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(* This file was originally generated by why.
It can be modified; only the generated parts will be overwritten. *)
Require Import Why.
(* ----------- PRELIMINAIRES ------------- *)
(* définition et propriétés de *)
Require Import Omega.
Require Import ZArithRing.
Ltac omega' := abstract omega.
Set Implicit Arguments.
Unset Strict Implicit.
(* Induction pour vérifier qu'on est le maximum *)
Inductive Maximize (t:array Z) (n m:Z) : Z -> Prop :=
Maxim_cons :
forall k:Z,
((k <= n)%Z -> (access t k <= m)%Z) ->
((k < n)%Z -> Maximize t n m (k + 1)%Z) -> Maximize t n m k.
(* Signification de ce prédicat: *)
Lemma Maximize_ext1 :
forall (t:array Z) (n m k i:Z),
Maximize t n m k -> (k <= i <= n)%Z -> (access t i <= m)%Z.
Proof.
intros t n m k i H1; elim H1; auto.
intros k0 H2 H3 HR H4; case (Z_eq_dec k0 i).
intros H; rewrite <- H; apply H2; omega'.
intros; apply HR; omega'.
Qed.
Lemma Maximize_ext2 :
forall (t:array Z) (n m k:Z),
(forall i:Z, (k <= i <= n)%Z -> (access t i <= m)%Z) ->
Maximize t n m k.
Proof.
intros t n m k.
refine
(well_founded_ind (Zwf_up_well_founded n)
(fun k:Z =>
(forall i:Z, (k <= i <= n)%Z -> (access t i <= m)%Z) ->
Maximize t n m k) _ _).
clear k; intros k HR H.
constructor 1.
intros; apply H; omega'.
intros; apply HR.
unfold Zwf_up; omega'.
intros; apply H; omega'.
Qed.
(* compatibilité de avec *)
Lemma Maximize_Zle :
forall (t:array Z) (n m1 m2 k:Z),
Maximize t n m1 k -> (k <= n)%Z -> (m1 <= m2)%Z -> Maximize t n m2 k.
Proof.
intros t n m1 m2 k H0; elim H0.
intros k0 H1 H2 H3 H4 H5; constructor 1.
omega'.
intros; apply H3; omega'.
Qed.
Set Strict Implicit.
Unset Implicit Arguments.
(* ----------- FIN PRELIMINAIRES ----------- *)
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
ArraySubst t0.
Save.
Proof.
intuition.
subst; auto with *.
Save.
(*Why type*) Definition farray: Set ->Set.
Admitted.
(*Why logic*) Definition access : forall (A1:Set), (array A1) -> Z -> A1.
Admitted.
Implicit Arguments access.
(*Why logic*) Definition update :
forall (A1:Set), (array A1) -> Z -> A1 -> (array A1).
Admitted.
Implicit Arguments update.
(*Why axiom*) Lemma access_update :
forall (A1:Set),
(forall (a:(array A1)),
(forall (i:Z), (forall (v:A1), (access (update a i v) i) = v))).
Admitted.
(*Why axiom*) Lemma access_update_neq :
forall (A1:Set),
(forall (a:(array A1)),
(forall (i:Z),
(forall (j:Z),
(forall (v:A1), (i <> j -> (access (update a i v) j) = (access a j)))))).
Admitted.
(*Why logic*) Definition array_length : forall (A1:Set), (array A1) -> Z.
Admitted.
Implicit Arguments array_length.
(*Why predicate*) Definition sorted_array (t:(array Z)) (i:Z) (j:Z)
:= (forall (k1:Z),
(forall (k2:Z),
((i <= k1 /\ k1 <= k2) /\ k2 <= j -> (access t k1) <= (access t k2)))).
(*Why predicate*) Definition exchange (A111:Set) (a1:(array A111)) (a2:(array A111)) (i:Z) (j:Z)
:= (array_length a1) = (array_length a2) /\
(access a1 i) = (access a2 j) /\ (access a2 i) = (access a1 j) /\
(forall (k:Z), (k <> i /\ k <> j -> (access a1 k) = (access a2 k))).
Implicit Arguments exchange.
(*Why logic*) Definition permut :
forall (A1:Set), (array A1) -> (array A1) -> Z -> Z -> Prop.
Admitted.
Implicit Arguments permut.
(*Why axiom*) Lemma permut_refl :
forall (A1:Set),
(forall (t:(array A1)), (forall (l:Z), (forall (u:Z), (permut t t l u)))).
Admitted.
(*Why axiom*) Lemma permut_sym :
forall (A1:Set),
(forall (t1:(array A1)),
(forall (t2:(array A1)),
(forall (l:Z), (forall (u:Z), ((permut t1 t2 l u) -> (permut t2 t1 l u)))))).
Admitted.
(*Why axiom*) Lemma permut_trans :
forall (A1:Set),
(forall (t1:(array A1)),
(forall (t2:(array A1)),
(forall (t3:(array A1)),
(forall (l:Z),
(forall (u:Z),
((permut t1 t2 l u) -> ((permut t2 t3 l u) -> (permut t1 t3 l u)))))))).
Admitted.
(*Why axiom*) Lemma permut_exchange :
forall (A1:Set),
(forall (a1:(array A1)),
(forall (a2:(array A1)),
(forall (l:Z),
(forall (u:Z),
(forall (i:Z),
(forall (j:Z),
(l <= i /\ i <= u ->
(l <= j /\ j <= u -> ((exchange a1 a2 i j) -> (permut a1 a2 l u)))))))))).
Admitted.
(*Why axiom*) Lemma exchange_upd :
forall (A1:Set),
(forall (a:(array A1)),
(forall (i:Z),
(forall (j:Z),
(exchange a (update (update a i (access a j)) j (access a i)) i j)))).
Admitted.
(*Why axiom*) Lemma permut_weakening :
forall (A1:Set),
(forall (a1:(array A1)),
(forall (a2:(array A1)),
(forall (l1:Z),
(forall (r1:Z),
(forall (l2:Z),
(forall (r2:Z),
((l1 <= l2 /\ l2 <= r2) /\ r2 <= r1 ->
((permut a1 a2 l2 r2) -> (permut a1 a2 l1 r1))))))))).
Admitted.
(*Why axiom*) Lemma permut_eq :
forall (A1:Set),
(forall (a1:(array A1)),
(forall (a2:(array A1)),
(forall (l:Z),
(forall (u:Z),
(l <= u ->
((permut a1 a2 l u) ->
(forall (i:Z), (i < l \/ u < i -> (access a2 i) = (access a1 i))))))))).
Admitted.
(*Why predicate*) Definition permutation (A120:Set) (a1:(array A120)) (a2:(array A120))
:= (permut a1 a2 0 ((array_length a1) - 1)).
Implicit Arguments permutation.
(*Why axiom*) Lemma array_length_update :
forall (A1:Set),
(forall (a:(array A1)),
(forall (i:Z),
(forall (v:A1), (array_length (update a i v)) = (array_length a)))).
Admitted.
(*Why axiom*) Lemma permut_array_length :
forall (A1:Set),
(forall (a1:(array A1)),
(forall (a2:(array A1)),
(forall (l:Z),
(forall (u:Z),
((permut a1 a2 l u) -> (array_length a1) = (array_length a2)))))).
Admitted.
(*Why predicate*) Definition Maximize (t:(array Z)) (n:Z) (x:Z) (k:Z)
:= (forall (i:Z), (k <= i /\ i <= n -> (access t k) <= x)).
Proof.
intuition.
subst; auto.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition; subst.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
Proof.
intuition.
Save.
(* Début: preuve de *)
Proof.
intros; split.
Omega'.
rewrite Test4 in Pre18; tauto.
Qed.
Proof.
intros; Omega'.
Qed.
Proof.
intros; Omega'.
Qed.
Proof.
repeat (split; [ Omega' | auto ]).
subst nk.
ring (k0 - 1 + 1)%Z; intros;
apply Maximize_Zle with (m1 := access t i0); Omega' || tauto.
Qed.
Proof.
intros; subst nk; unfold Zwf; Omega'.
Qed.
Proof.
intros; subst nk.
repeat (split; [ Omega' | auto ]); ring (k0 - 1 + 1)%Z; tauto.
Qed.
Proof.
intros; subst nk.
unfold Zwf; Omega'.
Qed.
(* fin preuve de maximum *)
Proof.
intros; split.
Omega'.
split.
Omega'.
split.
Omega'.
constructor 1.
Omega'.
intros H; absurd (i1 < i1)%Z; Omega'.
Qed.
Proof.
intros; Omega'.
Qed.
Proof.
intros; decompose [and] Pre8; clear Pre8; split.
ArrayLength.
split.
omega.
split.
(* post-condition 1 *)
unfold sorted_array in H0; unfold sorted_array.
intros C1 k C2 C3; case Post9.
intros Clength C4 C5 C6 C7 C8.
case (Z_eq_dec k i1).
intros C9; rewrite C9; rewrite C6; rewrite C8; try Omega'.
apply Maximize_ext1 with (n := i1) (k := 0%Z); try Omega'.
apply H5; Omega'.
intros C9; rewrite C8; try Omega'.
rewrite C8; try Omega'.
apply H0; try Omega'.
(* post-condition 2 *)
split.
apply permut_trans with (t' := t0); auto.
eapply exchange_is_permut; eauto.
(* post-condition 3 *)
decompose [and] Post7; clear Post7.
case Post9; clear Post9.
intros Clength C1 C2 C3 C4 C5 C5a; replace (i0 + 1)%Z with i1.
rewrite C3.
apply Maximize_ext2; intros i' C6.
case (Z_eq_dec i' r).
intros C7; rewrite C7; rewrite C4.
apply Maximize_ext1 with (n := i1) (k := 0%Z); try Omega';
auto.
intros; rewrite C5; try Omega'.
apply Maximize_ext1 with (n := i1) (k := 0%Z); try Omega';
auto.
omega.
unfold Zwf; omega.
Qed.
Proof.
intros; subst i; ring (array_length t - 1 + 1)%Z; split.
Omega'.
split.
unfold sorted_array; intros H;
absurd (array_length t <= array_length t - 1)%Z; [ Omega' | auto ].
split.
apply permut_refl.
intros H; absurd (array_length t < array_length t)%Z;
[ Omega' | auto ].
Qed.
Proof.
intros; cut ((i1 + 1)%Z = 0%Z);
[ intros H; rewrite H in Post2; split; tauto | Omega' ].
Qed.
Proof.
(* FILL PROOF HERE *)
Save.
|
