diff options
author | Enrico Tassi <gareuselesinge@debian.org> | 2016-01-26 16:56:33 +0100 |
---|---|---|
committer | Enrico Tassi <gareuselesinge@debian.org> | 2016-01-26 16:56:33 +0100 |
commit | 164c6861860e6b52818c031f901ffeff91fca16a (patch) | |
tree | 4f91d20c890c25915e7b28226c663b94a8cfb0d3 /theories/ZArith | |
parent | 91dbeab8eef959c3f64960909ca69d4e68c8198d (diff) |
Imported Upstream version 8.5upstream/8.5
Diffstat (limited to 'theories/ZArith')
32 files changed, 35 insertions, 35 deletions
diff --git a/theories/ZArith/BinInt.v b/theories/ZArith/BinInt.v index cb0c6880..5aa397f8 100644 --- a/theories/ZArith/BinInt.v +++ b/theories/ZArith/BinInt.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/BinIntDef.v b/theories/ZArith/BinIntDef.v index 9bb86fd5..8c2e7d94 100644 --- a/theories/ZArith/BinIntDef.v +++ b/theories/ZArith/BinIntDef.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Wf_Z.v b/theories/ZArith/Wf_Z.v index 09909bdb..4fbbac26 100644 --- a/theories/ZArith/Wf_Z.v +++ b/theories/ZArith/Wf_Z.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/ZArith.v b/theories/ZArith/ZArith.v index 04cccd04..f86a7e52 100644 --- a/theories/ZArith/ZArith.v +++ b/theories/ZArith/ZArith.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/ZArith_base.v b/theories/ZArith/ZArith_base.v index 4c93173b..a1da544d 100644 --- a/theories/ZArith/ZArith_base.v +++ b/theories/ZArith/ZArith_base.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/ZArith_dec.v b/theories/ZArith/ZArith_dec.v index ac69cebd..8947295e 100644 --- a/theories/ZArith/ZArith_dec.v +++ b/theories/ZArith/ZArith_dec.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zabs.v b/theories/ZArith/Zabs.v index 146009bc..df9db5b2 100644 --- a/theories/ZArith/Zabs.v +++ b/theories/ZArith/Zabs.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zbool.v b/theories/ZArith/Zbool.v index 61eb2a34..41d1b2b5 100644 --- a/theories/ZArith/Zbool.v +++ b/theories/ZArith/Zbool.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zcompare.v b/theories/ZArith/Zcompare.v index d4ac72e9..2627f174 100644 --- a/theories/ZArith/Zcompare.v +++ b/theories/ZArith/Zcompare.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zcomplements.v b/theories/ZArith/Zcomplements.v index 9604a06e..5bdb32fc 100644 --- a/theories/ZArith/Zcomplements.v +++ b/theories/ZArith/Zcomplements.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zdigits.v b/theories/ZArith/Zdigits.v index b5d04719..bc3694bc 100644 --- a/theories/ZArith/Zdigits.v +++ b/theories/ZArith/Zdigits.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zdiv.v b/theories/ZArith/Zdiv.v index d0d10891..2ba865bd 100644 --- a/theories/ZArith/Zdiv.v +++ b/theories/ZArith/Zdiv.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -279,7 +279,7 @@ Proof. intros; rewrite Z.div_exact; auto. Qed. Theorem Zmod_le: forall a b, 0 < b -> 0 <= a -> a mod b <= a. Proof. intros. apply Z.mod_le; auto. Qed. -(** Some additionnal inequalities about Z.div. *) +(** Some additional inequalities about Z.div. *) Theorem Zdiv_lt_upper_bound: forall a b q, 0 < b -> a < q*b -> a/b < q. diff --git a/theories/ZArith/Zeuclid.v b/theories/ZArith/Zeuclid.v index f5cacc7e..38a824cd 100644 --- a/theories/ZArith/Zeuclid.v +++ b/theories/ZArith/Zeuclid.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zeven.v b/theories/ZArith/Zeven.v index d88bf7a9..d4051ef7 100644 --- a/theories/ZArith/Zeven.v +++ b/theories/ZArith/Zeven.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zgcd_alt.v b/theories/ZArith/Zgcd_alt.v index 14286bde..3977ca9d 100644 --- a/theories/ZArith/Zgcd_alt.v +++ b/theories/ZArith/Zgcd_alt.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zhints.v b/theories/ZArith/Zhints.v index 1942c2ab..c4e201e3 100644 --- a/theories/ZArith/Zhints.v +++ b/theories/ZArith/Zhints.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zlogarithm.v b/theories/ZArith/Zlogarithm.v index 6e349569..fdfd71e1 100644 --- a/theories/ZArith/Zlogarithm.v +++ b/theories/ZArith/Zlogarithm.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zmax.v b/theories/ZArith/Zmax.v index c436b3ad..529e9f1d 100644 --- a/theories/ZArith/Zmax.v +++ b/theories/ZArith/Zmax.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zmin.v b/theories/ZArith/Zmin.v index 1cfa2e03..782a5158 100644 --- a/theories/ZArith/Zmin.v +++ b/theories/ZArith/Zmin.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zminmax.v b/theories/ZArith/Zminmax.v index 05a94a8e..ea9a5f86 100644 --- a/theories/ZArith/Zminmax.v +++ b/theories/ZArith/Zminmax.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zmisc.v b/theories/ZArith/Zmisc.v index b401e6b6..65831c78 100644 --- a/theories/ZArith/Zmisc.v +++ b/theories/ZArith/Zmisc.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Znat.v b/theories/ZArith/Znat.v index 20e7c2e8..06428a7c 100644 --- a/theories/ZArith/Znat.v +++ b/theories/ZArith/Znat.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v index f69cf315..ee6efb3c 100644 --- a/theories/ZArith/Znumtheory.v +++ b/theories/ZArith/Znumtheory.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v index e090302e..73dee0cf 100644 --- a/theories/ZArith/Zorder.v +++ b/theories/ZArith/Zorder.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zpow_alt.v b/theories/ZArith/Zpow_alt.v index 8f661a9c..79a5a555 100644 --- a/theories/ZArith/Zpow_alt.v +++ b/theories/ZArith/Zpow_alt.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -11,7 +11,7 @@ Local Open Scope Z_scope. (** An alternative power function for Z *) -(** This [Zpower_alt] is extensionnaly equal to [Z.pow], +(** This [Zpower_alt] is extensionally equal to [Z.pow], but not convertible with it. The number of multiplications is logarithmic instead of linear, but these multiplications are bigger. Experimentally, it seems diff --git a/theories/ZArith/Zpow_def.v b/theories/ZArith/Zpow_def.v index 740c45fd..9eafa076 100644 --- a/theories/ZArith/Zpow_def.v +++ b/theories/ZArith/Zpow_def.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zpow_facts.v b/theories/ZArith/Zpow_facts.v index ac41a98f..2930e677 100644 --- a/theories/ZArith/Zpow_facts.v +++ b/theories/ZArith/Zpow_facts.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zpower.v b/theories/ZArith/Zpower.v index 747bd4fd..6f3a89f1 100644 --- a/theories/ZArith/Zpower.v +++ b/theories/ZArith/Zpower.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zquot.v b/theories/ZArith/Zquot.v index 3ef11189..0d92f1d5 100644 --- a/theories/ZArith/Zquot.v +++ b/theories/ZArith/Zquot.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -243,7 +243,7 @@ Proof. intros. zero_or_not b. intuition. apply Z.quot_exact; auto. Qed. Theorem Zrem_le a b : 0 <= a -> 0 <= b -> Z.rem a b <= a. Proof. intros. zero_or_not b. apply Z.rem_le; auto with zarith. Qed. -(** Some additionnal inequalities about Zdiv. *) +(** Some additional inequalities about Zdiv. *) Theorem Zquot_le_upper_bound: forall a b q, 0 < b -> a <= q*b -> a÷b <= q. diff --git a/theories/ZArith/Zsqrt_compat.v b/theories/ZArith/Zsqrt_compat.v index 65959a69..b80eb445 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 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v index cba709e8..1ac00bdd 100644 --- a/theories/ZArith/Zwf.v +++ b/theories/ZArith/Zwf.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) diff --git a/theories/ZArith/auxiliary.v b/theories/ZArith/auxiliary.v index 25ef852a..c6c38976 100644 --- a/theories/ZArith/auxiliary.v +++ b/theories/ZArith/auxiliary.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) |