why3/examples/stdlib/array/array_ArrayPermut_exchange_permut_sub_1.v

(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below    *)
Require Import BuiltIn.
Require BuiltIn.
Require HighOrd.
Require int.Int.
Require map.Map.
Require map.Occ.
Require map.MapPermut.

Axiom array : forall (a:Type), Type.
Parameter array_WhyType : forall (a:Type) {a_WT:WhyType a},
  WhyType (array a).
Existing Instance array_WhyType.

Parameter elts: forall {a:Type} {a_WT:WhyType a}, (array a) -> (Z -> a).

Parameter length: forall {a:Type} {a_WT:WhyType a}, (array a) -> Z.

Axiom array'invariant : forall {a:Type} {a_WT:WhyType a}, forall (self:(array
  a)), (0%Z <= (length self))%Z.

(* Why3 assumption *)
Definition mixfix_lbrb {a:Type} {a_WT:WhyType a} (a1:(array a)) (i:Z): a :=
  ((elts a1) i).

Parameter mixfix_lblsmnrb: forall {a:Type} {a_WT:WhyType a}, (array a) ->
  Z -> a -> (array a).

Axiom mixfix_lblsmnrb_spec : forall {a:Type} {a_WT:WhyType a},
  forall (a1:(array a)) (i:Z) (v:a), ((length (mixfix_lblsmnrb a1 i
  v)) = (length a1)) /\ ((elts (mixfix_lblsmnrb a1 i
  v)) = (map.Map.set (elts a1) i v)).

(* Why3 assumption *)
Definition map_eq_sub {a:Type} {a_WT:WhyType a} (a1:(Z -> a)) (a2:(Z -> a))
  (l:Z) (u:Z): Prop := forall (i:Z), ((l <= i)%Z /\ (i < u)%Z) -> ((a1
  i) = (a2 i)).

(* Why3 assumption *)
Definition array_eq_sub {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array
  a)) (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\ (((0%Z <= l)%Z /\
  (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\ (u <= (length a1))%Z) /\
  (map_eq_sub (elts a1) (elts a2) l u))).

(* Why3 assumption *)
Definition array_eq {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array
  a)): Prop := ((length a1) = (length a2)) /\ (map_eq_sub (elts a1) (elts a2)
  0%Z (length a1)).

(* Why3 assumption *)
Definition exchange {a:Type} {a_WT:WhyType a} (a1:(Z -> a)) (a2:(Z -> a))
  (l:Z) (u:Z) (i:Z) (j:Z): Prop := ((l <= i)%Z /\ (i < u)%Z) /\
  (((l <= j)%Z /\ (j < u)%Z) /\ (((a1 i) = (a2 j)) /\ (((a1 j) = (a2 i)) /\
  forall (k:Z), ((l <= k)%Z /\ (k < u)%Z) -> ((~ (k = i)) -> ((~ (k = j)) ->
  ((a1 k) = (a2 k))))))).

Axiom exchange_set : forall {a:Type} {a_WT:WhyType a}, forall (a1:(Z -> a))
  (l:Z) (u:Z) (i:Z) (j:Z), ((l <= i)%Z /\ (i < u)%Z) -> (((l <= j)%Z /\
  (j < u)%Z) -> (exchange a1 (map.Map.set (map.Map.set a1 i (a1 j)) j (a1 i))
  l u i j)).

(* Why3 assumption *)
Definition exchange1 {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array a))
  (i:Z) (j:Z): Prop := ((length a1) = (length a2)) /\ (exchange (elts a1)
  (elts a2) 0%Z (length a1) i j).

(* Why3 assumption *)
Definition permut {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array a))
  (l:Z) (u:Z): Prop := ((length a1) = (length a2)) /\ (((0%Z <= l)%Z /\
  (l <= (length a1))%Z) /\ (((0%Z <= u)%Z /\ (u <= (length a1))%Z) /\
  (map.MapPermut.permut (elts a1) (elts a2) l u))).

(* Why3 assumption *)
Definition permut_sub {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array a))
  (l:Z) (u:Z): Prop := (map_eq_sub (elts a1) (elts a2) 0%Z l) /\ ((permut a1
  a2 l u) /\ (map_eq_sub (elts a1) (elts a2) u (length a1))).

(* Why3 assumption *)
Definition permut_all {a:Type} {a_WT:WhyType a} (a1:(array a)) (a2:(array
  a)): Prop := ((length a1) = (length a2)) /\ (map.MapPermut.permut (elts a1)
  (elts a2) 0%Z (length a1)).

(* Why3 goal *)
Theorem exchange_permut_sub : forall {a:Type} {a_WT:WhyType a},
  forall (a1:(array a)) (a2:(array a)) (i:Z) (j:Z) (l:Z) (u:Z), (exchange1 a1
  a2 i j) -> (((l <= i)%Z /\ (i < u)%Z) -> (((l <= j)%Z /\ (j < u)%Z) ->
  ((0%Z <= l)%Z -> ((u <= (length a1))%Z -> (permut_sub a1 a2 l u))))).
(* Why3 intros a1 a2 i j l u h1 (h2,h3) (h4,h5) h6 h7. *)
intros a1 a2 i j l u h1 (h2,h3) (h4,h5) h6 h7.
destruct h1 as (h11,h12).
destruct h12 as (ha,(hb,(hc,(hd,he)))).
red. repeat split.
(* eq_sub *)
red. intros. apply he; omega.
assumption. assumption. omega. omega. assumption.
(* permut *)
red. intro v.

assert (Occ.occ v (elts a1) i (i+1) + Occ.occ v (elts a1) j (j+1)
      = Occ.occ v (elts a2) i (i+1) + Occ.occ v (elts a2) j (j+1))%Z.
destruct (why_decidable_eq (elts a1 i) v).
rewrite Occ.occ_right_add. 2: omega. 2: ring_simplify (i+1-1)%Z; assumption.
rewrite (Occ.occ_right_add v (elts a2) j). 2: omega.
  2: ring_simplify (j+1-1)%Z; rewrite <- hc; assumption.
ring_simplify (i+1-1)%Z. ring_simplify (j+1-1)%Z.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) j). 2: omega.
destruct (why_decidable_eq (elts a1 j) v).
rewrite Occ.occ_right_add. 2: omega. 2: ring_simplify (j+1-1)%Z; assumption.
rewrite (Occ.occ_right_add v (elts a2) i). 2: omega.
  2: ring_simplify (i+1-1)%Z; rewrite <- hd; assumption.
ring_simplify (i+1-1)%Z. ring_simplify (j+1-1)%Z.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) i). 2: omega.
ring.
rewrite Occ.occ_right_no_add. 2: omega. 2: ring_simplify (j+1-1)%Z; assumption.
rewrite (Occ.occ_right_no_add v (elts a2) i). 2: omega.
  2: ring_simplify (i+1-1)%Z; rewrite <- hd; assumption.
ring_simplify (i+1-1)%Z. ring_simplify (j+1-1)%Z.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) i). 2: omega.
ring.
rewrite Occ.occ_right_no_add. 2: omega. 2: ring_simplify (i+1-1)%Z; assumption.
rewrite (Occ.occ_right_no_add v (elts a2) j). 2: omega.
  2: ring_simplify (j+1-1)%Z; rewrite <- hc; assumption.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) j). 2: omega.
destruct (why_decidable_eq (elts a1 j) v).
rewrite Occ.occ_right_add. 2: omega. 2: ring_simplify (j+1-1)%Z; assumption.
rewrite (Occ.occ_right_add v (elts a2) i). 2: omega.
  2: ring_simplify (i+1-1)%Z; rewrite <- hd; assumption.
ring_simplify (i+1-1)%Z. ring_simplify (j+1-1)%Z.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) i). 2: omega.
ring.
rewrite Occ.occ_right_no_add. 2: omega. 2: ring_simplify (j+1-1)%Z; assumption.
rewrite (Occ.occ_right_no_add v (elts a2) i). 2: omega.
  2: ring_simplify (i+1-1)%Z; rewrite <- hd; assumption.
ring_simplify (i+1-1)%Z. ring_simplify (j+1-1)%Z.
rewrite Occ.occ_empty. 2: omega. rewrite (Occ.occ_empty v (elts a2) i). 2: omega.
ring.

assert (c: (i < j \/ i = j \/ j < i)%Z) by omega. destruct c as [c|c].
(* i < j *)
assert (Occ.occ v (elts a1) l u = Occ.occ v (elts a1) l i + Occ.occ v (elts a1) i u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) i u = Occ.occ v (elts a1) i (i+1) + Occ.occ v (elts a1) (i+1) u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) (i+1) u = Occ.occ v (elts a1) (i+1) j + Occ.occ v (elts a1) j u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) j u = Occ.occ v (elts a1) j (j+1) + Occ.occ v (elts a1) (j+1) u)%Z.
  apply Occ.occ_append. omega.

assert (Occ.occ v (elts a2) l u = Occ.occ v (elts a2) l i + Occ.occ v (elts a2) i u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) i u = Occ.occ v (elts a2) i (i+1) + Occ.occ v (elts a2) (i+1) u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) (i+1) u = Occ.occ v (elts a2) (i+1) j + Occ.occ v (elts a2) j u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) j u = Occ.occ v (elts a2) j (j+1) + Occ.occ v (elts a2) (j+1) u)%Z.
  apply Occ.occ_append. omega.

assert (Occ.occ v (elts a1) l i = Occ.occ v (elts a2) l i).
  apply Occ.occ_eq. intros. apply he; omega.
assert (Occ.occ v (elts a1) (i+1) j = Occ.occ v (elts a2) (i+1) j).
  apply Occ.occ_eq. intros; apply he; omega.
assert (Occ.occ v (elts a1) (j+1) u = Occ.occ v (elts a2) (j+1) u).
  apply Occ.occ_eq. intros; apply he; omega.

omega.

(* i = j *)
destruct c.
subst j.
apply Occ.occ_eq.
intros k hk.
assert (c: (k=i \/ k <>i)%Z) by omega. destruct c.
subst k. assumption.
apply he. omega. assumption. assumption.

(* j < i *)
assert (Occ.occ v (elts a1) l u = Occ.occ v (elts a1) l j + Occ.occ v (elts a1) j u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) j u = Occ.occ v (elts a1) j (j+1) + Occ.occ v (elts a1) (j+1) u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) (j+1) u = Occ.occ v (elts a1) (j+1) i + Occ.occ v (elts a1) i u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a1) i u = Occ.occ v (elts a1) i (i+1) + Occ.occ v (elts a1) (i+1) u)%Z.
  apply Occ.occ_append. omega.

assert (Occ.occ v (elts a2) l u = Occ.occ v (elts a2) l j + Occ.occ v (elts a2) j u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) j u = Occ.occ v (elts a2) j (j+1) + Occ.occ v (elts a2) (j+1) u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) (j+1) u = Occ.occ v (elts a2) (j+1) i + Occ.occ v (elts a2) i u)%Z.
  apply Occ.occ_append. omega.
assert (Occ.occ v (elts a2) i u = Occ.occ v (elts a2) i (i+1) + Occ.occ v (elts a2) (i+1) u)%Z.
  apply Occ.occ_append. omega.

assert (Occ.occ v (elts a1) l j = Occ.occ v (elts a2) l j).
  apply Occ.occ_eq. intros. apply he; omega.
assert (Occ.occ v (elts a1) (j+1) i = Occ.occ v (elts a2) (j+1) i).
  apply Occ.occ_eq. intros; apply he; omega.
assert (Occ.occ v (elts a1) (i+1) u = Occ.occ v (elts a2) (i+1) u).
  apply Occ.occ_eq. intros; apply he; omega.

omega.

(* eq_sub *)
red. intros. apply he; omega.
Qed.