restored insertion sort proof (my mistake)

parent 105f6179
(* 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.
Axiom exchange_permut : forall (a:Type), forall (a1:(array a)) (a2:(array a))
(i:Z) (j:Z), (exchange1 a1 a2 i j) -> (((length a1) = (length a2)) ->
(((0%Z <= i)%Z /\ (i < (length a1))%Z) -> (((0%Z <= j)%Z /\
(j < (length a1))%Z) -> (permut a1 a2)))).
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 *)
(* 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.
Axiom exchange_permut : forall (a:Type), forall (a1:(array a)) (a2:(array a))
(i:Z) (j:Z), (exchange1 a1 a2 i j) -> (((length a1) = (length a2)) ->
(((0%Z <= i)%Z /\ (i < (length a1))%Z) -> (((0%Z <= j)%Z /\
(j < (length a1))%Z) -> (permut a1 a2)))).
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)), (a7 = (set a5 j (get a5
(j - 1%Z)%Z))) -> ((exchange match (set1 (mk_array a a7) (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) -> forall (k1:Z)
(k2:Z), (((0%Z <= k1)%Z /\ (k1 <= k2)%Z) /\ (k2 <= i)%Z) ->
((~ (k1 = j1)) -> ((~ (k2 = j1)) -> ((get a7 k1) <= (get a7
k2))%Z))))))))))))).
(* YOU MAY EDIT THE PROOF BELOW *)
intuition.
intuition.
Qed.
(* DO NOT EDIT BELOW *)
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