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(* Load Programs. *)(**************************************************************************)
(* *)
(* Proof of the Heapsort Algorithm. *)
(* *)
(* Jean-Christophe Fillitre (LRI, Universit Paris Sud) *)
(* March 1999 *)
(* *)
(**************************************************************************)
Require Import ZArith.
Require Import Why.
Require Import Omega.
Require Import heap.
Set Implicit Arguments.
Unset Strict Implicit.
(* We will also need another property about a heap, which is to have
* all his elements smaller or equal than a given value v.
*
* It is expressed by the predicate (inftree t n v k), inductively
* defined as follows:
*)
Inductive inftree (t:array Z) (n v:Z) : Z -> Prop :=
inftree_cons :
forall k:Z,
(0 <= k <= n)%Z ->
(access t k <= v)%Z ->
((2 * k + 1 <= n)%Z -> inftree t n v (2 * k + 1)%Z) ->
((2 * k + 2 <= n)%Z -> inftree t n v (2 * k + 2)%Z) ->
inftree t n v k.
(* Some lemmas about inftree *)
Lemma inftree_1 :
forall (t:array Z) (n v k:Z), inftree t n v k -> (access t k <= v)%Z.
Proof.
intros t n v k H.
elim H; auto.
Qed.
Lemma inftree_id :
forall (t1 t2:array Z) (n v k:Z),
inftree t1 n v k ->
(forall i:Z, (k <= i <= n)%Z -> access t1 i = access t2 i) ->
inftree t2 n v k.
Proof.
intros t1 t2 n v k H.
elim H; intros.
apply inftree_cons.
assumption.
rewrite <- (H6 k0).
assumption.
Omega'.
intro.
apply H3.
assumption.
intros i Hi.
apply H6; Omega'.
intro.
apply H5.
assumption.
intros i Hi.
apply H6; Omega'.
Qed.
Lemma inftree_2 :
forall (t1 t2:array Z) (n v k j:Z),
(n < array_length t1)%Z ->
inftree t1 n v j ->
exchange t2 t1 k j ->
(k < j)%Z -> (access t1 k <= access t1 j)%Z -> inftree t2 n v j.
Proof.
intros t1 t2 n v k j Hn H.
case H.
intros.
apply inftree_cons.
assumption.
decompose [exchange] H4.
Omega'.
intro.
apply inftree_id with (t1 := t1).
auto.
decompose [exchange] H4.
intros i Hi.
symmetry.
apply H13.
Omega'.
Omega'.
Omega'.
intro.
apply inftree_id with (t1 := t1).
auto.
decompose [exchange] H4.
intros i Hi.
symmetry.
apply H13.
Omega'.
Omega'.
Omega'.
Qed.
Lemma inftree_trans :
forall (t:array Z) (n k v v':Z),
(v <= v')%Z -> inftree t n v k -> inftree t n v' k.
Proof.
intros t n k v v' Hvv' H.
elim H; intros.
apply inftree_cons.
assumption.
Omega'.
auto.
auto.
Qed.
Lemma inftree_3 :
forall (t:array Z) (n k:Z), heap t n k -> inftree t n (access t k) k.
Proof.
intros t n k H.
elim H; intros.
apply inftree_cons.
assumption.
auto with zarith.
intro.
apply inftree_trans with
(v := access t (2 * k0 + 1)) (v' := access t k0).
Omega'.
auto.
intro.
apply inftree_trans with
(v := access t (2 * k0 + 2)) (v' := access t k0).
Omega'.
auto.
Qed.
Lemma inftree_all :
forall (t:array Z) (n v:Z),
inftree t n v 0 -> forall i:Z, (0 <= i <= n)%Z -> inftree t n v i.
Proof.
intros t n v H0 i H.
generalize H.
pattern i.
apply heap_induction.
auto.
intros.
elim (Z_le_gt_dec k n).
intro.
generalize H1 a.
case H2; intros.
intuition.
split.
intro; apply H5; omega.
intro; apply H6; omega.
intro.
split; intro; absurd (k > n)%Z; omega.
intuition.
Qed.
Lemma inftree_0_right :
forall (t:array Z) (n v:Z),
inftree t n v 0 ->
forall i:Z, (0 <= i <= n)%Z -> (access t i <= v)%Z.
Proof.
intros t n v H.
generalize (inftree_all H).
intros.
apply inftree_1 with (n := n).
exact (H0 i H1).
Qed.
Lemma inftree_0_left :
forall (t:array Z) (n v:Z),
(0 <= n)%Z ->
(forall i:Z, (0 <= i <= n)%Z -> (access t i <= v)%Z) ->
inftree t n v 0.
Proof.
intros.
cut (forall i:Z, (0 <= i <= n)%Z -> inftree t n v i).
intro.
apply H1; omega.
intros i Hi.
generalize Hi.
replace i with (n - (n - i))%Z.
pattern (n - i)%Z.
apply Z_lt_induction.
intros.
apply inftree_cons.
assumption.
apply H0; omega.
intro.
replace (2 * (n - x) + 1)%Z with (n - (n - (2 * (n - x) + 1)))%Z.
apply H1; omega.
omega.
intro.
replace (2 * (n - x) + 2)%Z with (n - (n - (2 * (n - x) + 2)))%Z.
apply H1; omega.
omega.
omega.
omega.
Qed.
Lemma inftree_exchange :
forall (t1 t2:array Z) (n v:Z),
(n < array_length t1)%Z ->
inftree t1 n v 0 -> exchange t2 t1 0 n -> inftree t2 n v 0.
Proof.
intros.
apply inftree_0_left.
decompose [exchange] H1.
omega.
generalize (inftree_0_right H0).
intros.
decompose [exchange] H1.
elim (Z_lt_ge_dec 0 i); intro.
elim (Z_lt_ge_dec i n); intro.
rewrite (H9 i).
apply H2; omega.
omega.
omega.
omega.
replace i with n.
rewrite H8.
apply H2; omega.
omega.
replace i with 0%Z.
rewrite H7.
apply H2; omega.
omega.
Qed.
Lemma inftree_weakening :
forall (t:array Z) (n v k:Z),
(1 <= n < array_length t)%Z ->
inftree t n v k -> (k <= n - 1)%Z -> inftree t (n - 1) v k.
Proof.
intros t n v k Hn Htree.
elim Htree; intros.
apply inftree_cons.
omega.
assumption.
intro; apply H2; omega.
intro; apply H4; omega.
Qed.
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