diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZAddOrder.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZAddOrder.v | 123 |
1 files changed, 54 insertions, 69 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v index 6ce54f88..423cdf58 100644 --- a/theories/Numbers/Integer/Abstract/ZAddOrder.v +++ b/theories/Numbers/Integer/Abstract/ZAddOrder.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -8,180 +8,173 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: ZAddOrder.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export ZLt. -Module ZAddOrderPropFunct (Import Z : ZAxiomsSig'). -Include ZOrderPropFunct Z. +Module ZAddOrderProp (Import Z : ZAxiomsMiniSig'). +Include ZOrderProp Z. (** Theorems that are either not valid on N or have different proofs on N and Z *) Theorem add_neg_neg : forall n m, n < 0 -> m < 0 -> n + m < 0. Proof. -intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_lt_mono. +intros. rewrite <- (add_0_l 0). now apply add_lt_mono. Qed. Theorem add_neg_nonpos : forall n m, n < 0 -> m <= 0 -> n + m < 0. Proof. -intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_lt_le_mono. +intros. rewrite <- (add_0_l 0). now apply add_lt_le_mono. Qed. Theorem add_nonpos_neg : forall n m, n <= 0 -> m < 0 -> n + m < 0. Proof. -intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_le_lt_mono. +intros. rewrite <- (add_0_l 0). now apply add_le_lt_mono. Qed. Theorem add_nonpos_nonpos : forall n m, n <= 0 -> m <= 0 -> n + m <= 0. Proof. -intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_le_mono. +intros. rewrite <- (add_0_l 0). now apply add_le_mono. Qed. (** Sub and order *) Theorem lt_0_sub : forall n m, 0 < m - n <-> n < m. Proof. -intros n m. stepl (0 + n < m - n + n) by symmetry; apply add_lt_mono_r. -rewrite add_0_l; now rewrite sub_simpl_r. +intros n m. now rewrite (add_lt_mono_r _ _ n), add_0_l, sub_simpl_r. Qed. Notation sub_pos := lt_0_sub (only parsing). Theorem le_0_sub : forall n m, 0 <= m - n <-> n <= m. Proof. -intros n m; stepl (0 + n <= m - n + n) by symmetry; apply add_le_mono_r. -rewrite add_0_l; now rewrite sub_simpl_r. +intros n m. now rewrite (add_le_mono_r _ _ n), add_0_l, sub_simpl_r. Qed. Notation sub_nonneg := le_0_sub (only parsing). Theorem lt_sub_0 : forall n m, n - m < 0 <-> n < m. Proof. -intros n m. stepl (n - m + m < 0 + m) by symmetry; apply add_lt_mono_r. -rewrite add_0_l; now rewrite sub_simpl_r. +intros n m. now rewrite (add_lt_mono_r _ _ m), add_0_l, sub_simpl_r. Qed. Notation sub_neg := lt_sub_0 (only parsing). Theorem le_sub_0 : forall n m, n - m <= 0 <-> n <= m. Proof. -intros n m. stepl (n - m + m <= 0 + m) by symmetry; apply add_le_mono_r. -rewrite add_0_l; now rewrite sub_simpl_r. +intros n m. now rewrite (add_le_mono_r _ _ m), add_0_l, sub_simpl_r. Qed. Notation sub_nonpos := le_sub_0 (only parsing). Theorem opp_lt_mono : forall n m, n < m <-> - m < - n. Proof. -intros n m. stepr (m + - m < m + - n) by symmetry; apply add_lt_mono_l. -do 2 rewrite add_opp_r. rewrite sub_diag. symmetry; apply lt_0_sub. +intros n m. now rewrite <- lt_0_sub, <- add_opp_l, <- sub_opp_r, lt_0_sub. Qed. Theorem opp_le_mono : forall n m, n <= m <-> - m <= - n. Proof. -intros n m. stepr (m + - m <= m + - n) by symmetry; apply add_le_mono_l. -do 2 rewrite add_opp_r. rewrite sub_diag. symmetry; apply le_0_sub. +intros n m. now rewrite <- le_0_sub, <- add_opp_l, <- sub_opp_r, le_0_sub. Qed. Theorem opp_pos_neg : forall n, 0 < - n <-> n < 0. Proof. -intro n; rewrite (opp_lt_mono n 0); now rewrite opp_0. +intro n; now rewrite (opp_lt_mono n 0), opp_0. Qed. Theorem opp_neg_pos : forall n, - n < 0 <-> 0 < n. Proof. -intro n. rewrite (opp_lt_mono 0 n). now rewrite opp_0. +intro n. now rewrite (opp_lt_mono 0 n), opp_0. Qed. Theorem opp_nonneg_nonpos : forall n, 0 <= - n <-> n <= 0. Proof. -intro n; rewrite (opp_le_mono n 0); now rewrite opp_0. +intro n; now rewrite (opp_le_mono n 0), opp_0. Qed. Theorem opp_nonpos_nonneg : forall n, - n <= 0 <-> 0 <= n. Proof. -intro n. rewrite (opp_le_mono 0 n). now rewrite opp_0. +intro n. now rewrite (opp_le_mono 0 n), opp_0. +Qed. + +Theorem lt_m1_0 : -1 < 0. +Proof. +apply opp_neg_pos, lt_0_1. Qed. Theorem sub_lt_mono_l : forall n m p, n < m <-> p - m < p - n. Proof. -intros n m p. do 2 rewrite <- add_opp_r. rewrite <- add_lt_mono_l. -apply opp_lt_mono. +intros. now rewrite <- 2 add_opp_r, <- add_lt_mono_l, opp_lt_mono. Qed. Theorem sub_lt_mono_r : forall n m p, n < m <-> n - p < m - p. Proof. -intros n m p; do 2 rewrite <- add_opp_r; apply add_lt_mono_r. +intros. now rewrite <- 2 add_opp_r, add_lt_mono_r. Qed. Theorem sub_lt_mono : forall n m p q, n < m -> q < p -> n - p < m - q. Proof. intros n m p q H1 H2. apply lt_trans with (m - p); -[now apply -> sub_lt_mono_r | now apply -> sub_lt_mono_l]. +[now apply sub_lt_mono_r | now apply sub_lt_mono_l]. Qed. Theorem sub_le_mono_l : forall n m p, n <= m <-> p - m <= p - n. Proof. -intros n m p; do 2 rewrite <- add_opp_r; rewrite <- add_le_mono_l; -apply opp_le_mono. +intros. now rewrite <- 2 add_opp_r, <- add_le_mono_l, opp_le_mono. Qed. Theorem sub_le_mono_r : forall n m p, n <= m <-> n - p <= m - p. Proof. -intros n m p; do 2 rewrite <- add_opp_r; apply add_le_mono_r. +intros. now rewrite <- 2 add_opp_r, add_le_mono_r. Qed. Theorem sub_le_mono : forall n m p q, n <= m -> q <= p -> n - p <= m - q. Proof. intros n m p q H1 H2. apply le_trans with (m - p); -[now apply -> sub_le_mono_r | now apply -> sub_le_mono_l]. +[now apply sub_le_mono_r | now apply sub_le_mono_l]. Qed. Theorem sub_lt_le_mono : forall n m p q, n < m -> q <= p -> n - p < m - q. Proof. intros n m p q H1 H2. apply lt_le_trans with (m - p); -[now apply -> sub_lt_mono_r | now apply -> sub_le_mono_l]. +[now apply sub_lt_mono_r | now apply sub_le_mono_l]. Qed. Theorem sub_le_lt_mono : forall n m p q, n <= m -> q < p -> n - p < m - q. Proof. intros n m p q H1 H2. apply le_lt_trans with (m - p); -[now apply -> sub_le_mono_r | now apply -> sub_lt_mono_l]. +[now apply sub_le_mono_r | now apply sub_lt_mono_l]. Qed. Theorem le_lt_sub_lt : forall n m p q, n <= m -> p - n < q - m -> p < q. Proof. intros n m p q H1 H2. apply (le_lt_add_lt (- m) (- n)); -[now apply -> opp_le_mono | now do 2 rewrite add_opp_r]. +[now apply -> opp_le_mono | now rewrite 2 add_opp_r]. Qed. Theorem lt_le_sub_lt : forall n m p q, n < m -> p - n <= q - m -> p < q. Proof. intros n m p q H1 H2. apply (lt_le_add_lt (- m) (- n)); -[now apply -> opp_lt_mono | now do 2 rewrite add_opp_r]. +[now apply -> opp_lt_mono | now rewrite 2 add_opp_r]. Qed. Theorem le_le_sub_lt : forall n m p q, n <= m -> p - n <= q - m -> p <= q. Proof. intros n m p q H1 H2. apply (le_le_add_le (- m) (- n)); -[now apply -> opp_le_mono | now do 2 rewrite add_opp_r]. +[now apply -> opp_le_mono | now rewrite 2 add_opp_r]. Qed. Theorem lt_add_lt_sub_r : forall n m p, n + p < m <-> n < m - p. Proof. -intros n m p. stepl (n + p - p < m - p) by symmetry; apply sub_lt_mono_r. -now rewrite add_simpl_r. +intros n m p. now rewrite (sub_lt_mono_r _ _ p), add_simpl_r. Qed. Theorem le_add_le_sub_r : forall n m p, n + p <= m <-> n <= m - p. Proof. -intros n m p. stepl (n + p - p <= m - p) by symmetry; apply sub_le_mono_r. -now rewrite add_simpl_r. +intros n m p. now rewrite (sub_le_mono_r _ _ p), add_simpl_r. Qed. Theorem lt_add_lt_sub_l : forall n m p, n + p < m <-> p < m - n. @@ -196,14 +189,12 @@ Qed. Theorem lt_sub_lt_add_r : forall n m p, n - p < m <-> n < m + p. Proof. -intros n m p. stepl (n - p + p < m + p) by symmetry; apply add_lt_mono_r. -now rewrite sub_simpl_r. +intros n m p. now rewrite (add_lt_mono_r _ _ p), sub_simpl_r. Qed. Theorem le_sub_le_add_r : forall n m p, n - p <= m <-> n <= m + p. Proof. -intros n m p. stepl (n - p + p <= m + p) by symmetry; apply add_le_mono_r. -now rewrite sub_simpl_r. +intros n m p. now rewrite (add_le_mono_r _ _ p), sub_simpl_r. Qed. Theorem lt_sub_lt_add_l : forall n m p, n - m < p <-> n < m + p. @@ -218,74 +209,68 @@ Qed. Theorem lt_sub_lt_add : forall n m p q, n - m < p - q <-> n + q < m + p. Proof. -intros n m p q. rewrite lt_sub_lt_add_l. rewrite add_sub_assoc. -now rewrite <- lt_add_lt_sub_r. +intros n m p q. now rewrite lt_sub_lt_add_l, add_sub_assoc, <- lt_add_lt_sub_r. Qed. Theorem le_sub_le_add : forall n m p q, n - m <= p - q <-> n + q <= m + p. Proof. -intros n m p q. rewrite le_sub_le_add_l. rewrite add_sub_assoc. -now rewrite <- le_add_le_sub_r. +intros n m p q. now rewrite le_sub_le_add_l, add_sub_assoc, <- le_add_le_sub_r. Qed. Theorem lt_sub_pos : forall n m, 0 < m <-> n - m < n. Proof. -intros n m. stepr (n - m < n - 0) by now rewrite sub_0_r. apply sub_lt_mono_l. +intros n m. now rewrite (sub_lt_mono_l _ _ n), sub_0_r. Qed. Theorem le_sub_nonneg : forall n m, 0 <= m <-> n - m <= n. Proof. -intros n m. stepr (n - m <= n - 0) by now rewrite sub_0_r. apply sub_le_mono_l. +intros n m. now rewrite (sub_le_mono_l _ _ n), sub_0_r. Qed. Theorem sub_lt_cases : forall n m p q, n - m < p - q -> n < m \/ q < p. Proof. -intros n m p q H. rewrite lt_sub_lt_add in H. now apply add_lt_cases. +intros. now apply add_lt_cases, lt_sub_lt_add. Qed. Theorem sub_le_cases : forall n m p q, n - m <= p - q -> n <= m \/ q <= p. Proof. -intros n m p q H. rewrite le_sub_le_add in H. now apply add_le_cases. +intros. now apply add_le_cases, le_sub_le_add. Qed. Theorem sub_neg_cases : forall n m, n - m < 0 -> n < 0 \/ 0 < m. Proof. -intros n m H; rewrite <- add_opp_r in H. -setoid_replace (0 < m) with (- m < 0) using relation iff by (symmetry; apply opp_neg_pos). -now apply add_neg_cases. +intros. +rewrite <- (opp_neg_pos m). apply add_neg_cases. now rewrite add_opp_r. Qed. Theorem sub_pos_cases : forall n m, 0 < n - m -> 0 < n \/ m < 0. Proof. -intros n m H; rewrite <- add_opp_r in H. -setoid_replace (m < 0) with (0 < - m) using relation iff by (symmetry; apply opp_pos_neg). -now apply add_pos_cases. +intros. +rewrite <- (opp_pos_neg m). apply add_pos_cases. now rewrite add_opp_r. Qed. Theorem sub_nonpos_cases : forall n m, n - m <= 0 -> n <= 0 \/ 0 <= m. Proof. -intros n m H; rewrite <- add_opp_r in H. -setoid_replace (0 <= m) with (- m <= 0) using relation iff by (symmetry; apply opp_nonpos_nonneg). -now apply add_nonpos_cases. +intros. +rewrite <- (opp_nonpos_nonneg m). apply add_nonpos_cases. now rewrite add_opp_r. Qed. Theorem sub_nonneg_cases : forall n m, 0 <= n - m -> 0 <= n \/ m <= 0. Proof. -intros n m H; rewrite <- add_opp_r in H. -setoid_replace (m <= 0) with (0 <= - m) using relation iff by (symmetry; apply opp_nonneg_nonpos). -now apply add_nonneg_cases. +intros. +rewrite <- (opp_nonneg_nonpos m). apply add_nonneg_cases. now rewrite add_opp_r. Qed. Section PosNeg. Variable P : Z.t -> Prop. -Hypothesis P_wd : Proper (Z.eq ==> iff) P. +Hypothesis P_wd : Proper (eq ==> iff) P. Theorem zero_pos_neg : P 0 -> (forall n, 0 < n -> P n /\ P (- n)) -> forall n, P n. Proof. intros H1 H2 n. destruct (lt_trichotomy n 0) as [H3 | [H3 | H3]]. -apply <- opp_pos_neg in H3. apply H2 in H3. destruct H3 as [_ H3]. +apply opp_pos_neg, H2 in H3. destruct H3 as [_ H3]. now rewrite opp_involutive in H3. now rewrite H3. apply H2 in H3; now destruct H3. @@ -295,6 +280,6 @@ End PosNeg. Ltac zero_pos_neg n := induction_maker n ltac:(apply zero_pos_neg). -End ZAddOrderPropFunct. +End ZAddOrderProp. |