From 164c6861860e6b52818c031f901ffeff91fca16a Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 26 Jan 2016 16:56:33 +0100 Subject: Imported Upstream version 8.5 --- theories/Numbers/BigNumPrelude.v | 2 +- theories/Numbers/BinNums.v | 2 +- theories/Numbers/Cyclic/Abstract/CyclicAxioms.v | 2 +- theories/Numbers/Cyclic/Abstract/NZCyclic.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v | 2 +- theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v | 2 +- theories/Numbers/Cyclic/Int31/Cyclic31.v | 2 +- theories/Numbers/Cyclic/Int31/Int31.v | 2 +- theories/Numbers/Cyclic/Int31/Ring31.v | 2 +- theories/Numbers/Cyclic/ZModulo/ZModulo.v | 2 +- theories/Numbers/Integer/Abstract/ZAdd.v | 2 +- theories/Numbers/Integer/Abstract/ZAddOrder.v | 2 +- theories/Numbers/Integer/Abstract/ZAxioms.v | 2 +- theories/Numbers/Integer/Abstract/ZBase.v | 2 +- theories/Numbers/Integer/Abstract/ZBits.v | 2 +- theories/Numbers/Integer/Abstract/ZDivEucl.v | 4 ++-- theories/Numbers/Integer/Abstract/ZDivFloor.v | 4 ++-- theories/Numbers/Integer/Abstract/ZDivTrunc.v | 4 ++-- theories/Numbers/Integer/Abstract/ZGcd.v | 2 +- theories/Numbers/Integer/Abstract/ZLcm.v | 2 +- theories/Numbers/Integer/Abstract/ZLt.v | 2 +- theories/Numbers/Integer/Abstract/ZMaxMin.v | 2 +- theories/Numbers/Integer/Abstract/ZMul.v | 2 +- theories/Numbers/Integer/Abstract/ZMulOrder.v | 2 +- theories/Numbers/Integer/Abstract/ZParity.v | 2 +- theories/Numbers/Integer/Abstract/ZPow.v | 2 +- theories/Numbers/Integer/Abstract/ZProperties.v | 2 +- theories/Numbers/Integer/Abstract/ZSgnAbs.v | 2 +- theories/Numbers/Integer/BigZ/BigZ.v | 2 +- theories/Numbers/Integer/BigZ/ZMake.v | 2 +- theories/Numbers/Integer/Binary/ZBinary.v | 2 +- theories/Numbers/Integer/NatPairs/ZNatPairs.v | 2 +- theories/Numbers/Integer/SpecViaZ/ZSig.v | 2 +- theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v | 2 +- theories/Numbers/NaryFunctions.v | 2 +- theories/Numbers/NatInt/NZAdd.v | 2 +- theories/Numbers/NatInt/NZAddOrder.v | 2 +- theories/Numbers/NatInt/NZAxioms.v | 2 +- theories/Numbers/NatInt/NZBase.v | 2 +- theories/Numbers/NatInt/NZBits.v | 2 +- theories/Numbers/NatInt/NZDiv.v | 4 ++-- theories/Numbers/NatInt/NZDomain.v | 2 +- theories/Numbers/NatInt/NZGcd.v | 2 +- theories/Numbers/NatInt/NZLog.v | 2 +- theories/Numbers/NatInt/NZMul.v | 2 +- theories/Numbers/NatInt/NZMulOrder.v | 2 +- theories/Numbers/NatInt/NZOrder.v | 2 +- theories/Numbers/NatInt/NZParity.v | 2 +- theories/Numbers/NatInt/NZPow.v | 2 +- theories/Numbers/NatInt/NZProperties.v | 2 +- theories/Numbers/NatInt/NZSqrt.v | 2 +- theories/Numbers/Natural/Abstract/NAdd.v | 2 +- theories/Numbers/Natural/Abstract/NAddOrder.v | 2 +- theories/Numbers/Natural/Abstract/NAxioms.v | 2 +- theories/Numbers/Natural/Abstract/NBase.v | 2 +- theories/Numbers/Natural/Abstract/NBits.v | 2 +- theories/Numbers/Natural/Abstract/NDefOps.v | 2 +- theories/Numbers/Natural/Abstract/NDiv.v | 4 ++-- theories/Numbers/Natural/Abstract/NGcd.v | 2 +- theories/Numbers/Natural/Abstract/NIso.v | 2 +- theories/Numbers/Natural/Abstract/NLcm.v | 2 +- theories/Numbers/Natural/Abstract/NLog.v | 2 +- theories/Numbers/Natural/Abstract/NMaxMin.v | 2 +- theories/Numbers/Natural/Abstract/NMulOrder.v | 2 +- theories/Numbers/Natural/Abstract/NOrder.v | 2 +- theories/Numbers/Natural/Abstract/NParity.v | 4 ++-- theories/Numbers/Natural/Abstract/NPow.v | 2 +- theories/Numbers/Natural/Abstract/NProperties.v | 2 +- theories/Numbers/Natural/Abstract/NSqrt.v | 2 +- theories/Numbers/Natural/Abstract/NStrongRec.v | 2 +- theories/Numbers/Natural/Abstract/NSub.v | 2 +- theories/Numbers/Natural/BigN/BigN.v | 2 +- theories/Numbers/Natural/BigN/NMake.v | 2 +- theories/Numbers/Natural/BigN/NMake_gen.ml | 2 +- theories/Numbers/Natural/BigN/Nbasic.v | 2 +- theories/Numbers/Natural/Binary/NBinary.v | 2 +- theories/Numbers/Natural/Peano/NPeano.v | 2 +- theories/Numbers/Natural/SpecViaZ/NSig.v | 2 +- theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 2 +- theories/Numbers/NumPrelude.v | 2 +- theories/Numbers/Rational/BigQ/BigQ.v | 2 +- theories/Numbers/Rational/BigQ/QMake.v | 2 +- theories/Numbers/Rational/SpecViaQ/QSig.v | 2 +- 90 files changed, 96 insertions(+), 96 deletions(-) (limited to 'theories/Numbers') diff --git a/theories/Numbers/BigNumPrelude.v b/theories/Numbers/BigNumPrelude.v index 6817947c..45a7527c 100644 --- a/theories/Numbers/BigNumPrelude.v +++ b/theories/Numbers/BigNumPrelude.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* a < b*q -> a/b < q. diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v index d5f3f4ad..310748dd 100644 --- a/theories/Numbers/Integer/Abstract/ZDivFloor.v +++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* a < b*q -> a/b < q. diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v index de2e99ec..04301077 100644 --- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* 0 a < b*q -> a÷b < q. diff --git a/theories/Numbers/Integer/Abstract/ZGcd.v b/theories/Numbers/Integer/Abstract/ZGcd.v index cf6ff79e..30adaeb4 100644 --- a/theories/Numbers/Integer/Abstract/ZGcd.v +++ b/theories/Numbers/Integer/Abstract/ZGcd.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* 0 a < b*q -> a/b < q. diff --git a/theories/Numbers/NatInt/NZDomain.v b/theories/Numbers/NatInt/NZDomain.v index ffb04f08..3881a27f 100644 --- a/theories/Numbers/NatInt/NZDomain.v +++ b/theories/Numbers/NatInt/NZDomain.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* (a == b*(a/b) <-> a mod b == 0). Proof. intros. apply div_exact; auto'. Qed. -(** Some additionnal inequalities about div. *) +(** Some additional inequalities about div. *) Theorem div_lt_upper_bound: forall a b q, b~=0 -> a < b*q -> a/b < q. diff --git a/theories/Numbers/Natural/Abstract/NGcd.v b/theories/Numbers/Natural/Abstract/NGcd.v index a1f4ddf8..1eac134d 100644 --- a/theories/Numbers/Natural/Abstract/NGcd.v +++ b/theories/Numbers/Natural/Abstract/NGcd.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(*