Fixed Coq proofs using realized MapInjection

examples/vacid_0_sparse_array/vacid_0_sparse_array_2_SparseArray_permutation_1.v

@@ -4,6 +4,7 @@ Require Import BuiltIn.
 Require BuiltIn.
 Require int.Int.
 Require map.Map.
+Require map.MapInjection.
 
 (* Why3 assumption *)
 Definition unit := unit.
@@ -99,24 +100,6 @@ Axiom value_def : forall {a:Type} {a_WT:WhyType a}, forall (a1:(@sparse_array
 Definition length1 {a:Type} {a_WT:WhyType a} (a1:(@sparse_array
   a a_WT)): Z := (length (values a1)).
 
-(* Why3 assumption *)
-Definition injective (a:(@map.Map.map Z _ Z _)) (n:Z): Prop := forall (i:Z)
-  (j:Z), ((0%Z <= i)%Z /\ (i < n)%Z) -> (((0%Z <= j)%Z /\ (j < n)%Z) ->
-  ((~ (i = j)) -> ~ ((map.Map.get a i) = (map.Map.get a j)))).
-
-(* Why3 assumption *)
-Definition surjective (a:(@map.Map.map Z _ Z _)) (n:Z): Prop := forall (i:Z),
-  ((0%Z <= i)%Z /\ (i < n)%Z) -> exists j:Z, ((0%Z <= j)%Z /\ (j < n)%Z) /\
-  ((map.Map.get a j) = i).
-
-(* Why3 assumption *)
-Definition range (a:(@map.Map.map Z _ Z _)) (n:Z): Prop := forall (i:Z),
-  ((0%Z <= i)%Z /\ (i < n)%Z) -> ((0%Z <= (map.Map.get a i))%Z /\
-  ((map.Map.get a i) < n)%Z).
-
-Axiom injective_surjective : forall (a:(@map.Map.map Z _ Z _)) (n:Z),
-  (injective a n) -> ((range a n) -> (surjective a n)).
-
 Require Import Why3.
 Ltac ae := why3 "alt-ergo".
 
@@ -138,9 +121,9 @@ unfold is_elt, length1, get; simpl.
 intro H; decompose [and] H; clear H.
 subst n1 n2.
 intros. subst a_card.
-assert (inj: injective a_back n0) by ae.
-assert (rng: range a_back n0) by ae.
-generalize (injective_surjective a_back n0 inj rng); intro surj.
+assert (inj: MapInjection.injective a_back n0) by ae.
+assert (rng: MapInjection.range a_back n0) by ae.
+generalize (MapInjection.injective_surjective a_back n0 inj rng); intro surj.
 destruct (surj i H1) as (j, (hj1, hj2)).
 ae.
 Qed.

examples/vacid_0_sparse_array/why3session.xml

@@ -106,7 +106,7 @@
      edited="vacid_0_sparse_array_2_SparseArray_permutation_1.v"
      obsolete="false"
      archived="false">
-     <result status="valid" time="1.22"/>
+     <result status="valid" time="1.24"/>
     </proof>
    </goal>
    <goal
@@ -128,11 +128,11 @@
       name="WP_parameter set.1"
       locfile="../vacid_0_sparse_array.mlw"
       loclnum="96" loccnumb="6" loccnume="9"
-      expl="1. precondition"
+      expl="1. index in array bounds"
       sum="88db21b5a3cc7ee52c64849f80745892"
       proved="true"
       expanded="false"
-      shape="preconditionainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V11V11Aainfix &lt;agetV5V11V0Aainfix &lt;=c0agetV5V11Iainfix &lt;V11V6Aainfix &lt;=c0V11FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
+      shape="index in array boundsainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V11V11Aainfix &lt;agetV5V11V0Aainfix &lt;=c0agetV5V11Iainfix &lt;V11V6Aainfix &lt;=c0V11FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
       <label
        name="expl:VC for set"/>
       <proof
@@ -244,11 +244,11 @@
       name="WP_parameter set.6"
       locfile="../vacid_0_sparse_array.mlw"
       loclnum="96" loccnumb="6" loccnume="9"
-      expl="6. precondition"
+      expl="6. index in array bounds"
       sum="7e779574092aeeba5588cc2cceb39bb5"
       proved="true"
       expanded="false"
-      shape="preconditionainfix &lt;V8V2Aainfix &lt;=c0V8Iainfix &lt;V6V0INainfix =V13aTrueIais_eltV12V8qainfix =V13aTrueFIainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V0Aainfix =agetV3agetV5V14V14Aainfix &lt;agetV5V14V0Aainfix &lt;=c0agetV5V14Iainfix &lt;V14V6Aainfix &lt;=c0V14FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Iainfix =V11asetV1V8V9Aainfix &lt;=c0V0Lamk sparse_arrayamk arrayV0V11amk arrayV2V3amk arrayV4V5V6V7FIainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V15V15Aainfix &lt;agetV5V15V0Aainfix &lt;=c0agetV5V15Iainfix &lt;V15V6Aainfix &lt;=c0V15FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
+      shape="index in array boundsainfix &lt;V8V2Aainfix &lt;=c0V8Iainfix &lt;V6V0INainfix =V13aTrueIais_eltV12V8qainfix =V13aTrueFIainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V0Aainfix =agetV3agetV5V14V14Aainfix &lt;agetV5V14V0Aainfix &lt;=c0agetV5V14Iainfix &lt;V14V6Aainfix &lt;=c0V14FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Iainfix =V11asetV1V8V9Aainfix &lt;=c0V0Lamk sparse_arrayamk arrayV0V11amk arrayV2V3amk arrayV4V5V6V7FIainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V15V15Aainfix &lt;agetV5V15V0Aainfix &lt;=c0agetV5V15Iainfix &lt;V15V6Aainfix &lt;=c0V15FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
       <label
        name="expl:VC for set"/>
       <proof
@@ -264,11 +264,11 @@
       name="WP_parameter set.7"
       locfile="../vacid_0_sparse_array.mlw"
       loclnum="96" loccnumb="6" loccnume="9"
-      expl="7. precondition"
+      expl="7. index in array bounds"
       sum="03a900ddb7ac18d46efbf1d31a08a243"
       proved="true"
       expanded="false"
-      shape="preconditionainfix &lt;V6V4Aainfix &lt;=c0V6Iainfix =V14asetV3V8V6Aainfix &lt;=c0V2FIainfix &lt;V8V2Aainfix &lt;=c0V8Iainfix &lt;V6V0INainfix =V13aTrueIais_eltV12V8qainfix =V13aTrueFIainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V0Aainfix =agetV3agetV5V15V15Aainfix &lt;agetV5V15V0Aainfix &lt;=c0agetV5V15Iainfix &lt;V15V6Aainfix &lt;=c0V15FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Iainfix =V11asetV1V8V9Aainfix &lt;=c0V0Lamk sparse_arrayamk arrayV0V11amk arrayV2V3amk arrayV4V5V6V7FIainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V16V16Aainfix &lt;agetV5V16V0Aainfix &lt;=c0agetV5V16Iainfix &lt;V16V6Aainfix &lt;=c0V16FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
+      shape="index in array boundsainfix &lt;V6V4Aainfix &lt;=c0V6Iainfix =V14asetV3V8V6Aainfix &lt;=c0V2FIainfix &lt;V8V2Aainfix &lt;=c0V8Iainfix &lt;V6V0INainfix =V13aTrueIais_eltV12V8qainfix =V13aTrueFIainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V0Aainfix =agetV3agetV5V15V15Aainfix &lt;agetV5V15V0Aainfix &lt;=c0agetV5V15Iainfix &lt;V15V6Aainfix &lt;=c0V15FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Iainfix =V11asetV1V8V9Aainfix &lt;=c0V0Lamk sparse_arrayamk arrayV0V11amk arrayV2V3amk arrayV4V5V6V7FIainfix &lt;V8V0Aainfix &lt;=c0V8Iainfix &lt;V8V0Aainfix &lt;=c0V8Aainfix &lt;=c0V4Aainfix &lt;=c0V2Aainfix &lt;=c0V0Aainfix =agetV3agetV5V16V16Aainfix &lt;agetV5V16V0Aainfix &lt;=c0agetV5V16Iainfix &lt;V16V6Aainfix &lt;=c0V16FAainfix =V2V4Aainfix =V0V2Aainfix &lt;=V0amaxlenAainfix &lt;=V6V0Aainfix &lt;=c0V6Lamk sparse_arrayamk arrayV0V1amk arrayV2V3amk arrayV4V5V6V7F">
       <label
        name="expl:VC for set"/>
       <proof

examples/vstte10_inverting/vstte10_inverting_WP_InvertingAnInjection_WP_parameter_inverting_1.v

-(* This file is generated by Why3's Coq driver *)
+(* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
 Require BuiltIn.
 Require int.Int.
 Require map.Map.
+Require map.MapInjection.
 
 (* Why3 assumption *)
 Definition unit := unit.
@@ -11,75 +12,63 @@ Definition unit := unit.
 (* Why3 assumption *)
 Inductive array
   (a:Type) {a_WT:WhyType a} :=
-  | mk_array : Z -> (map.Map.map Z a) -> array a.
+  | mk_array : Z -> (@map.Map.map Z _ a a_WT) -> array a.
 Axiom array_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (array a).
 Existing Instance array_WhyType.
 Implicit Arguments mk_array [[a] [a_WT]].
 
 (* Why3 assumption *)
-Definition elts {a:Type} {a_WT:WhyType a} (v:(array a)): (map.Map.map Z a) :=
-  match v with
+Definition elts {a:Type} {a_WT:WhyType a} (v:(@array a a_WT)): (@map.Map.map
+  Z _ a a_WT) := match v with
   | (mk_array x x1) => x1
   end.
 
 (* Why3 assumption *)
-Definition length {a:Type} {a_WT:WhyType a} (v:(array a)): Z :=
+Definition length {a:Type} {a_WT:WhyType a} (v:(@array a a_WT)): Z :=
   match v with
   | (mk_array x x1) => x
   end.
 
 (* Why3 assumption *)
-Definition get {a:Type} {a_WT:WhyType a} (a1:(array a)) (i:Z): a :=
+Definition get {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (i:Z): a :=
   (map.Map.get (elts a1) i).
 
 (* Why3 assumption *)
-Definition set {a:Type} {a_WT:WhyType a} (a1:(array a)) (i:Z) (v:a): (array
-  a) := (mk_array (length a1) (map.Map.set (elts a1) i v)).
+Definition set {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (i:Z)
+  (v:a): (@array a a_WT) := (mk_array (length a1) (map.Map.set (elts a1) i
+  v)).
 
 (* Why3 assumption *)
-Definition make {a:Type} {a_WT:WhyType a} (n:Z) (v:a): (array a) :=
-  (mk_array n (map.Map.const v:(map.Map.map Z a))).
+Definition make {a:Type} {a_WT:WhyType a} (n:Z) (v:a): (@array a a_WT) :=
+  (mk_array n (map.Map.const v:(@map.Map.map Z _ a a_WT))).
 
 (* Why3 assumption *)
-Definition injective (a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z) (j:Z),
-  ((0%Z <= i)%Z /\ (i < n)%Z) -> (((0%Z <= j)%Z /\ (j < n)%Z) ->
-  ((~ (i = j)) -> ~ ((map.Map.get a i) = (map.Map.get a j)))).
+Definition injective (a:(@array Z _)) (n:Z): Prop :=
+  (map.MapInjection.injective (elts a) n).
 
 (* Why3 assumption *)
-Definition surjective (a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z),
-  ((0%Z <= i)%Z /\ (i < n)%Z) -> exists j:Z, ((0%Z <= j)%Z /\ (j < n)%Z) /\
-  ((map.Map.get a j) = i).
+Definition surjective (a:(@array Z _)) (n:Z): Prop :=
+  (map.MapInjection.surjective (elts a) n).
 
 (* Why3 assumption *)
-Definition range (a:(map.Map.map Z Z)) (n:Z): Prop := forall (i:Z),
-  ((0%Z <= i)%Z /\ (i < n)%Z) -> ((0%Z <= (map.Map.get a i))%Z /\
-  ((map.Map.get a i) < n)%Z).
-
-Axiom injective_surjective : forall (a:(map.Map.map Z Z)) (n:Z), (injective a
-  n) -> ((range a n) -> (surjective a n)).
-
-(* Why3 assumption *)
-Definition injective1 (a:(array Z)) (n:Z): Prop := (injective (elts a) n).
-
-(* Why3 assumption *)
-Definition surjective1 (a:(array Z)) (n:Z): Prop := (surjective (elts a) n).
-
-(* Why3 assumption *)
-Definition range1 (a:(array Z)) (n:Z): Prop := (range (elts a) n).
+Definition range (a:(@array Z _)) (n:Z): Prop := (map.MapInjection.range
+  (elts a) n).
 
 (* Why3 goal *)
-Theorem WP_parameter_inverting : forall (a:Z) (b:Z) (n:Z),
-  forall (a1:(map.Map.map Z Z)), (((0%Z <= a)%Z /\ (0%Z <= b)%Z) /\
-  (((n = a) /\ (a = b)) /\ ((injective a1 n) /\ (range a1 n)))) -> let o :=
-  (n - 1%Z)%Z in ((0%Z <= o)%Z -> forall (b1:(map.Map.map Z Z)),
-  (forall (j:Z), ((0%Z <= j)%Z /\ (j < (o + 1%Z)%Z)%Z) -> ((map.Map.get b1
-  (map.Map.get a1 j)) = j)) -> ((0%Z <= b)%Z -> (injective b1 n))).
+Theorem WP_parameter_inverting : forall (a:Z) (a1:(@map.Map.map Z _ Z _))
+  (b:Z) (n:Z), (((0%Z <= a)%Z /\ (0%Z <= b)%Z) /\ (((n = a) /\ (a = b)) /\
+  ((map.MapInjection.injective a1 n) /\ (map.MapInjection.range a1 n)))) ->
+  let o := (n - 1%Z)%Z in ((0%Z <= o)%Z -> forall (b1:(@map.Map.map Z _
+  Z _)), (forall (j:Z), ((0%Z <= j)%Z /\ (j < (o + 1%Z)%Z)%Z) ->
+  ((map.Map.get b1 (map.Map.get a1 j)) = j)) -> ((0%Z <= b)%Z ->
+  (map.MapInjection.injective b1 n))).
+(* Why3 intros a a1 b n ((h1,h2),((h3,h4),(h5,h6))) o h7 b1 h8 h9. *)
 intuition.
 intuition.
 red; intros.
 unfold get; simpl.
-assert (surjective a1 n).
-apply injective_surjective; assumption.
+assert (MapInjection.surjective a1 n).
+apply MapInjection.injective_surjective; assumption.
 generalize (H11 i H8); unfold get; simpl; intros (i1, (Hi1,Hi2)).
 generalize (H11 j H9); unfold get; simpl; intros (j1, (Hj1,Hj2)).
 rewrite <- Hi2.
@@ -92,4 +81,3 @@ subst.
 auto.
 Qed.
 
-