insertion sort completed

examples/programs/insertion_sort.mlw

@@ -25,9 +25,12 @@ module InsertionSort
              0 <= k1 <= k2 <= i -> k1 <> !j -> k2 <> !j -> a[k1] <= a[k2]) and
           (forall k: int. !j+1 <= k <= i -> v < a[k]) }
         variant { !j }
+        label L1:
         a[!j] <- a[!j - 1];
+        assert { exchange a[!j-1 <- v] (at a L1)[!j <- v] (!j - 1) !j};
         j := !j - 1
       done;
+      assert { forall k: int. 0 <= k < !j -> a[k] <= v };
       a[!j] <- v
     done
     { sorted a and permut a (old a) }

examples/programs/insertion_sort/insertion_sort.mlw_InsertionSort_WP_parameter insertion_sort_4.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 label : Type.
+
+Parameter at1: forall (a:Type), a -> label  -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a  -> a.
+
+Implicit Arguments old.
+
+Inductive ref (a:Type) :=
+  | mk_ref : a -> ref a.
+Implicit Arguments mk_ref.
+
+Definition contents (a:Type)(u:(ref a)): a :=
+  match u with
+  | mk_ref contents1 => contents1
+  end.
+Implicit Arguments contents.
+
+Parameter map : forall (a:Type) (b:Type), Type.
+
+Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
+
+Implicit Arguments get.
+
+Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
+
+Implicit Arguments set.
+
+Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
+  forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1)
+  a2) = b1).
+
+Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
+  forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
+  a2) = (get m a2)).
+
+Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
+
+Set Contextual Implicit.
+Implicit Arguments const.
+Unset Contextual Implicit.
+
+Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
+  b1):(map a b)) a1) = b1).
+
+Inductive array (a:Type) :=
+  | mk_array : Z -> (map Z a) -> array a.
+Implicit Arguments mk_array.
+
+Definition elts (a:Type)(u:(array a)): (map Z a) :=
+  match u with
+  | mk_array _ elts1 => elts1
+  end.
+Implicit Arguments elts.
+
+Definition length (a:Type)(u:(array a)): Z :=
+  match u with
+  | mk_array length1 _ => length1
+  end.
+Implicit Arguments length.
+
+Definition get1 (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
+Implicit Arguments get1.
+
+Definition set1 (a:Type)(a1:(array a)) (i:Z) (v:a): (array a) :=
+  match a1 with
+  | mk_array xcl0 _ => (mk_array xcl0 (set (elts a1) i v))
+  end.
+Implicit Arguments set1.
+
+Definition sorted_sub(a:(map Z Z)) (l:Z) (u:Z): Prop := forall (i1:Z) (i2:Z),
+  (((l <= i1)%Z /\ (i1 <= i2)%Z) /\ (i2 <  u)%Z) -> ((get a i1) <= (get a
+  i2))%Z.
+
+Definition sorted_sub1(a:(array Z)) (l:Z) (u:Z): Prop := (sorted_sub (elts a)
+  l u).
+
+Definition sorted(a:(array Z)): Prop := (sorted_sub (elts a) 0%Z (length a)).
+
+Definition map_eq_sub (a:Type)(a1:(map Z a)) (a2:(map Z a)) (l:Z)
+  (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\ (i <  u)%Z) -> ((get a1
+  i) = (get a2 i)).
+Implicit Arguments map_eq_sub.
+
+Definition exchange (a:Type)(a1:(map Z a)) (a2:(map Z a)) (i:Z)
+  (j:Z): Prop := ((get a1 i) = (get a2 j)) /\ (((get a2 i) = (get a1 j)) /\
+  forall (k:Z), ((~ (k = i)) /\ ~ (k = j)) -> ((get a1 k) = (get a2 k))).
+Implicit Arguments exchange.
+
+Axiom exchange_set : forall (a:Type), forall (a1:(map Z a)), forall (i:Z)
+  (j:Z), (exchange a1 (set (set a1 i (get a1 j)) j (get a1 i)) i j).
+
+Inductive permut_sub{a:Type}  : (map Z a) -> (map Z a) -> Z -> Z -> Prop :=
+  | permut_refl : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
+      (map_eq_sub a1 a2 l u) -> (permut_sub a1 a2 l u)
+  | permut_sym : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z) (u:Z),
+      (permut_sub a1 a2 l u) -> (permut_sub a2 a1 l u)
+  | permut_trans : forall (a1:(map Z a)) (a2:(map Z a)) (a3:(map Z a)),
+      forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> ((permut_sub a2 a3 l
+      u) -> (permut_sub a1 a3 l u))
+  | permut_exchange : forall (a1:(map Z a)) (a2:(map Z a)), forall (l:Z)
+      (u:Z) (i:Z) (j:Z), ((l <= i)%Z /\ (i <  u)%Z) -> (((l <= j)%Z /\
+      (j <  u)%Z) -> ((exchange a1 a2 i j) -> (permut_sub a1 a2 l u))).
+Implicit Arguments permut_sub.
+
+Axiom permut_weakening : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z
+  a)), forall (l1:Z) (r1:Z) (l2:Z) (r2:Z), (((l1 <= l2)%Z /\ (l2 <= r2)%Z) /\
+  (r2 <= r1)%Z) -> ((permut_sub a1 a2 l2 r2) -> (permut_sub a1 a2 l1 r1)).
+
+Axiom permut_eq : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
+  forall (l:Z) (u:Z), (l <= u)%Z -> ((permut_sub a1 a2 l u) -> forall (i:Z),
+  ((i <  l)%Z \/ (u <= i)%Z) -> ((get a2 i) = (get a1 i))).
+
+Axiom permut_exists : forall (a:Type), forall (a1:(map Z a)) (a2:(map Z a)),
+  forall (l:Z) (u:Z), (permut_sub a1 a2 l u) -> forall (i:Z), ((l <= i)%Z /\
+  (i <  u)%Z) -> exists j:Z, ((l <= j)%Z /\ (j <  u)%Z) /\ ((get a2
+  i) = (get a1 j)).
+
+Definition exchange1 (a:Type)(a1:(array a)) (a2:(array a)) (i:Z)
+  (j:Z): Prop := (exchange (elts a1) (elts a2) i j).
+Implicit Arguments exchange1.
+
+Definition permut_sub1 (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
+  (u:Z): Prop := (permut_sub (elts a1) (elts a2) l u).
+Implicit Arguments permut_sub1.
+
+Definition permut (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
+  ((length a1) = (length a2)) /\ (permut_sub (elts a1) (elts a2) 0%Z
+  (length a1)).
+Implicit Arguments permut.
+
+Definition array_eq_sub (a:Type)(a1:(array a)) (a2:(array a)) (l:Z)
+  (u:Z): Prop := (map_eq_sub (elts a1) (elts a2) l u).
+Implicit Arguments array_eq_sub.
+
+Definition array_eq (a:Type)(a1:(array a)) (a2:(array a)): Prop :=
+  ((length a1) = (length a2)) /\ (array_eq_sub a1 a2 0%Z (length a1)).
+Implicit Arguments array_eq.
+
+Axiom array_eq_sub_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
+  a)) (l:Z) (u:Z), (array_eq_sub a1 a2 l u) -> (permut_sub1 a1 a2 l u).
+
+Axiom array_eq_permut : forall (a:Type), forall (a1:(array a)) (a2:(array
+  a)), (array_eq a1 a2) -> (permut a1 a2).
+
+Theorem WP_parameter_insertion_sort : forall (a:Z), forall (a1:(map Z Z)),
+  let a2 := (mk_array a a1) in ((1%Z <= (a - 1%Z)%Z)%Z -> forall (a3:(map Z
+  Z)), let a4 := (mk_array a a3) in forall (i:Z), ((1%Z <= i)%Z /\
+  (i <= (a - 1%Z)%Z)%Z) -> (((sorted_sub a3 0%Z i) /\ (permut a4 a2)) ->
+  (((0%Z <= i)%Z /\ (i <  a)%Z) -> let result := (get a3 i) in
+  ((((0%Z <= i)%Z /\ (i <= i)%Z) /\ ((permut (set1 a4 i result) a2) /\
+  ((forall (k1:Z) (k2:Z), (((0%Z <= k1)%Z /\ (k1 <= k2)%Z) /\ (k2 <= i)%Z) ->
+  ((~ (k1 = i)) -> ((~ (k2 = i)) -> ((get a3 k1) <= (get a3 k2))%Z))) /\
+  forall (k:Z), (((i + 1%Z)%Z <= k)%Z /\ (k <= i)%Z) -> (result <  (get a3
+  k))%Z))) -> forall (j:Z), forall (a5:(map Z Z)), let a6 := (mk_array a
+  a5) in ((((0%Z <= j)%Z /\ (j <= i)%Z) /\ ((permut (set1 a6 j result) a2) /\
+  ((forall (k1:Z) (k2:Z), (((0%Z <= k1)%Z /\ (k1 <= k2)%Z) /\ (k2 <= i)%Z) ->
+  ((~ (k1 = j)) -> ((~ (k2 = j)) -> ((get a5 k1) <= (get a5 k2))%Z))) /\
+  forall (k:Z), (((j + 1%Z)%Z <= k)%Z /\ (k <= i)%Z) -> (result <  (get a5
+  k))%Z))) -> ((0%Z <  j)%Z -> (((0%Z <= (j - 1%Z)%Z)%Z /\
+  ((j - 1%Z)%Z <  a)%Z) -> ((result <  (get a5 (j - 1%Z)%Z))%Z ->
+  (((0%Z <= (j - 1%Z)%Z)%Z /\ ((j - 1%Z)%Z <  a)%Z) -> (((0%Z <= j)%Z /\
+  (j <  a)%Z) -> forall (a7:(map Z Z)), let a8 := (mk_array a a7) in
+  ((a7 = (set a5 j (get a5 (j - 1%Z)%Z))) -> ((exchange match (set1 a8
+  (j - 1%Z)%Z result) with
+  | mk_array _ elts1 => elts1
+  end match (set1 a6 j result) with
+  | mk_array _ elts1 => elts1
+  end (j - 1%Z)%Z j) -> forall (j1:Z), (j1 = (j - 1%Z)%Z) -> (permut (set1 a8
+  j1 result) a2))))))))))))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intuition.
+intuition.
+unfold set1.
+unfold permut.
+split.
+simpl.
+auto.
+apply permut_trans with (elts (set1 (mk_array a a5) j (get a3 i))); auto.
+subst j1.
+apply permut_exchange with (j-1)%Z j.
+simpl; omega.
+simpl; omega.
+assumption.
+unfold permut in H17.
+intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+