).
+Implicit Arguments permut.
+
+Axiom Permut_refl : forall (a:Type), forall (l:(list a)), (permut l l).
+
+Axiom Permut_sym : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> (permut l2 l1).
+
+Axiom Permut_trans : forall (a:Type), forall (l1:(list a)) (l2:(list a))
+  (l3:(list a)), (permut l1 l2) -> ((permut l2 l3) -> (permut l1 l3)).
+
+Axiom Permut_cons : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list
+  a)), (permut l1 l2) -> (permut (Cons x l1) (Cons x l2)).
+
+Axiom Permut_swap : forall (a:Type), forall (x:a) (y:a) (l:(list a)),
+  (permut (Cons x (Cons y l)) (Cons y (Cons x l))).
+
+Axiom Permut_mem : forall (a:Type), forall (x:a) (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((mem x l1) -> (mem x l2)).
+
+Axiom Permut_length : forall (a:Type), forall (l1:(list a)) (l2:(list a)),
+  (permut l1 l2) -> ((length l1) = (length l2)).
+
+Axiom Permut_cons_append : forall (a:Type), forall (x:a) (l1:(list a))
+  (l2:(list a)), (permut (infix_plpl (Cons x l1) l2) (infix_plpl l1 (Cons x
+  l2))).
+
+Theorem WP_parameter_mergesort : forall (l:(list Z)),
+  match l with
+  | (Nil |Cons _ Nil ) => True
+  | _ => (2%Z <= (length l))%Z -> forall (result:(list Z)) (result1:(list
+      Z)), ((1%Z <= (length result))%Z /\ ((1%Z <= (length result1))%Z /\
+      (permut l (infix_plpl result result1)))) -> (((0%Z <= (length l))%Z /\
+      ((length result) <  (length l))%Z) -> forall (result2:(list Z)),
+      ((sorted result2) /\ (permut result result2)) ->
+      (((0%Z <= (length l))%Z /\ ((length result1) <  (length l))%Z) ->
+      forall (result3:(list Z)), ((sorted result3) /\ (permut result1
+      result3)) -> (((sorted result2) /\ (sorted result3)) ->
+      forall (result4:(list Z)), ((sorted result4) /\ (permut result4
+      (infix_plpl result2 result3))) -> (permut l result4))))
+  end.
+(* YOU MAY EDIT THE PROOF BELOW *)
+destruct l; try trivial.
+destruct l; try trivial.
+intuition.
+apply Permut_trans with (infix_plpl result2 result3).
+apply Permut_trans with (infix_plpl result result1); auto.
+apply Permut_append; auto.
+apply Permut_sym; auto.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/mergesort_list/why3session.xml

+<?xml version="1.0" encoding="UTF-8"?>
+<!DOCTYPE why3session SYSTEM "why3session.dtd">
+<why3session name="examples/programs/mergesort_list/why3session.xml">
+ <file name="../mergesort_list.mlw" verified="true" expanded="true">
+  <theory name="WP M" verified="true" expanded="true">
+   <goal name="Permut_cons_append" sum="0df043e8309df0e1689f1f7a88d0d180" proved="true" expanded="true">
+    <proof prover="coq" timelimit="10" edited="mergesort_list_WP_M_Permut_cons_append_1.v" obsolete="false">
+     <result status="valid" time="0.61"/>
+    </proof>
+   </goal>
+   <goal name="Permut_append" sum="d4e21514e0240fc5a8bd445d2283b9bf" proved="true" expanded="true">
+    <proof prover="coq" timelimit="10" edited="mergesort_list_WP_M_Permut_append_1.v" obsolete="false">
+     <result status="valid" time="1.22"/>
+    </proof>
+   </goal>
+   <goal name="WP_parameter split" expl="correctness of parameter split" sum="9ff7c8dbe7d79e1015cb9368a03c1daf" proved="true" expanded="true">
+    <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+     <result status="valid" time="6.84"/>
+    </proof>
+   </goal>
+   <goal name="WP_parameter merge" expl="correctness of parameter merge" sum="71611dbb934bd53af15c4a8a78858ad6" proved="true" expanded="true">
+    <transf name="split_goal" proved="true" expanded="true">
+     <goal name="WP_parameter merge.1" expl="correctness of parameter merge" sum="fe8bc9f9df8d9a6921a7196e1d60e387" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.05"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.2" expl="correctness of parameter merge" sum="9fa15d85897c3aa78c4380dbaf90ebc6" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="1.57"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.3" expl="correctness of parameter merge" sum="d16c4bd22b087a32b97aee68e2501bed" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.07"/>
+      </proof>
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.04"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.4" expl="correctness of parameter merge" sum="020cc4887c52566b6403bdfa06601ab1" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="7.53"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.5" expl="correctness of parameter merge" sum="fd637999f4d66653db893dca4f5c92b2" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.6" expl="correctness of parameter merge" sum="75fd590235fcb4f15729660f43f7a752" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter merge.7" expl="correctness of parameter merge" sum="72e65cedafc98047e6ecb81d4eaa9ffe" proved="true" expanded="true">
+      <proof prover="cvc3" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+    </transf>
+   </goal>
+   <goal name="WP_parameter mergesort" expl="correctness of parameter mergesort" sum="5f4885e9f401b40502a2e0d9345496aa" proved="true" expanded="true">
+    <transf name="split_goal" proved="true" expanded="true">
+     <goal name="WP_parameter mergesort.1" expl="correctness of parameter mergesort" sum="5d04401ec7eba9af0405180837439499" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter mergesort.2" expl="correctness of parameter mergesort" sum="0ee613aaf02d097f3831905d71ecf617" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.58"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter mergesort.3" expl="correctness of parameter mergesort" sum="0bcc5da55f2144e02064f3bc161146e7" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter mergesort.4" expl="correctness of parameter mergesort" sum="9112f2883b791fdb9a930833c95a5d59" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.04"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter mergesort.5" expl="correctness of parameter mergesort" sum="373dd0ea919f1e26bad2eed80d46fffa" proved="true" expanded="true">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter mergesort.6" expl="correctness of parameter mergesort" sum="f5c5275c34f9944f6ee03bbf8548a67e" proved="true" expanded="true">
+      <transf name="split_goal" proved="true" expanded="true">
+       <goal name="WP_parameter mergesort.6.1" expl="correctness of parameter mergesort" sum="743e45f9bf536ddd5da2fe6b550656e3" proved="true" expanded="true">
+        <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+         <result status="valid" time="0.02"/>
+        </proof>
+       </goal>
+       <goal name="WP_parameter mergesort.6.2" expl="correctness of parameter mergesort" sum="a5fdc432b28f1622e6a1095d7dd4bb3f" proved="true" expanded="true">
+        <proof prover="coq" timelimit="10" edited="mergesort_list_WP_M_WP_parameter_mergesort_1.v" obsolete="false">
+         <result status="valid" time="0.64"/>
+        </proof>
+       </goal>
+      </transf>
+     </goal>
+    </transf>
+   </goal>
+  </theory>
+ </file>