diff options
Diffstat (limited to 'theories/ZArith/Zsqrt_compat.v')
| -rw-r--r-- | theories/ZArith/Zsqrt_compat.v | 63 |
1 files changed, 31 insertions, 32 deletions
diff --git a/theories/ZArith/Zsqrt_compat.v b/theories/ZArith/Zsqrt_compat.v index 4584c3f8..a6c83241 100644 --- a/theories/ZArith/Zsqrt_compat.v +++ b/theories/ZArith/Zsqrt_compat.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -9,7 +9,7 @@ Require Import ZArithRing. Require Import Omega. Require Export ZArith_base. -Open Local Scope Z_scope. +Local Open Scope Z_scope. (** THIS FILE IS DEPRECATED @@ -32,12 +32,12 @@ Ltac compute_POS := | |- context [(Zpos (xI ?X1))] => match constr:X1 with | context [1%positive] => fail 1 - | _ => rewrite (BinInt.Zpos_xI X1) + | _ => rewrite (Pos2Z.inj_xI X1) end | |- context [(Zpos (xO ?X1))] => match constr:X1 with | context [1%positive] => fail 1 - | _ => rewrite (BinInt.Zpos_xO X1) + | _ => rewrite (Pos2Z.inj_xO X1) end end. @@ -115,7 +115,7 @@ Definition Zsqrt : fun h => match sqrtrempos p with | c_sqrt s r Heq Hint => - existS + existT (fun s:Z => {r : Z | Zpos p = s * s + r /\ s * s <= Zpos p < (s + 1) * (s + 1)}) @@ -131,10 +131,10 @@ Definition Zsqrt : {s : Z & {r : Z | Zneg p = s * s + r /\ s * s <= Zneg p < (s + 1) * (s + 1)}} - (h (refl_equal Datatypes.Gt)) + (h (eq_refl Datatypes.Gt)) | Z0 => fun h => - existS + existT (fun s:Z => {r : Z | 0 = s * s + r /\ s * s <= 0 < (s + 1) * (s + 1)}) 0 (exist @@ -149,8 +149,8 @@ Defined. Definition Zsqrt_plain (x:Z) : Z := match x with | Zpos p => - match Zsqrt (Zpos p) (Zorder.Zle_0_pos p) with - | existS s _ => s + match Zsqrt (Zpos p) (Pos2Z.is_nonneg p) with + | existT s _ => s end | Zneg p => 0 | Z0 => 0 @@ -164,12 +164,11 @@ Theorem Zsqrt_interval : Zsqrt_plain n * Zsqrt_plain n <= n < (Zsqrt_plain n + 1) * (Zsqrt_plain n + 1). Proof. - intros x; case x. - unfold Zsqrt_plain in |- *; omega. - intros p; unfold Zsqrt_plain in |- *; - case (Zsqrt (Zpos p) (Zorder.Zle_0_pos p)). - intros s [r [Heq Hint]] Hle; assumption. - intros p Hle; elim Hle; auto. + intros [|p|p] Hp. + - now compute. + - unfold Zsqrt_plain. + now destruct Zsqrt as (s & r & Heq & Hint). + - now elim Hp. Qed. (** Positivity *) @@ -177,9 +176,9 @@ Qed. Theorem Zsqrt_plain_is_pos: forall n, 0 <= n -> 0 <= Zsqrt_plain n. Proof. intros n m; case (Zsqrt_interval n); auto with zarith. - intros H1 H2; case (Zle_or_lt 0 (Zsqrt_plain n)); auto. - intros H3; contradict H2; auto; apply Zle_not_lt. - apply Zle_trans with ( 2 := H1 ). + intros H1 H2; case (Z.le_gt_cases 0 (Zsqrt_plain n)); auto. + intros H3; contradict H2; auto; apply Z.le_ngt. + apply Z.le_trans with ( 2 := H1 ). replace ((Zsqrt_plain n + 1) * (Zsqrt_plain n + 1)) with (Zsqrt_plain n * Zsqrt_plain n + (2 * Zsqrt_plain n + 1)); auto with zarith. @@ -194,13 +193,13 @@ Proof. generalize (Zsqrt_plain_is_pos (a * a)); auto with zarith; intros Haa. case (Zsqrt_interval (a * a)); auto with zarith. intros H1 H2. - case (Zle_or_lt a (Zsqrt_plain (a * a))); intros H3; auto. - case Zle_lt_or_eq with (1:=H3); auto; clear H3; intros H3. - contradict H1; auto; apply Zlt_not_le; auto with zarith. - apply Zle_lt_trans with (a * Zsqrt_plain (a * a)); auto with zarith. - apply Zmult_lt_compat_r; auto with zarith. - contradict H2; auto; apply Zle_not_lt; auto with zarith. - apply Zmult_le_compat; auto with zarith. + case (Z.le_gt_cases a (Zsqrt_plain (a * a))); intros H3. + - Z.le_elim H3; auto. + contradict H1; auto; apply Z.lt_nge; auto with zarith. + apply Z.le_lt_trans with (a * Zsqrt_plain (a * a)); auto with zarith. + apply Z.mul_lt_mono_pos_r; auto with zarith. + - contradict H2; auto; apply Z.le_ngt; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. Qed. (** [Zsqrt_plain] is increasing *) @@ -208,16 +207,16 @@ Qed. Theorem Zsqrt_le: forall p q, 0 <= p <= q -> Zsqrt_plain p <= Zsqrt_plain q. Proof. - intros p q [H1 H2]; case Zle_lt_or_eq with (1:=H2); clear H2; intros H2; - [ | subst q; auto with zarith]. - case (Zle_or_lt (Zsqrt_plain p) (Zsqrt_plain q)); auto; intros H3. + intros p q [H1 H2]. + Z.le_elim H2; [ | subst q; auto with zarith]. + case (Z.le_gt_cases (Zsqrt_plain p) (Zsqrt_plain q)); auto; intros H3. assert (Hp: (0 <= Zsqrt_plain q)). - apply Zsqrt_plain_is_pos; auto with zarith. + { apply Zsqrt_plain_is_pos; auto with zarith. } absurd (q <= p); auto with zarith. - apply Zle_trans with ((Zsqrt_plain q + 1) * (Zsqrt_plain q + 1)). + apply Z.le_trans with ((Zsqrt_plain q + 1) * (Zsqrt_plain q + 1)). case (Zsqrt_interval q); auto with zarith. - apply Zle_trans with (Zsqrt_plain p * Zsqrt_plain p); auto with zarith. - apply Zmult_le_compat; auto with zarith. + apply Z.le_trans with (Zsqrt_plain p * Zsqrt_plain p); auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. case (Zsqrt_interval p); auto with zarith. Qed. |