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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Library about sorted (sub-)arrays / Nicolas Magaud, July 1998 *)
(* $Id: Sorted.v,v 1.7.2.1 2004/07/16 19:30:00 herbelin Exp $ *)
Require Export Arrays.
Require Import ArrayPermut.
Require Import ZArithRing.
Require Import Omega.
Open Local Scope Z_scope.
Set Implicit Arguments.
(* Definition *)
Definition sorted_array (N:Z) (A:array N Z) (deb fin:Z) :=
deb <= fin -> forall x:Z, x >= deb -> x < fin -> #A [x] <= #A [x + 1].
(* Elements of a sorted sub-array are in increasing order *)
(* one element and the next one *)
Lemma sorted_elements_1 :
forall (N:Z) (A:array N Z) (n m:Z),
sorted_array A n m ->
forall k:Z,
k >= n -> forall i:Z, 0 <= i -> k + i <= m -> #A [k] <= #A [k + i].
Proof.
intros N A n m H_sorted k H_k i H_i.
pattern i in |- *. apply natlike_ind.
intro.
replace (k + 0) with k; omega. (*** Ring `k+0` => BUG ***)
intros.
apply Zle_trans with (m := #A [k + x]).
apply H0; omega.
unfold Zsucc in |- *.
replace (k + (x + 1)) with (k + x + 1).
unfold sorted_array in H_sorted.
apply H_sorted; omega.
omega.
assumption.
Qed.
(* one element and any of the following *)
Lemma sorted_elements :
forall (N:Z) (A:array N Z) (n m k l:Z),
sorted_array A n m ->
k >= n -> l < N -> k <= l -> l <= m -> #A [k] <= #A [l].
Proof.
intros.
replace l with (k + (l - k)).
apply sorted_elements_1 with (n := n) (m := m);
[ assumption | omega | omega | omega ].
omega.
Qed.
Hint Resolve sorted_elements: datatypes v62.
(* A sub-array of a sorted array is sorted *)
Lemma sub_sorted_array :
forall (N:Z) (A:array N Z) (deb fin i j:Z),
sorted_array A deb fin ->
i >= deb -> j <= fin -> i <= j -> sorted_array A i j.
Proof.
unfold sorted_array in |- *.
intros.
apply H; omega.
Qed.
Hint Resolve sub_sorted_array: datatypes v62.
(* Extension on the left of the property of being sorted *)
Lemma left_extension :
forall (N:Z) (A:array N Z) (i j:Z),
i > 0 ->
j < N ->
sorted_array A i j -> #A [i - 1] <= #A [i] -> sorted_array A (i - 1) j.
Proof.
intros; unfold sorted_array in |- *; intros.
elim (Z_ge_lt_dec x i). (* (`x >= i`) + (`x < i`) *)
intro Hcut.
apply H1; omega.
intro Hcut.
replace x with (i - 1).
replace (i - 1 + 1) with i; [ assumption | omega ].
omega.
Qed.
(* Extension on the right *)
Lemma right_extension :
forall (N:Z) (A:array N Z) (i j:Z),
i >= 0 ->
j < N - 1 ->
sorted_array A i j -> #A [j] <= #A [j + 1] -> sorted_array A i (j + 1).
Proof.
intros; unfold sorted_array in |- *; intros.
elim (Z_lt_ge_dec x j).
intro Hcut.
apply H1; omega.
intro HCut.
replace x with j; [ assumption | omega ].
Qed.
(* Substitution of the leftmost value by a smaller value *)
Lemma left_substitution :
forall (N:Z) (A:array N Z) (i j v:Z),
i >= 0 ->
j < N ->
sorted_array A i j -> v <= #A [i] -> sorted_array (store A i v) i j.
Proof.
intros N A i j v H_i H_j H_sorted H_v.
unfold sorted_array in |- *; intros.
cut (x = i \/ x > i).
intro Hcut; elim Hcut; clear Hcut; intro.
rewrite H2.
rewrite store_def_1; try omega.
rewrite store_def_2; try omega.
apply Zle_trans with (m := #A [i]); [ assumption | apply H_sorted; omega ].
rewrite store_def_2; try omega.
rewrite store_def_2; try omega.
apply H_sorted; omega.
omega.
Qed.
(* Substitution of the rightmost value by a larger value *)
Lemma right_substitution :
forall (N:Z) (A:array N Z) (i j v:Z),
i >= 0 ->
j < N ->
sorted_array A i j -> #A [j] <= v -> sorted_array (store A j v) i j.
Proof.
intros N A i j v H_i H_j H_sorted H_v.
unfold sorted_array in |- *; intros.
cut (x = j - 1 \/ x < j - 1).
intro Hcut; elim Hcut; clear Hcut; intro.
rewrite H2.
replace (j - 1 + 1) with j; [ idtac | omega ]. (*** Ring `j-1+1`. => BUG ***)
rewrite store_def_2; try omega.
rewrite store_def_1; try omega.
apply Zle_trans with (m := #A [j]).
apply sorted_elements with (n := i) (m := j); try omega; assumption.
assumption.
rewrite store_def_2; try omega.
rewrite store_def_2; try omega.
apply H_sorted; omega.
omega.
Qed.
(* Affectation outside of the sorted region *)
Lemma no_effect :
forall (N:Z) (A:array N Z) (i j k v:Z),
i >= 0 ->
j < N ->
sorted_array A i j ->
0 <= k < i \/ j < k < N -> sorted_array (store A k v) i j.
Proof.
intros.
unfold sorted_array in |- *; intros.
rewrite store_def_2; try omega.
rewrite store_def_2; try omega.
apply H1; assumption.
Qed.
Lemma sorted_array_id :
forall (N:Z) (t1 t2:array N Z) (g d:Z),
sorted_array t1 g d -> array_id t1 t2 g d -> sorted_array t2 g d.
Proof.
intros N t1 t2 g d Hsorted Hid.
unfold array_id in Hid.
unfold sorted_array in Hsorted. unfold sorted_array in |- *.
intros Hgd x H1x H2x.
rewrite <- (Hid x); [ idtac | omega ].
rewrite <- (Hid (x + 1)); [ idtac | omega ].
apply Hsorted; assumption.
Qed.
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