(* 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.

(* Why3 assumption *)
Inductive ref (a:Type) :=
  | mk_ref : a -> ref a.
Axiom ref_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (ref a).
Existing Instance ref_WhyType.
Implicit Arguments mk_ref [[a]].

(* Why3 assumption *)
Definition contents {a:Type} {a_WT:WhyType a} (v:(ref a)): a :=
  match v with
  | (mk_ref x) => x
  end.

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)).

Axiom char : Type.
Parameter char_WhyType : WhyType char.
Existing Instance char_WhyType.

(* Why3 assumption *)
Definition matches (a1:(array char)) (i1:Z) (a2:(array char)) (i2:Z)
  (n:Z): Prop := ((0%Z <= i1)%Z /\ (i1 <= ((length a1) - n)%Z)%Z) /\
  (((0%Z <= i2)%Z /\ (i2 <= ((length a2) - n)%Z)%Z) /\ forall (i:Z),
  ((0%Z <= i)%Z /\ (i < n)%Z) -> ((mixfix_lbrb a1
  (i1 + i)%Z) = (mixfix_lbrb a2 (i2 + i)%Z))).

Axiom matches_empty : forall (a1:(array char)) (a2:(array char)) (i1:Z)
  (i2:Z), ((0%Z <= i1)%Z /\ (i1 <= (length a1))%Z) -> (((0%Z <= i2)%Z /\
  (i2 <= (length a2))%Z) -> (matches a1 i1 a2 i2 0%Z)).

Axiom matches_right_extension : forall (a1:(array char)) (a2:(array char))
  (i1:Z) (i2:Z) (n:Z), (matches a1 i1 a2 i2 n) ->
  ((i1 <= (((length a1) - n)%Z - 1%Z)%Z)%Z ->
  ((i2 <= (((length a2) - n)%Z - 1%Z)%Z)%Z -> (((mixfix_lbrb a1
  (i1 + n)%Z) = (mixfix_lbrb a2 (i2 + n)%Z)) -> (matches a1 i1 a2 i2
  (n + 1%Z)%Z)))).

Axiom matches_contradiction_at_first : forall (a1:(array char)) (a2:(array
  char)) (i1:Z) (i2:Z) (n:Z), (0%Z < n)%Z -> ((~ ((mixfix_lbrb a1
  i1) = (mixfix_lbrb a2 i2))) -> ~ (matches a1 i1 a2 i2 n)).

Axiom matches_contradiction_at_i : forall (a1:(array char)) (a2:(array char))
  (i1:Z) (i2:Z) (i:Z) (n:Z), (0%Z < n)%Z -> (((0%Z <= i)%Z /\ (i < n)%Z) ->
  ((~ ((mixfix_lbrb a1 (i1 + i)%Z) = (mixfix_lbrb a2 (i2 + i)%Z))) ->
  ~ (matches a1 i1 a2 i2 n))).

Axiom matches_right_weakening : forall (a1:(array char)) (a2:(array char))
  (i1:Z) (i2:Z) (n:Z) (n':Z), (matches a1 i1 a2 i2 n) -> ((n' < n)%Z ->
  (matches a1 i1 a2 i2 n')).

Axiom matches_left_weakening : forall (a1:(array char)) (a2:(array char))
  (i1:Z) (i2:Z) (n:Z) (n':Z), (matches a1 (i1 - (n - n')%Z)%Z a2
  (i2 - (n - n')%Z)%Z n) -> ((n' < n)%Z -> (matches a1 i1 a2 i2 n')).

Axiom matches_sym : forall (a1:(array char)) (a2:(array char)) (i1:Z) (i2:Z)
  (n:Z), (matches a1 i1 a2 i2 n) -> (matches a2 i2 a1 i1 n).

Axiom matches_trans : forall (a1:(array char)) (a2:(array char)) (a3:(array
  char)) (i1:Z) (i2:Z) (i3:Z) (n:Z), (matches a1 i1 a2 i2 n) -> ((matches a2
  i2 a3 i3 n) -> (matches a1 i1 a3 i3 n)).

(* Why3 assumption *)
Definition is_next (p:(array char)) (j:Z) (n:Z): Prop := ((0%Z <= n)%Z /\
  (n < j)%Z) /\ ((matches p (j - n)%Z p 0%Z n) /\ forall (z:Z), ((n < z)%Z /\
  (z < j)%Z) -> ~ (matches p (j - z)%Z p 0%Z z)).

Axiom next_iteration : forall (p:(array char)) (a:(array char)) (i:Z) (j:Z)
  (n:Z), ((0%Z < j)%Z /\ (j < (length p))%Z) -> (((j <= i)%Z /\
  (i <= (length a))%Z) -> ((matches a (i - j)%Z p 0%Z j) -> ((is_next p j
  n) -> (matches a (i - n)%Z p 0%Z n)))).

Axiom next_is_maximal : forall (p:(array char)) (a:(array char)) (i:Z) (j:Z)
  (n:Z) (k:Z), ((0%Z < j)%Z /\ (j < (length p))%Z) -> (((j <= i)%Z /\
  (i <= (length a))%Z) -> ((((i - j)%Z < k)%Z /\ (k < (i - n)%Z)%Z) ->
  ((matches a (i - j)%Z p 0%Z j) -> ((is_next p j n) -> ~ (matches a k p 0%Z
  (length p)))))).

Axiom next_1_0 : forall (p:(array char)), (1%Z <= (length p))%Z -> (is_next p
  1%Z 0%Z).

(* Why3 goal *)
Theorem VC_initnext : forall (p:(array char)), (1%Z <= (length p))%Z ->
  let m := (length p) in forall (next:(array Z)), ((forall (i:Z),
  ((0%Z <= i)%Z /\ (i < m)%Z) -> ((mixfix_lbrb next i) = 0%Z)) /\
  ((length next) = m)) -> ((1%Z < m)%Z -> forall (next1:(array Z)),
  ((length next1) = (length next)) ->
  (((elts next1) = (map.Map.set (elts next) 1%Z 0%Z)) -> forall (j:Z) (i:Z)
  (next2:(array Z)), ((length next2) = (length next1)) -> ((((0%Z <= j)%Z /\
  ((j < i)%Z /\ (i <= m)%Z)) /\ ((matches p (i - j)%Z p 0%Z j) /\
  ((forall (z:Z), (((j + 1%Z)%Z < z)%Z /\ (z < (i + 1%Z)%Z)%Z) -> ~ (matches
  p ((i + 1%Z)%Z - z)%Z p 0%Z z)) /\ forall (k:Z), ((0%Z < k)%Z /\
  (k <= i)%Z) -> (is_next p k (mixfix_lbrb next2 k))))) ->
  ((i < (m - 1%Z)%Z)%Z -> ((~ ((mixfix_lbrb p i) = (mixfix_lbrb p j))) ->
  ((j = 0%Z) -> forall (i1:Z), (i1 = (i + 1%Z)%Z) -> forall (next3:(array
  Z)), ((length next3) = (length next2)) ->
  (((elts next3) = (map.Map.set (elts next2) i1 0%Z)) -> (((0%Z <= j)%Z /\
  ((j < i1)%Z /\ (i1 <= m)%Z)) -> ((matches p (i1 - j)%Z p 0%Z j) ->
  forall (z:Z), (((j + 1%Z)%Z < z)%Z /\ (z < (i1 + 1%Z)%Z)%Z) -> ~ (matches p
  ((i1 + 1%Z)%Z - z)%Z p 0%Z z)))))))))).
intros p h1 m next (h2,h3) h4 next1 h5 h6 j i next2 h7
((h8,(h9,h10)),(h11,(h12,h13))) h14 h15 h16 i1 h17 next3 h18 h19
(h20,(h21,h22)) h23 z (h24,h25).
(*
intros n p1.
intro p3. unfold p3. clear p3.
intros (_,hn) _ next next1 _ _ _ _ _.
intros j i next4 ((hj,hi),(h0,(h1,h2))).
intros hi' _ _ neq j0. subst j.
intros i1 hi1; subst i1.
intros _ _ _ z hz.
*)
red; intro h. unfold matches in h. simpl in h.
destruct h as (hy1, (hy2, hy3)).

assert (case: (z = 2 \/ 2 < z)%Z) by omega. destruct case.
subst.
absurd (elts p i = elts p 0%Z); auto.
generalize (hy3 0%Z).
ring_simplify (i + 1 + 1 - 2 + 0)%Z.
intuition.

apply h12 with (z-1)%Z.
omega.
red; simpl.
repeat split; try omega.
intros; subst i1.
replace (i + 1 - (z - 1) + i0)%Z with (i + 1 + 1 - z + i0)%Z by omega.
apply hy3.
omega.
Qed.

