diff options
Diffstat (limited to 'theories/ZArith/Zdiv.v')
| -rw-r--r-- | theories/ZArith/Zdiv.v | 35 |
1 files changed, 22 insertions, 13 deletions
diff --git a/theories/ZArith/Zdiv.v b/theories/ZArith/Zdiv.v index 2ba865bd..15d0e487 100644 --- a/theories/ZArith/Zdiv.v +++ b/theories/ZArith/Zdiv.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) *) (************************************************************************) (** * Euclidean Division *) @@ -18,16 +20,16 @@ Local Open Scope Z_scope. (** The definition of the division is now in [BinIntDef], the initial specifications and properties are in [BinInt]. *) -Notation Zdiv_eucl_POS := Z.pos_div_eucl (compat "8.3"). -Notation Zdiv_eucl := Z.div_eucl (compat "8.3"). -Notation Zdiv := Z.div (compat "8.3"). -Notation Zmod := Z.modulo (compat "8.3"). +Notation Zdiv_eucl_POS := Z.pos_div_eucl (only parsing). +Notation Zdiv_eucl := Z.div_eucl (compat "8.6"). +Notation Zdiv := Z.div (compat "8.6"). +Notation Zmod := Z.modulo (only parsing). -Notation Zdiv_eucl_eq := Z.div_eucl_eq (compat "8.3"). -Notation Z_div_mod_eq_full := Z.div_mod (compat "8.3"). -Notation Zmod_POS_bound := Z.pos_div_eucl_bound (compat "8.3"). -Notation Zmod_pos_bound := Z.mod_pos_bound (compat "8.3"). -Notation Zmod_neg_bound := Z.mod_neg_bound (compat "8.3"). +Notation Zdiv_eucl_eq := Z.div_eucl_eq (compat "8.6"). +Notation Z_div_mod_eq_full := Z.div_mod (only parsing). +Notation Zmod_POS_bound := Z.pos_div_eucl_bound (only parsing). +Notation Zmod_pos_bound := Z.mod_pos_bound (only parsing). +Notation Zmod_neg_bound := Z.mod_neg_bound (only parsing). (** * Main division theorems *) @@ -508,6 +510,13 @@ Qed. (** Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) *) +Lemma Zmod_div : forall a b, a mod b / b = 0. +Proof. + intros a b. + zero_or_not b. + auto using Z.mod_div. +Qed. + (** A last inequality: *) Theorem Zdiv_mult_le: |