diff options
Diffstat (limited to 'theories/ZArith/Zorder.v')
| -rw-r--r-- | theories/ZArith/Zorder.v | 102 |
1 files changed, 52 insertions, 50 deletions
diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v index 73dee0cf..a1ec4b35 100644 --- a/theories/ZArith/Zorder.v +++ b/theories/ZArith/Zorder.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (** Binary Integers : results about order predicates *) @@ -38,9 +40,9 @@ Qed. (**********************************************************************) (** * Decidability of equality and order on Z *) -Notation dec_eq := Z.eq_decidable (compat "8.3"). -Notation dec_Zle := Z.le_decidable (compat "8.3"). -Notation dec_Zlt := Z.lt_decidable (compat "8.3"). +Notation dec_eq := Z.eq_decidable (only parsing). +Notation dec_Zle := Z.le_decidable (only parsing). +Notation dec_Zlt := Z.lt_decidable (only parsing). Theorem dec_Zne n m : decidable (Zne n m). Proof. @@ -64,12 +66,12 @@ Qed. (** * Relating strict and large orders *) -Notation Zgt_lt := Z.gt_lt (compat "8.3"). -Notation Zlt_gt := Z.lt_gt (compat "8.3"). -Notation Zge_le := Z.ge_le (compat "8.3"). -Notation Zle_ge := Z.le_ge (compat "8.3"). -Notation Zgt_iff_lt := Z.gt_lt_iff (compat "8.3"). -Notation Zge_iff_le := Z.ge_le_iff (compat "8.3"). +Notation Zgt_lt := Z.gt_lt (compat "8.6"). +Notation Zlt_gt := Z.lt_gt (compat "8.6"). +Notation Zge_le := Z.ge_le (compat "8.6"). +Notation Zle_ge := Z.le_ge (compat "8.6"). +Notation Zgt_iff_lt := Z.gt_lt_iff (only parsing). +Notation Zge_iff_le := Z.ge_le_iff (only parsing). Lemma Zle_not_lt n m : n <= m -> ~ m < n. Proof. @@ -121,18 +123,18 @@ Qed. (** Reflexivity *) -Notation Zle_refl := Z.le_refl (compat "8.3"). -Notation Zeq_le := Z.eq_le_incl (compat "8.3"). +Notation Zle_refl := Z.le_refl (compat "8.6"). +Notation Zeq_le := Z.eq_le_incl (only parsing). Hint Resolve Z.le_refl: zarith. (** Antisymmetry *) -Notation Zle_antisym := Z.le_antisymm (compat "8.3"). +Notation Zle_antisym := Z.le_antisymm (only parsing). (** Asymmetry *) -Notation Zlt_asym := Z.lt_asymm (compat "8.3"). +Notation Zlt_asym := Z.lt_asymm (only parsing). Lemma Zgt_asym n m : n > m -> ~ m > n. Proof. @@ -141,8 +143,8 @@ Qed. (** Irreflexivity *) -Notation Zlt_irrefl := Z.lt_irrefl (compat "8.3"). -Notation Zlt_not_eq := Z.lt_neq (compat "8.3"). +Notation Zlt_irrefl := Z.lt_irrefl (compat "8.6"). +Notation Zlt_not_eq := Z.lt_neq (only parsing). Lemma Zgt_irrefl n : ~ n > n. Proof. @@ -151,8 +153,8 @@ Qed. (** Large = strict or equal *) -Notation Zlt_le_weak := Z.lt_le_incl (compat "8.3"). -Notation Zle_lt_or_eq_iff := Z.lt_eq_cases (compat "8.3"). +Notation Zlt_le_weak := Z.lt_le_incl (only parsing). +Notation Zle_lt_or_eq_iff := Z.lt_eq_cases (only parsing). Lemma Zle_lt_or_eq n m : n <= m -> n < m \/ n = m. Proof. @@ -161,11 +163,11 @@ Qed. (** Dichotomy *) -Notation Zle_or_lt := Z.le_gt_cases (compat "8.3"). +Notation Zle_or_lt := Z.le_gt_cases (only parsing). (** Transitivity of strict orders *) -Notation Zlt_trans := Z.lt_trans (compat "8.3"). +Notation Zlt_trans := Z.lt_trans (compat "8.6"). Lemma Zgt_trans n m p : n > m -> m > p -> n > p. Proof. @@ -174,8 +176,8 @@ Qed. (** Mixed transitivity *) -Notation Zlt_le_trans := Z.lt_le_trans (compat "8.3"). -Notation Zle_lt_trans := Z.le_lt_trans (compat "8.3"). +Notation Zlt_le_trans := Z.lt_le_trans (compat "8.6"). +Notation Zle_lt_trans := Z.le_lt_trans (compat "8.6"). Lemma Zle_gt_trans n m p : m <= n -> m > p -> n > p. Proof. @@ -189,7 +191,7 @@ Qed. (** Transitivity of large orders *) -Notation Zle_trans := Z.le_trans (compat "8.3"). +Notation Zle_trans := Z.le_trans (compat "8.6"). Lemma Zge_trans n m p : n >= m -> m >= p -> n >= p. Proof. @@ -240,8 +242,8 @@ Qed. (** Special base instances of order *) -Notation Zlt_succ := Z.lt_succ_diag_r (compat "8.3"). -Notation Zlt_pred := Z.lt_pred_l (compat "8.3"). +Notation Zlt_succ := Z.lt_succ_diag_r (only parsing). +Notation Zlt_pred := Z.lt_pred_l (only parsing). Lemma Zgt_succ n : Z.succ n > n. Proof. @@ -255,8 +257,8 @@ Qed. (** Relating strict and large order using successor or predecessor *) -Notation Zlt_succ_r := Z.lt_succ_r (compat "8.3"). -Notation Zle_succ_l := Z.le_succ_l (compat "8.3"). +Notation Zlt_succ_r := Z.lt_succ_r (compat "8.6"). +Notation Zle_succ_l := Z.le_succ_l (compat "8.6"). Lemma Zgt_le_succ n m : m > n -> Z.succ n <= m. Proof. @@ -295,10 +297,10 @@ Qed. (** Weakening order *) -Notation Zle_succ := Z.le_succ_diag_r (compat "8.3"). -Notation Zle_pred := Z.le_pred_l (compat "8.3"). -Notation Zlt_lt_succ := Z.lt_lt_succ_r (compat "8.3"). -Notation Zle_le_succ := Z.le_le_succ_r (compat "8.3"). +Notation Zle_succ := Z.le_succ_diag_r (only parsing). +Notation Zle_pred := Z.le_pred_l (only parsing). +Notation Zlt_lt_succ := Z.lt_lt_succ_r (only parsing). +Notation Zle_le_succ := Z.le_le_succ_r (only parsing). Lemma Zle_succ_le n m : Z.succ n <= m -> n <= m. Proof. @@ -334,12 +336,12 @@ Qed. (** Special cases of ordered integers *) -Notation Zlt_0_1 := Z.lt_0_1 (compat "8.3"). -Notation Zle_0_1 := Z.le_0_1 (compat "8.3"). +Notation Zlt_0_1 := Z.lt_0_1 (compat "8.6"). +Notation Zle_0_1 := Z.le_0_1 (compat "8.6"). Lemma Zle_neg_pos : forall p q:positive, Zneg p <= Zpos q. Proof. - easy. + exact Pos2Z.neg_le_pos. Qed. Lemma Zgt_pos_0 : forall p:positive, Zpos p > 0. @@ -350,12 +352,12 @@ Qed. (* weaker but useful (in [Z.pow] for instance) *) Lemma Zle_0_pos : forall p:positive, 0 <= Zpos p. Proof. - easy. + exact Pos2Z.pos_is_nonneg. Qed. Lemma Zlt_neg_0 : forall p:positive, Zneg p < 0. Proof. - easy. + exact Pos2Z.neg_is_neg. Qed. Lemma Zle_0_nat : forall n:nat, 0 <= Z.of_nat n. @@ -375,10 +377,10 @@ Qed. (** ** Addition *) (** Compatibility of addition wrt to order *) -Notation Zplus_lt_le_compat := Z.add_lt_le_mono (compat "8.3"). -Notation Zplus_le_lt_compat := Z.add_le_lt_mono (compat "8.3"). -Notation Zplus_le_compat := Z.add_le_mono (compat "8.3"). -Notation Zplus_lt_compat := Z.add_lt_mono (compat "8.3"). +Notation Zplus_lt_le_compat := Z.add_lt_le_mono (only parsing). +Notation Zplus_le_lt_compat := Z.add_le_lt_mono (only parsing). +Notation Zplus_le_compat := Z.add_le_mono (only parsing). +Notation Zplus_lt_compat := Z.add_lt_mono (only parsing). Lemma Zplus_gt_compat_l n m p : n > m -> p + n > p + m. Proof. @@ -412,7 +414,7 @@ Qed. (** Compatibility of addition wrt to being positive *) -Notation Zplus_le_0_compat := Z.add_nonneg_nonneg (compat "8.3"). +Notation Zplus_le_0_compat := Z.add_nonneg_nonneg (only parsing). (** Simplification of addition wrt to order *) @@ -570,9 +572,9 @@ Qed. (** Compatibility of multiplication by a positive wrt to being positive *) -Notation Zmult_le_0_compat := Z.mul_nonneg_nonneg (compat "8.3"). -Notation Zmult_lt_0_compat := Z.mul_pos_pos (compat "8.3"). -Notation Zmult_lt_O_compat := Z.mul_pos_pos (compat "8.3"). +Notation Zmult_le_0_compat := Z.mul_nonneg_nonneg (only parsing). +Notation Zmult_lt_0_compat := Z.mul_pos_pos (only parsing). +Notation Zmult_lt_O_compat := Z.mul_pos_pos (only parsing). Lemma Zmult_gt_0_compat n m : n > 0 -> m > 0 -> n * m > 0. Proof. @@ -624,9 +626,9 @@ Qed. (** * Equivalence between inequalities *) -Notation Zle_plus_swap := Z.le_add_le_sub_r (compat "8.3"). -Notation Zlt_plus_swap := Z.lt_add_lt_sub_r (compat "8.3"). -Notation Zlt_minus_simpl_swap := Z.lt_sub_pos (compat "8.3"). +Notation Zle_plus_swap := Z.le_add_le_sub_r (only parsing). +Notation Zlt_plus_swap := Z.lt_add_lt_sub_r (only parsing). +Notation Zlt_minus_simpl_swap := Z.lt_sub_pos (only parsing). Lemma Zeq_plus_swap n m p : n + p = m <-> n = m - p. Proof. |