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 | |
| parent | 91dbeab8eef959c3f64960909ca69d4e68c8198d (diff) | |
Imported Upstream version 8.5upstream/8.5
Diffstat (limited to 'theories')
375 files changed, 515 insertions, 7918 deletions
diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v index 620a4201..953f7e4d 100644 --- a/theories/Arith/Arith.v +++ b/theories/Arith/Arith.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/Arith/Arith_base.v b/theories/Arith/Arith_base.v index a99c4113..b378828e 100644 --- a/theories/Arith/Arith_base.v +++ b/theories/Arith/Arith_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/Arith/Between.v b/theories/Arith/Between.v index 06723541..f998e861 100644 --- a/theories/Arith/Between.v +++ b/theories/Arith/Between.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/Arith/Bool_nat.v b/theories/Arith/Bool_nat.v index f91f3340..602555b6 100644 --- a/theories/Arith/Bool_nat.v +++ b/theories/Arith/Bool_nat.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/Arith/Compare.v b/theories/Arith/Compare.v index 400f2d81..610e9a69 100644 --- a/theories/Arith/Compare.v +++ b/theories/Arith/Compare.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/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v index a97cf6dc..976507b5 100644 --- a/theories/Arith/Compare_dec.v +++ b/theories/Arith/Compare_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/Arith/Div2.v b/theories/Arith/Div2.v index 1c65a192..016cb85e 100644 --- a/theories/Arith/Div2.v +++ b/theories/Arith/Div2.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/Arith/EqNat.v b/theories/Arith/EqNat.v index 2771670e..206fc0ab 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.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/Arith/Euclid.v b/theories/Arith/Euclid.v index eaacab02..fbe98d17 100644 --- a/theories/Arith/Euclid.v +++ b/theories/Arith/Euclid.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/Arith/Even.v b/theories/Arith/Even.v index 0f94a8ed..3c8c250a 100644 --- a/theories/Arith/Even.v +++ b/theories/Arith/Even.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/Arith/Factorial.v b/theories/Arith/Factorial.v index 7d29f23c..b119bb00 100644 --- a/theories/Arith/Factorial.v +++ b/theories/Arith/Factorial.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/Arith/Gt.v b/theories/Arith/Gt.v index e406ff0d..dfd57694 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.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/Arith/Le.v b/theories/Arith/Le.v index 875863e4..ceb91187 100644 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.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/Arith/Lt.v b/theories/Arith/Lt.v index b783ca33..f824ee6f 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.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/Arith/Max.v b/theories/Arith/Max.v index 26875373..65534b2e 100644 --- a/theories/Arith/Max.v +++ b/theories/Arith/Max.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/Arith/Min.v b/theories/Arith/Min.v index f2fa3aec..38e59b7b 100644 --- a/theories/Arith/Min.v +++ b/theories/Arith/Min.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/Arith/Minus.v b/theories/Arith/Minus.v index 6e312e4f..bc3a318c 100644 --- a/theories/Arith/Minus.v +++ b/theories/Arith/Minus.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/Arith/Mult.v b/theories/Arith/Mult.v index 2d82920b..96581243 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.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/Arith/PeanoNat.v b/theories/Arith/PeanoNat.v index 799031a2..e3240bb7 100644 --- a/theories/Arith/PeanoNat.v +++ b/theories/Arith/PeanoNat.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/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index a7ede3fc..340a7968 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_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/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index 64764830..94bbd50a 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.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/Bool/Bool.v b/theories/Bool/Bool.v index cc12cf47..721ab693 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.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/Bool/BoolEq.v b/theories/Bool/BoolEq.v index 11af2fd1..aec4f0bb 100644 --- a/theories/Bool/BoolEq.v +++ b/theories/Bool/BoolEq.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/Bool/Bvector.v b/theories/Bool/Bvector.v index 7c63f069..09f643c8 100644 --- a/theories/Bool/Bvector.v +++ b/theories/Bool/Bvector.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/Bool/DecBool.v b/theories/Bool/DecBool.v index e0b8ec9b..501366ce 100644 --- a/theories/Bool/DecBool.v +++ b/theories/Bool/DecBool.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/Bool/IfProp.v b/theories/Bool/IfProp.v index a0acbe8c..11f3d1d6 100644 --- a/theories/Bool/IfProp.v +++ b/theories/Bool/IfProp.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/Bool/Sumbool.v b/theories/Bool/Sumbool.v index c2e9183b..fd7f42e2 100644 --- a/theories/Bool/Sumbool.v +++ b/theories/Bool/Sumbool.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/Bool/Zerob.v b/theories/Bool/Zerob.v index e146f25f..16e47ac5 100644 --- a/theories/Bool/Zerob.v +++ b/theories/Bool/Zerob.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/Classes/CEquivalence.v b/theories/Classes/CEquivalence.v index 65353ed2..bd01de47 100644 --- a/theories/Classes/CEquivalence.v +++ b/theories/Classes/CEquivalence.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/Classes/CMorphisms.v b/theories/Classes/CMorphisms.v index fdedbf67..c41eb2fa 100644 --- a/theories/Classes/CMorphisms.v +++ b/theories/Classes/CMorphisms.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 *) @@ -269,16 +269,6 @@ Section GenericInstances. Unset Strict Universe Declaration. (** The complement of a crelation conserves its proper elements. *) - Program Definition complement_proper (A : Type@{k}) (RA : crelation A) - `(mR : Proper (A -> A -> Prop) (RA ==> RA ==> iff) R) : - Proper (RA ==> RA ==> iff) (complement@{i j Prop} R) := _. - - Next Obligation. - Proof. - unfold complement. - pose (mR x y X x0 y0 X0). - intuition. - Qed. (** The [flip] too, actually the [flip] instance is a bit more general. *) Program Definition flip_proper @@ -521,8 +511,8 @@ Ltac proper_reflexive := Hint Extern 1 (subrelation (flip _) _) => class_apply @flip1 : typeclass_instances. Hint Extern 1 (subrelation _ (flip _)) => class_apply @flip2 : typeclass_instances. -Hint Extern 1 (Proper _ (complement _)) => apply @complement_proper - : typeclass_instances. +(* Hint Extern 1 (Proper _ (complement _)) => apply @complement_proper *) +(* : typeclass_instances. *) Hint Extern 1 (Proper _ (flip _)) => apply @flip_proper : typeclass_instances. Hint Extern 2 (@Proper _ (flip _) _) => class_apply @proper_flip_proper diff --git a/theories/Classes/CRelationClasses.v b/theories/Classes/CRelationClasses.v index 35b2b8a3..3d7ef01f 100644 --- a/theories/Classes/CRelationClasses.v +++ b/theories/Classes/CRelationClasses.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/Classes/DecidableClass.v b/theories/Classes/DecidableClass.v index 9fe3d0fe..a3b7e311 100644 --- a/theories/Classes/DecidableClass.v +++ b/theories/Classes/DecidableClass.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/Classes/EquivDec.v b/theories/Classes/EquivDec.v index 59e800c2..52313735 100644 --- a/theories/Classes/EquivDec.v +++ b/theories/Classes/EquivDec.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/Classes/Equivalence.v b/theories/Classes/Equivalence.v index c281af80..c4588947 100644 --- a/theories/Classes/Equivalence.v +++ b/theories/Classes/Equivalence.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/Classes/Init.v b/theories/Classes/Init.v index 9574cf85..c13b36fd 100644 --- a/theories/Classes/Init.v +++ b/theories/Classes/Init.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/Classes/Morphisms.v b/theories/Classes/Morphisms.v index 1bdce654..8d942d90 100644 --- a/theories/Classes/Morphisms.v +++ b/theories/Classes/Morphisms.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/Classes/Morphisms_Prop.v b/theories/Classes/Morphisms_Prop.v index 096c96e5..15900177 100644 --- a/theories/Classes/Morphisms_Prop.v +++ b/theories/Classes/Morphisms_Prop.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/Classes/Morphisms_Relations.v b/theories/Classes/Morphisms_Relations.v index 68a8c06a..6048fe06 100644 --- a/theories/Classes/Morphisms_Relations.v +++ b/theories/Classes/Morphisms_Relations.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/Classes/RelationClasses.v b/theories/Classes/RelationClasses.v index 15cb02d3..11c204da 100644 --- a/theories/Classes/RelationClasses.v +++ b/theories/Classes/RelationClasses.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 *) @@ -208,6 +208,10 @@ Hint Extern 4 (subrelation (flip _) _) => class_apply @subrelation_symmetric : typeclass_instances. Arguments irreflexivity {A R Irreflexive} [x] _. +Arguments symmetry {A} {R} {_} [x] [y] _. +Arguments asymmetry {A} {R} {_} [x] [y] _ _. +Arguments transitivity {A} {R} {_} [x] [y] [z] _ _. +Arguments Antisymmetric A eqA {_} _. Hint Resolve irreflexivity : ord. diff --git a/theories/Classes/SetoidClass.v b/theories/Classes/SetoidClass.v index f20100fe..4b133a4d 100644 --- a/theories/Classes/SetoidClass.v +++ b/theories/Classes/SetoidClass.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/Classes/SetoidDec.v b/theories/Classes/SetoidDec.v index bf05934e..7201c0b1 100644 --- a/theories/Classes/SetoidDec.v +++ b/theories/Classes/SetoidDec.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/Classes/SetoidTactics.v b/theories/Classes/SetoidTactics.v index 8ca93341..145d451f 100644 --- a/theories/Classes/SetoidTactics.v +++ b/theories/Classes/SetoidTactics.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/Compat/AdmitAxiom.v b/theories/Compat/AdmitAxiom.v new file mode 100644 index 00000000..4d9f55cf --- /dev/null +++ b/theories/Compat/AdmitAxiom.v @@ -0,0 +1,15 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <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 *) +(************************************************************************) + +(** Compatibility file for making the admit tactic act similar to Coq v8.4. In +8.4, [admit] created a new axiom; in 8.5, it just shelves the goal. This +compatibility definition is not in the Coq84.v file to avoid loading an +inconsistent axiom implicitly. *) + +Axiom proof_admitted : False. +Ltac admit := clear; abstract case proof_admitted. diff --git a/theories/Compat/Coq84.v b/theories/Compat/Coq84.v index b04d5168..90083b00 100644 --- a/theories/Compat/Coq84.v +++ b/theories/Compat/Coq84.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 *) @@ -18,10 +18,6 @@ Global Set Asymmetric Patterns. (** See bug 3545 *) Global Set Universal Lemma Under Conjunction. -(** In 8.4, [admit] created a new axiom; in 8.5, it just shelves the goal. *) -Axiom proof_admitted : False. -Ltac admit := clear; abstract case proof_admitted. - (** In 8.5, [refine] leaves over dependent subgoals. *) Tactic Notation "refine" uconstr(term) := refine term; shelve_unifiable. diff --git a/theories/Compat/Coq85.v b/theories/Compat/Coq85.v index 1622f2ae..6e2b3564 100644 --- a/theories/Compat/Coq85.v +++ b/theories/Compat/Coq85.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/Compat/vo.itarget b/theories/Compat/vo.itarget index c0c40ab1..43b19700 100644 --- a/theories/Compat/vo.itarget +++ b/theories/Compat/vo.itarget @@ -1,2 +1,3 @@ +AdmitAxiom.vo Coq84.vo Coq85.vo diff --git a/theories/FSets/FMapFacts.v b/theories/FSets/FMapFacts.v index 8c6f4b64..eaeb2914 100644 --- a/theories/FSets/FMapFacts.v +++ b/theories/FSets/FMapFacts.v @@ -2143,7 +2143,7 @@ Module OrdProperties (M:S). Section Fold_properties. (** The following lemma has already been proved on Weak Maps, - but with one additionnal hypothesis (some [transpose] fact). *) + but with one additional hypothesis (some [transpose] fact). *) Lemma fold_Equal : forall m1 m2 (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA) (f:key->elt->A->A)(i:A), diff --git a/theories/FSets/FMapPositive.v b/theories/FSets/FMapPositive.v index 3eac15b0..9e59f0c5 100644 --- a/theories/FSets/FMapPositive.v +++ b/theories/FSets/FMapPositive.v @@ -1061,7 +1061,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. End PositiveMap. -(** Here come some additionnal facts about this implementation. +(** Here come some additional facts about this implementation. Most are facts that cannot be derivable from the general interface. *) diff --git a/theories/Init/Datatypes.v b/theories/Init/Datatypes.v index de615301..4850c9ca 100644 --- a/theories/Init/Datatypes.v +++ b/theories/Init/Datatypes.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/Init/Logic.v b/theories/Init/Logic.v index 50f853f0..85123cc4 100644 --- a/theories/Init/Logic.v +++ b/theories/Init/Logic.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/Init/Logic_Type.v b/theories/Init/Logic_Type.v index 1e126463..4a5f2ad6 100644 --- a/theories/Init/Logic_Type.v +++ b/theories/Init/Logic_Type.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/Init/Nat.v b/theories/Init/Nat.v index afb46436..b8920586 100644 --- a/theories/Init/Nat.v +++ b/theories/Init/Nat.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/Init/Notations.v b/theories/Init/Notations.v index a7bdba90..ab6bf472 100644 --- a/theories/Init/Notations.v +++ b/theories/Init/Notations.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/Init/Peano.v b/theories/Init/Peano.v index 7a14ab39..3749baf6 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.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/Init/Prelude.v b/theories/Init/Prelude.v index 0efb8744..04a263ad 100644 --- a/theories/Init/Prelude.v +++ b/theories/Init/Prelude.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/Init/Specif.v b/theories/Init/Specif.v index 1384901b..6c022185 100644 --- a/theories/Init/Specif.v +++ b/theories/Init/Specif.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/Init/Tactics.v b/theories/Init/Tactics.v index a7d3f806..59fdbb42 100644 --- a/theories/Init/Tactics.v +++ b/theories/Init/Tactics.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/Init/Wf.v b/theories/Init/Wf.v index 6501b1e1..985ecaf2 100644 --- a/theories/Init/Wf.v +++ b/theories/Init/Wf.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/Lists/List.v b/theories/Lists/List.v index fe18686e..cc7586fe 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.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 *) @@ -970,6 +970,7 @@ Section Map. Lemma in_flat_map : forall (f:A->list B)(l:list A)(y:B), In y (flat_map f l) <-> exists x, In x l /\ In y (f x). Proof using A B. + clear Hfinjective. induction l; simpl; split; intros. contradiction. destruct H as (x,(H,_)); contradiction. diff --git a/theories/Lists/ListDec.v b/theories/Lists/ListDec.v index 8bd2daaf..3e2eeac0 100644 --- a/theories/Lists/ListDec.v +++ b/theories/Lists/ListDec.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/Lists/ListSet.v b/theories/Lists/ListSet.v index 0a0bf0de..fd0464fb 100644 --- a/theories/Lists/ListSet.v +++ b/theories/Lists/ListSet.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/Lists/ListTactics.v b/theories/Lists/ListTactics.v index f19d95a9..537d5f68 100644 --- a/theories/Lists/ListTactics.v +++ b/theories/Lists/ListTactics.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/Lists/StreamMemo.v b/theories/Lists/StreamMemo.v index 74d464c5..5a16cc43 100644 --- a/theories/Lists/StreamMemo.v +++ b/theories/Lists/StreamMemo.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/Lists/Streams.v b/theories/Lists/Streams.v index cc4fb179..7ec3d250 100644 --- a/theories/Lists/Streams.v +++ b/theories/Lists/Streams.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/Logic/Berardi.v b/theories/Logic/Berardi.v index d72f4072..1e0bd0fe 100644 --- a/theories/Logic/Berardi.v +++ b/theories/Logic/Berardi.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/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index d2327498..1420a000 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.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/Logic/Classical.v b/theories/Logic/Classical.v index 600db472..14d83501 100644 --- a/theories/Logic/Classical.v +++ b/theories/Logic/Classical.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/Logic/ClassicalChoice.v b/theories/Logic/ClassicalChoice.v index 07153b35..7041ee40 100644 --- a/theories/Logic/ClassicalChoice.v +++ b/theories/Logic/ClassicalChoice.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/Logic/ClassicalDescription.v b/theories/Logic/ClassicalDescription.v index bdad50e2..9e6d07b2 100644 --- a/theories/Logic/ClassicalDescription.v +++ b/theories/Logic/ClassicalDescription.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/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v index 2d9a1ffd..0e91613d 100644 --- a/theories/Logic/ClassicalEpsilon.v +++ b/theories/Logic/ClassicalEpsilon.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/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v index 6f736e45..c947062a 100644 --- a/theories/Logic/ClassicalFacts.v +++ b/theories/Logic/ClassicalFacts.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 *) @@ -442,10 +442,10 @@ Section Proof_irrelevance_WEM_CC. Theorem wproof_irrelevance_cc : ~~(b1 = b2). Proof. intros h. - refine (let NB := exist (fun P=>~~P -> P) B _ in _). + unshelve (refine (let NB := exist (fun P=>~~P -> P) B _ in _)). { exact (fun _ => b1). } pose proof (NoRetractToNegativeProp.paradox NB p2b b2p (wp2p2 h) wp2p1) as paradox. - refine (let F := exist (fun P=>~~P->P) False _ in _). + unshelve (refine (let F := exist (fun P=>~~P->P) False _ in _)). { auto. } exact (paradox F). Qed. @@ -658,4 +658,3 @@ Proof. exists x; intro; exact Hx. exists x0; exact Hnot. Qed. - diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v index 4b0ec15e..57f367e5 100644 --- a/theories/Logic/ClassicalUniqueChoice.v +++ b/theories/Logic/ClassicalUniqueChoice.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/Logic/Classical_Pred_Type.v b/theories/Logic/Classical_Pred_Type.v index 8468ced3..6665798d 100644 --- a/theories/Logic/Classical_Pred_Type.v +++ b/theories/Logic/Classical_Pred_Type.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/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v index be75c4e9..2c69d4f0 100644 --- a/theories/Logic/Classical_Prop.v +++ b/theories/Logic/Classical_Prop.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/Logic/ConstructiveEpsilon.v b/theories/Logic/ConstructiveEpsilon.v index 6f5bfae4..a304dd24 100644 --- a/theories/Logic/ConstructiveEpsilon.v +++ b/theories/Logic/ConstructiveEpsilon.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/Logic/Decidable.v b/theories/Logic/Decidable.v index 545f92bd..2ba7253c 100644 --- a/theories/Logic/Decidable.v +++ b/theories/Logic/Decidable.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/Logic/Description.v b/theories/Logic/Description.v index 70cc0787..0239222e 100644 --- a/theories/Logic/Description.v +++ b/theories/Logic/Description.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/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v index 64517354..23af5afc 100644 --- a/theories/Logic/Diaconescu.v +++ b/theories/Logic/Diaconescu.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/Logic/Epsilon.v b/theories/Logic/Epsilon.v index fe17cde4..ffbb5758 100644 --- a/theories/Logic/Epsilon.v +++ b/theories/Logic/Epsilon.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/Logic/Eqdep.v b/theories/Logic/Eqdep.v index d9ffe68d..f3a2783e 100644 --- a/theories/Logic/Eqdep.v +++ b/theories/Logic/Eqdep.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/Logic/EqdepFacts.v b/theories/Logic/EqdepFacts.v index 34aba486..30e26c7c 100644 --- a/theories/Logic/EqdepFacts.v +++ b/theories/Logic/EqdepFacts.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/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v index 65011e8e..b7b4dec2 100644 --- a/theories/Logic/Eqdep_dec.v +++ b/theories/Logic/Eqdep_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/Logic/ExtensionalityFacts.v b/theories/Logic/ExtensionalityFacts.v index 61ee9eb9..0e34e7e9 100644 --- a/theories/Logic/ExtensionalityFacts.v +++ b/theories/Logic/ExtensionalityFacts.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/Logic/FinFun.v b/theories/Logic/FinFun.v index 670aa219..06466801 100644 --- a/theories/Logic/FinFun.v +++ b/theories/Logic/FinFun.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/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v index eb50a3aa..04d9a670 100644 --- a/theories/Logic/FunctionalExtensionality.v +++ b/theories/Logic/FunctionalExtensionality.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/Logic/Hurkens.v b/theories/Logic/Hurkens.v index ede51f57..8ded7476 100644 --- a/theories/Logic/Hurkens.v +++ b/theories/Logic/Hurkens.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 *) @@ -266,7 +266,7 @@ End Paradox. (** The [paradox] tactic can be called as a shortcut to use the paradox. *) Ltac paradox h := - refine ((fun h => _) (paradox _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ));cycle 1. + unshelve (refine ((fun h => _) (paradox _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ))). End Generic. @@ -319,77 +319,31 @@ Proof. + cbn. exact (fun u F => forall x:u, F x). + cbn. exact (fun _ _ x => x). + cbn. exact (fun _ _ x => x). - + cbn. easy. + + cbn. exact (fun F => u22u1 (forall x, F x)). + cbn. exact (fun _ x => u22u1_unit _ x). + cbn. exact (fun _ x => u22u1_counit _ x). - + cbn. intros **. now rewrite u22u1_coherent. (** Small universe *) + exact U0. (** The interpretation of the small universe is the image of [U0] in [U1]. *) + cbn. exact (fun X => u02u1 X). + cbn. exact (fun u F => u12u0 (forall x:(u02u1 u), u02u1 (F x))). - + cbn. intros * x. exact (u12u0_unit _ x). - + cbn. intros * x. exact (u12u0_counit _ x). + cbn. exact (fun u F => u12u0 (forall x:u, u02u1 (F x))). - + cbn. intros * x. exact (u12u0_unit _ x). - + cbn. intros * x. exact (u12u0_counit _ x). + cbn. exact (u12u0 F). + cbn in h. exact (u12u0_counit _ h). -Qed. - -End Paradox. - -End NoRetractToImpredicativeUniverse. - -(** * Prop is not a retract *) - -(** The existence in the pure Calculus of Constructions of a retract - from [Prop] into a small type of [Prop] is inconsistent. This is a - special case of the previous result. *) - -Module NoRetractFromSmallPropositionToProp. - -Section Paradox. - -(** ** Retract of [Prop] in a small type *) - -(** The retract is axiomatized using logical equivalence as the - equality on propositions. *) - -Variable bool : Prop. -Variable p2b : Prop -> bool. -Variable b2p : bool -> Prop. -Hypothesis p2p1 : forall A:Prop, b2p (p2b A) -> A. -Hypothesis p2p2 : forall A:Prop, A -> b2p (p2b A). - -(** ** Paradox *) - -Theorem paradox : forall B:Prop, B. -Proof. - intros B. - pose proof - (NoRetractToImpredicativeUniverse.paradox@{Type Prop}) as P. - refine (P _ _ _ _ _ _ _ _ _ _);clear P. - + exact bool. - + exact (fun x => forall P:Prop, (x->P)->P). - + cbn. exact (fun _ x P k => k x). - + cbn. intros F P x. - apply P. - intros f. - exact (f x). + cbn. easy. - + exact b2p. - + exact p2b. - + exact p2p2. - + exact p2p1. + + cbn. intros **. now rewrite u22u1_coherent. + + cbn. intros * x. exact (u12u0_unit _ x). + + cbn. intros * x. exact (u12u0_counit _ x). + + cbn. intros * x. exact (u12u0_unit _ x). + + cbn. intros * x. exact (u12u0_counit _ x). Qed. End Paradox. -End NoRetractFromSmallPropositionToProp. +End NoRetractToImpredicativeUniverse. (** * Modal fragments of [Prop] are not retracts *) @@ -428,7 +382,7 @@ Qed. Definition Forall {A:Type} (P:A->MProp) : MProp. Proof. - refine (exist _ _ _). + unshelve (refine (exist _ _ _)). + exact (forall x:A, El (P x)). + intros h x. eapply strength in h. @@ -458,27 +412,27 @@ Proof. + exact (fun _ => Forall). + cbn. exact (fun _ _ f => f). + cbn. exact (fun _ _ f => f). - + cbn. easy. + exact Forall. + cbn. exact (fun _ f => f). + cbn. exact (fun _ f => f). - + cbn. easy. (** Small universe *) + exact bool. + exact (fun b => El (b2p b)). + cbn. exact (fun _ F => p2b (Forall (fun x => b2p (F x)))). + + exact (fun _ F => p2b (Forall (fun x => b2p (F x)))). + + apply p2b. + exact B. + + cbn in h. auto. + + cbn. easy. + + cbn. easy. + cbn. auto. + cbn. intros * f. apply p2p1 in f. cbn in f. exact f. - + exact (fun _ F => p2b (Forall (fun x => b2p (F x)))). + cbn. auto. + cbn. intros * f. apply p2p1 in f. cbn in f. exact f. - + apply p2b. - exact B. - + cbn in h. auto. Qed. End Paradox. @@ -516,23 +470,97 @@ Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). Theorem paradox : forall B:NProp, El B. Proof. intros B. - refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _));cycle 1. + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + exact (fun P => ~~P). + + exact bool. + + exact p2b. + + exact b2p. + + exact B. + + exact h. + cbn. auto. + cbn. auto. + cbn. auto. + + auto. + + auto. +Qed. + +End Paradox. + +End NoRetractToNegativeProp. + +(** * Prop is not a retract *) + +(** The existence in the pure Calculus of Constructions of a retract + from [Prop] into a small type of [Prop] is inconsistent. This is a + special case of the previous result. *) + +Module NoRetractFromSmallPropositionToProp. + +(** ** The universe of propositions. *) + +Definition NProp := { P:Prop | P -> P}. +Definition El : NProp -> Prop := @proj1_sig _ _. + +Section MParadox. + +(** ** Retract of [Prop] in a small type, using the identity modality. *) + +Variable bool : NProp. +Variable p2b : NProp -> El bool. +Variable b2p : El bool -> NProp. +Hypothesis p2p1 : forall A:NProp, El (b2p (p2b A)) -> El A. +Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)). + +(** ** Paradox *) + +Theorem mparadox : forall B:NProp, El B. +Proof. + intros B. + unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _ _ _))). + + exact (fun P => P). + exact bool. + exact p2b. + exact b2p. - + auto. - + auto. + exact B. + exact h. + + cbn. auto. + + cbn. auto. + + cbn. auto. + + auto. + + auto. +Qed. + +End MParadox. + +Section Paradox. + +(** ** Retract of [Prop] in a small type *) + +(** The retract is axiomatized using logical equivalence as the + equality on propositions. *) +Variable bool : Prop. +Variable p2b : Prop -> bool. +Variable b2p : bool -> Prop. +Hypothesis p2p1 : forall A:Prop, b2p (p2b A) -> A. +Hypothesis p2p2 : forall A:Prop, A -> b2p (p2b A). + +(** ** Paradox *) + +Theorem paradox : forall B:Prop, B. +Proof. + intros B. + unshelve (refine (mparadox (exist _ bool (fun x => x)) _ _ _ _ + (exist _ B (fun x => x)))). + + intros p. red. red. exact (p2b (El p)). + + cbn. intros b. red. exists (b2p b). exact (fun x => x). + + cbn. intros [A H]. cbn. apply p2p1. + + cbn. intros [A H]. cbn. apply p2p2. Qed. End Paradox. -End NoRetractToNegativeProp. +End NoRetractFromSmallPropositionToProp. + (** * Large universes are no retracts of [Prop]. *) @@ -569,7 +597,6 @@ Proof. + cbn. exact (fun u F => forall x, F x). + cbn. exact (fun _ _ x => x). + cbn. exact (fun _ _ x => x). - + cbn. easy. + exact (fun F => forall A:Prop, F(up A)). + cbn. exact (fun F f A => f (up A)). + cbn. @@ -577,20 +604,21 @@ Proof. specialize (f (down A)). rewrite up_down in f. exact f. + + exact Prop. + + cbn. exact (fun X => X). + + cbn. exact (fun A P => forall x:A, P x). + + cbn. exact (fun A P => forall x:A, P x). + + cbn. exact P. + + exact h. + + cbn. easy. + cbn. intros F f A. destruct (up_down A). cbn. reflexivity. - + exact Prop. - + cbn. exact (fun X => X). - + cbn. exact (fun A P => forall x:A, P x). + cbn. exact (fun _ _ x => x). + cbn. exact (fun _ _ x => x). - + cbn. exact (fun A P => forall x:A, P x). + cbn. exact (fun _ _ x => x). + cbn. exact (fun _ _ x => x). - + cbn. exact P. - + exact h. Qed. End Paradox. @@ -637,37 +665,37 @@ Proof. + cbn. exact (fun X F => forall x:X, F x). + cbn. exact (fun _ _ x => x). + cbn. exact (fun _ _ x => x). - + cbn. easy. + exact (fun F => forall x:A, F (up x)). + cbn. exact (fun _ f => fun x:A => f (up x)). + cbn. intros * f X. specialize (f (down X)). rewrite up_down in f. exact f. - + cbn. intros ? f X. - destruct (up_down X). cbn. - reflexivity. (** Small universe *) + exact A. (** The interpretation of [A] as a universe is [U]. *) + cbn. exact up. + cbn. exact (fun _ F => down (forall x, up (F x))). + + cbn. exact (fun _ F => down (forall x, up (F x))). + + cbn. exact (down False). + + rewrite up_down in p. + exact p. + + cbn. easy. + + cbn. intros ? f X. + destruct (up_down X). cbn. + reflexivity. + cbn. intros ? ? f. rewrite up_down. exact f. + cbn. intros ? ? f. rewrite up_down in f. exact f. - + cbn. exact (fun _ F => down (forall x, up (F x))). + cbn. intros ? ? f. rewrite up_down. exact f. + cbn. intros ? ? f. rewrite up_down in f. exact f. - + cbn. exact (down False). - + rewrite up_down in p. - exact p. Qed. End Paradox. @@ -683,7 +711,7 @@ Module PropNeqType. Theorem paradox : Prop <> Type. Proof. intros h. - refine (TypeNeqSmallType.paradox _ _). + unshelve (refine (TypeNeqSmallType.paradox _ _)). + exact Prop. + easy. Qed. diff --git a/theories/Logic/IndefiniteDescription.v b/theories/Logic/IndefiniteDescription.v index 9875710e..21be5032 100644 --- a/theories/Logic/IndefiniteDescription.v +++ b/theories/Logic/IndefiniteDescription.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/Logic/JMeq.v b/theories/Logic/JMeq.v index 98cddf0a..2f95856b 100644 --- a/theories/Logic/JMeq.v +++ b/theories/Logic/JMeq.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/Logic/ProofIrrelevance.v b/theories/Logic/ProofIrrelevance.v index eb00dedd..305839cd 100644 --- a/theories/Logic/ProofIrrelevance.v +++ b/theories/Logic/ProofIrrelevance.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/Logic/ProofIrrelevanceFacts.v b/theories/Logic/ProofIrrelevanceFacts.v index 6ab6abcf..19b3e9e6 100644 --- a/theories/Logic/ProofIrrelevanceFacts.v +++ b/theories/Logic/ProofIrrelevanceFacts.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/Logic/RelationalChoice.v b/theories/Logic/RelationalChoice.v index 61598130..d16835f8 100644 --- a/theories/Logic/RelationalChoice.v +++ b/theories/Logic/RelationalChoice.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/Logic/SetIsType.v b/theories/Logic/SetIsType.v index f110237e..33fce6cc 100644 --- a/theories/Logic/SetIsType.v +++ b/theories/Logic/SetIsType.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/Logic/WKL.v b/theories/Logic/WKL.v index 408eca4a..95f3e83f 100644 --- a/theories/Logic/WKL.v +++ b/theories/Logic/WKL.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 *) @@ -40,7 +40,7 @@ Proposition is_path_from_characterization P n l : Proof. intros. split. - induction 1 as [|* HP _ (l'&Hl'&HPl')|* HP _ (l'&Hl'&HPl')]. - + exists []. split. reflexivity. intros n <-/le_n_0_eq. assumption. + + exists []. split. reflexivity. intros n <-%le_n_0_eq. assumption. + exists (true :: l'). split. apply eq_S, Hl'. intros [|] H. * assumption. * simpl. rewrite <- app_assoc. apply HPl', le_S_n, H. @@ -51,10 +51,10 @@ intros. split. + constructor. apply (HPl' 0). apply le_0_n. + eapply next_left. * apply (HPl' 0), le_0_n. - * fold (length l'). apply IHl'. intros n' H/le_n_S. apply HPl' in H. simpl in H. rewrite <- app_assoc in H. assumption. + * fold (length l'). apply IHl'. intros n' H%le_n_S. apply HPl' in H. simpl in H. rewrite <- app_assoc in H. assumption. + apply next_right. * apply (HPl' 0), le_0_n. - * fold (length l'). apply IHl'. intros n' H/le_n_S. apply HPl' in H. simpl in H. rewrite <- app_assoc in H. assumption. + * fold (length l'). apply IHl'. intros n' H%le_n_S. apply HPl' in H. simpl in H. rewrite <- app_assoc in H. assumption. Qed. (** [infinite_from P l] means that we can find arbitrary long paths diff --git a/theories/Logic/WeakFan.v b/theories/Logic/WeakFan.v index 2f84ebe5..4416d38d 100644 --- a/theories/Logic/WeakFan.v +++ b/theories/Logic/WeakFan.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 *) @@ -89,7 +89,7 @@ Qed. Theorem WeakFanTheorem : forall P, barred P -> inductively_barred P []. Proof. intros P Hbar. -destruct Hbar with (X P) as (l,(Hd/Y_approx,HP)). +destruct Hbar with (X P) as (l,(Hd%Y_approx,HP)). assert (inductively_barred P l) by (apply (now P l), HP). clear Hbar HP. induction l as [|a l]. diff --git a/theories/MMaps/MMapAVL.v b/theories/MMaps/MMapAVL.v deleted file mode 100644 index d840f1f3..00000000 --- a/theories/MMaps/MMapAVL.v +++ /dev/null @@ -1,2158 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(* Finite map library. *) - -(** * MMapAVL *) - -(** This module implements maps using AVL trees. - It follows the implementation from Ocaml's standard library. - - See the comments at the beginning of MSetAVL for more details. -*) - -Require Import Bool PeanoNat BinInt Int MMapInterface MMapList. -Require Import Orders OrdersFacts OrdersLists. - -Set Implicit Arguments. -Unset Strict Implicit. -(* For nicer extraction, we create inductive principles - only when needed *) -Local Unset Elimination Schemes. - -(** Notations and helper lemma about pairs *) - -Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope. -Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope. - -(** * The Raw functor - - Functor of pure functions + separate proofs of invariant - preservation *) - -Module Raw (Import I:Int)(X: OrderedType). -Local Open Scope pair_scope. -Local Open Scope lazy_bool_scope. -Local Open Scope Int_scope. -Local Notation int := I.t. - -Definition key := X.t. -Hint Transparent key. - -(** * Trees *) - -Section Elt. - -Variable elt : Type. - -(** * Trees - - The fifth field of [Node] is the height of the tree *) - -Inductive tree := - | Leaf : tree - | Node : tree -> key -> elt -> tree -> int -> tree. - -Notation t := tree. - -(** * Basic functions on trees: height and cardinal *) - -Definition height (m : t) : int := - match m with - | Leaf => 0 - | Node _ _ _ _ h => h - end. - -Fixpoint cardinal (m : t) : nat := - match m with - | Leaf => 0%nat - | Node l _ _ r _ => S (cardinal l + cardinal r) - end. - -(** * Empty Map *) - -Definition empty := Leaf. - -(** * Emptyness test *) - -Definition is_empty m := match m with Leaf => true | _ => false end. - -(** * Membership *) - -(** The [mem] function is deciding membership. It exploits the [Bst] property - to achieve logarithmic complexity. *) - -Fixpoint mem x m : bool := - match m with - | Leaf => false - | Node l y _ r _ => - match X.compare x y with - | Eq => true - | Lt => mem x l - | Gt => mem x r - end - end. - -Fixpoint find x m : option elt := - match m with - | Leaf => None - | Node l y d r _ => - match X.compare x y with - | Eq => Some d - | Lt => find x l - | Gt => find x r - end - end. - -(** * Helper functions *) - -(** [create l x r] creates a node, assuming [l] and [r] - to be balanced and [|height l - height r| <= 2]. *) - -Definition create l x e r := - Node l x e r (max (height l) (height r) + 1). - -(** [bal l x e r] acts as [create], but performs one step of - rebalancing if necessary, i.e. assumes [|height l - height r| <= 3]. *) - -Definition assert_false := create. - -Fixpoint bal l x d r := - let hl := height l in - let hr := height r in - if (hr+2) <? hl then - match l with - | Leaf => assert_false l x d r - | Node ll lx ld lr _ => - if (height lr) <=? (height ll) then - create ll lx ld (create lr x d r) - else - match lr with - | Leaf => assert_false l x d r - | Node lrl lrx lrd lrr _ => - create (create ll lx ld lrl) lrx lrd (create lrr x d r) - end - end - else - if (hl+2) <? hr then - match r with - | Leaf => assert_false l x d r - | Node rl rx rd rr _ => - if (height rl) <=? (height rr) then - create (create l x d rl) rx rd rr - else - match rl with - | Leaf => assert_false l x d r - | Node rll rlx rld rlr _ => - create (create l x d rll) rlx rld (create rlr rx rd rr) - end - end - else - create l x d r. - -(** * Insertion *) - -Fixpoint add x d m := - match m with - | Leaf => Node Leaf x d Leaf 1 - | Node l y d' r h => - match X.compare x y with - | Eq => Node l y d r h - | Lt => bal (add x d l) y d' r - | Gt => bal l y d' (add x d r) - end - end. - -(** * Extraction of minimum binding - - Morally, [remove_min] is to be applied to a non-empty tree - [t = Node l x e r h]. Since we can't deal here with [assert false] - for [t=Leaf], we pre-unpack [t] (and forget about [h]). -*) - -Fixpoint remove_min l x d r : t*(key*elt) := - match l with - | Leaf => (r,(x,d)) - | Node ll lx ld lr lh => - let (l',m) := remove_min ll lx ld lr in - (bal l' x d r, m) - end. - -(** * Merging two trees - - [merge0 t1 t2] builds the union of [t1] and [t2] assuming all elements - of [t1] to be smaller than all elements of [t2], and - [|height t1 - height t2| <= 2]. -*) - -Definition merge0 s1 s2 := - match s1,s2 with - | Leaf, _ => s2 - | _, Leaf => s1 - | _, Node l2 x2 d2 r2 h2 => - let '(s2',(x,d)) := remove_min l2 x2 d2 r2 in - bal s1 x d s2' - end. - -(** * Deletion *) - -Fixpoint remove x m := match m with - | Leaf => Leaf - | Node l y d r h => - match X.compare x y with - | Eq => merge0 l r - | Lt => bal (remove x l) y d r - | Gt => bal l y d (remove x r) - end - end. - -(** * join - - Same as [bal] but does not assume anything regarding heights of [l] - and [r]. -*) - -Fixpoint join l : key -> elt -> t -> t := - match l with - | Leaf => add - | Node ll lx ld lr lh => fun x d => - fix join_aux (r:t) : t := match r with - | Leaf => add x d l - | Node rl rx rd rr rh => - if rh+2 <? lh then bal ll lx ld (join lr x d r) - else if lh+2 <? rh then bal (join_aux rl) rx rd rr - else create l x d r - end - end. - -(** * Splitting - - [split x m] returns a triple [(l, o, r)] where - - [l] is the set of elements of [m] that are [< x] - - [r] is the set of elements of [m] that are [> x] - - [o] is the result of [find x m]. -*) - -Record triple := mktriple { t_left:t; t_opt:option elt; t_right:t }. -Notation "〚 l , b , r 〛" := (mktriple l b r) (at level 9). - -Fixpoint split x m : triple := match m with - | Leaf => 〚 Leaf, None, Leaf 〛 - | Node l y d r h => - match X.compare x y with - | Lt => let (ll,o,rl) := split x l in 〚 ll, o, join rl y d r 〛 - | Eq => 〚 l, Some d, r 〛 - | Gt => let (rl,o,rr) := split x r in 〚 join l y d rl, o, rr 〛 - end - end. - -(** * Concatenation - - Same as [merge] but does not assume anything about heights. -*) - -Definition concat m1 m2 := - match m1, m2 with - | Leaf, _ => m2 - | _ , Leaf => m1 - | _, Node l2 x2 d2 r2 _ => - let (m2',xd) := remove_min l2 x2 d2 r2 in - join m1 xd#1 xd#2 m2' - end. - -(** * Bindings *) - -(** [bindings_aux acc t] catenates the bindings of [t] in infix - order to the list [acc] *) - -Fixpoint bindings_aux (acc : list (key*elt)) m : list (key*elt) := - match m with - | Leaf => acc - | Node l x d r _ => bindings_aux ((x,d) :: bindings_aux acc r) l - end. - -(** then [bindings] is an instantiation with an empty [acc] *) - -Definition bindings := bindings_aux nil. - -(** * Fold *) - -Fixpoint fold {A} (f : key -> elt -> A -> A) (m : t) : A -> A := - fun a => match m with - | Leaf => a - | Node l x d r _ => fold f r (f x d (fold f l a)) - end. - -(** * Comparison *) - -Variable cmp : elt->elt->bool. - -(** ** Enumeration of the elements of a tree *) - -Inductive enumeration := - | End : enumeration - | More : key -> elt -> t -> enumeration -> enumeration. - -(** [cons m e] adds the elements of tree [m] on the head of - enumeration [e]. *) - -Fixpoint cons m e : enumeration := - match m with - | Leaf => e - | Node l x d r h => cons l (More x d r e) - end. - -(** One step of comparison of elements *) - -Definition equal_more x1 d1 (cont:enumeration->bool) e2 := - match e2 with - | End => false - | More x2 d2 r2 e2 => - match X.compare x1 x2 with - | Eq => cmp d1 d2 &&& cont (cons r2 e2) - | _ => false - end - end. - -(** Comparison of left tree, middle element, then right tree *) - -Fixpoint equal_cont m1 (cont:enumeration->bool) e2 := - match m1 with - | Leaf => cont e2 - | Node l1 x1 d1 r1 _ => - equal_cont l1 (equal_more x1 d1 (equal_cont r1 cont)) e2 - end. - -(** Initial continuation *) - -Definition equal_end e2 := match e2 with End => true | _ => false end. - -(** The complete comparison *) - -Definition equal m1 m2 := equal_cont m1 equal_end (cons m2 End). - -End Elt. -Notation t := tree. -Notation "〚 l , b , r 〛" := (mktriple l b r) (at level 9). -Notation "t #l" := (t_left t) (at level 9, format "t '#l'"). -Notation "t #o" := (t_opt t) (at level 9, format "t '#o'"). -Notation "t #r" := (t_right t) (at level 9, format "t '#r'"). - - -(** * Map *) - -Fixpoint map (elt elt' : Type)(f : elt -> elt')(m : t elt) : t elt' := - match m with - | Leaf _ => Leaf _ - | Node l x d r h => Node (map f l) x (f d) (map f r) h - end. - -(* * Mapi *) - -Fixpoint mapi (elt elt' : Type)(f : key -> elt -> elt')(m : t elt) : t elt' := - match m with - | Leaf _ => Leaf _ - | Node l x d r h => Node (mapi f l) x (f x d) (mapi f r) h - end. - -(** * Map with removal *) - -Fixpoint mapo (elt elt' : Type)(f : key -> elt -> option elt')(m : t elt) - : t elt' := - match m with - | Leaf _ => Leaf _ - | Node l x d r h => - match f x d with - | Some d' => join (mapo f l) x d' (mapo f r) - | None => concat (mapo f l) (mapo f r) - end - end. - -(** * Generalized merge - - Suggestion by B. Gregoire: a [merge] function with specialized - arguments that allows bypassing some tree traversal. Instead of one - [f0] of type [key -> option elt -> option elt' -> option elt''], - we ask here for: - - [f] which is a specialisation of [f0] when first option isn't [None] - - [mapl] treats a [tree elt] with [f0] when second option is [None] - - [mapr] treats a [tree elt'] with [f0] when first option is [None] - - The idea is that [mapl] and [mapr] can be instantaneous (e.g. - the identity or some constant function). -*) - -Section GMerge. -Variable elt elt' elt'' : Type. -Variable f : key -> elt -> option elt' -> option elt''. -Variable mapl : t elt -> t elt''. -Variable mapr : t elt' -> t elt''. - -Fixpoint gmerge m1 m2 := - match m1, m2 with - | Leaf _, _ => mapr m2 - | _, Leaf _ => mapl m1 - | Node l1 x1 d1 r1 h1, _ => - let (l2',o2,r2') := split x1 m2 in - match f x1 d1 o2 with - | Some e => join (gmerge l1 l2') x1 e (gmerge r1 r2') - | None => concat (gmerge l1 l2') (gmerge r1 r2') - end - end. - -End GMerge. - -(** * Merge - - The [merge] function of the Map interface can be implemented - via [gmerge] and [mapo]. -*) - -Section Merge. -Variable elt elt' elt'' : Type. -Variable f : key -> option elt -> option elt' -> option elt''. - -Definition merge : t elt -> t elt' -> t elt'' := - gmerge - (fun k d o => f k (Some d) o) - (mapo (fun k d => f k (Some d) None)) - (mapo (fun k d' => f k None (Some d'))). - -End Merge. - - - -(** * Invariants *) - -Section Invariants. -Variable elt : Type. - -(** ** Occurrence in a tree *) - -Inductive MapsTo (x : key)(e : elt) : t elt -> Prop := - | MapsRoot : forall l r h y, - X.eq x y -> MapsTo x e (Node l y e r h) - | MapsLeft : forall l r h y e', - MapsTo x e l -> MapsTo x e (Node l y e' r h) - | MapsRight : forall l r h y e', - MapsTo x e r -> MapsTo x e (Node l y e' r h). - -Inductive In (x : key) : t elt -> Prop := - | InRoot : forall l r h y e, - X.eq x y -> In x (Node l y e r h) - | InLeft : forall l r h y e', - In x l -> In x (Node l y e' r h) - | InRight : forall l r h y e', - In x r -> In x (Node l y e' r h). - -Definition In0 k m := exists e:elt, MapsTo k e m. - -(** ** Binary search trees *) - -(** [Above x m] : [x] is strictly greater than any key in [m]. - [Below x m] : [x] is strictly smaller than any key in [m]. *) - -Inductive Above (x:key) : t elt -> Prop := - | AbLeaf : Above x (Leaf _) - | AbNode l r h y e : Above x l -> X.lt y x -> Above x r -> - Above x (Node l y e r h). - -Inductive Below (x:key) : t elt -> Prop := - | BeLeaf : Below x (Leaf _) - | BeNode l r h y e : Below x l -> X.lt x y -> Below x r -> - Below x (Node l y e r h). - -Definition Apart (m1 m2 : t elt) : Prop := - forall x1 x2, In x1 m1 -> In x2 m2 -> X.lt x1 x2. - -(** Alternative statements, equivalent with [LtTree] and [GtTree] *) - -Definition lt_tree x m := forall y, In y m -> X.lt y x. -Definition gt_tree x m := forall y, In y m -> X.lt x y. - -(** [Bst t] : [t] is a binary search tree *) - -Inductive Bst : t elt -> Prop := - | BSLeaf : Bst (Leaf _) - | BSNode : forall x e l r h, Bst l -> Bst r -> - Above x l -> Below x r -> Bst (Node l x e r h). - -End Invariants. - - -(** * Correctness proofs, isolated in a sub-module *) - -Module Proofs. - Module MX := OrderedTypeFacts X. - Module PX := KeyOrderedType X. - Module L := MMapList.Raw X. - -Local Infix "∈" := In (at level 70). -Local Infix "==" := X.eq (at level 70). -Local Infix "<" := X.lt (at level 70). -Local Infix "<<" := Below (at level 70). -Local Infix ">>" := Above (at level 70). -Local Infix "<<<" := Apart (at level 70). - -Scheme tree_ind := Induction for tree Sort Prop. -Scheme Bst_ind := Induction for Bst Sort Prop. -Scheme MapsTo_ind := Induction for MapsTo Sort Prop. -Scheme In_ind := Induction for In Sort Prop. -Scheme Above_ind := Induction for Above Sort Prop. -Scheme Below_ind := Induction for Below Sort Prop. - -Functional Scheme mem_ind := Induction for mem Sort Prop. -Functional Scheme find_ind := Induction for find Sort Prop. -Functional Scheme bal_ind := Induction for bal Sort Prop. -Functional Scheme add_ind := Induction for add Sort Prop. -Functional Scheme remove_min_ind := Induction for remove_min Sort Prop. -Functional Scheme merge0_ind := Induction for merge0 Sort Prop. -Functional Scheme remove_ind := Induction for remove Sort Prop. -Functional Scheme concat_ind := Induction for concat Sort Prop. -Functional Scheme split_ind := Induction for split Sort Prop. -Functional Scheme mapo_ind := Induction for mapo Sort Prop. -Functional Scheme gmerge_ind := Induction for gmerge Sort Prop. - -(** * Automation and dedicated tactics. *) - -Local Hint Constructors tree MapsTo In Bst Above Below. -Local Hint Unfold lt_tree gt_tree Apart. -Local Hint Immediate MX.eq_sym. -Local Hint Resolve MX.eq_refl MX.eq_trans MX.lt_trans. - -Tactic Notation "factornode" ident(s) := - try clear s; - match goal with - | |- context [Node ?l ?x ?e ?r ?h] => - set (s:=Node l x e r h) in *; clearbody s; clear l x e r h - | _ : context [Node ?l ?x ?e ?r ?h] |- _ => - set (s:=Node l x e r h) in *; clearbody s; clear l x e r h - end. - -(** A tactic for cleaning hypothesis after use of functional induction. *) - -Ltac cleanf := - match goal with - | H : X.compare _ _ = Eq |- _ => - rewrite ?H; apply MX.compare_eq in H; cleanf - | H : X.compare _ _ = Lt |- _ => - rewrite ?H; apply MX.compare_lt_iff in H; cleanf - | H : X.compare _ _ = Gt |- _ => - rewrite ?H; apply MX.compare_gt_iff in H; cleanf - | _ => idtac - end. - - -(** A tactic to repeat [inversion_clear] on all hyps of the - form [(f (Node ...))] *) - -Ltac inv f := - match goal with - | H:f (Leaf _) |- _ => inversion_clear H; inv f - | H:f _ (Leaf _) |- _ => inversion_clear H; inv f - | H:f _ _ (Leaf _) |- _ => inversion_clear H; inv f - | H:f _ _ _ (Leaf _) |- _ => inversion_clear H; inv f - | H:f (Node _ _ _ _ _) |- _ => inversion_clear H; inv f - | H:f _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f - | H:f _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f - | H:f _ _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f - | _ => idtac - end. - -Ltac inv_all f := - match goal with - | H: f _ |- _ => inversion_clear H; inv f - | H: f _ _ |- _ => inversion_clear H; inv f - | H: f _ _ _ |- _ => inversion_clear H; inv f - | H: f _ _ _ _ |- _ => inversion_clear H; inv f - | _ => idtac - end. - -Ltac intuition_in := repeat (intuition; inv In; inv MapsTo). - -(* Function/Functional Scheme can't deal with internal fix. - Let's do its job by hand: *) - -Ltac join_tac l x d r := - revert x d r; - induction l as [| ll _ lx ld lr Hlr lh]; - [ | intros x d r; induction r as [| rl Hrl rx rd rr _ rh]; unfold join; - [ | destruct (rh+2 <? lh) eqn:LT; - [ match goal with |- context [ bal ?u ?v ?w ?z ] => - replace (bal u v w z) - with (bal ll lx ld (join lr x d (Node rl rx rd rr rh))); [ | auto] - end - | destruct (lh+2 <? rh) eqn:LT'; - [ match goal with |- context [ bal ?u ?v ?w ?z ] => - replace (bal u v w z) - with (bal (join (Node ll lx ld lr lh) x d rl) rx rd rr); [ | auto] - end - | ] ] ] ]; intros. - -Ltac cleansplit := - simpl; cleanf; inv Bst; - match goal with - | E:split _ _ = 〚 ?l, ?o, ?r 〛 |- _ => - change l with (〚l,o,r〛#l); rewrite <- ?E; - change o with (〚l,o,r〛#o); rewrite <- ?E; - change r with (〚l,o,r〛#r); rewrite <- ?E - | _ => idtac - end. - -(** * Basic results about [MapsTo], [In], [lt_tree], [gt_tree], [height] *) - -(** Facts about [MapsTo] and [In]. *) - -Lemma MapsTo_In {elt} k (e:elt) m : MapsTo k e m -> k ∈ m. -Proof. - induction 1; auto. -Qed. -Local Hint Resolve MapsTo_In. - -Lemma In_MapsTo {elt} k m : k ∈ m -> exists (e:elt), MapsTo k e m. -Proof. - induction 1; try destruct IHIn as (e,He); exists e; auto. -Qed. - -Lemma In_alt {elt} k (m:t elt) : In0 k m <-> k ∈ m. -Proof. - split. - intros (e,H); eauto. - unfold In0; apply In_MapsTo; auto. -Qed. - -Lemma MapsTo_1 {elt} m x y (e:elt) : - x == y -> MapsTo x e m -> MapsTo y e m. -Proof. - induction m; simpl; intuition_in; eauto. -Qed. -Hint Immediate MapsTo_1. - -Instance MapsTo_compat {elt} : - Proper (X.eq==>Logic.eq==>Logic.eq==>iff) (@MapsTo elt). -Proof. - intros x x' Hx e e' He m m' Hm. subst. - split; now apply MapsTo_1. -Qed. - -Instance In_compat {elt} : - Proper (X.eq==>Logic.eq==>iff) (@In elt). -Proof. - intros x x' H m m' <-. - induction m; simpl; intuition_in; eauto. -Qed. - -Lemma In_node_iff {elt} l x (e:elt) r h y : - y ∈ (Node l x e r h) <-> y ∈ l \/ y == x \/ y ∈ r. -Proof. - intuition_in. -Qed. - -(** Results about [Above] and [Below] *) - -Lemma above {elt} (m:t elt) x : - x >> m <-> forall y, y ∈ m -> y < x. -Proof. - split. - - induction 1; intuition_in; MX.order. - - induction m; constructor; auto. -Qed. - -Lemma below {elt} (m:t elt) x : - x << m <-> forall y, y ∈ m -> x < y. -Proof. - split. - - induction 1; intuition_in; MX.order. - - induction m; constructor; auto. -Qed. - -Lemma AboveLt {elt} (m:t elt) x y : x >> m -> y ∈ m -> y < x. -Proof. - rewrite above; intuition. -Qed. - -Lemma BelowGt {elt} (m:t elt) x y : x << m -> y ∈ m -> x < y. -Proof. - rewrite below; intuition. -Qed. - -Lemma Above_not_In {elt} (m:t elt) x : x >> m -> ~ x ∈ m. -Proof. - induction 1; intuition_in; MX.order. -Qed. - -Lemma Below_not_In {elt} (m:t elt) x : x << m -> ~ x ∈ m. -Proof. - induction 1; intuition_in; MX.order. -Qed. - -Lemma Above_trans {elt} (m:t elt) x y : x < y -> x >> m -> y >> m. -Proof. - induction 2; constructor; trivial; MX.order. -Qed. - -Lemma Below_trans {elt} (m:t elt) x y : y < x -> x << m -> y << m. -Proof. - induction 2; constructor; trivial; MX.order. -Qed. - -Local Hint Resolve - AboveLt Above_not_In Above_trans - BelowGt Below_not_In Below_trans. - -(** Helper tactic concerning order of elements. *) - -Ltac order := match goal with - | U: _ >> ?m, V: _ ∈ ?m |- _ => - generalize (AboveLt U V); clear U; order - | U: _ << ?m, V: _ ∈ ?m |- _ => - generalize (BelowGt U V); clear U; order - | U: _ >> ?m, V: MapsTo _ _ ?m |- _ => - generalize (AboveLt U (MapsTo_In V)); clear U; order - | U: _ << ?m, V: MapsTo _ _ ?m |- _ => - generalize (BelowGt U (MapsTo_In V)); clear U; order - | _ => MX.order -end. - -Lemma between {elt} (m m':t elt) x : - x >> m -> x << m' -> m <<< m'. -Proof. - intros H H' y y' Hy Hy'. order. -Qed. - -Section Elt. -Variable elt:Type. -Implicit Types m r : t elt. - -(** * Membership *) - -Lemma find_1 m x e : Bst m -> MapsTo x e m -> find x m = Some e. -Proof. - functional induction (find x m); cleanf; - intros; inv Bst; intuition_in; order. -Qed. - -Lemma find_2 m x e : find x m = Some e -> MapsTo x e m. -Proof. - functional induction (find x m); cleanf; subst; intros; auto. - - discriminate. - - injection H as ->. auto. -Qed. - -Lemma find_spec m x e : Bst m -> - (find x m = Some e <-> MapsTo x e m). -Proof. - split; auto using find_1, find_2. -Qed. - -Lemma find_in m x : find x m <> None -> x ∈ m. -Proof. - destruct (find x m) eqn:F; intros H. - - apply MapsTo_In with e. now apply find_2. - - now elim H. -Qed. - -Lemma in_find m x : Bst m -> x ∈ m -> find x m <> None. -Proof. - intros H H'. - destruct (In_MapsTo H') as (d,Hd). - now rewrite (find_1 H Hd). -Qed. - -Lemma find_in_iff m x : Bst m -> - (find x m <> None <-> x ∈ m). -Proof. - split; auto using find_in, in_find. -Qed. - -Lemma not_find_iff m x : Bst m -> - (find x m = None <-> ~ x ∈ m). -Proof. - intros H. rewrite <- find_in_iff; trivial. - destruct (find x m); split; try easy. now destruct 1. -Qed. - -Lemma eq_option_alt (o o':option elt) : - o=o' <-> (forall e, o=Some e <-> o'=Some e). -Proof. -split; intros. -- now subst. -- destruct o, o'; rewrite ?H; auto. symmetry; now apply H. -Qed. - -Lemma find_mapsto_equiv : forall m m' x, Bst m -> Bst m' -> - (find x m = find x m' <-> - (forall d, MapsTo x d m <-> MapsTo x d m')). -Proof. - intros m m' x Hm Hm'. rewrite eq_option_alt. - split; intros H d. now rewrite <- 2 find_spec. now rewrite 2 find_spec. -Qed. - -Lemma find_in_equiv : forall m m' x, Bst m -> Bst m' -> - find x m = find x m' -> - (x ∈ m <-> x ∈ m'). -Proof. - split; intros; apply find_in; [ rewrite <- H1 | rewrite H1 ]; - apply in_find; auto. -Qed. - -Lemma find_compat m x x' : Bst m -> X.eq x x' -> find x m = find x' m. -Proof. - intros B E. - destruct (find x' m) eqn:H. - - apply find_1; trivial. rewrite E. now apply find_2. - - rewrite not_find_iff in *; trivial. now rewrite E. -Qed. - -Lemma mem_spec m x : Bst m -> mem x m = true <-> x ∈ m. -Proof. - functional induction (mem x m); auto; intros; cleanf; - inv Bst; intuition_in; try discriminate; order. -Qed. - -(** * Empty map *) - -Lemma empty_bst : Bst (empty elt). -Proof. - constructor. -Qed. - -Lemma empty_spec x : find x (empty elt) = None. -Proof. - reflexivity. -Qed. - -(** * Emptyness test *) - -Lemma is_empty_spec m : is_empty m = true <-> forall x, find x m = None. -Proof. - destruct m as [|r x e l h]; simpl; split; try easy. - intros H. specialize (H x). now rewrite MX.compare_refl in H. -Qed. - -(** * Helper functions *) - -Lemma create_bst l x e r : - Bst l -> Bst r -> x >> l -> x << r -> Bst (create l x e r). -Proof. - unfold create; auto. -Qed. -Hint Resolve create_bst. - -Lemma create_in l x e r y : - y ∈ (create l x e r) <-> y == x \/ y ∈ l \/ y ∈ r. -Proof. - unfold create; split; [ inversion_clear 1 | ]; intuition. -Qed. - -Lemma bal_bst l x e r : Bst l -> Bst r -> - x >> l -> x << r -> Bst (bal l x e r). -Proof. - functional induction (bal l x e r); intros; cleanf; - inv Bst; inv Above; inv Below; - repeat apply create_bst; auto; unfold create; constructor; eauto. -Qed. -Hint Resolve bal_bst. - -Lemma bal_in l x e r y : - y ∈ (bal l x e r) <-> y == x \/ y ∈ l \/ y ∈ r. -Proof. - functional induction (bal l x e r); intros; cleanf; - rewrite !create_in; intuition_in. -Qed. - -Lemma bal_mapsto l x e r y e' : - MapsTo y e' (bal l x e r) <-> MapsTo y e' (create l x e r). -Proof. - functional induction (bal l x e r); intros; cleanf; - unfold assert_false, create; intuition_in. -Qed. - -Lemma bal_find l x e r y : - Bst l -> Bst r -> x >> l -> x << r -> - find y (bal l x e r) = find y (create l x e r). -Proof. - functional induction (bal l x e r); intros; cleanf; trivial; - inv Bst; inv Above; inv Below; - simpl; repeat case X.compare_spec; intuition; order. -Qed. - -(** * Insertion *) - -Lemma add_in m x y e : - y ∈ (add x e m) <-> y == x \/ y ∈ m. -Proof. - functional induction (add x e m); auto; intros; cleanf; - rewrite ?bal_in; intuition_in. setoid_replace y with x; auto. -Qed. - -Lemma add_lt m x e y : y >> m -> x < y -> y >> add x e m. -Proof. - intros. apply above. intros z. rewrite add_in. destruct 1; order. -Qed. - -Lemma add_gt m x e y : y << m -> y < x -> y << add x e m. -Proof. - intros. apply below. intros z. rewrite add_in. destruct 1; order. -Qed. - -Lemma add_bst m x e : Bst m -> Bst (add x e m). -Proof. - functional induction (add x e m); intros; cleanf; - inv Bst; try apply bal_bst; auto using add_lt, add_gt. -Qed. -Hint Resolve add_lt add_gt add_bst. - -Lemma add_spec1 m x e : Bst m -> find x (add x e m) = Some e. -Proof. - functional induction (add x e m); simpl; intros; cleanf; trivial. - - now rewrite MX.compare_refl. - - inv Bst. rewrite bal_find; auto. - simpl. case X.compare_spec; try order; auto. - - inv Bst. rewrite bal_find; auto. - simpl. case X.compare_spec; try order; auto. -Qed. - -Lemma add_spec2 m x y e : Bst m -> ~ x == y -> - find y (add x e m) = find y m. -Proof. - functional induction (add x e m); simpl; intros; cleanf; trivial. - - case X.compare_spec; trivial; order. - - case X.compare_spec; trivial; order. - - inv Bst. rewrite bal_find by auto. simpl. now rewrite IHt. - - inv Bst. rewrite bal_find by auto. simpl. now rewrite IHt. -Qed. - -Lemma add_find m x y e : Bst m -> - find y (add x e m) = - match X.compare y x with Eq => Some e | _ => find y m end. -Proof. - intros. - case X.compare_spec; intros. - - apply find_spec; auto. rewrite H0. apply find_spec; auto. - now apply add_spec1. - - apply add_spec2; trivial; order. - - apply add_spec2; trivial; order. -Qed. - -(** * Extraction of minimum binding *) - -Definition RemoveMin m res := - match m with - | Leaf _ => False - | Node l x e r h => remove_min l x e r = res - end. - -Lemma RemoveMin_step l x e r h m' p : - RemoveMin (Node l x e r h) (m',p) -> - (l = Leaf _ /\ m' = r /\ p = (x,e) \/ - exists m0, RemoveMin l (m0,p) /\ m' = bal m0 x e r). -Proof. - simpl. destruct l as [|ll lx le lr lh]; simpl. - - intros [= -> ->]. now left. - - destruct (remove_min ll lx le lr) as (l',p'). - intros [= <- <-]. right. now exists l'. -Qed. - -Lemma remove_min_mapsto m m' p : RemoveMin m (m',p) -> - forall y e, - MapsTo y e m <-> (y == p#1 /\ e = p#2) \/ MapsTo y e m'. -Proof. - revert m'. - induction m as [|l IH x d r _ h]; [destruct 1|]. - intros m' R. apply RemoveMin_step in R. - destruct R as [(->,(->,->))|[m0 (R,->)]]; intros y e; simpl. - - intuition_in. subst. now constructor. - - rewrite bal_mapsto. unfold create. specialize (IH _ R y e). - intuition_in. -Qed. - -Lemma remove_min_in m m' p : RemoveMin m (m',p) -> - forall y, y ∈ m <-> y == p#1 \/ y ∈ m'. -Proof. - revert m'. - induction m as [|l IH x e r _ h]; [destruct 1|]. - intros m' R y. apply RemoveMin_step in R. - destruct R as [(->,(->,->))|[m0 (R,->)]]. - + intuition_in. - + rewrite bal_in, In_node_iff, (IH _ R); intuition. -Qed. - -Lemma remove_min_lt m m' p : RemoveMin m (m',p) -> - forall y, y >> m -> y >> m'. -Proof. - intros R y L. apply above. intros z Hz. - apply (AboveLt L). - apply (remove_min_in R). now right. -Qed. - -Lemma remove_min_gt m m' p : RemoveMin m (m',p) -> - Bst m -> p#1 << m'. -Proof. - revert m'. - induction m as [|l IH x e r _ h]; [destruct 1|]. - intros m' R H. inv Bst. apply RemoveMin_step in R. - destruct R as [(_,(->,->))|[m0 (R,->)]]; auto. - assert (p#1 << m0) by now apply IH. - assert (In p#1 l) by (apply (remove_min_in R); now left). - apply below. intros z. rewrite bal_in. - intuition_in; order. -Qed. - -Lemma remove_min_bst m m' p : RemoveMin m (m',p) -> - Bst m -> Bst m'. -Proof. - revert m'. - induction m as [|l IH x e r _ h]; [destruct 1|]. - intros m' R H. inv Bst. apply RemoveMin_step in R. - destruct R as [(_,(->,->))|[m0 (R,->)]]; auto. - apply bal_bst; eauto using remove_min_lt. -Qed. - -Lemma remove_min_find m m' p : RemoveMin m (m',p) -> - Bst m -> - forall y, - find y m = - match X.compare y p#1 with - | Eq => Some p#2 - | Lt => None - | Gt => find y m' - end. -Proof. - revert m'. - induction m as [|l IH x e r _ h]; [destruct 1|]. - intros m' R B y. inv Bst. apply RemoveMin_step in R. - destruct R as [(->,(->,->))|[m0 (R,->)]]; auto. - assert (Bst m0) by now apply (remove_min_bst R). - assert (p#1 << m0) by now apply (remove_min_gt R). - assert (x >> m0) by now apply (remove_min_lt R). - assert (In p#1 l) by (apply (remove_min_in R); now left). - simpl in *. - rewrite (IH _ R), bal_find by trivial. clear IH. simpl. - do 2 case X.compare_spec; trivial; try order. -Qed. - -(** * Merging two trees *) - -Ltac factor_remove_min m R := match goal with - | h:int, H:remove_min ?l ?x ?e ?r = ?p |- _ => - assert (R:RemoveMin (Node l x e r h) p) by exact H; - set (m:=Node l x e r h) in *; clearbody m; clear H l x e r -end. - -Lemma merge0_in m1 m2 y : - y ∈ (merge0 m1 m2) <-> y ∈ m1 \/ y ∈ m2. -Proof. - functional induction (merge0 m1 m2); intros; try factornode m1. - - intuition_in. - - intuition_in. - - factor_remove_min l R. rewrite bal_in, (remove_min_in R). - simpl; intuition. -Qed. - -Lemma merge0_mapsto m1 m2 y e : - MapsTo y e (merge0 m1 m2) <-> MapsTo y e m1 \/ MapsTo y e m2. -Proof. - functional induction (merge0 m1 m2); intros; try factornode m1. - - intuition_in. - - intuition_in. - - factor_remove_min l R. rewrite bal_mapsto, (remove_min_mapsto R). - simpl. unfold create; intuition_in. subst. now constructor. -Qed. - -Lemma merge0_bst m1 m2 : Bst m1 -> Bst m2 -> m1 <<< m2 -> - Bst (merge0 m1 m2). -Proof. - functional induction (merge0 m1 m2); intros B1 B2 B12; trivial. - factornode m1. factor_remove_min l R. - apply bal_bst; auto. - - eapply remove_min_bst; eauto. - - apply above. intros z Hz. apply B12; trivial. - rewrite (remove_min_in R). now left. - - now apply (remove_min_gt R). -Qed. -Hint Resolve merge0_bst. - -(** * Deletion *) - -Lemma remove_in m x y : Bst m -> - (y ∈ remove x m <-> ~ y == x /\ y ∈ m). -Proof. - functional induction (remove x m); simpl; intros; cleanf; inv Bst; - rewrite ?merge0_in, ?bal_in, ?IHt; intuition_in; order. -Qed. - -Lemma remove_lt m x y : Bst m -> y >> m -> y >> remove x m. -Proof. - intros. apply above. intro. rewrite remove_in by trivial. - destruct 1; order. -Qed. - -Lemma remove_gt m x y : Bst m -> y << m -> y << remove x m. -Proof. - intros. apply below. intro. rewrite remove_in by trivial. - destruct 1; order. -Qed. - -Lemma remove_bst m x : Bst m -> Bst (remove x m). -Proof. - functional induction (remove x m); simpl; intros; cleanf; inv Bst. - - trivial. - - apply merge0_bst; eauto. - - apply bal_bst; auto using remove_lt. - - apply bal_bst; auto using remove_gt. -Qed. -Hint Resolve remove_bst remove_gt remove_lt. - -Lemma remove_spec1 m x : Bst m -> find x (remove x m) = None. -Proof. - intros. apply not_find_iff; auto. rewrite remove_in; intuition. -Qed. - -Lemma remove_spec2 m x y : Bst m -> ~ x == y -> - find y (remove x m) = find y m. -Proof. - functional induction (remove x m); simpl; intros; cleanf; inv Bst. - - trivial. - - case X.compare_spec; intros; try order; - rewrite find_mapsto_equiv; auto. - + intros. rewrite merge0_mapsto; intuition; order. - + apply merge0_bst; auto. red; intros; transitivity y0; order. - + intros. rewrite merge0_mapsto; intuition; order. - + apply merge0_bst; auto. now apply between with y0. - - rewrite bal_find by auto. simpl. case X.compare_spec; auto. - - rewrite bal_find by auto. simpl. case X.compare_spec; auto. -Qed. - -(** * join *) - -Lemma join_in l x d r y : - y ∈ (join l x d r) <-> y == x \/ y ∈ l \/ y ∈ r. -Proof. - join_tac l x d r. - - simpl join. rewrite add_in. intuition_in. - - rewrite add_in. intuition_in. - - rewrite bal_in, Hlr. clear Hlr Hrl. intuition_in. - - rewrite bal_in, Hrl; clear Hlr Hrl; intuition_in. - - apply create_in. -Qed. - -Lemma join_bst l x d r : - Bst (create l x d r) -> Bst (join l x d r). -Proof. - join_tac l x d r; unfold create in *; - inv Bst; inv Above; inv Below; auto. - - simpl. auto. - - apply bal_bst; auto. - apply below. intro. rewrite join_in. intuition_in; order. - - apply bal_bst; auto. - apply above. intro. rewrite join_in. intuition_in; order. -Qed. -Hint Resolve join_bst. - -Lemma join_find l x d r y : - Bst (create l x d r) -> - find y (join l x d r) = find y (create l x d r). -Proof. - unfold create at 1. - join_tac l x d r; trivial. - - simpl in *. inv Bst. - rewrite add_find; trivial. - case X.compare_spec; intros; trivial. - apply not_find_iff; auto. intro. order. - - clear Hlr. factornode l. simpl. inv Bst. - rewrite add_find by auto. - case X.compare_spec; intros; trivial. - apply not_find_iff; auto. intro. order. - - clear Hrl LT. factornode r. inv Bst; inv Above; inv Below. - rewrite bal_find; auto; simpl. - + rewrite Hlr; auto; simpl. - repeat (case X.compare_spec; trivial; try order). - + apply below. intro. rewrite join_in. intuition_in; order. - - clear Hlr LT LT'. factornode l. inv Bst; inv Above; inv Below. - rewrite bal_find; auto; simpl. - + rewrite Hrl; auto; simpl. - repeat (case X.compare_spec; trivial; try order). - + apply above. intro. rewrite join_in. intuition_in; order. -Qed. - -(** * split *) - -Lemma split_in_l0 m x y : y ∈ (split x m)#l -> y ∈ m. -Proof. - functional induction (split x m); cleansplit; - rewrite ?join_in; intuition. -Qed. - -Lemma split_in_r0 m x y : y ∈ (split x m)#r -> y ∈ m. -Proof. - functional induction (split x m); cleansplit; - rewrite ?join_in; intuition. -Qed. - -Lemma split_in_l m x y : Bst m -> - (y ∈ (split x m)#l <-> y ∈ m /\ y < x). -Proof. - functional induction (split x m); intros; cleansplit; - rewrite ?join_in, ?IHt; intuition_in; order. -Qed. - -Lemma split_in_r m x y : Bst m -> - (y ∈ (split x m)#r <-> y ∈ m /\ x < y). -Proof. - functional induction (split x m); intros; cleansplit; - rewrite ?join_in, ?IHt; intuition_in; order. -Qed. - -Lemma split_in_o m x : (split x m)#o = find x m. -Proof. - functional induction (split x m); intros; cleansplit; auto. -Qed. - -Lemma split_lt_l m x : Bst m -> x >> (split x m)#l. -Proof. - intro. apply above. intro. rewrite split_in_l; intuition; order. -Qed. - -Lemma split_lt_r m x y : y >> m -> y >> (split x m)#r. -Proof. - intro. apply above. intros z Hz. apply split_in_r0 in Hz. order. -Qed. - -Lemma split_gt_r m x : Bst m -> x << (split x m)#r. -Proof. - intro. apply below. intro. rewrite split_in_r; intuition; order. -Qed. - -Lemma split_gt_l m x y : y << m -> y << (split x m)#l. -Proof. - intro. apply below. intros z Hz. apply split_in_l0 in Hz. order. -Qed. -Hint Resolve split_lt_l split_lt_r split_gt_l split_gt_r. - -Lemma split_bst_l m x : Bst m -> Bst (split x m)#l. -Proof. - functional induction (split x m); intros; cleansplit; intuition; - auto using join_bst. -Qed. - -Lemma split_bst_r m x : Bst m -> Bst (split x m)#r. -Proof. - functional induction (split x m); intros; cleansplit; intuition; - auto using join_bst. -Qed. -Hint Resolve split_bst_l split_bst_r. - -Lemma split_find m x y : Bst m -> - find y m = match X.compare y x with - | Eq => (split x m)#o - | Lt => find y (split x m)#l - | Gt => find y (split x m)#r - end. -Proof. - functional induction (split x m); intros; cleansplit. - - now case X.compare. - - repeat case X.compare_spec; trivial; order. - - simpl in *. rewrite join_find, IHt; auto. - simpl. repeat case X.compare_spec; trivial; order. - - rewrite join_find, IHt; auto. - simpl; repeat case X.compare_spec; trivial; order. -Qed. - -(** * Concatenation *) - -Lemma concat_in m1 m2 y : - y ∈ (concat m1 m2) <-> y ∈ m1 \/ y ∈ m2. -Proof. - functional induction (concat m1 m2); intros; try factornode m1. - - intuition_in. - - intuition_in. - - factor_remove_min m2 R. - rewrite join_in, (remove_min_in R); simpl; intuition. -Qed. - -Lemma concat_bst m1 m2 : Bst m1 -> Bst m2 -> m1 <<< m2 -> - Bst (concat m1 m2). -Proof. - functional induction (concat m1 m2); intros B1 B2 LT; auto; - try factornode m1. - factor_remove_min m2 R. - apply join_bst, create_bst; auto. - - now apply (remove_min_bst R). - - apply above. intros y Hy. apply LT; trivial. - rewrite (remove_min_in R); now left. - - now apply (remove_min_gt R). -Qed. -Hint Resolve concat_bst. - -Definition oelse {A} (o1 o2:option A) := - match o1 with - | Some x => Some x - | None => o2 - end. - -Lemma concat_find m1 m2 y : Bst m1 -> Bst m2 -> m1 <<< m2 -> - find y (concat m1 m2) = oelse (find y m2) (find y m1). -Proof. - functional induction (concat m1 m2); intros B1 B2 B; auto; try factornode m1. - - destruct (find y m2); auto. - - factor_remove_min m2 R. - assert (xd#1 >> m1). - { apply above. intros z Hz. apply B; trivial. - rewrite (remove_min_in R). now left. } - rewrite join_find; simpl; auto. - + rewrite (remove_min_find R B2 y). - case X.compare_spec; intros; auto. - destruct (find y m2'); trivial. - simpl. symmetry. apply not_find_iff; eauto. - + apply create_bst; auto. - * now apply (remove_min_bst R). - * now apply (remove_min_gt R). -Qed. - - -(** * Elements *) - -Notation eqk := (PX.eqk (elt:= elt)). -Notation eqke := (PX.eqke (elt:= elt)). -Notation ltk := (PX.ltk (elt:= elt)). - -Lemma bindings_aux_mapsto : forall (s:t elt) acc x e, - InA eqke (x,e) (bindings_aux acc s) <-> MapsTo x e s \/ InA eqke (x,e) acc. -Proof. - induction s as [ | l Hl x e r Hr h ]; simpl; auto. - intuition. - inversion H0. - intros. - rewrite Hl. - destruct (Hr acc x0 e0); clear Hl Hr. - intuition; inversion_clear H3; intuition. - compute in H0. destruct H0; simpl in *; subst; intuition. -Qed. - -Lemma bindings_mapsto : forall (s:t elt) x e, - InA eqke (x,e) (bindings s) <-> MapsTo x e s. -Proof. - intros; generalize (bindings_aux_mapsto s nil x e); intuition. - inversion_clear H0. -Qed. - -Lemma bindings_in : forall (s:t elt) x, L.PX.In x (bindings s) <-> x ∈ s. -Proof. - intros. - unfold L.PX.In. - rewrite <- In_alt; unfold In0. - split; intros (y,H); exists y. - - now rewrite <- bindings_mapsto. - - unfold L.PX.MapsTo; now rewrite bindings_mapsto. -Qed. - -Lemma bindings_aux_sort : forall (s:t elt) acc, - Bst s -> sort ltk acc -> - (forall x e y, InA eqke (x,e) acc -> y ∈ s -> y < x) -> - sort ltk (bindings_aux acc s). -Proof. - induction s as [ | l Hl y e r Hr h]; simpl; intuition. - inv Bst. - apply Hl; auto. - - constructor. - + apply Hr; eauto. - + clear Hl Hr. - apply InA_InfA with (eqA:=eqke); auto with *. - intros (y',e') Hy'. - apply bindings_aux_mapsto in Hy'. compute. intuition; eauto. - - clear Hl Hr. intros x e' y' Hx Hy'. - inversion_clear Hx. - + compute in H. destruct H; simpl in *. order. - + apply bindings_aux_mapsto in H. intuition eauto. -Qed. - -Lemma bindings_sort : forall s : t elt, Bst s -> sort ltk (bindings s). -Proof. - intros; unfold bindings; apply bindings_aux_sort; auto. - intros; inversion H0. -Qed. -Hint Resolve bindings_sort. - -Lemma bindings_nodup : forall s : t elt, Bst s -> NoDupA eqk (bindings s). -Proof. - intros; apply PX.Sort_NoDupA; auto. -Qed. - -Lemma bindings_aux_cardinal m acc : - (length acc + cardinal m)%nat = length (bindings_aux acc m). -Proof. - revert acc. induction m; simpl; intuition. - rewrite <- IHm1; simpl. - rewrite <- IHm2. rewrite Nat.add_succ_r, <- Nat.add_assoc. - f_equal. f_equal. apply Nat.add_comm. -Qed. - -Lemma bindings_cardinal m : cardinal m = length (bindings m). -Proof. - exact (bindings_aux_cardinal m nil). -Qed. - -Lemma bindings_app : - forall (s:t elt) acc, bindings_aux acc s = bindings s ++ acc. -Proof. - induction s; simpl; intros; auto. - rewrite IHs1, IHs2. - unfold bindings; simpl. - rewrite 2 IHs1, IHs2, !app_nil_r, !app_ass; auto. -Qed. - -Lemma bindings_node : - forall (t1 t2:t elt) x e z l, - bindings t1 ++ (x,e) :: bindings t2 ++ l = - bindings (Node t1 x e t2 z) ++ l. -Proof. - unfold bindings; simpl; intros. - rewrite !bindings_app, !app_nil_r, !app_ass; auto. -Qed. - -(** * Fold *) - -Definition fold' {A} (f : key -> elt -> A -> A)(s : t elt) := - L.fold f (bindings s). - -Lemma fold_equiv_aux {A} (s : t elt) (f : key -> elt -> A -> A) (a : A) acc : - L.fold f (bindings_aux acc s) a = L.fold f acc (fold f s a). -Proof. - revert a acc. - induction s; simpl; trivial. - intros. rewrite IHs1. simpl. apply IHs2. -Qed. - -Lemma fold_equiv {A} (s : t elt) (f : key -> elt -> A -> A) (a : A) : - fold f s a = fold' f s a. -Proof. - unfold fold', bindings. now rewrite fold_equiv_aux. -Qed. - -Lemma fold_spec (s:t elt)(Hs:Bst s){A}(i:A)(f : key -> elt -> A -> A) : - fold f s i = fold_left (fun a p => f p#1 p#2 a) (bindings s) i. -Proof. - rewrite fold_equiv. unfold fold'. now rewrite L.fold_spec. -Qed. - -(** * Comparison *) - -(** [flatten_e e] returns the list of bindings of the enumeration [e] - i.e. the list of bindings actually compared *) - -Fixpoint flatten_e (e : enumeration elt) : list (key*elt) := match e with - | End _ => nil - | More x e t r => (x,e) :: bindings t ++ flatten_e r - end. - -Lemma flatten_e_bindings : - forall (l:t elt) r x d z e, - bindings l ++ flatten_e (More x d r e) = - bindings (Node l x d r z) ++ flatten_e e. -Proof. - intros; apply bindings_node. -Qed. - -Lemma cons_1 : forall (s:t elt) e, - flatten_e (cons s e) = bindings s ++ flatten_e e. -Proof. - induction s; auto; intros. - simpl flatten_e; rewrite IHs1; apply flatten_e_bindings; auto. -Qed. - -(** Proof of correction for the comparison *) - -Variable cmp : elt->elt->bool. - -Definition IfEq b l1 l2 := L.equal cmp l1 l2 = b. - -Lemma cons_IfEq : forall b x1 x2 d1 d2 l1 l2, - X.eq x1 x2 -> cmp d1 d2 = true -> - IfEq b l1 l2 -> - IfEq b ((x1,d1)::l1) ((x2,d2)::l2). -Proof. - unfold IfEq; destruct b; simpl; intros; case X.compare_spec; simpl; - try rewrite H0; auto; order. -Qed. - -Lemma equal_end_IfEq : forall e2, - IfEq (equal_end e2) nil (flatten_e e2). -Proof. - destruct e2; red; auto. -Qed. - -Lemma equal_more_IfEq : - forall x1 d1 (cont:enumeration elt -> bool) x2 d2 r2 e2 l, - IfEq (cont (cons r2 e2)) l (bindings r2 ++ flatten_e e2) -> - IfEq (equal_more cmp x1 d1 cont (More x2 d2 r2 e2)) ((x1,d1)::l) - (flatten_e (More x2 d2 r2 e2)). -Proof. - unfold IfEq; simpl; intros; destruct X.compare; simpl; auto. - rewrite <-andb_lazy_alt; f_equal; auto. -Qed. - -Lemma equal_cont_IfEq : forall m1 cont e2 l, - (forall e, IfEq (cont e) l (flatten_e e)) -> - IfEq (equal_cont cmp m1 cont e2) (bindings m1 ++ l) (flatten_e e2). -Proof. - induction m1 as [|l1 Hl1 x1 d1 r1 Hr1 h1]; intros; auto. - rewrite <- bindings_node; simpl. - apply Hl1; auto. - clear e2; intros [|x2 d2 r2 e2]. - simpl; red; auto. - apply equal_more_IfEq. - rewrite <- cons_1; auto. -Qed. - -Lemma equal_IfEq : forall (m1 m2:t elt), - IfEq (equal cmp m1 m2) (bindings m1) (bindings m2). -Proof. - intros; unfold equal. - rewrite <- (app_nil_r (bindings m1)). - replace (bindings m2) with (flatten_e (cons m2 (End _))) - by (rewrite cons_1; simpl; rewrite app_nil_r; auto). - apply equal_cont_IfEq. - intros. - apply equal_end_IfEq; auto. -Qed. - -Definition Equivb m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true). - -Lemma Equivb_bindings : forall s s', - Equivb s s' <-> L.Equivb cmp (bindings s) (bindings s'). -Proof. -unfold Equivb, L.Equivb; split; split; intros. -do 2 rewrite bindings_in; firstorder. -destruct H. -apply (H2 k); rewrite <- bindings_mapsto; auto. -do 2 rewrite <- bindings_in; firstorder. -destruct H. -apply (H2 k); unfold L.PX.MapsTo; rewrite bindings_mapsto; auto. -Qed. - -Lemma equal_Equivb : forall (s s': t elt), Bst s -> Bst s' -> - (equal cmp s s' = true <-> Equivb s s'). -Proof. - intros s s' B B'. - rewrite Equivb_bindings, <- equal_IfEq. - split; [apply L.equal_2|apply L.equal_1]; auto. -Qed. - -End Elt. - -Section Map. -Variable elt elt' : Type. -Variable f : elt -> elt'. - -Lemma map_spec m x : - find x (map f m) = option_map f (find x m). -Proof. -induction m; simpl; trivial. case X.compare_spec; auto. -Qed. - -Lemma map_in m x : x ∈ (map f m) <-> x ∈ m. -Proof. -induction m; simpl; intuition_in. -Qed. - -Lemma map_bst m : Bst m -> Bst (map f m). -Proof. -induction m; simpl; auto. intros; inv Bst; constructor; auto. -- apply above. intro. rewrite map_in. intros. order. -- apply below. intro. rewrite map_in. intros. order. -Qed. - -End Map. -Section Mapi. -Variable elt elt' : Type. -Variable f : key -> elt -> elt'. - -Lemma mapi_spec m x : - exists y:key, - X.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). -Proof. - induction m; simpl. - - now exists x. - - case X.compare_spec; simpl; auto. intros. now exists k. -Qed. - -Lemma mapi_in m x : x ∈ (mapi f m) <-> x ∈ m. -Proof. -induction m; simpl; intuition_in. -Qed. - -Lemma mapi_bst m : Bst m -> Bst (mapi f m). -Proof. -induction m; simpl; auto. intros; inv Bst; constructor; auto. -- apply above. intro. rewrite mapi_in. intros. order. -- apply below. intro. rewrite mapi_in. intros. order. -Qed. - -End Mapi. - -Section Mapo. -Variable elt elt' : Type. -Variable f : key -> elt -> option elt'. - -Lemma mapo_in m x : - x ∈ (mapo f m) -> - exists y d, X.eq y x /\ MapsTo x d m /\ f y d <> None. -Proof. -functional induction (mapo f m); simpl; auto; intro H. -- inv In. -- rewrite join_in in H; destruct H as [H|[H|H]]. - + exists x0, d. do 2 (split; auto). congruence. - + destruct (IHt H) as (y & e & ? & ? & ?). exists y, e. auto. - + destruct (IHt0 H) as (y & e & ? & ? & ?). exists y, e. auto. -- rewrite concat_in in H; destruct H as [H|H]. - + destruct (IHt H) as (y & e & ? & ? & ?). exists y, e. auto. - + destruct (IHt0 H) as (y & e & ? & ? & ?). exists y, e. auto. -Qed. - -Lemma mapo_lt m x : x >> m -> x >> mapo f m. -Proof. - intros H. apply above. intros y Hy. - destruct (mapo_in Hy) as (y' & e & ? & ? & ?). order. -Qed. - -Lemma mapo_gt m x : x << m -> x << mapo f m. -Proof. - intros H. apply below. intros y Hy. - destruct (mapo_in Hy) as (y' & e & ? & ? & ?). order. -Qed. -Hint Resolve mapo_lt mapo_gt. - -Lemma mapo_bst m : Bst m -> Bst (mapo f m). -Proof. -functional induction (mapo f m); simpl; auto; intro H; inv Bst. -- apply join_bst, create_bst; auto. -- apply concat_bst; auto. apply between with x; auto. -Qed. -Hint Resolve mapo_bst. - -Ltac nonify e := - replace e with (@None elt) by - (symmetry; rewrite not_find_iff; auto; intro; order). - -Definition obind {A B} (o:option A) (f:A->option B) := - match o with Some a => f a | None => None end. - -Lemma mapo_find m x : - Bst m -> - exists y, X.eq y x /\ - find x (mapo f m) = obind (find x m) (f y). -Proof. -functional induction (mapo f m); simpl; auto; intros B; - inv Bst. -- now exists x. -- rewrite join_find; auto. - + simpl. case X.compare_spec; simpl; intros. - * now exists x0. - * destruct IHt as (y' & ? & ?); auto. - exists y'; split; trivial. - * destruct IHt0 as (y' & ? & ?); auto. - exists y'; split; trivial. - + constructor; auto using mapo_lt, mapo_gt. -- rewrite concat_find; auto. - + destruct IHt0 as (y' & ? & ->); auto. - destruct IHt as (y'' & ? & ->); auto. - case X.compare_spec; simpl; intros. - * nonify (find x r). nonify (find x l). simpl. now exists x0. - * nonify (find x r). now exists y''. - * nonify (find x l). exists y'. split; trivial. - destruct (find x r); simpl; trivial. - now destruct (f y' e). - + apply between with x0; auto. -Qed. - -End Mapo. - -Section Gmerge. -Variable elt elt' elt'' : Type. -Variable f0 : key -> option elt -> option elt' -> option elt''. -Variable f : key -> elt -> option elt' -> option elt''. -Variable mapl : t elt -> t elt''. -Variable mapr : t elt' -> t elt''. -Hypothesis f0_f : forall x d o, f x d o = f0 x (Some d) o. -Hypothesis mapl_bst : forall m, Bst m -> Bst (mapl m). -Hypothesis mapr_bst : forall m', Bst m' -> Bst (mapr m'). -Hypothesis mapl_f0 : forall x m, Bst m -> - exists y, X.eq y x /\ - find x (mapl m) = obind (find x m) (fun d => f0 y (Some d) None). -Hypothesis mapr_f0 : forall x m, Bst m -> - exists y, X.eq y x /\ - find x (mapr m) = obind (find x m) (fun d => f0 y None (Some d)). - -Notation gmerge := (gmerge f mapl mapr). - -Lemma gmerge_in m m' y : Bst m -> Bst m' -> - y ∈ (gmerge m m') -> y ∈ m \/ y ∈ m'. -Proof. - functional induction (gmerge m m'); intros B1 B2 H; - try factornode m2; inv Bst. - - right. apply find_in. - generalize (in_find (mapr_bst B2) H). - destruct (@mapr_f0 y m2) as (y' & ? & ->); trivial. - intros A B. rewrite B in A. now elim A. - - left. apply find_in. - generalize (in_find (mapl_bst B1) H). - destruct (@mapl_f0 y m2) as (y' & ? & ->); trivial. - intros A B. rewrite B in A. now elim A. - - rewrite join_in in *. revert IHt1 IHt0 H. cleansplit. - generalize (split_bst_l x1 B2) (split_bst_r x1 B2). - rewrite split_in_r, split_in_l; intuition_in. - - rewrite concat_in in *. revert IHt1 IHt0 H; cleansplit. - generalize (split_bst_l x1 B2) (split_bst_r x1 B2). - rewrite split_in_r, split_in_l; intuition_in. -Qed. - -Lemma gmerge_lt m m' x : Bst m -> Bst m' -> - x >> m -> x >> m' -> x >> gmerge m m'. -Proof. - intros. apply above. intros y Hy. - apply gmerge_in in Hy; intuition_in; order. -Qed. - -Lemma gmerge_gt m m' x : Bst m -> Bst m' -> - x << m -> x << m' -> x << gmerge m m'. -Proof. - intros. apply below. intros y Hy. - apply gmerge_in in Hy; intuition_in; order. -Qed. -Hint Resolve gmerge_lt gmerge_gt. -Hint Resolve split_bst_l split_bst_r split_lt_l split_gt_r. - -Lemma gmerge_bst m m' : Bst m -> Bst m' -> Bst (gmerge m m'). -Proof. - functional induction (gmerge m m'); intros B1 B2; auto; - factornode m2; inv Bst; - (apply join_bst, create_bst || apply concat_bst); - revert IHt1 IHt0; cleansplit; intuition. - apply between with x1; auto. -Qed. -Hint Resolve gmerge_bst. - -Lemma oelse_none_r {A} (o:option A) : oelse o None = o. -Proof. now destruct o. Qed. - -Ltac nonify e := - let E := fresh "E" in - assert (E : e = None); - [ rewrite not_find_iff; auto; intro U; - try apply gmerge_in in U; intuition_in; order - | rewrite E; clear E ]. - -Lemma gmerge_find m m' x : Bst m -> Bst m' -> - In x m \/ In x m' -> - exists y, X.eq y x /\ - find x (gmerge m m') = f0 y (find x m) (find x m'). -Proof. - functional induction (gmerge m m'); intros B1 B2 H; - try factornode m2; inv Bst. - - destruct H; [ intuition_in | ]. - destruct (@mapr_f0 x m2) as (y,(Hy,E)); trivial. - exists y; split; trivial. - rewrite E. simpl. apply in_find in H; trivial. - destruct (find x m2); simpl; intuition. - - destruct H; [ | intuition_in ]. - destruct (@mapl_f0 x m2) as (y,(Hy,E)); trivial. - exists y; split; trivial. - rewrite E. simpl. apply in_find in H; trivial. - destruct (find x m2); simpl; intuition. - - generalize (split_bst_l x1 B2) (split_bst_r x1 B2). - rewrite (split_find x1 x B2). - rewrite e1 in *; simpl in *. intros. - rewrite join_find by (cleansplit; constructor; auto). - simpl. case X.compare_spec; intros. - + exists x1. split; auto. now rewrite <- e3, f0_f. - + apply IHt1; auto. clear IHt1 IHt0. - cleansplit; rewrite split_in_l; trivial. - intuition_in; order. - + apply IHt0; auto. clear IHt1 IHt0. - cleansplit; rewrite split_in_r; trivial. - intuition_in; order. - - generalize (split_bst_l x1 B2) (split_bst_r x1 B2). - rewrite (split_find x1 x B2). - pose proof (split_lt_l x1 B2). - pose proof (split_gt_r x1 B2). - rewrite e1 in *; simpl in *. intros. - rewrite concat_find by (try apply between with x1; auto). - case X.compare_spec; intros. - + clear IHt0 IHt1. - exists x1. split; auto. rewrite <- f0_f, e2. - nonify (find x (gmerge r1 r2')). - nonify (find x (gmerge l1 l2')). trivial. - + nonify (find x (gmerge r1 r2')). - simpl. apply IHt1; auto. clear IHt1 IHt0. - intuition_in; try order. - right. cleansplit. now apply split_in_l. - + nonify (find x (gmerge l1 l2')). simpl. - rewrite oelse_none_r. - apply IHt0; auto. clear IHt1 IHt0. - intuition_in; try order. - right. cleansplit. now apply split_in_r. -Qed. - -End Gmerge. - -Section Merge. -Variable elt elt' elt'' : Type. -Variable f : key -> option elt -> option elt' -> option elt''. - -Lemma merge_bst m m' : Bst m -> Bst m' -> Bst (merge f m m'). -Proof. -unfold merge; intros. -apply gmerge_bst with f; - auto using mapo_bst, mapo_find. -Qed. - -Lemma merge_spec1 m m' x : Bst m -> Bst m' -> - In x m \/ In x m' -> - exists y, X.eq y x /\ - find x (merge f m m') = f y (find x m) (find x m'). -Proof. - unfold merge; intros. - edestruct (gmerge_find (f0:=f)) as (y,(Hy,E)); - eauto using mapo_bst. - - reflexivity. - - intros. now apply mapo_find. - - intros. now apply mapo_find. -Qed. - -Lemma merge_spec2 m m' x : Bst m -> Bst m' -> - In x (merge f m m') -> In x m \/ In x m'. -Proof. -unfold merge; intros. -eapply gmerge_in with (f0:=f); try eassumption; - auto using mapo_bst, mapo_find. -Qed. - -End Merge. -End Proofs. -End Raw. - -(** * Encapsulation - - Now, in order to really provide a functor implementing [S], we - need to encapsulate everything into a type of balanced binary search trees. *) - -Module IntMake (I:Int)(X: OrderedType) <: S with Module E := X. - - Module E := X. - Module Raw := Raw I X. - Import Raw.Proofs. - - Record tree (elt:Type) := - Mk {this :> Raw.tree elt; is_bst : Raw.Bst this}. - - Definition t := tree. - Definition key := E.t. - - Section Elt. - Variable elt elt' elt'': Type. - - Implicit Types m : t elt. - Implicit Types x y : key. - Implicit Types e : elt. - - Definition empty : t elt := Mk (empty_bst elt). - Definition is_empty m : bool := Raw.is_empty m.(this). - Definition add x e m : t elt := Mk (add_bst x e m.(is_bst)). - Definition remove x m : t elt := Mk (remove_bst x m.(is_bst)). - Definition mem x m : bool := Raw.mem x m.(this). - Definition find x m : option elt := Raw.find x m.(this). - Definition map f m : t elt' := Mk (map_bst f m.(is_bst)). - Definition mapi (f:key->elt->elt') m : t elt' := - Mk (mapi_bst f m.(is_bst)). - Definition merge f m (m':t elt') : t elt'' := - Mk (merge_bst f m.(is_bst) m'.(is_bst)). - Definition bindings m : list (key*elt) := Raw.bindings m.(this). - Definition cardinal m := Raw.cardinal m.(this). - Definition fold {A} (f:key->elt->A->A) m i := Raw.fold (A:=A) f m.(this) i. - Definition equal cmp m m' : bool := Raw.equal cmp m.(this) m'.(this). - - Definition MapsTo x e m : Prop := Raw.MapsTo x e m.(this). - Definition In x m : Prop := Raw.In0 x m.(this). - - Definition eq_key : (key*elt) -> (key*elt) -> Prop := @PX.eqk elt. - Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop := @PX.eqke elt. - Definition lt_key : (key*elt) -> (key*elt) -> Prop := @PX.ltk elt. - - Instance MapsTo_compat : - Proper (E.eq==>Logic.eq==>Logic.eq==>iff) MapsTo. - Proof. - intros k k' Hk e e' He m m' Hm. unfold MapsTo; simpl. - now rewrite Hk, He, Hm. - Qed. - - Lemma find_spec m x e : find x m = Some e <-> MapsTo x e m. - Proof. apply find_spec. apply is_bst. Qed. - - Lemma mem_spec m x : mem x m = true <-> In x m. - Proof. - unfold In, mem; rewrite In_alt. apply mem_spec. apply is_bst. - Qed. - - Lemma empty_spec x : find x empty = None. - Proof. apply empty_spec. Qed. - - Lemma is_empty_spec m : is_empty m = true <-> forall x, find x m = None. - Proof. apply is_empty_spec. Qed. - - Lemma add_spec1 m x e : find x (add x e m) = Some e. - Proof. apply add_spec1. apply is_bst. Qed. - Lemma add_spec2 m x y e : ~ E.eq x y -> find y (add x e m) = find y m. - Proof. apply add_spec2. apply is_bst. Qed. - - Lemma remove_spec1 m x : find x (remove x m) = None. - Proof. apply remove_spec1. apply is_bst. Qed. - Lemma remove_spec2 m x y : ~E.eq x y -> find y (remove x m) = find y m. - Proof. apply remove_spec2. apply is_bst. Qed. - - Lemma bindings_spec1 m x e : - InA eq_key_elt (x,e) (bindings m) <-> MapsTo x e m. - Proof. apply bindings_mapsto. Qed. - - Lemma bindings_spec2 m : sort lt_key (bindings m). - Proof. apply bindings_sort. apply is_bst. Qed. - - Lemma bindings_spec2w m : NoDupA eq_key (bindings m). - Proof. apply bindings_nodup. apply is_bst. Qed. - - Lemma fold_spec m {A} (i : A) (f : key -> elt -> A -> A) : - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. - Proof. apply fold_spec. apply is_bst. Qed. - - Lemma cardinal_spec m : cardinal m = length (bindings m). - Proof. apply bindings_cardinal. Qed. - - Definition Equal m m' := forall y, find y m = find y m'. - Definition Equiv (eq_elt:elt->elt->Prop) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). - Definition Equivb cmp := Equiv (Cmp cmp). - - Lemma Equivb_Equivb cmp m m' : - Equivb cmp m m' <-> Raw.Proofs.Equivb cmp m m'. - Proof. - unfold Equivb, Equiv, Raw.Proofs.Equivb, In. intuition. - generalize (H0 k); do 2 rewrite In_alt; intuition. - generalize (H0 k); do 2 rewrite In_alt; intuition. - generalize (H0 k); do 2 rewrite <- In_alt; intuition. - generalize (H0 k); do 2 rewrite <- In_alt; intuition. - Qed. - - Lemma equal_spec m m' cmp : - equal cmp m m' = true <-> Equivb cmp m m'. - Proof. rewrite Equivb_Equivb. apply equal_Equivb; apply is_bst. Qed. - - End Elt. - - Lemma map_spec {elt elt'} (f:elt->elt') m x : - find x (map f m) = option_map f (find x m). - Proof. apply map_spec. Qed. - - Lemma mapi_spec {elt elt'} (f:key->elt->elt') m x : - exists y:key, E.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). - Proof. apply mapi_spec. Qed. - - Lemma merge_spec1 {elt elt' elt''} - (f:key->option elt->option elt'->option elt'') m m' x : - In x m \/ In x m' -> - exists y:key, E.eq y x /\ - find x (merge f m m') = f y (find x m) (find x m'). - Proof. - unfold In. rewrite !In_alt. apply merge_spec1; apply is_bst. - Qed. - - Lemma merge_spec2 {elt elt' elt''} - (f:key -> option elt->option elt'->option elt'') m m' x : - In x (merge f m m') -> In x m \/ In x m'. - Proof. - unfold In. rewrite !In_alt. apply merge_spec2; apply is_bst. - Qed. - -End IntMake. - - -Module IntMake_ord (I:Int)(X: OrderedType)(D : OrderedType) <: - Sord with Module Data := D - with Module MapS.E := X. - - Module Data := D. - Module Import MapS := IntMake(I)(X). - Module LO := MMapList.Make_ord(X)(D). - Module R := Raw. - Module P := Raw.Proofs. - - Definition t := MapS.t D.t. - - Definition cmp e e' := - match D.compare e e' with Eq => true | _ => false end. - - (** One step of comparison of bindings *) - - Definition compare_more x1 d1 (cont:R.enumeration D.t -> comparison) e2 := - match e2 with - | R.End _ => Gt - | R.More x2 d2 r2 e2 => - match X.compare x1 x2 with - | Eq => match D.compare d1 d2 with - | Eq => cont (R.cons r2 e2) - | Lt => Lt - | Gt => Gt - end - | Lt => Lt - | Gt => Gt - end - end. - - (** Comparison of left tree, middle element, then right tree *) - - Fixpoint compare_cont s1 (cont:R.enumeration D.t -> comparison) e2 := - match s1 with - | R.Leaf _ => cont e2 - | R.Node l1 x1 d1 r1 _ => - compare_cont l1 (compare_more x1 d1 (compare_cont r1 cont)) e2 - end. - - (** Initial continuation *) - - Definition compare_end (e2:R.enumeration D.t) := - match e2 with R.End _ => Eq | _ => Lt end. - - (** The complete comparison *) - - Definition compare m1 m2 := - compare_cont m1.(this) compare_end (R.cons m2 .(this) (Raw.End _)). - - (** Correctness of this comparison *) - - Definition Cmp c := - match c with - | Eq => LO.eq_list - | Lt => LO.lt_list - | Gt => (fun l1 l2 => LO.lt_list l2 l1) - end. - - Lemma cons_Cmp c x1 x2 d1 d2 l1 l2 : - X.eq x1 x2 -> D.eq d1 d2 -> - Cmp c l1 l2 -> Cmp c ((x1,d1)::l1) ((x2,d2)::l2). - Proof. - destruct c; simpl; intros; case X.compare_spec; auto; try P.MX.order. - intros. right. split; auto. now symmetry. - Qed. - Hint Resolve cons_Cmp. - - Lemma compare_end_Cmp e2 : - Cmp (compare_end e2) nil (P.flatten_e e2). - Proof. - destruct e2; simpl; auto. - Qed. - - Lemma compare_more_Cmp x1 d1 cont x2 d2 r2 e2 l : - Cmp (cont (R.cons r2 e2)) l (R.bindings r2 ++ P.flatten_e e2) -> - Cmp (compare_more x1 d1 cont (R.More x2 d2 r2 e2)) ((x1,d1)::l) - (P.flatten_e (R.More x2 d2 r2 e2)). - Proof. - simpl; case X.compare_spec; simpl; - try case D.compare_spec; simpl; auto; - case X.compare_spec; try P.MX.order; auto. - Qed. - - Lemma compare_cont_Cmp : forall s1 cont e2 l, - (forall e, Cmp (cont e) l (P.flatten_e e)) -> - Cmp (compare_cont s1 cont e2) (R.bindings s1 ++ l) (P.flatten_e e2). - Proof. - induction s1 as [|l1 Hl1 x1 d1 r1 Hr1 h1] using P.tree_ind; - intros; auto. - rewrite <- P.bindings_node; simpl. - apply Hl1; auto. clear e2. intros [|x2 d2 r2 e2]. - simpl; auto. - apply compare_more_Cmp. - rewrite <- P.cons_1; auto. - Qed. - - Lemma compare_Cmp m1 m2 : - Cmp (compare m1 m2) (bindings m1) (bindings m2). - Proof. - destruct m1 as (s1,H1), m2 as (s2,H2). - unfold compare, bindings; simpl. - rewrite <- (app_nil_r (R.bindings s1)). - replace (R.bindings s2) with (P.flatten_e (R.cons s2 (R.End _))) by - (rewrite P.cons_1; simpl; rewrite app_nil_r; auto). - auto using compare_cont_Cmp, compare_end_Cmp. - Qed. - - Definition eq (m1 m2 : t) := LO.eq_list (bindings m1) (bindings m2). - Definition lt (m1 m2 : t) := LO.lt_list (bindings m1) (bindings m2). - - Lemma compare_spec m1 m2 : CompSpec eq lt m1 m2 (compare m1 m2). - Proof. - assert (H := compare_Cmp m1 m2). - unfold Cmp in H. - destruct (compare m1 m2); auto. - Qed. - - (* Proofs about [eq] and [lt] *) - - Definition sbindings (m1 : t) := - LO.MapS.Mk (P.bindings_sort m1.(is_bst)). - - Definition seq (m1 m2 : t) := LO.eq (sbindings m1) (sbindings m2). - Definition slt (m1 m2 : t) := LO.lt (sbindings m1) (sbindings m2). - - Lemma eq_seq : forall m1 m2, eq m1 m2 <-> seq m1 m2. - Proof. - unfold eq, seq, sbindings, bindings, LO.eq; intuition. - Qed. - - Lemma lt_slt : forall m1 m2, lt m1 m2 <-> slt m1 m2. - Proof. - unfold lt, slt, sbindings, bindings, LO.lt; intuition. - Qed. - - Lemma eq_spec m m' : eq m m' <-> Equivb cmp m m'. - Proof. - rewrite eq_seq; unfold seq. - rewrite Equivb_Equivb. - rewrite P.Equivb_bindings. apply LO.eq_spec. - Qed. - - Instance eq_equiv : Equivalence eq. - Proof. - constructor; red; [intros x|intros x y| intros x y z]; - rewrite !eq_seq; apply LO.eq_equiv. - Qed. - - Instance lt_compat : Proper (eq ==> eq ==> iff) lt. - Proof. - intros m1 m2 H1 m1' m2' H2. rewrite !lt_slt. rewrite eq_seq in *. - now apply LO.lt_compat. - Qed. - - Instance lt_strorder : StrictOrder lt. - Proof. - constructor; red; [intros x; red|intros x y z]; - rewrite !lt_slt; apply LO.lt_strorder. - Qed. - -End IntMake_ord. - -(* For concrete use inside Coq, we propose an instantiation of [Int] by [Z]. *) - -Module Make (X: OrderedType) <: S with Module E := X - :=IntMake(Z_as_Int)(X). - -Module Make_ord (X: OrderedType)(D: OrderedType) - <: Sord with Module Data := D - with Module MapS.E := X - :=IntMake_ord(Z_as_Int)(X)(D). diff --git a/theories/MMaps/MMapFacts.v b/theories/MMaps/MMapFacts.v deleted file mode 100644 index 69066a7b..00000000 --- a/theories/MMaps/MMapFacts.v +++ /dev/null @@ -1,2434 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(** * Finite maps library *) - -(** This functor derives additional facts from [MMapInterface.S]. These - facts are mainly the specifications of [MMapInterface.S] written using - different styles: equivalence and boolean equalities. -*) - -Require Import Bool Equalities Orders OrdersFacts OrdersLists. -Require Import Morphisms Permutation SetoidPermutation. -Require Export MMapInterface. -Set Implicit Arguments. -Unset Strict Implicit. - -Lemma eq_bool_alt b b' : b=b' <-> (b=true <-> b'=true). -Proof. - destruct b, b'; intuition. -Qed. - -Lemma eq_option_alt {elt}(o o':option elt) : - o=o' <-> (forall e, o=Some e <-> o'=Some e). -Proof. -split; intros. -- now subst. -- destruct o, o'; rewrite ?H; auto. - symmetry; now apply H. -Qed. - -Lemma option_map_some {A B}(f:A->B) o : - option_map f o <> None <-> o <> None. -Proof. - destruct o; simpl. now split. split; now destruct 1. -Qed. - -(** * Properties about weak maps *) - -Module WProperties_fun (E:DecidableType)(Import M:WSfun E). - -Definition Empty {elt}(m : t elt) := forall x e, ~MapsTo x e m. - -(** A few things about E.eq *) - -Lemma eq_refl x : E.eq x x. Proof. apply E.eq_equiv. Qed. -Lemma eq_sym x y : E.eq x y -> E.eq y x. Proof. apply E.eq_equiv. Qed. -Lemma eq_trans x y z : E.eq x y -> E.eq y z -> E.eq x z. -Proof. apply E.eq_equiv. Qed. -Hint Immediate eq_refl eq_sym : map. -Hint Resolve eq_trans eq_equivalence E.eq_equiv : map. - -Definition eqb x y := if E.eq_dec x y then true else false. - -Lemma eqb_eq x y : eqb x y = true <-> E.eq x y. -Proof. - unfold eqb; case E.eq_dec; now intuition. -Qed. - -Lemma eqb_sym x y : eqb x y = eqb y x. -Proof. - apply eq_bool_alt. rewrite !eqb_eq. split; apply E.eq_equiv. -Qed. - -(** Initial results about MapsTo and In *) - -Lemma mapsto_fun {elt} m x (e e':elt) : - MapsTo x e m -> MapsTo x e' m -> e=e'. -Proof. -rewrite <- !find_spec. congruence. -Qed. - -Lemma in_find {elt} (m : t elt) x : In x m <-> find x m <> None. -Proof. - unfold In. split. - - intros (e,H). rewrite <-find_spec in H. congruence. - - destruct (find x m) as [e|] eqn:H. - + exists e. now apply find_spec. - + now destruct 1. -Qed. - -Lemma not_in_find {elt} (m : t elt) x : ~In x m <-> find x m = None. -Proof. - rewrite in_find. split; auto. - intros; destruct (find x m); trivial. now destruct H. -Qed. - -Notation in_find_iff := in_find (only parsing). -Notation not_find_in_iff := not_in_find (only parsing). - -(** * [Equal] is a setoid equality. *) - -Infix "==" := Equal (at level 30). - -Lemma Equal_refl {elt} (m : t elt) : m == m. -Proof. red; reflexivity. Qed. - -Lemma Equal_sym {elt} (m m' : t elt) : m == m' -> m' == m. -Proof. unfold Equal; auto. Qed. - -Lemma Equal_trans {elt} (m m' m'' : t elt) : - m == m' -> m' == m'' -> m == m''. -Proof. unfold Equal; congruence. Qed. - -Instance Equal_equiv {elt} : Equivalence (@Equal elt). -Proof. -constructor; [exact Equal_refl | exact Equal_sym | exact Equal_trans]. -Qed. - -Arguments Equal {elt} m m'. - -Instance MapsTo_m {elt} : - Proper (E.eq==>Logic.eq==>Equal==>iff) (@MapsTo elt). -Proof. -intros k k' Hk e e' <- m m' Hm. rewrite <- Hk. -now rewrite <- !find_spec, Hm. -Qed. - -Instance In_m {elt} : - Proper (E.eq==>Equal==>iff) (@In elt). -Proof. -intros k k' Hk m m' Hm. unfold In. -split; intros (e,H); exists e; revert H; - now rewrite Hk, <- !find_spec, Hm. -Qed. - -Instance find_m {elt} : Proper (E.eq==>Equal==>Logic.eq) (@find elt). -Proof. -intros k k' Hk m m' <-. -rewrite eq_option_alt. intros. now rewrite !find_spec, Hk. -Qed. - -Instance mem_m {elt} : Proper (E.eq==>Equal==>Logic.eq) (@mem elt). -Proof. -intros k k' Hk m m' Hm. now rewrite eq_bool_alt, !mem_spec, Hk, Hm. -Qed. - -Instance Empty_m {elt} : Proper (Equal==>iff) (@Empty elt). -Proof. -intros m m' Hm. unfold Empty. now setoid_rewrite Hm. -Qed. - -Instance is_empty_m {elt} : Proper (Equal ==> Logic.eq) (@is_empty elt). -Proof. -intros m m' Hm. rewrite eq_bool_alt, !is_empty_spec. - now setoid_rewrite Hm. -Qed. - -Instance add_m {elt} : Proper (E.eq==>Logic.eq==>Equal==>Equal) (@add elt). -Proof. -intros k k' Hk e e' <- m m' Hm y. -destruct (E.eq_dec k y) as [H|H]. -- rewrite <-H, add_spec1. now rewrite Hk, add_spec1. -- rewrite !add_spec2; trivial. now rewrite <- Hk. -Qed. - -Instance remove_m {elt} : Proper (E.eq==>Equal==>Equal) (@remove elt). -Proof. -intros k k' Hk m m' Hm y. -destruct (E.eq_dec k y) as [H|H]. -- rewrite <-H, remove_spec1. now rewrite Hk, remove_spec1. -- rewrite !remove_spec2; trivial. now rewrite <- Hk. -Qed. - -Instance map_m {elt elt'} : - Proper ((Logic.eq==>Logic.eq)==>Equal==>Equal) (@map elt elt'). -Proof. -intros f f' Hf m m' Hm y. rewrite !map_spec, Hm. -destruct (find y m'); simpl; trivial. f_equal. now apply Hf. -Qed. - -Instance mapi_m {elt elt'} : - Proper ((E.eq==>Logic.eq==>Logic.eq)==>Equal==>Equal) (@mapi elt elt'). -Proof. -intros f f' Hf m m' Hm y. -destruct (mapi_spec f m y) as (x,(Hx,->)). -destruct (mapi_spec f' m' y) as (x',(Hx',->)). -rewrite <- Hm. destruct (find y m); trivial. simpl. -f_equal. apply Hf; trivial. now rewrite Hx, Hx'. -Qed. - -Instance merge_m {elt elt' elt''} : - Proper ((E.eq==>Logic.eq==>Logic.eq==>Logic.eq)==>Equal==>Equal==>Equal) - (@merge elt elt' elt''). -Proof. -intros f f' Hf m1 m1' Hm1 m2 m2' Hm2 y. -destruct (find y m1) as [e1|] eqn:H1. -- apply find_spec in H1. - assert (H : In y m1 \/ In y m2) by (left; now exists e1). - destruct (merge_spec1 f H) as (y1,(Hy1,->)). - rewrite Hm1,Hm2 in H. - destruct (merge_spec1 f' H) as (y2,(Hy2,->)). - rewrite <- Hm1, <- Hm2. apply Hf; trivial. now transitivity y. -- destruct (find y m2) as [e2|] eqn:H2. - + apply find_spec in H2. - assert (H : In y m1 \/ In y m2) by (right; now exists e2). - destruct (merge_spec1 f H) as (y1,(Hy1,->)). - rewrite Hm1,Hm2 in H. - destruct (merge_spec1 f' H) as (y2,(Hy2,->)). - rewrite <- Hm1, <- Hm2. apply Hf; trivial. now transitivity y. - + apply not_in_find in H1. apply not_in_find in H2. - assert (H : ~In y (merge f m1 m2)). - { intro H. apply merge_spec2 in H. intuition. } - apply not_in_find in H. rewrite H. - symmetry. apply not_in_find. intro H'. - apply merge_spec2 in H'. rewrite <- Hm1, <- Hm2 in H'. - intuition. -Qed. - -(* Later: compatibility for cardinal, fold, ... *) - -(** ** Earlier specifications (cf. FMaps) *) - -Section OldSpecs. -Variable elt: Type. -Implicit Type m: t elt. -Implicit Type x y z: key. -Implicit Type e: elt. - -Lemma MapsTo_1 m x y e : E.eq x y -> MapsTo x e m -> MapsTo y e m. -Proof. - now intros ->. -Qed. - -Lemma find_1 m x e : MapsTo x e m -> find x m = Some e. -Proof. apply find_spec. Qed. - -Lemma find_2 m x e : find x m = Some e -> MapsTo x e m. -Proof. apply find_spec. Qed. - -Lemma mem_1 m x : In x m -> mem x m = true. -Proof. apply mem_spec. Qed. - -Lemma mem_2 m x : mem x m = true -> In x m. -Proof. apply mem_spec. Qed. - -Lemma empty_1 : Empty (@empty elt). -Proof. - intros x e. now rewrite <- find_spec, empty_spec. -Qed. - -Lemma is_empty_1 m : Empty m -> is_empty m = true. -Proof. - unfold Empty; rewrite is_empty_spec. setoid_rewrite <- find_spec. - intros H x. specialize (H x). - destruct (find x m) as [e|]; trivial. - now destruct (H e). -Qed. - -Lemma is_empty_2 m : is_empty m = true -> Empty m. -Proof. - rewrite is_empty_spec. intros H x e. now rewrite <- find_spec, H. -Qed. - -Lemma add_1 m x y e : E.eq x y -> MapsTo y e (add x e m). -Proof. - intros <-. rewrite <-find_spec. apply add_spec1. -Qed. - -Lemma add_2 m x y e e' : - ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m). -Proof. - intro. now rewrite <- !find_spec, add_spec2. -Qed. - -Lemma add_3 m x y e e' : - ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m. -Proof. - intro. rewrite <- !find_spec, add_spec2; trivial. -Qed. - -Lemma remove_1 m x y : E.eq x y -> ~ In y (remove x m). -Proof. - intros <-. apply not_in_find. apply remove_spec1. -Qed. - -Lemma remove_2 m x y e : - ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m). -Proof. - intro. now rewrite <- !find_spec, remove_spec2. -Qed. - -Lemma remove_3bis m x y e : - find y (remove x m) = Some e -> find y m = Some e. -Proof. - destruct (E.eq_dec x y) as [<-|H]. - - now rewrite remove_spec1. - - now rewrite remove_spec2. -Qed. - -Lemma remove_3 m x y e : MapsTo y e (remove x m) -> MapsTo y e m. -Proof. - rewrite <-!find_spec. apply remove_3bis. -Qed. - -Lemma bindings_1 m x e : - MapsTo x e m -> InA eq_key_elt (x,e) (bindings m). -Proof. apply bindings_spec1. Qed. - -Lemma bindings_2 m x e : - InA eq_key_elt (x,e) (bindings m) -> MapsTo x e m. -Proof. apply bindings_spec1. Qed. - -Lemma bindings_3w m : NoDupA eq_key (bindings m). -Proof. apply bindings_spec2w. Qed. - -Lemma cardinal_1 m : cardinal m = length (bindings m). -Proof. apply cardinal_spec. Qed. - -Lemma fold_1 m (A : Type) (i : A) (f : key -> elt -> A -> A) : - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. -Proof. apply fold_spec. Qed. - -Lemma equal_1 m m' cmp : Equivb cmp m m' -> equal cmp m m' = true. -Proof. apply equal_spec. Qed. - -Lemma equal_2 m m' cmp : equal cmp m m' = true -> Equivb cmp m m'. -Proof. apply equal_spec. Qed. - -End OldSpecs. - -Lemma map_1 {elt elt'}(m: t elt)(x:key)(e:elt)(f:elt->elt') : - MapsTo x e m -> MapsTo x (f e) (map f m). -Proof. - rewrite <- !find_spec, map_spec. now intros ->. -Qed. - -Lemma map_2 {elt elt'}(m: t elt)(x:key)(f:elt->elt') : - In x (map f m) -> In x m. -Proof. - rewrite !in_find, map_spec. apply option_map_some. -Qed. - -Lemma mapi_1 {elt elt'}(m: t elt)(x:key)(e:elt)(f:key->elt->elt') : - MapsTo x e m -> - exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m). -Proof. - destruct (mapi_spec f m x) as (y,(Hy,Eq)). - intro H. exists y; split; trivial. - rewrite <-find_spec in *. now rewrite Eq, H. -Qed. - -Lemma mapi_2 {elt elt'}(m: t elt)(x:key)(f:key->elt->elt') : - In x (mapi f m) -> In x m. -Proof. - destruct (mapi_spec f m x) as (y,(Hy,Eq)). - rewrite !in_find. intro H; contradict H. now rewrite Eq, H. -Qed. - -(** The ancestor [map2] of the current [merge] was dealing with functions - on datas only, not on keys. *) - -Definition map2 {elt elt' elt''} (f:option elt->option elt'->option elt'') - := merge (fun _ => f). - -Lemma map2_1 {elt elt' elt''}(m: t elt)(m': t elt') - (x:key)(f:option elt->option elt'->option elt'') : - In x m \/ In x m' -> - find x (map2 f m m') = f (find x m) (find x m'). -Proof. - intros. unfold map2. - now destruct (merge_spec1 (fun _ => f) H) as (y,(_,->)). -Qed. - -Lemma map2_2 {elt elt' elt''}(m: t elt)(m': t elt') - (x:key)(f:option elt->option elt'->option elt'') : - In x (map2 f m m') -> In x m \/ In x m'. -Proof. apply merge_spec2. Qed. - -Hint Immediate MapsTo_1 mem_2 is_empty_2 - map_2 mapi_2 add_3 remove_3 find_2 : map. -Hint Resolve mem_1 is_empty_1 is_empty_2 add_1 add_2 remove_1 - remove_2 find_1 fold_1 map_1 mapi_1 mapi_2 : map. - -(** ** Specifications written using equivalences *) - -Section IffSpec. -Variable elt: Type. -Implicit Type m: t elt. -Implicit Type x y z: key. -Implicit Type e: elt. - -Lemma in_iff m x y : E.eq x y -> (In x m <-> In y m). -Proof. now intros ->. Qed. - -Lemma mapsto_iff m x y e : E.eq x y -> (MapsTo x e m <-> MapsTo y e m). -Proof. now intros ->. Qed. - -Lemma mem_in_iff m x : In x m <-> mem x m = true. -Proof. symmetry. apply mem_spec. Qed. - -Lemma not_mem_in_iff m x : ~In x m <-> mem x m = false. -Proof. -rewrite mem_in_iff; destruct (mem x m); intuition. -Qed. - -Lemma mem_find m x : mem x m = true <-> find x m <> None. -Proof. - rewrite <- mem_in_iff. apply in_find. -Qed. - -Lemma not_mem_find m x : mem x m = false <-> find x m = None. -Proof. - rewrite <- not_mem_in_iff. apply not_in_find. -Qed. - -Lemma In_dec m x : { In x m } + { ~ In x m }. -Proof. - generalize (mem_in_iff m x). - destruct (mem x m); [left|right]; intuition. -Qed. - -Lemma find_mapsto_iff m x e : MapsTo x e m <-> find x m = Some e. -Proof. symmetry. apply find_spec. Qed. - -Lemma equal_iff m m' cmp : Equivb cmp m m' <-> equal cmp m m' = true. -Proof. symmetry. apply equal_spec. Qed. - -Lemma empty_mapsto_iff x e : MapsTo x e empty <-> False. -Proof. -rewrite <- find_spec, empty_spec. now split. -Qed. - -Lemma not_in_empty x : ~In x (@empty elt). -Proof. -intros (e,H). revert H. apply empty_mapsto_iff. -Qed. - -Lemma empty_in_iff x : In x (@empty elt) <-> False. -Proof. -split; [ apply not_in_empty | destruct 1 ]. -Qed. - -Lemma is_empty_iff m : Empty m <-> is_empty m = true. -Proof. split; [apply is_empty_1 | apply is_empty_2 ]. Qed. - -Lemma add_mapsto_iff m x y e e' : - MapsTo y e' (add x e m) <-> - (E.eq x y /\ e=e') \/ - (~E.eq x y /\ MapsTo y e' m). -Proof. -split. -- intros H. destruct (E.eq_dec x y); [left|right]; split; trivial. - + symmetry. apply (mapsto_fun H); auto with map. - + now apply add_3 with x e. -- destruct 1 as [(H,H')|(H,H')]; subst; auto with map. -Qed. - -Lemma add_mapsto_new m x y e e' : ~In x m -> - MapsTo y e' (add x e m) <-> (E.eq x y /\ e=e') \/ MapsTo y e' m. -Proof. - intros. - rewrite add_mapsto_iff. intuition. - right; split; trivial. contradict H. exists e'. now rewrite H. -Qed. - -Lemma in_add m x y e : In y m -> In y (add x e m). -Proof. - destruct (E.eq_dec x y) as [<-|H']. - - now rewrite !in_find, add_spec1. - - now rewrite !in_find, add_spec2. -Qed. - -Lemma add_in_iff m x y e : In y (add x e m) <-> E.eq x y \/ In y m. -Proof. -split. -- intros H. destruct (E.eq_dec x y); [now left|right]. - rewrite in_find, add_spec2 in H; trivial. now apply in_find. -- intros [<-|H]. - + exists e. now apply add_1. - + now apply in_add. -Qed. - -Lemma add_neq_mapsto_iff m x y e e' : - ~ E.eq x y -> (MapsTo y e' (add x e m) <-> MapsTo y e' m). -Proof. -split; [apply add_3|apply add_2]; auto. -Qed. - -Lemma add_neq_in_iff m x y e : - ~ E.eq x y -> (In y (add x e m) <-> In y m). -Proof. -split; intros (e',H0); exists e'. -- now apply add_3 with x e. -- now apply add_2. -Qed. - -Lemma remove_mapsto_iff m x y e : - MapsTo y e (remove x m) <-> ~E.eq x y /\ MapsTo y e m. -Proof. -split; [split|destruct 1]. -- intro E. revert H. now rewrite <-E, <- find_spec, remove_spec1. -- now apply remove_3 with x. -- now apply remove_2. -Qed. - -Lemma remove_in_iff m x y : In y (remove x m) <-> ~E.eq x y /\ In y m. -Proof. -unfold In; split; [ intros (e,H) | intros (E,(e,H)) ]. -- apply remove_mapsto_iff in H. destruct H; split; trivial. - now exists e. -- exists e. now apply remove_2. -Qed. - -Lemma remove_neq_mapsto_iff : forall m x y e, - ~ E.eq x y -> (MapsTo y e (remove x m) <-> MapsTo y e m). -Proof. -split; [apply remove_3|apply remove_2]; auto. -Qed. - -Lemma remove_neq_in_iff : forall m x y, - ~ E.eq x y -> (In y (remove x m) <-> In y m). -Proof. -split; intros (e',H0); exists e'. -- now apply remove_3 with x. -- now apply remove_2. -Qed. - -Lemma bindings_mapsto_iff m x e : - MapsTo x e m <-> InA eq_key_elt (x,e) (bindings m). -Proof. symmetry. apply bindings_spec1. Qed. - -Lemma bindings_in_iff m x : - In x m <-> exists e, InA eq_key_elt (x,e) (bindings m). -Proof. -unfold In; split; intros (e,H); exists e; now apply bindings_spec1. -Qed. - -End IffSpec. - -Lemma map_mapsto_iff {elt elt'} m x b (f : elt -> elt') : - MapsTo x b (map f m) <-> exists a, b = f a /\ MapsTo x a m. -Proof. -rewrite <-find_spec, map_spec. setoid_rewrite <- find_spec. -destruct (find x m); simpl; split. -- injection 1. now exists e. -- intros (a,(->,H)). now injection H as ->. -- discriminate. -- intros (a,(_,H)); discriminate. -Qed. - -Lemma map_in_iff {elt elt'} m x (f : elt -> elt') : - In x (map f m) <-> In x m. -Proof. -rewrite !in_find, map_spec. apply option_map_some. -Qed. - -Lemma mapi_in_iff {elt elt'} m x (f:key->elt->elt') : - In x (mapi f m) <-> In x m. -Proof. -rewrite !in_find. destruct (mapi_spec f m x) as (y,(_,->)). -apply option_map_some. -Qed. - -(** Unfortunately, we don't have simple equivalences for [mapi] - and [MapsTo]. The only correct one needs compatibility of [f]. *) - -Lemma mapi_inv {elt elt'} m x b (f : key -> elt -> elt') : - MapsTo x b (mapi f m) -> - exists a y, E.eq y x /\ b = f y a /\ MapsTo x a m. -Proof. -rewrite <- find_spec. setoid_rewrite <- find_spec. -destruct (mapi_spec f m x) as (y,(E,->)). -destruct (find x m); simpl. -- injection 1 as <-. now exists e, y. -- discriminate. -Qed. - -Lemma mapi_spec' {elt elt'} (f:key->elt->elt') : - Proper (E.eq==>Logic.eq==>Logic.eq) f -> - forall m x, - find x (mapi f m) = option_map (f x) (find x m). -Proof. - intros. destruct (mapi_spec f m x) as (y,(Hy,->)). - destruct (find x m); simpl; trivial. - now rewrite Hy. -Qed. - -Lemma mapi_1bis {elt elt'} m x e (f:key->elt->elt') : - Proper (E.eq==>Logic.eq==>Logic.eq) f -> - MapsTo x e m -> MapsTo x (f x e) (mapi f m). -Proof. -intros. destruct (mapi_1 f H0) as (y,(->,H2)). trivial. -Qed. - -Lemma mapi_mapsto_iff {elt elt'} m x b (f:key->elt->elt') : - Proper (E.eq==>Logic.eq==>Logic.eq) f -> - (MapsTo x b (mapi f m) <-> exists a, b = f x a /\ MapsTo x a m). -Proof. -rewrite <-find_spec. setoid_rewrite <-find_spec. -intros Pr. rewrite mapi_spec' by trivial. -destruct (find x m); simpl; split. -- injection 1 as <-. now exists e. -- intros (a,(->,H)). now injection H as <-. -- discriminate. -- intros (a,(_,H)). discriminate. -Qed. - -(** Things are even worse for [merge] : we don't try to state any - equivalence, see instead boolean results below. *) - -(** Useful tactic for simplifying expressions like - [In y (add x e (remove z m))] *) - -Ltac map_iff := - repeat (progress ( - rewrite add_mapsto_iff || rewrite add_in_iff || - rewrite remove_mapsto_iff || rewrite remove_in_iff || - rewrite empty_mapsto_iff || rewrite empty_in_iff || - rewrite map_mapsto_iff || rewrite map_in_iff || - rewrite mapi_in_iff)). - -(** ** Specifications written using boolean predicates *) - -Section BoolSpec. - -Lemma mem_find_b {elt}(m:t elt)(x:key) : - mem x m = if find x m then true else false. -Proof. -apply eq_bool_alt. rewrite mem_find. destruct (find x m). -- now split. -- split; (discriminate || now destruct 1). -Qed. - -Variable elt elt' elt'' : Type. -Implicit Types m : t elt. -Implicit Types x y z : key. -Implicit Types e : elt. - -Lemma mem_b m x y : E.eq x y -> mem x m = mem y m. -Proof. now intros ->. Qed. - -Lemma find_o m x y : E.eq x y -> find x m = find y m. -Proof. now intros ->. Qed. - -Lemma empty_o x : find x (@empty elt) = None. -Proof. apply empty_spec. Qed. - -Lemma empty_a x : mem x (@empty elt) = false. -Proof. apply not_mem_find. apply empty_spec. Qed. - -Lemma add_eq_o m x y e : - E.eq x y -> find y (add x e m) = Some e. -Proof. - intros <-. apply add_spec1. -Qed. - -Lemma add_neq_o m x y e : - ~ E.eq x y -> find y (add x e m) = find y m. -Proof. apply add_spec2. Qed. -Hint Resolve add_neq_o : map. - -Lemma add_o m x y e : - find y (add x e m) = if E.eq_dec x y then Some e else find y m. -Proof. -destruct (E.eq_dec x y); auto with map. -Qed. - -Lemma add_eq_b m x y e : - E.eq x y -> mem y (add x e m) = true. -Proof. -intros <-. apply mem_spec, add_in_iff. now left. -Qed. - -Lemma add_neq_b m x y e : - ~E.eq x y -> mem y (add x e m) = mem y m. -Proof. -intros. now rewrite !mem_find_b, add_neq_o. -Qed. - -Lemma add_b m x y e : - mem y (add x e m) = eqb x y || mem y m. -Proof. -rewrite !mem_find_b, add_o. unfold eqb. -now destruct (E.eq_dec x y). -Qed. - -Lemma remove_eq_o m x y : - E.eq x y -> find y (remove x m) = None. -Proof. intros ->. apply remove_spec1. Qed. - -Lemma remove_neq_o m x y : - ~ E.eq x y -> find y (remove x m) = find y m. -Proof. apply remove_spec2. Qed. - -Hint Resolve remove_eq_o remove_neq_o : map. - -Lemma remove_o m x y : - find y (remove x m) = if E.eq_dec x y then None else find y m. -Proof. -destruct (E.eq_dec x y); auto with map. -Qed. - -Lemma remove_eq_b m x y : - E.eq x y -> mem y (remove x m) = false. -Proof. -intros <-. now rewrite mem_find_b, remove_eq_o. -Qed. - -Lemma remove_neq_b m x y : - ~ E.eq x y -> mem y (remove x m) = mem y m. -Proof. -intros. now rewrite !mem_find_b, remove_neq_o. -Qed. - -Lemma remove_b m x y : - mem y (remove x m) = negb (eqb x y) && mem y m. -Proof. -rewrite !mem_find_b, remove_o; unfold eqb. -now destruct (E.eq_dec x y). -Qed. - -Lemma map_o m x (f:elt->elt') : - find x (map f m) = option_map f (find x m). -Proof. apply map_spec. Qed. - -Lemma map_b m x (f:elt->elt') : - mem x (map f m) = mem x m. -Proof. -rewrite !mem_find_b, map_o. now destruct (find x m). -Qed. - -Lemma mapi_b m x (f:key->elt->elt') : - mem x (mapi f m) = mem x m. -Proof. -apply eq_bool_alt; rewrite !mem_spec. apply mapi_in_iff. -Qed. - -Lemma mapi_o m x (f:key->elt->elt') : - Proper (E.eq==>Logic.eq==>Logic.eq) f -> - find x (mapi f m) = option_map (f x) (find x m). -Proof. intros; now apply mapi_spec'. Qed. - -Lemma merge_spec1' (f:key->option elt->option elt'->option elt'') : - Proper (E.eq==>Logic.eq==>Logic.eq==>Logic.eq) f -> - forall (m:t elt)(m':t elt') x, - In x m \/ In x m' -> - find x (merge f m m') = f x (find x m) (find x m'). -Proof. - intros Hf m m' x H. - now destruct (merge_spec1 f H) as (y,(->,->)). -Qed. - -Lemma merge_spec1_none (f:key->option elt->option elt'->option elt'') : - (forall x, f x None None = None) -> - forall (m: t elt)(m': t elt') x, - exists y, E.eq y x /\ find x (merge f m m') = f y (find x m) (find x m'). -Proof. -intros Hf m m' x. -destruct (find x m) as [e|] eqn:Hm. -- assert (H : In x m \/ In x m') by (left; exists e; now apply find_spec). - destruct (merge_spec1 f H) as (y,(Hy,->)). - exists y; split; trivial. now rewrite Hm. -- destruct (find x m') as [e|] eqn:Hm'. - + assert (H : In x m \/ In x m') by (right; exists e; now apply find_spec). - destruct (merge_spec1 f H) as (y,(Hy,->)). - exists y; split; trivial. now rewrite Hm, Hm'. - + exists x. split. reflexivity. rewrite Hf. - apply not_in_find. intro H. - apply merge_spec2 in H. apply not_in_find in Hm. apply not_in_find in Hm'. - intuition. -Qed. - -Lemma merge_spec1'_none (f:key->option elt->option elt'->option elt'') : - Proper (E.eq==>Logic.eq==>Logic.eq==>Logic.eq) f -> - (forall x, f x None None = None) -> - forall (m: t elt)(m': t elt') x, - find x (merge f m m') = f x (find x m) (find x m'). -Proof. - intros Hf Hf' m m' x. - now destruct (merge_spec1_none Hf' m m' x) as (y,(->,->)). -Qed. - -Lemma bindings_o : forall m x, - find x m = findA (eqb x) (bindings m). -Proof. -intros. rewrite eq_option_alt. intro e. -rewrite <- find_mapsto_iff, bindings_mapsto_iff. -unfold eqb. -rewrite <- findA_NoDupA; dintuition; try apply bindings_3w; eauto. -Qed. - -Lemma bindings_b : forall m x, - mem x m = existsb (fun p => eqb x (fst p)) (bindings m). -Proof. -intros. -apply eq_bool_alt. -rewrite mem_spec, bindings_in_iff, existsb_exists. -split. -- intros (e,H). - rewrite InA_alt in H. - destruct H as ((k,e'),((H1,H2),H')); simpl in *; subst e'. - exists (k, e); split; trivial. simpl. now apply eqb_eq. -- intros ((k,e),(H,H')); simpl in *. apply eqb_eq in H'. - exists e. rewrite InA_alt. exists (k,e). now repeat split. -Qed. - -End BoolSpec. - -Section Equalities. -Variable elt:Type. - -(** A few basic equalities *) - -Lemma eq_empty (m: t elt) : m == empty <-> is_empty m = true. -Proof. - unfold Equal. rewrite is_empty_spec. now setoid_rewrite empty_spec. -Qed. - -Lemma add_id (m: t elt) x e : add x e m == m <-> find x m = Some e. -Proof. - split. - - intros H. rewrite <- (H x). apply add_spec1. - - intros H y. rewrite !add_o. now destruct E.eq_dec as [<-|E]. -Qed. - -Lemma add_add_1 (m: t elt) x e : - add x e (add x e m) == add x e m. -Proof. - intros y. rewrite !add_o. destruct E.eq_dec; auto. -Qed. - -Lemma add_add_2 (m: t elt) x x' e e' : - ~E.eq x x' -> add x e (add x' e' m) == add x' e' (add x e m). -Proof. - intros H y. rewrite !add_o. - do 2 destruct E.eq_dec; auto. - elim H. now transitivity y. -Qed. - -Lemma remove_id (m: t elt) x : remove x m == m <-> ~In x m. -Proof. - rewrite not_in_find. split. - - intros H. rewrite <- (H x). apply remove_spec1. - - intros H y. rewrite !remove_o. now destruct E.eq_dec as [<-|E]. -Qed. - -Lemma remove_remove_1 (m: t elt) x : - remove x (remove x m) == remove x m. -Proof. - intros y. rewrite !remove_o. destruct E.eq_dec; auto. -Qed. - -Lemma remove_remove_2 (m: t elt) x x' : - remove x (remove x' m) == remove x' (remove x m). -Proof. - intros y. rewrite !remove_o. do 2 destruct E.eq_dec; auto. -Qed. - -Lemma remove_add_1 (m: t elt) x e : - remove x (add x e m) == remove x m. -Proof. - intro y. rewrite !remove_o, !add_o. now destruct E.eq_dec. -Qed. - -Lemma remove_add_2 (m: t elt) x x' e : - ~E.eq x x' -> remove x' (add x e m) == add x e (remove x' m). -Proof. - intros H y. rewrite !remove_o, !add_o. - do 2 destruct E.eq_dec; auto. - - elim H; now transitivity y. - - symmetry. now apply remove_eq_o. - - symmetry. now apply remove_neq_o. -Qed. - -Lemma add_remove_1 (m: t elt) x e : - add x e (remove x m) == add x e m. -Proof. - intro y. rewrite !add_o, !remove_o. now destruct E.eq_dec. -Qed. - -(** Another characterisation of [Equal] *) - -Lemma Equal_mapsto_iff : forall m1 m2 : t elt, - m1 == m2 <-> (forall k e, MapsTo k e m1 <-> MapsTo k e m2). -Proof. -intros m1 m2. split; [intros Heq k e|intros Hiff]. -rewrite 2 find_mapsto_iff, Heq. split; auto. -intro k. rewrite eq_option_alt. intro e. -rewrite <- 2 find_mapsto_iff; auto. -Qed. - -(** * Relations between [Equal], [Equiv] and [Equivb]. *) - -(** First, [Equal] is [Equiv] with Leibniz on elements. *) - -Lemma Equal_Equiv : forall (m m' : t elt), - m == m' <-> Equiv Logic.eq m m'. -Proof. -intros. rewrite Equal_mapsto_iff. split; intros. -- split. - + split; intros (e,Hin); exists e; [rewrite <- H|rewrite H]; auto. - + intros; apply mapsto_fun with m k; auto; rewrite H; auto. -- split; intros H'. - + destruct H. - assert (Hin : In k m') by (rewrite <- H; exists e; auto). - destruct Hin as (e',He'). - rewrite (H0 k e e'); auto. - + destruct H. - assert (Hin : In k m) by (rewrite H; exists e; auto). - destruct Hin as (e',He'). - rewrite <- (H0 k e' e); auto. -Qed. - -(** [Equivb] and [Equiv] and equivalent when [eq_elt] and [cmp] - are related. *) - -Section Cmp. -Variable eq_elt : elt->elt->Prop. -Variable cmp : elt->elt->bool. - -Definition compat_cmp := - forall e e', cmp e e' = true <-> eq_elt e e'. - -Lemma Equiv_Equivb : compat_cmp -> - forall m m', Equiv eq_elt m m' <-> Equivb cmp m m'. -Proof. - unfold Equivb, Equiv, Cmp; intuition. - red in H; rewrite H; eauto. - red in H; rewrite <-H; eauto. -Qed. -End Cmp. - -(** Composition of the two last results: relation between [Equal] - and [Equivb]. *) - -Lemma Equal_Equivb : forall cmp, - (forall e e', cmp e e' = true <-> e = e') -> - forall (m m':t elt), m == m' <-> Equivb cmp m m'. -Proof. - intros; rewrite Equal_Equiv. - apply Equiv_Equivb; auto. -Qed. - -Lemma Equal_Equivb_eqdec : - forall eq_elt_dec : (forall e e', { e = e' } + { e <> e' }), - let cmp := fun e e' => if eq_elt_dec e e' then true else false in - forall (m m':t elt), m == m' <-> Equivb cmp m m'. -Proof. -intros; apply Equal_Equivb. -unfold cmp; clear cmp; intros. -destruct eq_elt_dec; now intuition. -Qed. - -End Equalities. - -(** * Results about [fold], [bindings], induction principles... *) - -Section Elt. - Variable elt:Type. - - Definition Add x (e:elt) m m' := m' == (add x e m). - - Notation eqke := (@eq_key_elt elt). - Notation eqk := (@eq_key elt). - - Instance eqk_equiv : Equivalence eqk. - Proof. unfold eq_key. destruct E.eq_equiv. constructor; eauto. Qed. - - Instance eqke_equiv : Equivalence eqke. - Proof. - unfold eq_key_elt; split; repeat red; intuition; simpl in *; - etransitivity; eauto. - Qed. - - (** Complements about InA, NoDupA and findA *) - - Lemma InA_eqke_eqk k k' e e' l : - E.eq k k' -> InA eqke (k,e) l -> InA eqk (k',e') l. - Proof. - intros Hk. rewrite 2 InA_alt. - intros ((k'',e'') & (Hk'',He'') & H); simpl in *; subst e''. - exists (k'',e); split; auto. red; simpl. now transitivity k. - Qed. - - Lemma NoDupA_incl {A} (R R':relation A) : - (forall x y, R x y -> R' x y) -> - forall l, NoDupA R' l -> NoDupA R l. - Proof. - intros Incl. - induction 1 as [ | a l E _ IH ]; constructor; auto. - contradict E. revert E. rewrite 2 InA_alt. firstorder. - Qed. - - Lemma NoDupA_eqk_eqke l : NoDupA eqk l -> NoDupA eqke l. - Proof. - apply NoDupA_incl. now destruct 1. - Qed. - - Lemma findA_rev l k : NoDupA eqk l -> - findA (eqb k) l = findA (eqb k) (rev l). - Proof. - intros H. apply eq_option_alt. intros e. unfold eqb. - rewrite <- !findA_NoDupA, InA_rev; eauto with map. reflexivity. - change (NoDupA eqk (rev l)). apply NoDupA_rev; auto using eqk_equiv. - Qed. - - (** * Bindings *) - - Lemma bindings_Empty (m:t elt) : Empty m <-> bindings m = nil. - Proof. - unfold Empty. split; intros H. - - assert (H' : forall a, ~ List.In a (bindings m)). - { intros (k,e) H'. apply (H k e). - rewrite bindings_mapsto_iff, InA_alt. - exists (k,e); repeat split; auto with map. } - destruct (bindings m) as [|p l]; trivial. - destruct (H' p); simpl; auto. - - intros x e. rewrite bindings_mapsto_iff, InA_alt. - rewrite H. now intros (y,(E,H')). - Qed. - - Lemma bindings_empty : bindings (@empty elt) = nil. - Proof. - rewrite <-bindings_Empty; apply empty_1. - Qed. - - (** * Conversions between maps and association lists. *) - - Definition uncurry {U V W : Type} (f : U -> V -> W) : U*V -> W := - fun p => f (fst p) (snd p). - - Definition of_list := - List.fold_right (uncurry (@add _)) (@empty elt). - - Definition to_list := bindings. - - Lemma of_list_1 : forall l k e, - NoDupA eqk l -> - (MapsTo k e (of_list l) <-> InA eqke (k,e) l). - Proof. - induction l as [|(k',e') l IH]; simpl; intros k e Hnodup. - - rewrite empty_mapsto_iff, InA_nil; intuition. - - unfold uncurry; simpl. - inversion_clear Hnodup as [| ? ? Hnotin Hnodup']. - specialize (IH k e Hnodup'); clear Hnodup'. - rewrite add_mapsto_iff, InA_cons, <- IH. - unfold eq_key_elt at 1; simpl. - split; destruct 1 as [H|H]; try (intuition;fail). - destruct (E.eq_dec k k'); [left|right]; split; auto with map. - contradict Hnotin. - apply InA_eqke_eqk with k e; intuition. - Qed. - - Lemma of_list_1b : forall l k, - NoDupA eqk l -> - find k (of_list l) = findA (eqb k) l. - Proof. - induction l as [|(k',e') l IH]; simpl; intros k Hnodup. - apply empty_o. - unfold uncurry; simpl. - inversion_clear Hnodup as [| ? ? Hnotin Hnodup']. - specialize (IH k Hnodup'); clear Hnodup'. - rewrite add_o, IH, eqb_sym. unfold eqb; now destruct E.eq_dec. - Qed. - - Lemma of_list_2 : forall l, NoDupA eqk l -> - equivlistA eqke l (to_list (of_list l)). - Proof. - intros l Hnodup (k,e). - rewrite <- bindings_mapsto_iff, of_list_1; intuition. - Qed. - - Lemma of_list_3 : forall s, Equal (of_list (to_list s)) s. - Proof. - intros s k. - rewrite of_list_1b, bindings_o; auto. - apply bindings_3w. - Qed. - - (** * Fold *) - - (** Alternative specification via [fold_right] *) - - Lemma fold_spec_right m (A:Type)(i:A)(f : key -> elt -> A -> A) : - fold f m i = List.fold_right (uncurry f) i (rev (bindings m)). - Proof. - rewrite fold_1. symmetry. apply fold_left_rev_right. - Qed. - - (** ** Induction principles about fold contributed by S. Lescuyer *) - - (** In the following lemma, the step hypothesis is deliberately restricted - to the precise map m we are considering. *) - - Lemma fold_rec : - forall (A:Type)(P : t elt -> A -> Type)(f : key -> elt -> A -> A), - forall (i:A)(m:t elt), - (forall m, Empty m -> P m i) -> - (forall k e a m' m'', MapsTo k e m -> ~In k m' -> - Add k e m' m'' -> P m' a -> P m'' (f k e a)) -> - P m (fold f m i). - Proof. - intros A P f i m Hempty Hstep. - rewrite fold_spec_right. - set (F:=uncurry f). - set (l:=rev (bindings m)). - assert (Hstep' : forall k e a m' m'', InA eqke (k,e) l -> ~In k m' -> - Add k e m' m'' -> P m' a -> P m'' (F (k,e) a)). - { - intros k e a m' m'' H ? ? ?; eapply Hstep; eauto. - revert H; unfold l; rewrite InA_rev, bindings_mapsto_iff; auto with *. } - assert (Hdup : NoDupA eqk l). - { unfold l. apply NoDupA_rev; try red; unfold eq_key ; eauto with *. - apply bindings_3w. } - assert (Hsame : forall k, find k m = findA (eqb k) l). - { intros k. unfold l. rewrite bindings_o, findA_rev; auto. - apply bindings_3w. } - clearbody l. clearbody F. clear Hstep f. revert m Hsame. induction l. - - (* empty *) - intros m Hsame; simpl. - apply Hempty. intros k e. - rewrite find_mapsto_iff, Hsame; simpl; discriminate. - - (* step *) - intros m Hsame; destruct a as (k,e); simpl. - apply Hstep' with (of_list l); auto. - + rewrite InA_cons; left; red; auto with map. - + inversion_clear Hdup. contradict H. destruct H as (e',He'). - apply InA_eqke_eqk with k e'; auto with map. - rewrite <- of_list_1; auto. - + intro k'. rewrite Hsame, add_o, of_list_1b. simpl. - rewrite eqb_sym. unfold eqb. now destruct E.eq_dec. - inversion_clear Hdup; auto with map. - + apply IHl. - * intros; eapply Hstep'; eauto. - * inversion_clear Hdup; auto. - * intros; apply of_list_1b. inversion_clear Hdup; auto. - Qed. - - (** Same, with [empty] and [add] instead of [Empty] and [Add]. In this - case, [P] must be compatible with equality of sets *) - - Theorem fold_rec_bis : - forall (A:Type)(P : t elt -> A -> Type)(f : key -> elt -> A -> A), - forall (i:A)(m:t elt), - (forall m m' a, Equal m m' -> P m a -> P m' a) -> - (P empty i) -> - (forall k e a m', MapsTo k e m -> ~In k m' -> - P m' a -> P (add k e m') (f k e a)) -> - P m (fold f m i). - Proof. - intros A P f i m Pmorphism Pempty Pstep. - apply fold_rec; intros. - apply Pmorphism with empty; auto. intro k. rewrite empty_o. - case_eq (find k m0); auto; intros e'; rewrite <- find_mapsto_iff. - intro H'; elim (H k e'); auto. - apply Pmorphism with (add k e m'); try intro; auto. - Qed. - - Lemma fold_rec_nodep : - forall (A:Type)(P : A -> Type)(f : key -> elt -> A -> A)(i:A)(m:t elt), - P i -> (forall k e a, MapsTo k e m -> P a -> P (f k e a)) -> - P (fold f m i). - Proof. - intros; apply fold_rec_bis with (P:=fun _ => P); auto. - Qed. - - (** [fold_rec_weak] is a weaker principle than [fold_rec_bis] : - the step hypothesis must here be applicable anywhere. - At the same time, it looks more like an induction principle, - and hence can be easier to use. *) - - Lemma fold_rec_weak : - forall (A:Type)(P : t elt -> A -> Type)(f : key -> elt -> A -> A)(i:A), - (forall m m' a, Equal m m' -> P m a -> P m' a) -> - P empty i -> - (forall k e a m, ~In k m -> P m a -> P (add k e m) (f k e a)) -> - forall m, P m (fold f m i). - Proof. - intros; apply fold_rec_bis; auto. - Qed. - - Lemma fold_rel : - forall (A B:Type)(R : A -> B -> Type) - (f : key -> elt -> A -> A)(g : key -> elt -> B -> B)(i : A)(j : B) - (m : t elt), - R i j -> - (forall k e a b, MapsTo k e m -> R a b -> R (f k e a) (g k e b)) -> - R (fold f m i) (fold g m j). - Proof. - intros A B R f g i j m Rempty Rstep. - rewrite 2 fold_spec_right. set (l:=rev (bindings m)). - assert (Rstep' : forall k e a b, InA eqke (k,e) l -> - R a b -> R (f k e a) (g k e b)). - { intros; apply Rstep; auto. - rewrite bindings_mapsto_iff, <- InA_rev; auto with map. } - clearbody l; clear Rstep m. - induction l; simpl; auto. - apply Rstep'; auto. - destruct a; simpl; rewrite InA_cons; left; red; auto with map. - Qed. - - (** From the induction principle on [fold], we can deduce some general - induction principles on maps. *) - - Lemma map_induction : - forall P : t elt -> Type, - (forall m, Empty m -> P m) -> - (forall m m', P m -> forall x e, ~In x m -> Add x e m m' -> P m') -> - forall m, P m. - Proof. - intros. apply (@fold_rec _ (fun s _ => P s) (fun _ _ _ => tt) tt m); eauto. - Qed. - - Lemma map_induction_bis : - forall P : t elt -> Type, - (forall m m', Equal m m' -> P m -> P m') -> - P empty -> - (forall x e m, ~In x m -> P m -> P (add x e m)) -> - forall m, P m. - Proof. - intros. - apply (@fold_rec_bis _ (fun s _ => P s) (fun _ _ _ => tt) tt m); eauto. - Qed. - - (** [fold] can be used to reconstruct the same initial set. *) - - Lemma fold_identity : forall m : t elt, Equal (fold (@add _) m empty) m. - Proof. - intros. - apply fold_rec with (P:=fun m acc => Equal acc m); auto with map. - intros m' Heq k'. - rewrite empty_o. - case_eq (find k' m'); auto; intros e'; rewrite <- find_mapsto_iff. - intro; elim (Heq k' e'); auto. - intros k e a m' m'' _ _ Hadd Heq k'. - red in Heq. rewrite Hadd, 2 add_o, Heq; auto. - Qed. - - Section Fold_More. - - (** ** Additional properties of fold *) - - (** When a function [f] is compatible and allows transpositions, we can - compute [fold f] in any order. *) - - Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA). - - Lemma fold_Empty (f:key->elt->A->A) : - forall m i, Empty m -> eqA (fold f m i) i. - Proof. - intros. apply fold_rec_nodep with (P:=fun a => eqA a i). - reflexivity. - intros. elim (H k e); auto. - Qed. - - Lemma fold_init (f:key->elt->A->A) : - Proper (E.eq==>eq==>eqA==>eqA) f -> - forall m i i', eqA i i' -> eqA (fold f m i) (fold f m i'). - Proof. - intros Hf m i i' Hi. apply fold_rel with (R:=eqA); auto. - intros. now apply Hf. - Qed. - - (** Transpositions of f (a.k.a diamond property). - Could we swap two sequential calls to f, i.e. do we have: - - f k e (f k' e' a) == f k' e' (f k e a) - - First, we do no need this equation for all keys, but only - when k and k' aren't equal, as suggested by Pierre Castéran. - Think for instance of [f] being [M.add] : in general, we don't have - [M.add k e (M.add k e' m) == M.add k e' (M.add k e m)]. - Fortunately, we will never encounter this situation during a real - [fold], since the keys received by this [fold] are unique. - NB: without this condition, this condition would be - [SetoidList.transpose2]. - - Secondly, instead of the equation above, we now use a statement - with more basic equalities, allowing to prove [fold_commutes] even - when [f] isn't a morphism. - NB: When [f] is a morphism, [Diamond f] gives back the equation above. -*) - - Definition Diamond (f:key->elt->A->A) := - forall k k' e e' a b b', ~E.eq k k' -> - eqA (f k e a) b -> eqA (f k' e' a) b' -> eqA (f k e b') (f k' e' b). - - Lemma fold_commutes (f:key->elt->A->A) : - Diamond f -> - forall i m k e, ~In k m -> - eqA (fold f m (f k e i)) (f k e (fold f m i)). - Proof. - intros Hf i m k e H. - apply fold_rel with (R:= fun a b => eqA a (f k e b)); auto. - - reflexivity. - - intros k' e' b a Hm E. - apply Hf with a; try easy. - contradict H; rewrite <- H. now exists e'. - Qed. - - Hint Resolve NoDupA_eqk_eqke NoDupA_rev bindings_3w : map. - - Lemma fold_Proper (f:key->elt->A->A) : - Proper (E.eq==>eq==>eqA==>eqA) f -> - Diamond f -> - Proper (Equal==>eqA==>eqA) (fold f). - Proof. - intros Hf Hf' m1 m2 Hm i j Hi. - rewrite 2 fold_spec_right. - assert (NoDupA eqk (rev (bindings m1))) by (auto with * ). - assert (NoDupA eqk (rev (bindings m2))) by (auto with * ). - apply fold_right_equivlistA_restr2 with (R:=complement eqk)(eqA:=eqke) - ; auto with *. - - intros (k1,e1) (k2,e2) (Hk,He) a1 a2 Ha; simpl in *. now apply Hf. - - unfold complement, eq_key, eq_key_elt; repeat red. intuition eauto with map. - - intros (k,e) (k',e') z z' h h'; unfold eq_key, uncurry;simpl; auto. - rewrite h'. eapply Hf'; now eauto. - - rewrite <- NoDupA_altdef; auto. - - intros (k,e). - rewrite 2 InA_rev, <- 2 bindings_mapsto_iff, 2 find_mapsto_iff, Hm; - auto with *. - Qed. - - Lemma fold_Equal (f:key->elt->A->A) : - Proper (E.eq==>eq==>eqA==>eqA) f -> - Diamond f -> - forall m1 m2 i, - Equal m1 m2 -> - eqA (fold f m1 i) (fold f m2 i). - Proof. - intros. now apply fold_Proper. - Qed. - - Lemma fold_Add (f:key->elt->A->A) : - Proper (E.eq==>eq==>eqA==>eqA) f -> - Diamond f -> - forall m1 m2 k e i, ~In k m1 -> Add k e m1 m2 -> - eqA (fold f m2 i) (f k e (fold f m1 i)). - Proof. - intros Hf Hf' m1 m2 k e i Hm1 Hm2. - rewrite 2 fold_spec_right. - set (f':=uncurry f). - change (f k e (fold_right f' i (rev (bindings m1)))) - with (f' (k,e) (fold_right f' i (rev (bindings m1)))). - assert (NoDupA eqk (rev (bindings m1))) by (auto with * ). - assert (NoDupA eqk (rev (bindings m2))) by (auto with * ). - apply fold_right_add_restr with - (R:=complement eqk)(eqA:=eqke); auto with *. - - intros (k1,e1) (k2,e2) (Hk,He) a a' Ha; unfold f'; simpl in *. now apply Hf. - - unfold complement, eq_key_elt, eq_key; repeat red; intuition eauto with map. - - intros (k1,e1) (k2,e2) z1 z2; unfold eq_key, f', uncurry; simpl. - eapply Hf'; now eauto. - - rewrite <- NoDupA_altdef; auto. - - rewrite InA_rev, <- bindings_mapsto_iff by (auto with * ). firstorder. - - intros (a,b). - rewrite InA_cons, 2 InA_rev, <- 2 bindings_mapsto_iff, - 2 find_mapsto_iff by (auto with * ). - unfold eq_key_elt; simpl. - rewrite Hm2, !find_spec, add_mapsto_new; intuition. - Qed. - - Lemma fold_add (f:key->elt->A->A) : - Proper (E.eq==>eq==>eqA==>eqA) f -> - Diamond f -> - forall m k e i, ~In k m -> - eqA (fold f (add k e m) i) (f k e (fold f m i)). - Proof. - intros. now apply fold_Add. - Qed. - - End Fold_More. - - (** * Cardinal *) - - Lemma cardinal_fold (m : t elt) : - cardinal m = fold (fun _ _ => S) m 0. - Proof. - rewrite cardinal_1, fold_1. - symmetry; apply fold_left_length; auto. - Qed. - - Lemma cardinal_Empty : forall m : t elt, - Empty m <-> cardinal m = 0. - Proof. - intros. - rewrite cardinal_1, bindings_Empty. - destruct (bindings m); intuition; discriminate. - Qed. - - Lemma Equal_cardinal (m m' : t elt) : - Equal m m' -> cardinal m = cardinal m'. - Proof. - intro. rewrite 2 cardinal_fold. - apply fold_Equal with (eqA:=eq); try congruence; auto with map. - Qed. - - Lemma cardinal_0 (m : t elt) : Empty m -> cardinal m = 0. - Proof. - intros; rewrite <- cardinal_Empty; auto. - Qed. - - Lemma cardinal_S m m' x e : - ~ In x m -> Add x e m m' -> cardinal m' = S (cardinal m). - Proof. - intros. rewrite 2 cardinal_fold. - change S with ((fun _ _ => S) x e). - apply fold_Add with (eqA:=eq); try congruence; auto with map. - Qed. - - Lemma cardinal_inv_1 : forall m : t elt, - cardinal m = 0 -> Empty m. - Proof. - intros; rewrite cardinal_Empty; auto. - Qed. - Hint Resolve cardinal_inv_1 : map. - - Lemma cardinal_inv_2 : - forall m n, cardinal m = S n -> { p : key*elt | MapsTo (fst p) (snd p) m }. - Proof. - intros; rewrite M.cardinal_spec in *. - generalize (bindings_mapsto_iff m). - destruct (bindings m); try discriminate. - exists p; auto. - rewrite H0; destruct p; simpl; auto. - constructor; red; auto with map. - Qed. - - Lemma cardinal_inv_2b : - forall m, cardinal m <> 0 -> { p : key*elt | MapsTo (fst p) (snd p) m }. - Proof. - intros. - generalize (@cardinal_inv_2 m); destruct cardinal. - elim H;auto. - eauto. - Qed. - - Lemma not_empty_mapsto (m : t elt) : - ~Empty m -> exists k e, MapsTo k e m. - Proof. - intro. - destruct (@cardinal_inv_2b m) as ((k,e),H'). - contradict H. now apply cardinal_inv_1. - exists k; now exists e. - Qed. - - Lemma not_empty_in (m:t elt) : - ~Empty m -> exists k, In k m. - Proof. - intro. destruct (not_empty_mapsto H) as (k,Hk). - now exists k. - Qed. - - (** * Additional notions over maps *) - - Definition Disjoint (m m' : t elt) := - forall k, ~(In k m /\ In k m'). - - Definition Partition (m m1 m2 : t elt) := - Disjoint m1 m2 /\ - (forall k e, MapsTo k e m <-> MapsTo k e m1 \/ MapsTo k e m2). - - (** * Emulation of some functions lacking in the interface *) - - Definition filter (f : key -> elt -> bool)(m : t elt) := - fold (fun k e m => if f k e then add k e m else m) m empty. - - Definition for_all (f : key -> elt -> bool)(m : t elt) := - fold (fun k e b => if f k e then b else false) m true. - - Definition exists_ (f : key -> elt -> bool)(m : t elt) := - fold (fun k e b => if f k e then true else b) m false. - - Definition partition (f : key -> elt -> bool)(m : t elt) := - (filter f m, filter (fun k e => negb (f k e)) m). - - (** [update] adds to [m1] all the bindings of [m2]. It can be seen as - an [union] operator which gives priority to its 2nd argument - in case of binding conflit. *) - - Definition update (m1 m2 : t elt) := fold (@add _) m2 m1. - - (** [restrict] keeps from [m1] only the bindings whose key is in [m2]. - It can be seen as an [inter] operator, with priority to its 1st argument - in case of binding conflit. *) - - Definition restrict (m1 m2 : t elt) := filter (fun k _ => mem k m2) m1. - - (** [diff] erases from [m1] all bindings whose key is in [m2]. *) - - Definition diff (m1 m2 : t elt) := filter (fun k _ => negb (mem k m2)) m1. - - (** Properties of these abbreviations *) - - Lemma filter_iff (f : key -> elt -> bool) : - Proper (E.eq==>eq==>eq) f -> - forall m k e, - MapsTo k e (filter f m) <-> MapsTo k e m /\ f k e = true. - Proof. - unfold filter. - set (f':=fun k e m => if f k e then add k e m else m). - intros Hf m. pattern m, (fold f' m empty). apply fold_rec. - - - intros m' Hm' k e. rewrite empty_mapsto_iff. intuition. - elim (Hm' k e); auto. - - - intros k e acc m1 m2 Hke Hn Hadd IH k' e'. - change (Equal m2 (add k e m1)) in Hadd; rewrite Hadd. - unfold f'; simpl. - rewrite add_mapsto_new by trivial. - case_eq (f k e); intros Hfke; simpl; - rewrite ?add_mapsto_iff, IH; clear IH; intuition. - + rewrite <- Hfke; apply Hf; auto with map. - + right. repeat split; trivial. contradict Hn. rewrite Hn. now exists e'. - + assert (f k e = f k' e') by (apply Hf; auto). congruence. - Qed. - - Lemma for_all_filter f m : - for_all f m = is_empty (filter (fun k e => negb (f k e)) m). - Proof. - unfold for_all, filter. - eapply fold_rel with (R:=fun x y => x = is_empty y). - - symmetry. apply is_empty_iff. apply empty_1. - - intros; subst. destruct (f k e); simpl; trivial. - symmetry. apply not_true_is_false. rewrite is_empty_spec. - intros H'. specialize (H' k). now rewrite add_spec1 in H'. - Qed. - - Lemma exists_filter f m : - exists_ f m = negb (is_empty (filter f m)). - Proof. - unfold for_all, filter. - eapply fold_rel with (R:=fun x y => x = negb (is_empty y)). - - symmetry. rewrite negb_false_iff. apply is_empty_iff. apply empty_1. - - intros; subst. destruct (f k e); simpl; trivial. - symmetry. rewrite negb_true_iff. apply not_true_is_false. - rewrite is_empty_spec. - intros H'. specialize (H' k). now rewrite add_spec1 in H'. - Qed. - - Lemma for_all_iff f m : - Proper (E.eq==>eq==>eq) f -> - (for_all f m = true <-> (forall k e, MapsTo k e m -> f k e = true)). - Proof. - intros Hf. - rewrite for_all_filter. - rewrite <- is_empty_iff. unfold Empty. - split; intros H k e; specialize (H k e); - rewrite filter_iff in * by solve_proper; intuition. - - destruct (f k e); auto. - - now rewrite H0 in H2. - Qed. - - Lemma exists_iff f m : - Proper (E.eq==>eq==>eq) f -> - (exists_ f m = true <-> - (exists k e, MapsTo k e m /\ f k e = true)). - Proof. - intros Hf. - rewrite exists_filter. rewrite negb_true_iff. - rewrite <- not_true_iff_false, <- is_empty_iff. - split. - - intros H. apply not_empty_mapsto in H. now setoid_rewrite filter_iff in H. - - unfold Empty. setoid_rewrite filter_iff; trivial. firstorder. - Qed. - - Lemma Disjoint_alt : forall m m', - Disjoint m m' <-> - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> False). - Proof. - unfold Disjoint; split. - intros H k v v' H1 H2. - apply H with k; split. - exists v; trivial. - exists v'; trivial. - intros H k ((v,Hv),(v',Hv')). - eapply H; eauto. - Qed. - - Section Partition. - Variable f : key -> elt -> bool. - Hypothesis Hf : Proper (E.eq==>eq==>eq) f. - - Lemma partition_iff_1 : forall m m1 k e, - m1 = fst (partition f m) -> - (MapsTo k e m1 <-> MapsTo k e m /\ f k e = true). - Proof. - unfold partition; simpl; intros. subst m1. - apply filter_iff; auto. - Qed. - - Lemma partition_iff_2 : forall m m2 k e, - m2 = snd (partition f m) -> - (MapsTo k e m2 <-> MapsTo k e m /\ f k e = false). - Proof. - unfold partition; simpl; intros. subst m2. - rewrite filter_iff. - split; intros (H,H'); split; auto. - destruct (f k e); simpl in *; auto. - rewrite H'; auto. - repeat red; intros. f_equal. apply Hf; auto. - Qed. - - Lemma partition_Partition : forall m m1 m2, - partition f m = (m1,m2) -> Partition m m1 m2. - Proof. - intros. split. - rewrite Disjoint_alt. intros k e e'. - rewrite (@partition_iff_1 m m1), (@partition_iff_2 m m2) - by (rewrite H; auto). - intros (U,V) (W,Z). rewrite <- (mapsto_fun U W) in Z; congruence. - intros k e. - rewrite (@partition_iff_1 m m1), (@partition_iff_2 m m2) - by (rewrite H; auto). - destruct (f k e); intuition. - Qed. - - End Partition. - - Lemma Partition_In : forall m m1 m2 k, - Partition m m1 m2 -> In k m -> {In k m1}+{In k m2}. - Proof. - intros m m1 m2 k Hm Hk. - destruct (In_dec m1 k) as [H|H]; [left|right]; auto. - destruct Hm as (Hm,Hm'). - destruct Hk as (e,He); rewrite Hm' in He; destruct He. - elim H; exists e; auto. - exists e; auto. - Defined. - - Lemma Disjoint_sym : forall m1 m2, Disjoint m1 m2 -> Disjoint m2 m1. - Proof. - intros m1 m2 H k (H1,H2). elim (H k); auto. - Qed. - - Lemma Partition_sym : forall m m1 m2, - Partition m m1 m2 -> Partition m m2 m1. - Proof. - intros m m1 m2 (H,H'); split. - apply Disjoint_sym; auto. - intros; rewrite H'; intuition. - Qed. - - Lemma Partition_Empty : forall m m1 m2, Partition m m1 m2 -> - (Empty m <-> (Empty m1 /\ Empty m2)). - Proof. - intros m m1 m2 (Hdisj,Heq). split. - intro He. - split; intros k e Hke; elim (He k e); rewrite Heq; auto. - intros (He1,He2) k e Hke. rewrite Heq in Hke. destruct Hke. - elim (He1 k e); auto. - elim (He2 k e); auto. - Qed. - - Lemma Partition_Add : - forall m m' x e , ~In x m -> Add x e m m' -> - forall m1 m2, Partition m' m1 m2 -> - exists m3, (Add x e m3 m1 /\ Partition m m3 m2 \/ - Add x e m3 m2 /\ Partition m m1 m3). - Proof. - unfold Partition. intros m m' x e Hn Hadd m1 m2 (Hdisj,Hor). - assert (Heq : Equal m (remove x m')). - { change (Equal m' (add x e m)) in Hadd. rewrite Hadd. - intro k. rewrite remove_o, add_o. - destruct E.eq_dec as [He|Hne]; auto. - rewrite <- He, <- not_find_in_iff; auto. } - assert (H : MapsTo x e m'). - { change (Equal m' (add x e m)) in Hadd; rewrite Hadd. - apply add_1; auto with map. } - rewrite Hor in H; destruct H. - - - (* first case : x in m1 *) - exists (remove x m1); left. split; [|split]. - + (* add *) - change (Equal m1 (add x e (remove x m1))). - intro k. - rewrite add_o, remove_o. - destruct E.eq_dec as [He|Hne]; auto. - rewrite <- He; apply find_1; auto. - + (* disjoint *) - intros k (H1,H2). elim (Hdisj k). split; auto. - rewrite remove_in_iff in H1; destruct H1; auto. - + (* mapsto *) - intros k' e'. - rewrite Heq, 2 remove_mapsto_iff, Hor. - intuition. - elim (Hdisj x); split; [exists e|exists e']; auto. - apply MapsTo_1 with k'; auto with map. - - - (* second case : x in m2 *) - exists (remove x m2); right. split; [|split]. - + (* add *) - change (Equal m2 (add x e (remove x m2))). - intro k. - rewrite add_o, remove_o. - destruct E.eq_dec as [He|Hne]; auto. - rewrite <- He; apply find_1; auto. - + (* disjoint *) - intros k (H1,H2). elim (Hdisj k). split; auto. - rewrite remove_in_iff in H2; destruct H2; auto. - + (* mapsto *) - intros k' e'. - rewrite Heq, 2 remove_mapsto_iff, Hor. - intuition. - elim (Hdisj x); split; [exists e'|exists e]; auto. - apply MapsTo_1 with k'; auto with map. - Qed. - - Lemma Partition_fold : - forall (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA)(f:key->elt->A->A), - Proper (E.eq==>eq==>eqA==>eqA) f -> - Diamond eqA f -> - forall m m1 m2 i, - Partition m m1 m2 -> - eqA (fold f m i) (fold f m1 (fold f m2 i)). - Proof. - intros A eqA st f Comp Tra. - induction m as [m Hm|m m' IH k e Hn Hadd] using map_induction. - - - intros m1 m2 i Hp. rewrite (fold_Empty (eqA:=eqA)); auto. - rewrite (Partition_Empty Hp) in Hm. destruct Hm. - rewrite 2 (fold_Empty (eqA:=eqA)); auto. reflexivity. - - - intros m1 m2 i Hp. - destruct (Partition_Add Hn Hadd Hp) as (m3,[(Hadd',Hp')|(Hadd',Hp')]). - + (* fst case: m3 is (k,e)::m1 *) - assert (~In k m3). - { contradict Hn. destruct Hn as (e',He'). - destruct Hp' as (Hp1,Hp2). exists e'. rewrite Hp2; auto. } - transitivity (f k e (fold f m i)). - apply fold_Add with (eqA:=eqA); auto. - symmetry. - transitivity (f k e (fold f m3 (fold f m2 i))). - apply fold_Add with (eqA:=eqA); auto. - apply Comp; auto with map. - symmetry; apply IH; auto. - + (* snd case: m3 is (k,e)::m2 *) - assert (~In k m3). - { contradict Hn. destruct Hn as (e',He'). - destruct Hp' as (Hp1,Hp2). exists e'. rewrite Hp2; auto. } - assert (~In k m1). - { contradict Hn. destruct Hn as (e',He'). - destruct Hp' as (Hp1,Hp2). exists e'. rewrite Hp2; auto. } - transitivity (f k e (fold f m i)). - apply fold_Add with (eqA:=eqA); auto. - transitivity (f k e (fold f m1 (fold f m3 i))). - apply Comp; auto using IH with map. - transitivity (fold f m1 (f k e (fold f m3 i))). - symmetry. - apply fold_commutes with (eqA:=eqA); auto. - apply fold_init with (eqA:=eqA); auto. - symmetry. - apply fold_Add with (eqA:=eqA); auto. - Qed. - - Lemma Partition_cardinal : forall m m1 m2, Partition m m1 m2 -> - cardinal m = cardinal m1 + cardinal m2. - Proof. - intros. - rewrite (cardinal_fold m), (cardinal_fold m1). - set (f:=fun (_:key)(_:elt)=>S). - setoid_replace (fold f m 0) with (fold f m1 (fold f m2 0)). - rewrite <- cardinal_fold. - apply fold_rel with (R:=fun u v => u = v + cardinal m2); simpl; auto. - apply Partition_fold with (eqA:=eq); compute; auto with map. congruence. - Qed. - - Lemma Partition_partition : forall m m1 m2, Partition m m1 m2 -> - let f := fun k (_:elt) => mem k m1 in - Equal m1 (fst (partition f m)) /\ Equal m2 (snd (partition f m)). - Proof. - intros m m1 m2 Hm f. - assert (Hf : Proper (E.eq==>eq==>eq) f). - intros k k' Hk e e' _; unfold f; rewrite Hk; auto. - set (m1':= fst (partition f m)). - set (m2':= snd (partition f m)). - split; rewrite Equal_mapsto_iff; intros k e. - rewrite (@partition_iff_1 f Hf m m1') by auto. - unfold f. - rewrite <- mem_in_iff. - destruct Hm as (Hm,Hm'). - rewrite Hm'. - intuition. - exists e; auto. - elim (Hm k); split; auto; exists e; auto. - rewrite (@partition_iff_2 f Hf m m2') by auto. - unfold f. - rewrite <- not_mem_in_iff. - destruct Hm as (Hm,Hm'). - rewrite Hm'. - intuition. - elim (Hm k); split; auto; exists e; auto. - elim H1; exists e; auto. - Qed. - - Lemma update_mapsto_iff : forall m m' k e, - MapsTo k e (update m m') <-> - (MapsTo k e m' \/ (MapsTo k e m /\ ~In k m')). - Proof. - unfold update. - intros m m'. - pattern m', (fold (@add _) m' m). apply fold_rec. - - - intros m0 Hm0 k e. - assert (~In k m0) by (intros (e0,He0); apply (Hm0 k e0); auto). - intuition. - elim (Hm0 k e); auto. - - - intros k e m0 m1 m2 _ Hn Hadd IH k' e'. - change (Equal m2 (add k e m1)) in Hadd. - rewrite Hadd, 2 add_mapsto_iff, IH, add_in_iff. clear IH. intuition. - Qed. - - Lemma update_dec : forall m m' k e, MapsTo k e (update m m') -> - { MapsTo k e m' } + { MapsTo k e m /\ ~In k m'}. - Proof. - intros m m' k e H. rewrite update_mapsto_iff in H. - destruct (In_dec m' k) as [H'|H']; [left|right]; intuition. - elim H'; exists e; auto. - Defined. - - Lemma update_in_iff : forall m m' k, - In k (update m m') <-> In k m \/ In k m'. - Proof. - intros m m' k. split. - intros (e,H); rewrite update_mapsto_iff in H. - destruct H; [right|left]; exists e; intuition. - destruct (In_dec m' k) as [H|H]. - destruct H as (e,H). intros _; exists e. - rewrite update_mapsto_iff; left; auto. - destruct 1 as [H'|H']; [|elim H; auto]. - destruct H' as (e,H'). exists e. - rewrite update_mapsto_iff; right; auto. - Qed. - - Lemma diff_mapsto_iff : forall m m' k e, - MapsTo k e (diff m m') <-> MapsTo k e m /\ ~In k m'. - Proof. - intros m m' k e. - unfold diff. - rewrite filter_iff. - intuition. - rewrite mem_1 in *; auto; discriminate. - intros ? ? Hk _ _ _; rewrite Hk; auto. - Qed. - - Lemma diff_in_iff : forall m m' k, - In k (diff m m') <-> In k m /\ ~In k m'. - Proof. - intros m m' k. split. - intros (e,H); rewrite diff_mapsto_iff in H. - destruct H; split; auto. exists e; auto. - intros ((e,H),H'); exists e; rewrite diff_mapsto_iff; auto. - Qed. - - Lemma restrict_mapsto_iff : forall m m' k e, - MapsTo k e (restrict m m') <-> MapsTo k e m /\ In k m'. - Proof. - intros m m' k e. - unfold restrict. - rewrite filter_iff. - intuition. - intros ? ? Hk _ _ _; rewrite Hk; auto. - Qed. - - Lemma restrict_in_iff : forall m m' k, - In k (restrict m m') <-> In k m /\ In k m'. - Proof. - intros m m' k. split. - intros (e,H); rewrite restrict_mapsto_iff in H. - destruct H; split; auto. exists e; auto. - intros ((e,H),H'); exists e; rewrite restrict_mapsto_iff; auto. - Qed. - - (** specialized versions analyzing only keys (resp. bindings) *) - - Definition filter_dom (f : key -> bool) := filter (fun k _ => f k). - Definition filter_range (f : elt -> bool) := filter (fun _ => f). - Definition for_all_dom (f : key -> bool) := for_all (fun k _ => f k). - Definition for_all_range (f : elt -> bool) := for_all (fun _ => f). - Definition exists_dom (f : key -> bool) := exists_ (fun k _ => f k). - Definition exists_range (f : elt -> bool) := exists_ (fun _ => f). - Definition partition_dom (f : key -> bool) := partition (fun k _ => f k). - Definition partition_range (f : elt -> bool) := partition (fun _ => f). - - End Elt. - - Instance cardinal_m {elt} : Proper (Equal ==> Logic.eq) (@cardinal elt). - Proof. intros m m'. apply Equal_cardinal. Qed. - - Instance Disjoint_m {elt} : Proper (Equal ==> Equal ==> iff) (@Disjoint elt). - Proof. - intros m1 m1' Hm1 m2 m2' Hm2. unfold Disjoint. split; intros. - rewrite <- Hm1, <- Hm2; auto. - rewrite Hm1, Hm2; auto. - Qed. - - Instance Partition_m {elt} : - Proper (Equal ==> Equal ==> Equal ==> iff) (@Partition elt). - Proof. - intros m1 m1' Hm1 m2 m2' Hm2 m3 m3' Hm3. unfold Partition. - rewrite <- Hm2, <- Hm3. - split; intros (H,H'); split; auto; intros. - rewrite <- Hm1, <- Hm2, <- Hm3; auto. - rewrite Hm1, Hm2, Hm3; auto. - Qed. - -(* - Instance filter_m0 {elt} (f:key->elt->bool) : - Proper (E.eq==>Logic.eq==>Logic.eq) f -> - Proper (Equal==>Equal) (filter f). - Proof. - intros Hf m m' Hm. apply Equal_mapsto_iff. intros. - now rewrite !filter_iff, Hm. - Qed. -*) - - Instance filter_m {elt} : - Proper ((E.eq==>Logic.eq==>Logic.eq)==>Equal==>Equal) (@filter elt). - Proof. - intros f f' Hf m m' Hm. unfold filter. - rewrite 2 fold_spec_right. - set (l := rev (bindings m)). - set (l' := rev (bindings m')). - set (op := fun (f:key->elt->bool) => - uncurry (fun k e acc => if f k e then add k e acc else acc)). - change (Equal (fold_right (op f) empty l) (fold_right (op f') empty l')). - assert (Hl : NoDupA eq_key l). - { apply NoDupA_rev. apply eqk_equiv. apply bindings_spec2w. } - assert (Hl' : NoDupA eq_key l'). - { apply NoDupA_rev. apply eqk_equiv. apply bindings_spec2w. } - assert (H : PermutationA eq_key_elt l l'). - { apply NoDupA_equivlistA_PermutationA. - - apply eqke_equiv. - - now apply NoDupA_eqk_eqke. - - now apply NoDupA_eqk_eqke. - - intros (k,e); unfold l, l'. rewrite 2 InA_rev, 2 bindings_spec1. - rewrite Equal_mapsto_iff in Hm. apply Hm. } - destruct (PermutationA_decompose (eqke_equiv _) H) as (l0,(P,E)). - transitivity (fold_right (op f) empty l0). - - apply fold_right_equivlistA_restr2 - with (eqA:=Logic.eq)(R:=complement eq_key); auto with *. - + intros p p' <- acc acc' Hacc. - destruct p as (k,e); unfold op, uncurry; simpl. - destruct (f k e); now rewrite Hacc. - + intros (k,e) (k',e') z z'. - unfold op, complement, uncurry, eq_key; simpl. - intros Hk Hz. - destruct (f k e), (f k' e'); rewrite <- Hz; try reflexivity. - now apply add_add_2. - + apply NoDupA_incl with eq_key; trivial. intros; subst; now red. - + apply PermutationA_preserves_NoDupA with l; auto with *. - apply Permutation_PermutationA; auto with *. - apply NoDupA_incl with eq_key; trivial. intros; subst; now red. - + apply NoDupA_altdef. apply NoDupA_rev. apply eqk_equiv. - apply bindings_spec2w. - + apply PermutationA_equivlistA; auto with *. - apply Permutation_PermutationA; auto with *. - - clearbody l'. clear l Hl Hl' H P m m' Hm. - induction E. - + reflexivity. - + simpl. destruct x as (k,e), x' as (k',e'). - unfold op, uncurry at 1 3; simpl. - destruct H; simpl in *. rewrite <- (Hf _ _ H _ _ H0). - destruct (f k e); trivial. now f_equiv. - Qed. - - Instance for_all_m {elt} : - Proper ((E.eq==>Logic.eq==>Logic.eq)==>Equal==>Logic.eq) (@for_all elt). - Proof. - intros f f' Hf m m' Hm. rewrite 2 for_all_filter. - (* Strange: we cannot rewrite Hm here... *) - f_equiv. f_equiv; trivial. - intros k k' Hk e e' He. f_equal. now apply Hf. - Qed. - - Instance exists_m {elt} : - Proper ((E.eq==>Logic.eq==>Logic.eq)==>Equal==>Logic.eq) (@exists_ elt). - Proof. - intros f f' Hf m m' Hm. rewrite 2 exists_filter. - f_equal. now apply is_empty_m, filter_m. - Qed. - - Fact diamond_add {elt} : Diamond Equal (@add elt). - Proof. - intros k k' e e' a b b' Hk <- <-. now apply add_add_2. - Qed. - - Instance update_m {elt} : Proper (Equal ==> Equal ==> Equal) (@update elt). - Proof. - intros m1 m1' Hm1 m2 m2' Hm2. - unfold update. - apply fold_Proper; auto using diamond_add with *. - Qed. - - Instance restrict_m {elt} : Proper (Equal==>Equal==>Equal) (@restrict elt). - Proof. - intros m1 m1' Hm1 m2 m2' Hm2 y. - unfold restrict. - apply eq_option_alt. intros e. - rewrite !find_spec, !filter_iff, Hm1, Hm2. reflexivity. - clear. intros x x' Hx e e' He. now rewrite Hx. - clear. intros x x' Hx e e' He. now rewrite Hx. - Qed. - - Instance diff_m {elt} : Proper (Equal==>Equal==>Equal) (@diff elt). - Proof. - intros m1 m1' Hm1 m2 m2' Hm2 y. - unfold diff. - apply eq_option_alt. intros e. - rewrite !find_spec, !filter_iff, Hm1, Hm2. reflexivity. - clear. intros x x' Hx e e' He. now rewrite Hx. - clear. intros x x' Hx e e' He. now rewrite Hx. - Qed. - -End WProperties_fun. - -(** * Same Properties for self-contained weak maps and for full maps *) - -Module WProperties (M:WS) := WProperties_fun M.E M. -Module Properties := WProperties. - -(** * Properties specific to maps with ordered keys *) - -Module OrdProperties (M:S). - Module Import ME := OrderedTypeFacts M.E. - Module Import O:=KeyOrderedType M.E. - Module Import P:=Properties M. - Import M. - - Section Elt. - Variable elt:Type. - - Definition Above x (m:t elt) := forall y, In y m -> E.lt y x. - Definition Below x (m:t elt) := forall y, In y m -> E.lt x y. - - Section Bindings. - - Lemma sort_equivlistA_eqlistA : forall l l' : list (key*elt), - sort ltk l -> sort ltk l' -> equivlistA eqke l l' -> eqlistA eqke l l'. - Proof. - apply SortA_equivlistA_eqlistA; eauto with *. - Qed. - - Ltac klean := unfold O.eqke, O.ltk, RelCompFun in *; simpl in *. - Ltac keauto := klean; intuition; eauto. - - Definition gtb (p p':key*elt) := - match E.compare (fst p) (fst p') with Gt => true | _ => false end. - Definition leb p := fun p' => negb (gtb p p'). - - Definition bindings_lt p m := List.filter (gtb p) (bindings m). - Definition bindings_ge p m := List.filter (leb p) (bindings m). - - Lemma gtb_1 : forall p p', gtb p p' = true <-> ltk p' p. - Proof. - intros (x,e) (y,e'); unfold gtb; klean. - case E.compare_spec; intuition; try discriminate; ME.order. - Qed. - - Lemma leb_1 : forall p p', leb p p' = true <-> ~ltk p' p. - Proof. - intros (x,e) (y,e'); unfold leb, gtb; klean. - case E.compare_spec; intuition; try discriminate; ME.order. - Qed. - - Instance gtb_compat : forall p, Proper (eqke==>eq) (gtb p). - Proof. - red; intros (x,e) (a,e') (b,e'') H; red in H; simpl in *; destruct H. - generalize (gtb_1 (x,e) (a,e'))(gtb_1 (x,e) (b,e'')); - destruct (gtb (x,e) (a,e')); destruct (gtb (x,e) (b,e'')); klean; auto. - - intros. symmetry; rewrite H2. rewrite <-H, <-H1; auto. - - intros. rewrite H1. rewrite H, <- H2; auto. - Qed. - - Instance leb_compat : forall p, Proper (eqke==>eq) (leb p). - Proof. - intros x a b H. unfold leb; f_equal; apply gtb_compat; auto. - Qed. - - Hint Resolve gtb_compat leb_compat bindings_spec2 : map. - - Lemma bindings_split : forall p m, - bindings m = bindings_lt p m ++ bindings_ge p m. - Proof. - unfold bindings_lt, bindings_ge, leb; intros. - apply filter_split with (eqA:=eqk) (ltA:=ltk); eauto with *. - intros; destruct x; destruct y; destruct p. - rewrite gtb_1 in H; klean. - apply not_true_iff_false in H0. rewrite gtb_1 in H0. klean. ME.order. - Qed. - - Lemma bindings_Add : forall m m' x e, ~In x m -> Add x e m m' -> - eqlistA eqke (bindings m') - (bindings_lt (x,e) m ++ (x,e):: bindings_ge (x,e) m). - Proof. - intros; unfold bindings_lt, bindings_ge. - apply sort_equivlistA_eqlistA; auto with *. - - apply (@SortA_app _ eqke); auto with *. - + apply (@filter_sort _ eqke); auto with *; keauto. - + constructor; auto with map. - * apply (@filter_sort _ eqke); auto with *; keauto. - * rewrite (@InfA_alt _ eqke); auto with *; try (keauto; fail). - { intros. - rewrite filter_InA in H1; auto with *; destruct H1. - rewrite leb_1 in H2. - destruct y; klean. - rewrite <- bindings_mapsto_iff in H1. - assert (~E.eq x t0). - { contradict H. - exists e0; apply MapsTo_1 with t0; auto. - ME.order. } - ME.order. } - { apply (@filter_sort _ eqke); auto with *; keauto. } - + intros. - rewrite filter_InA in H1; auto with *; destruct H1. - rewrite gtb_1 in H3. - destruct y; destruct x0; klean. - inversion_clear H2. - * red in H4; klean; destruct H4; simpl in *. ME.order. - * rewrite filter_InA in H4; auto with *; destruct H4. - rewrite leb_1 in H4. klean; ME.order. - - intros (k,e'). - rewrite InA_app_iff, InA_cons, 2 filter_InA, - <-2 bindings_mapsto_iff, leb_1, gtb_1, - find_mapsto_iff, (H0 k), <- find_mapsto_iff, - add_mapsto_iff by (auto with * ). - change (eqke (k,e') (x,e)) with (E.eq k x /\ e' = e). - klean. - split. - + intros [(->,->)|(Hk,Hm)]. - * right; now left. - * destruct (lt_dec k x); intuition. - + intros [(Hm,LT)|[(->,->)|(Hm,EQ)]]. - * right; split; trivial; ME.order. - * now left. - * destruct (eq_dec x k) as [Hk|Hk]. - elim H. exists e'. now rewrite Hk. - right; auto. - Qed. - - Lemma bindings_Add_Above : forall m m' x e, - Above x m -> Add x e m m' -> - eqlistA eqke (bindings m') (bindings m ++ (x,e)::nil). - Proof. - intros. - apply sort_equivlistA_eqlistA; auto with *. - apply (@SortA_app _ eqke); auto with *. - intros. - inversion_clear H2. - destruct x0; destruct y. - rewrite <- bindings_mapsto_iff in H1. - destruct H3; klean. - rewrite H2. - apply H; firstorder. - inversion H3. - red; intros a; destruct a. - rewrite InA_app_iff, InA_cons, InA_nil, <- 2 bindings_mapsto_iff, - find_mapsto_iff, (H0 t0), <- find_mapsto_iff, - add_mapsto_iff by (auto with *). - change (eqke (t0,e0) (x,e)) with (E.eq t0 x /\ e0 = e). - intuition. - destruct (E.eq_dec x t0) as [Heq|Hneq]; auto. - exfalso. - assert (In t0 m) by (exists e0; auto). - generalize (H t0 H1). - ME.order. - Qed. - - Lemma bindings_Add_Below : forall m m' x e, - Below x m -> Add x e m m' -> - eqlistA eqke (bindings m') ((x,e)::bindings m). - Proof. - intros. - apply sort_equivlistA_eqlistA; auto with *. - change (sort ltk (((x,e)::nil) ++ bindings m)). - apply (@SortA_app _ eqke); auto with *. - intros. - inversion_clear H1. - destruct y; destruct x0. - rewrite <- bindings_mapsto_iff in H2. - destruct H3; klean. - rewrite H1. - apply H; firstorder. - inversion H3. - red; intros a; destruct a. - rewrite InA_cons, <- 2 bindings_mapsto_iff, - find_mapsto_iff, (H0 t0), <- find_mapsto_iff, - add_mapsto_iff by (auto with * ). - change (eqke (t0,e0) (x,e)) with (E.eq t0 x /\ e0 = e). - intuition. - destruct (E.eq_dec x t0) as [Heq|Hneq]; auto. - exfalso. - assert (In t0 m) by (exists e0; auto). - generalize (H t0 H1). - ME.order. - Qed. - - Lemma bindings_Equal_eqlistA : forall (m m': t elt), - Equal m m' -> eqlistA eqke (bindings m) (bindings m'). - Proof. - intros. - apply sort_equivlistA_eqlistA; auto with *. - red; intros. - destruct x; do 2 rewrite <- bindings_mapsto_iff. - do 2 rewrite find_mapsto_iff; rewrite H; split; auto. - Qed. - - End Bindings. - - Section Min_Max_Elt. - - (** We emulate two [max_elt] and [min_elt] functions. *) - - Fixpoint max_elt_aux (l:list (key*elt)) := match l with - | nil => None - | (x,e)::nil => Some (x,e) - | (x,e)::l => max_elt_aux l - end. - Definition max_elt m := max_elt_aux (bindings m). - - Lemma max_elt_Above : - forall m x e, max_elt m = Some (x,e) -> Above x (remove x m). - Proof. - red; intros. - rewrite remove_in_iff in H0. - destruct H0. - rewrite bindings_in_iff in H1. - destruct H1. - unfold max_elt in *. - generalize (bindings_spec2 m). - revert x e H y x0 H0 H1. - induction (bindings m). - simpl; intros; try discriminate. - intros. - destruct a; destruct l; simpl in *. - injection H; clear H; intros; subst. - inversion_clear H1. - red in H; simpl in *; intuition. - now elim H0. - inversion H. - change (max_elt_aux (p::l) = Some (x,e)) in H. - generalize (IHl x e H); clear IHl; intros IHl. - inversion_clear H1; [ | inversion_clear H2; eauto ]. - red in H3; simpl in H3; destruct H3. - destruct p as (p1,p2). - destruct (E.eq_dec p1 x) as [Heq|Hneq]. - rewrite <- Heq; auto. - inversion_clear H2. - inversion_clear H5. - red in H2; simpl in H2; ME.order. - transitivity p1; auto. - inversion_clear H2. - inversion_clear H5. - red in H2; simpl in H2; ME.order. - eapply IHl; eauto with *. - econstructor; eauto. - red; eauto with *. - inversion H2; auto. - Qed. - - Lemma max_elt_MapsTo : - forall m x e, max_elt m = Some (x,e) -> MapsTo x e m. - Proof. - intros. - unfold max_elt in *. - rewrite bindings_mapsto_iff. - induction (bindings m). - simpl; try discriminate. - destruct a; destruct l; simpl in *. - injection H; intros; subst; constructor; red; auto with *. - constructor 2; auto. - Qed. - - Lemma max_elt_Empty : - forall m, max_elt m = None -> Empty m. - Proof. - intros. - unfold max_elt in *. - rewrite bindings_Empty. - induction (bindings m); auto. - destruct a; destruct l; simpl in *; try discriminate. - assert (H':=IHl H); discriminate. - Qed. - - Definition min_elt m : option (key*elt) := match bindings m with - | nil => None - | (x,e)::_ => Some (x,e) - end. - - Lemma min_elt_Below : - forall m x e, min_elt m = Some (x,e) -> Below x (remove x m). - Proof. - unfold min_elt, Below; intros. - rewrite remove_in_iff in H0; destruct H0. - rewrite bindings_in_iff in H1. - destruct H1. - generalize (bindings_spec2 m). - destruct (bindings m). - try discriminate. - destruct p; injection H; intros; subst. - inversion_clear H1. - red in H2; destruct H2; simpl in *; ME.order. - inversion_clear H4. - rewrite (@InfA_alt _ eqke) in H3; eauto with *. - apply (H3 (y,x0)); auto. - Qed. - - Lemma min_elt_MapsTo : - forall m x e, min_elt m = Some (x,e) -> MapsTo x e m. - Proof. - intros. - unfold min_elt in *. - rewrite bindings_mapsto_iff. - destruct (bindings m). - simpl; try discriminate. - destruct p; simpl in *. - injection H; intros; subst; constructor; red; auto with *. - Qed. - - Lemma min_elt_Empty : - forall m, min_elt m = None -> Empty m. - Proof. - intros. - unfold min_elt in *. - rewrite bindings_Empty. - destruct (bindings m); auto. - destruct p; simpl in *; discriminate. - Qed. - - End Min_Max_Elt. - - Section Induction_Principles. - - Lemma map_induction_max : - forall P : t elt -> Type, - (forall m, Empty m -> P m) -> - (forall m m', P m -> forall x e, Above x m -> Add x e m m' -> P m') -> - forall m, P m. - Proof. - intros; remember (cardinal m) as n; revert m Heqn; induction n; intros. - apply X; apply cardinal_inv_1; auto. - - case_eq (max_elt m); intros. - destruct p. - assert (Add k e (remove k m) m). - { apply max_elt_MapsTo, find_spec, add_id in H. - unfold Add. symmetry. now rewrite add_remove_1. } - apply X0 with (remove k m) k e; auto with map. - apply IHn. - assert (S n = S (cardinal (remove k m))). - { rewrite Heqn. - eapply cardinal_S; eauto with map. } - inversion H1; auto. - eapply max_elt_Above; eauto. - - apply X; apply max_elt_Empty; auto. - Qed. - - Lemma map_induction_min : - forall P : t elt -> Type, - (forall m, Empty m -> P m) -> - (forall m m', P m -> forall x e, Below x m -> Add x e m m' -> P m') -> - forall m, P m. - Proof. - intros; remember (cardinal m) as n; revert m Heqn; induction n; intros. - apply X; apply cardinal_inv_1; auto. - - case_eq (min_elt m); intros. - destruct p. - assert (Add k e (remove k m) m). - { apply min_elt_MapsTo, find_spec, add_id in H. - unfold Add. symmetry. now rewrite add_remove_1. } - apply X0 with (remove k m) k e; auto. - apply IHn. - assert (S n = S (cardinal (remove k m))). - { rewrite Heqn. - eapply cardinal_S; eauto with map. } - inversion H1; auto. - eapply min_elt_Below; eauto. - - apply X; apply min_elt_Empty; auto. - Qed. - - End Induction_Principles. - - Section Fold_properties. - - (** The following lemma has already been proved on Weak Maps, - but with one additionnal hypothesis (some [transpose] fact). *) - - Lemma fold_Equal : forall m1 m2 (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA) - (f:key->elt->A->A)(i:A), - Proper (E.eq==>eq==>eqA==>eqA) f -> - Equal m1 m2 -> - eqA (fold f m1 i) (fold f m2 i). - Proof. - intros m1 m2 A eqA st f i Hf Heq. - rewrite 2 fold_spec_right. - apply fold_right_eqlistA with (eqA:=eqke) (eqB:=eqA); auto. - intros (k,e) (k',e') (Hk,He) a a' Ha; simpl in *; apply Hf; auto. - apply eqlistA_rev. apply bindings_Equal_eqlistA. auto. - Qed. - - Lemma fold_Add_Above : forall m1 m2 x e (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA) - (f:key->elt->A->A)(i:A) (P:Proper (E.eq==>eq==>eqA==>eqA) f), - Above x m1 -> Add x e m1 m2 -> - eqA (fold f m2 i) (f x e (fold f m1 i)). - Proof. - intros. rewrite 2 fold_spec_right. set (f':=uncurry f). - transitivity (fold_right f' i (rev (bindings m1 ++ (x,e)::nil))). - apply fold_right_eqlistA with (eqA:=eqke) (eqB:=eqA); auto. - intros (k1,e1) (k2,e2) (Hk,He) a1 a2 Ha; unfold f'; simpl in *. apply P; auto. - apply eqlistA_rev. - apply bindings_Add_Above; auto. - rewrite distr_rev; simpl. - reflexivity. - Qed. - - Lemma fold_Add_Below : forall m1 m2 x e (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA) - (f:key->elt->A->A)(i:A) (P:Proper (E.eq==>eq==>eqA==>eqA) f), - Below x m1 -> Add x e m1 m2 -> - eqA (fold f m2 i) (fold f m1 (f x e i)). - Proof. - intros. rewrite 2 fold_spec_right. set (f':=uncurry f). - transitivity (fold_right f' i (rev (((x,e)::nil)++bindings m1))). - apply fold_right_eqlistA with (eqA:=eqke) (eqB:=eqA); auto. - intros (k1,e1) (k2,e2) (Hk,He) a1 a2 Ha; unfold f'; simpl in *; apply P; auto. - apply eqlistA_rev. - simpl; apply bindings_Add_Below; auto. - rewrite distr_rev; simpl. - rewrite fold_right_app. - reflexivity. - Qed. - - End Fold_properties. - - End Elt. - -End OrdProperties. diff --git a/theories/MMaps/MMapInterface.v b/theories/MMaps/MMapInterface.v deleted file mode 100644 index 05c5e5d8..00000000 --- a/theories/MMaps/MMapInterface.v +++ /dev/null @@ -1,292 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(** * Finite map library *) - -(** This file proposes interfaces for finite maps *) - -Require Export Bool Equalities Orders SetoidList. -Set Implicit Arguments. -Unset Strict Implicit. - -(** When compared with Ocaml Map, this signature has been split in - several parts : - - - The first parts [WSfun] and [WS] propose signatures for weak - maps, which are maps with no ordering on the key type nor the - data type. [WSfun] and [WS] are almost identical, apart from the - fact that [WSfun] is expressed in a functorial way whereas [WS] - is self-contained. For obtaining an instance of such signatures, - a decidable equality on keys in enough (see for example - [FMapWeakList]). These signatures contain the usual operators - (add, find, ...). The only function that asks for more is - [equal], whose first argument should be a comparison on data. - - - Then comes [Sfun] and [S], that extend [WSfun] and [WS] to the - case where the key type is ordered. The main novelty is that - [bindings] is required to produce sorted lists. - - - Finally, [Sord] extends [S] with a complete comparison function. For - that, the data type should have a decidable total ordering as well. - - If unsure, what you're looking for is probably [S]: apart from [Sord], - all other signatures are subsets of [S]. - - Some additional differences with Ocaml: - - - no [iter] function, useless since Coq is purely functional - - [option] types are used instead of [Not_found] exceptions - -*) - - -Definition Cmp {elt:Type}(cmp:elt->elt->bool) e1 e2 := cmp e1 e2 = true. - -(** ** Weak signature for maps - - No requirements for an ordering on keys nor elements, only decidability - of equality on keys. First, a functorial signature: *) - -Module Type WSfun (E : DecidableType). - - Definition key := E.t. - Hint Transparent key. - - Definition eq_key {elt} (p p':key*elt) := E.eq (fst p) (fst p'). - - Definition eq_key_elt {elt} (p p':key*elt) := - E.eq (fst p) (fst p') /\ (snd p) = (snd p'). - - Parameter t : Type -> Type. - (** the abstract type of maps *) - - Section Ops. - - Parameter empty : forall {elt}, t elt. - (** The empty map. *) - - Variable elt:Type. - - Parameter is_empty : t elt -> bool. - (** Test whether a map is empty or not. *) - - Parameter add : key -> elt -> t elt -> t elt. - (** [add x y m] returns a map containing the same bindings as [m], - plus a binding of [x] to [y]. If [x] was already bound in [m], - its previous binding disappears. *) - - Parameter find : key -> t elt -> option elt. - (** [find x m] returns the current binding of [x] in [m], - or [None] if no such binding exists. *) - - Parameter remove : key -> t elt -> t elt. - (** [remove x m] returns a map containing the same bindings as [m], - except for [x] which is unbound in the returned map. *) - - Parameter mem : key -> t elt -> bool. - (** [mem x m] returns [true] if [m] contains a binding for [x], - and [false] otherwise. *) - - Parameter bindings : t elt -> list (key*elt). - (** [bindings m] returns an assoc list corresponding to the bindings - of [m], in any order. *) - - Parameter cardinal : t elt -> nat. - (** [cardinal m] returns the number of bindings in [m]. *) - - Parameter fold : forall A: Type, (key -> elt -> A -> A) -> t elt -> A -> A. - (** [fold f m a] computes [(f kN dN ... (f k1 d1 a)...)], - where [k1] ... [kN] are the keys of all bindings in [m] - (in any order), and [d1] ... [dN] are the associated data. *) - - Parameter equal : (elt -> elt -> bool) -> t elt -> t elt -> bool. - (** [equal cmp m1 m2] tests whether the maps [m1] and [m2] are equal, - that is, contain equal keys and associate them with equal data. - [cmp] is the equality predicate used to compare the data associated - with the keys. *) - - Variable elt' elt'' : Type. - - Parameter map : (elt -> elt') -> t elt -> t elt'. - (** [map f m] returns a map with same domain as [m], where the associated - value a of all bindings of [m] has been replaced by the result of the - application of [f] to [a]. Since Coq is purely functional, the order - in which the bindings are passed to [f] is irrelevant. *) - - Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'. - (** Same as [map], but the function receives as arguments both the - key and the associated value for each binding of the map. *) - - Parameter merge : (key -> option elt -> option elt' -> option elt'') -> - t elt -> t elt' -> t elt''. - (** [merge f m m'] creates a new map whose bindings belong to the ones - of either [m] or [m']. The presence and value for a key [k] is - determined by [f k e e'] where [e] and [e'] are the (optional) - bindings of [k] in [m] and [m']. *) - - End Ops. - Section Specs. - - Variable elt:Type. - - Parameter MapsTo : key -> elt -> t elt -> Prop. - - Definition In (k:key)(m: t elt) : Prop := exists e:elt, MapsTo k e m. - - Global Declare Instance MapsTo_compat : - Proper (E.eq==>Logic.eq==>Logic.eq==>iff) MapsTo. - - Variable m m' : t elt. - Variable x y : key. - Variable e : elt. - - Parameter find_spec : find x m = Some e <-> MapsTo x e m. - Parameter mem_spec : mem x m = true <-> In x m. - Parameter empty_spec : find x (@empty elt) = None. - Parameter is_empty_spec : is_empty m = true <-> forall x, find x m = None. - Parameter add_spec1 : find x (add x e m) = Some e. - Parameter add_spec2 : ~E.eq x y -> find y (add x e m) = find y m. - Parameter remove_spec1 : find x (remove x m) = None. - Parameter remove_spec2 : ~E.eq x y -> find y (remove x m) = find y m. - - (** Specification of [bindings] *) - Parameter bindings_spec1 : - InA eq_key_elt (x,e) (bindings m) <-> MapsTo x e m. - (** When compared with ordered maps, here comes the only - property that is really weaker: *) - Parameter bindings_spec2w : NoDupA eq_key (bindings m). - - (** Specification of [cardinal] *) - Parameter cardinal_spec : cardinal m = length (bindings m). - - (** Specification of [fold] *) - Parameter fold_spec : - forall {A} (i : A) (f : key -> elt -> A -> A), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. - - (** Equality of maps *) - - (** Caveat: there are at least three distinct equality predicates on maps. - - The simpliest (and maybe most natural) way is to consider keys up to - their equivalence [E.eq], but elements up to Leibniz equality, in - the spirit of [eq_key_elt] above. This leads to predicate [Equal]. - - Unfortunately, this [Equal] predicate can't be used to describe - the [equal] function, since this function (for compatibility with - ocaml) expects a boolean comparison [cmp] that may identify more - elements than Leibniz. So logical specification of [equal] is done - via another predicate [Equivb] - - This predicate [Equivb] is quite ad-hoc with its boolean [cmp], - it can be generalized in a [Equiv] expecting a more general - (possibly non-decidable) equality predicate on elements *) - - Definition Equal (m m':t elt) := forall y, find y m = find y m'. - Definition Equiv (eq_elt:elt->elt->Prop) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). - Definition Equivb (cmp: elt->elt->bool) := Equiv (Cmp cmp). - - (** Specification of [equal] *) - Parameter equal_spec : forall cmp : elt -> elt -> bool, - equal cmp m m' = true <-> Equivb cmp m m'. - - End Specs. - Section SpecMaps. - - Variables elt elt' elt'' : Type. - - Parameter map_spec : forall (f:elt->elt') m x, - find x (map f m) = option_map f (find x m). - - Parameter mapi_spec : forall (f:key->elt->elt') m x, - exists y:key, E.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). - - Parameter merge_spec1 : - forall (f:key->option elt->option elt'->option elt'') m m' x, - In x m \/ In x m' -> - exists y:key, E.eq y x /\ - find x (merge f m m') = f y (find x m) (find x m'). - - Parameter merge_spec2 : - forall (f:key -> option elt->option elt'->option elt'') m m' x, - In x (merge f m m') -> In x m \/ In x m'. - - End SpecMaps. -End WSfun. - -(** ** Static signature for Weak Maps - - Similar to [WSfun] but expressed in a self-contained way. *) - -Module Type WS. - Declare Module E : DecidableType. - Include WSfun E. -End WS. - - - -(** ** Maps on ordered keys, functorial signature *) - -Module Type Sfun (E : OrderedType). - Include WSfun E. - - Definition lt_key {elt} (p p':key*elt) := E.lt (fst p) (fst p'). - - (** Additional specification of [bindings] *) - - Parameter bindings_spec2 : forall {elt}(m : t elt), sort lt_key (bindings m). - - (** Remark: since [fold] is specified via [bindings], this stronger - specification of [bindings] has an indirect impact on [fold], - which can now be proved to receive bindings in increasing order. *) - -End Sfun. - - -(** ** Maps on ordered keys, self-contained signature *) - -Module Type S. - Declare Module E : OrderedType. - Include Sfun E. -End S. - - - -(** ** Maps with ordering both on keys and datas *) - -Module Type Sord. - - Declare Module Data : OrderedType. - Declare Module MapS : S. - Import MapS. - - Definition t := MapS.t Data.t. - - Include HasEq <+ HasLt <+ IsEq <+ IsStrOrder. - - Definition cmp e e' := - match Data.compare e e' with Eq => true | _ => false end. - - Parameter eq_spec : forall m m', eq m m' <-> Equivb cmp m m'. - - Parameter compare : t -> t -> comparison. - - Parameter compare_spec : forall m1 m2, CompSpec eq lt m1 m2 (compare m1 m2). - -End Sord. - - -(* TODO: provides filter + partition *) - -(* TODO: provide split - Parameter split : key -> t elt -> t elt * option elt * t elt. - - Parameter split_spec k m : - split k m = (filter (fun x -> E.compare x k) m, find k m, filter ...) - - min_binding, max_binding, choose ? -*) diff --git a/theories/MMaps/MMapList.v b/theories/MMaps/MMapList.v deleted file mode 100644 index c521178c..00000000 --- a/theories/MMaps/MMapList.v +++ /dev/null @@ -1,1144 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(** * Finite map library *) - -(** This file proposes an implementation of the non-dependant interface - [MMapInterface.S] using lists of pairs ordered (increasing) with respect to - left projection. *) - -Require Import MMapInterface OrdersFacts OrdersLists. - -Set Implicit Arguments. -Unset Strict Implicit. - -Module Raw (X:OrderedType). - -Module Import MX := OrderedTypeFacts X. -Module Import PX := KeyOrderedType X. - -Definition key := X.t. -Definition t (elt:Type) := list (X.t * elt). - -Local Notation Sort := (sort ltk). -Local Notation Inf := (lelistA (ltk)). - -Section Elt. -Variable elt : Type. - -Ltac SortLt := - match goal with - | H1 : Sort ?m, H2:Inf (?k',?e') ?m, H3:In ?k ?m |- _ => - assert (X.lt k' k); - [let e := fresh "e" in destruct H3 as (e,H3); - change (ltk (k',e') (k,e)); - apply (Sort_Inf_In H1 H2 (InA_eqke_eqk H3)) | ] - | H1:Sort ?m, H2:Inf (?k',?e') ?m, H3:MapsTo ?k ?e ?m |- _ => - assert (X.lt k' k); - [change (ltk (k',e') (k,e)); - apply (Sort_Inf_In H1 H2 (InA_eqke_eqk H3)) | ] - | H1:Sort ?m, H2:Inf (?k',?e') ?m, H3:InA eqke (?k,?e) ?m |- _ => - assert (X.lt k' k); - [change (ltk (k',e') (k,e)); - apply (Sort_Inf_In H1 H2 (InA_eqke_eqk H3)) | ] - end. - -(** * [find] *) - -Fixpoint find (k:key) (m: t elt) : option elt := - match m with - | nil => None - | (k',x)::m' => - match X.compare k k' with - | Lt => None - | Eq => Some x - | Gt => find k m' - end - end. - -Lemma find_spec m (Hm:Sort m) x e : - find x m = Some e <-> MapsTo x e m. -Proof. - induction m as [|(k,e') m IH]; simpl. - - split. discriminate. inversion 1. - - inversion_clear Hm. - unfold MapsTo in *. rewrite InA_cons, eqke_def. - case X.compare_spec; intros. - + split. injection 1 as ->; auto. - intros [(_,<-)|IN]; trivial. SortLt. MX.order. - + split. discriminate. - intros [(E,<-)|IN]; trivial; try SortLt; MX.order. - + rewrite IH; trivial. split; auto. - intros [(E,<-)|IN]; trivial. MX.order. -Qed. - -(** * [mem] *) - -Fixpoint mem (k : key) (m : t elt) : bool := - match m with - | nil => false - | (k',_) :: l => - match X.compare k k' with - | Lt => false - | Eq => true - | Gt => mem k l - end - end. - -Lemma mem_spec m (Hm:Sort m) x : mem x m = true <-> In x m. -Proof. - induction m as [|(k,e') m IH]; simpl. - - split. discriminate. inversion 1. inversion_clear H0. - - inversion_clear Hm. - rewrite In_cons; simpl. - case X.compare_spec; intros. - + intuition. - + split. discriminate. intros [E|(e,IN)]. MX.order. - SortLt. MX.order. - + rewrite IH; trivial. split; auto. intros [E|IN]; trivial. - MX.order. -Qed. - -(** * [empty] *) - -Definition empty : t elt := nil. - -Lemma empty_spec x : find x empty = None. -Proof. - reflexivity. -Qed. - -Lemma empty_sorted : Sort empty. -Proof. - unfold empty; auto. -Qed. - -(** * [is_empty] *) - -Definition is_empty (l : t elt) : bool := if l then true else false. - -Lemma is_empty_spec m : - is_empty m = true <-> forall x, find x m = None. -Proof. - destruct m as [|(k,e) m]; simpl; split; trivial; try discriminate. - intros H. specialize (H k). now rewrite compare_refl in H. -Qed. - -(** * [add] *) - -Fixpoint add (k : key) (x : elt) (s : t elt) : t elt := - match s with - | nil => (k,x) :: nil - | (k',y) :: l => - match X.compare k k' with - | Lt => (k,x)::s - | Eq => (k,x)::l - | Gt => (k',y) :: add k x l - end - end. - -Lemma add_spec1 m x e : find x (add x e m) = Some e. -Proof. - induction m as [|(k,e') m IH]; simpl. - - now rewrite compare_refl. - - case X.compare_spec; simpl; rewrite ?compare_refl; trivial. - rewrite <- compare_gt_iff. now intros ->. -Qed. - -Lemma add_spec2 m x y e : ~X.eq x y -> find y (add x e m) = find y m. -Proof. - induction m as [|(k,e') m IH]; simpl. - - case X.compare_spec; trivial; MX.order. - - case X.compare_spec; simpl; intros; trivial. - + rewrite <-H. case X.compare_spec; trivial; MX.order. - + do 2 (case X.compare_spec; trivial; try MX.order). - + now rewrite IH. -Qed. - -Lemma add_Inf : forall (m:t elt)(x x':key)(e e':elt), - Inf (x',e') m -> ltk (x',e') (x,e) -> Inf (x',e') (add x e m). -Proof. - induction m. - simpl; intuition. - intros. - destruct a as (x'',e''). - inversion_clear H. - compute in H0,H1. - simpl; case X.compare; intuition. -Qed. -Hint Resolve add_Inf. - -Lemma add_sorted : forall m (Hm:Sort m) x e, Sort (add x e m). -Proof. - induction m. - simpl; intuition. - intros. - destruct a as (x',e'). - simpl; case (X.compare_spec x x'); intuition; inversion_clear Hm; auto. - constructor; auto. - apply Inf_eq with (x',e'); auto. -Qed. - -(** * [remove] *) - -Fixpoint remove (k : key) (s : t elt) : t elt := - match s with - | nil => nil - | (k',x) :: l => - match X.compare k k' with - | Lt => s - | Eq => l - | Gt => (k',x) :: remove k l - end - end. - -Lemma remove_spec1 m (Hm:Sort m) x : find x (remove x m) = None. -Proof. - induction m as [|(k,e') m IH]; simpl; trivial. - inversion_clear Hm. - case X.compare_spec; simpl. - - intros E. rewrite <- E in H0. - apply Sort_Inf_NotIn in H0; trivial. unfold In in H0. - setoid_rewrite <- find_spec in H0; trivial. - destruct (find x m); trivial. - elim H0; now exists e. - - rewrite <- compare_lt_iff. now intros ->. - - rewrite <- compare_gt_iff. intros ->; auto. -Qed. - -Lemma remove_spec2 m (Hm:Sort m) x y : - ~X.eq x y -> find y (remove x m) = find y m. -Proof. - induction m as [|(k,e') m IH]; simpl; trivial. - inversion_clear Hm. - case X.compare_spec; simpl; intros E E'; try rewrite IH; auto. - case X.compare_spec; simpl; trivial; try MX.order. - intros. rewrite <- E in H0,H1. clear E E'. - destruct (find y m) eqn:F; trivial. - apply find_spec in F; trivial. - SortLt. MX.order. -Qed. - -Lemma remove_Inf : forall (m:t elt)(Hm : Sort m)(x x':key)(e':elt), - Inf (x',e') m -> Inf (x',e') (remove x m). -Proof. - induction m. - simpl; intuition. - intros. - destruct a as (x'',e''). - inversion_clear H. - compute in H0. - simpl; case X.compare; intuition. - inversion_clear Hm. - apply Inf_lt with (x'',e''); auto. -Qed. -Hint Resolve remove_Inf. - -Lemma remove_sorted : forall m (Hm:Sort m) x, Sort (remove x m). -Proof. - induction m. - simpl; intuition. - intros. - destruct a as (x',e'). - simpl; case X.compare_spec; intuition; inversion_clear Hm; auto. -Qed. - -(** * [bindings] *) - -Definition bindings (m: t elt) := m. - -Lemma bindings_spec1 m x e : - InA eqke (x,e) (bindings m) <-> MapsTo x e m. -Proof. - reflexivity. -Qed. - -Lemma bindings_spec2 m (Hm:Sort m) : sort ltk (bindings m). -Proof. - auto. -Qed. - -Lemma bindings_spec2w m (Hm:Sort m) : NoDupA eqk (bindings m). -Proof. - now apply Sort_NoDupA. -Qed. - -(** * [fold] *) - -Fixpoint fold (A:Type)(f:key->elt->A->A)(m:t elt) (acc:A) : A := - match m with - | nil => acc - | (k,e)::m' => fold f m' (f k e acc) - end. - -Lemma fold_spec m : forall (A:Type)(i:A)(f:key->elt->A->A), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. -Proof. - induction m as [|(k,e) m IH]; simpl; auto. -Qed. - -(** * [equal] *) - -Fixpoint equal (cmp:elt->elt->bool)(m m' : t elt) : bool := - match m, m' with - | nil, nil => true - | (x,e)::l, (x',e')::l' => - match X.compare x x' with - | Eq => cmp e e' && equal cmp l l' - | _ => false - end - | _, _ => false - end. - -Definition Equivb (cmp:elt->elt->bool) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true). - -Lemma equal_1 : forall m (Hm:Sort m) m' (Hm': Sort m') cmp, - Equivb cmp m m' -> equal cmp m m' = true. -Proof. - induction m as [|(k,e) m IH]; destruct m' as [|(k',e') m']; simpl. - - trivial. - - intros _ cmp (H,_). - exfalso. apply (@In_nil elt k'). rewrite H, In_cons. now left. - - intros _ cmp (H,_). - exfalso. apply (@In_nil elt k). rewrite <- H, In_cons. now left. - - intros Hm' cmp E. - inversion_clear Hm; inversion_clear Hm'. - case X.compare_spec; intros E'. - + apply andb_true_intro; split. - * eapply E; eauto. apply InA_cons; now left. - * apply IH; clear IH; trivial. - destruct E as (E1,E2). split. - { intros x. clear E2. - split; intros; SortLt. - specialize (E1 x); rewrite 2 In_cons in E1; simpl in E1. - destruct E1 as ([E1|E1],_); eauto. MX.order. - specialize (E1 x); rewrite 2 In_cons in E1; simpl in E1. - destruct E1 as (_,[E1|E1]); eauto. MX.order. } - { intros x xe xe' Hx HX'. eapply E2; eauto. } - + assert (IN : In k ((k',e')::m')). - { apply E. apply In_cons; now left. } - apply In_cons in IN. simpl in IN. destruct IN as [?|IN]. MX.order. - SortLt. MX.order. - + assert (IN : In k' ((k,e)::m)). - { apply E. apply In_cons; now left. } - apply In_cons in IN. simpl in IN. destruct IN as [?|IN]. MX.order. - SortLt. MX.order. -Qed. - -Lemma equal_2 m (Hm:Sort m) m' (Hm':Sort m') cmp : - equal cmp m m' = true -> Equivb cmp m m'. -Proof. - revert m' Hm'. - induction m as [|(k,e) m IH]; destruct m' as [|(k',e') m']; simpl; - try discriminate. - - split. reflexivity. inversion 1. - - intros Hm'. case X.compare_spec; try discriminate. - rewrite andb_true_iff. intros E (C,EQ). - inversion_clear Hm; inversion_clear Hm'. - apply IH in EQ; trivial. - destruct EQ as (E1,E2). - split. - + intros x. rewrite 2 In_cons; simpl. rewrite <- E1. - intuition; now left; MX.order. - + intros x ex ex'. unfold MapsTo in *. rewrite 2 InA_cons, 2 eqke_def. - intuition; subst. - * trivial. - * SortLt. MX.order. - * SortLt. MX.order. - * eapply E2; eauto. -Qed. - -Lemma equal_spec m (Hm:Sort m) m' (Hm':Sort m') cmp : - equal cmp m m' = true <-> Equivb cmp m m'. -Proof. - split. now apply equal_2. now apply equal_1. -Qed. - -(** This lemma isn't part of the spec of [Equivb], but is used in [MMapAVL] *) - -Lemma equal_cons : forall cmp l1 l2 x y, Sort (x::l1) -> Sort (y::l2) -> - eqk x y -> cmp (snd x) (snd y) = true -> - (Equivb cmp l1 l2 <-> Equivb cmp (x :: l1) (y :: l2)). -Proof. - intros. - inversion H; subst. - inversion H0; subst. - destruct x; destruct y; compute in H1, H2. - split; intros. - apply equal_2; auto. - simpl. - case X.compare_spec; intros; try MX.order. - rewrite H2; simpl. - apply equal_1; auto. - apply equal_2; auto. - generalize (equal_1 H H0 H3). - simpl. - case X.compare_spec; try discriminate. - rewrite andb_true_iff. intuition. -Qed. - -Variable elt':Type. - -(** * [map] and [mapi] *) - -Fixpoint map (f:elt -> elt') (m:t elt) : t elt' := - match m with - | nil => nil - | (k,e)::m' => (k,f e) :: map f m' - end. - -Fixpoint mapi (f: key -> elt -> elt') (m:t elt) : t elt' := - match m with - | nil => nil - | (k,e)::m' => (k,f k e) :: mapi f m' - end. - -End Elt. -Arguments find {elt} k m. -Section Elt2. -Variable elt elt' : Type. - -(** Specification of [map] *) - -Lemma map_spec (f:elt->elt') m x : - find x (map f m) = option_map f (find x m). -Proof. - induction m as [|(k,e) m IH]; simpl; trivial. - now case X.compare_spec. -Qed. - -Lemma map_Inf (f:elt->elt') m x e e' : - Inf (x,e) m -> Inf (x,e') (map f m). -Proof. - induction m as [|(x0,e0) m IH]; simpl; auto. - inversion_clear 1; auto. -Qed. -Hint Resolve map_Inf. - -Lemma map_sorted (f:elt->elt')(m: t elt)(Hm : Sort m) : - Sort (map f m). -Proof. - induction m as [|(x,e) m IH]; simpl; auto. - inversion_clear Hm. constructor; eauto. -Qed. - -(** Specification of [mapi] *) - -Lemma mapi_spec (f:key->elt->elt') m x : - exists y, X.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). -Proof. - induction m as [|(k,e) m IH]; simpl. - - now exists x. - - elim X.compare_spec; intros; simpl. - + now exists k. - + now exists x. - + apply IH. -Qed. - -Lemma mapi_Inf (f:key->elt->elt') m x e : - Inf (x,e) m -> Inf (x,f x e) (mapi f m). -Proof. - induction m as [|(x0,e0) m IH]; simpl; auto. - inversion_clear 1; auto. -Qed. -Hint Resolve mapi_Inf. - -Lemma mapi_sorted (f:key->elt->elt') m (Hm : Sort m) : - Sort (mapi f m). -Proof. - induction m as [|(x,e) m IH]; simpl; auto. - inversion_clear Hm; auto. -Qed. - -End Elt2. -Section Elt3. - -(** * [merge] *) - -Variable elt elt' elt'' : Type. -Variable f : key -> option elt -> option elt' -> option elt''. - -Definition option_cons {A}(k:key)(o:option A)(l:list (key*A)) := - match o with - | Some e => (k,e)::l - | None => l - end. - -Fixpoint merge_l (m : t elt) : t elt'' := - match m with - | nil => nil - | (k,e)::l => option_cons k (f k (Some e) None) (merge_l l) - end. - -Fixpoint merge_r (m' : t elt') : t elt'' := - match m' with - | nil => nil - | (k,e')::l' => option_cons k (f k None (Some e')) (merge_r l') - end. - -Fixpoint merge (m : t elt) : t elt' -> t elt'' := - match m with - | nil => merge_r - | (k,e) :: l => - fix merge_aux (m' : t elt') : t elt'' := - match m' with - | nil => merge_l m - | (k',e') :: l' => - match X.compare k k' with - | Lt => option_cons k (f k (Some e) None) (merge l m') - | Eq => option_cons k (f k (Some e) (Some e')) (merge l l') - | Gt => option_cons k' (f k' None (Some e')) (merge_aux l') - end - end - end. - -Notation oee' := (option elt * option elt')%type. - -Fixpoint combine (m : t elt) : t elt' -> t oee' := - match m with - | nil => map (fun e' => (None,Some e')) - | (k,e) :: l => - fix combine_aux (m':t elt') : list (key * oee') := - match m' with - | nil => map (fun e => (Some e,None)) m - | (k',e') :: l' => - match X.compare k k' with - | Lt => (k,(Some e, None))::combine l m' - | Eq => (k,(Some e, Some e'))::combine l l' - | Gt => (k',(None,Some e'))::combine_aux l' - end - end - end. - -Definition fold_right_pair {A B C}(f: A->B->C->C)(l:list (A*B))(i:C) := - List.fold_right (fun p => f (fst p) (snd p)) i l. - -Definition merge' m m' := - let m0 : t oee' := combine m m' in - let m1 : t (option elt'') := mapi (fun k p => f k (fst p) (snd p)) m0 in - fold_right_pair (option_cons (A:=elt'')) m1 nil. - -Lemma merge_equiv : forall m m', merge' m m' = merge m m'. -Proof. - unfold merge'. - induction m as [|(k,e) m IHm]; intros. - - (* merge_r *) - simpl. - induction m' as [|(k',e') m' IHm']; simpl; rewrite ?IHm'; auto. - - induction m' as [|(k',e') m' IHm']; simpl. - + f_equal. - (* merge_l *) - clear k e IHm. - induction m as [|(k,e) m IHm]; simpl; rewrite ?IHm; auto. - + elim X.compare_spec; intros; simpl; f_equal. - * apply IHm. - * apply IHm. - * apply IHm'. -Qed. - -Lemma combine_Inf : - forall m m' (x:key)(e:elt)(e':elt')(e'':oee'), - Inf (x,e) m -> - Inf (x,e') m' -> - Inf (x,e'') (combine m m'). -Proof. - induction m. - - intros. simpl. eapply map_Inf; eauto. - - induction m'; intros. - + destruct a. - replace (combine ((t0, e0) :: m) nil) with - (map (fun e => (Some e,None (A:=elt'))) ((t0,e0)::m)); auto. - eapply map_Inf; eauto. - + simpl. - destruct a as (k,e0); destruct a0 as (k',e0'). - elim X.compare_spec. - * inversion_clear H; auto. - * inversion_clear H; auto. - * inversion_clear H0; auto. -Qed. -Hint Resolve combine_Inf. - -Lemma combine_sorted m (Hm : Sort m) m' (Hm' : Sort m') : - Sort (combine m m'). -Proof. - revert m' Hm'. - induction m. - - intros; clear Hm. simpl. apply map_sorted; auto. - - induction m'; intros. - + clear Hm'. - destruct a. - replace (combine ((t0, e) :: m) nil) with - (map (fun e => (Some e,None (A:=elt'))) ((t0,e)::m)); auto. - apply map_sorted; auto. - + simpl. - destruct a as (k,e); destruct a0 as (k',e'). - inversion_clear Hm; inversion_clear Hm'. - case X.compare_spec; [intros Heq| intros Hlt| intros Hlt]; - constructor; auto. - * assert (Inf (k, e') m') by (apply Inf_eq with (k',e'); auto). - exact (combine_Inf _ H0 H3). - * assert (Inf (k, e') ((k',e')::m')) by auto. - exact (combine_Inf _ H0 H3). - * assert (Inf (k', e) ((k,e)::m)) by auto. - exact (combine_Inf _ H3 H2). -Qed. - -Lemma merge_sorted m (Hm : Sort m) m' (Hm' : Sort m') : - Sort (merge m m'). -Proof. - intros. - rewrite <- merge_equiv. - unfold merge'. - assert (Hmm':=combine_sorted Hm Hm'). - set (l0:=combine m m') in *; clearbody l0. - set (f':= fun k p => f k (fst p) (snd p)). - assert (H1:=mapi_sorted f' Hmm'). - set (l1:=mapi f' l0) in *; clearbody l1. - clear f' f Hmm' l0 Hm Hm' m m'. - (* Sort fold_right_pair *) - induction l1. - - simpl; auto. - - inversion_clear H1. - destruct a; destruct o; auto. - simpl. - constructor; auto. - clear IHl1. - (* Inf fold_right_pair *) - induction l1. - + simpl; auto. - + destruct a; destruct o; simpl; auto. - * inversion_clear H0; auto. - * inversion_clear H0. inversion_clear H. - compute in H1. - apply IHl1; auto. - apply Inf_lt with (t1, None); auto. -Qed. - -Definition at_least_one (o:option elt)(o':option elt') := - match o, o' with - | None, None => None - | _, _ => Some (o,o') - end. - -Lemma combine_spec m (Hm : Sort m) m' (Hm' : Sort m') (x:key) : - find x (combine m m') = at_least_one (find x m) (find x m'). -Proof. - revert m' Hm'. - induction m. - intros. - simpl. - induction m'. - intros; simpl; auto. - simpl; destruct a. - simpl; destruct (X.compare x t0); simpl; auto. - inversion_clear Hm'; auto. - induction m'. - (* m' = nil *) - intros; destruct a; simpl. - destruct (X.compare_spec x t0) as [ |Hlt|Hlt]; simpl; auto. - inversion_clear Hm; clear H0 Hlt Hm' IHm t0. - induction m; simpl; auto. - inversion_clear H. - destruct a. - simpl; destruct (X.compare x t0); simpl; auto. - (* m' <> nil *) - intros. - destruct a as (k,e); destruct a0 as (k',e'); simpl. - inversion Hm; inversion Hm'; subst. - destruct (X.compare_spec k k'); simpl; - destruct (X.compare_spec x k); - MX.order || destruct (X.compare_spec x k'); - simpl; try MX.order; auto. - - rewrite IHm; auto; simpl. elim X.compare_spec; auto; MX.order. - - rewrite IHm; auto; simpl. elim X.compare_spec; auto; MX.order. - - rewrite IHm; auto; simpl. elim X.compare_spec; auto; MX.order. - - change (find x (combine ((k, e) :: m) m') = Some (Some e, find x m')). - rewrite IHm'; auto; simpl. elim X.compare_spec; auto; MX.order. - - change (find x (combine ((k, e) :: m) m') = at_least_one None (find x m')). - rewrite IHm'; auto; simpl. elim X.compare_spec; auto; MX.order. - - change (find x (combine ((k, e) :: m) m') = - at_least_one (find x m) (find x m')). - rewrite IHm'; auto; simpl. elim X.compare_spec; auto; MX.order. -Qed. - -Definition at_least_one_then_f k (o:option elt)(o':option elt') := - match o, o' with - | None, None => None - | _, _ => f k o o' - end. - -Lemma merge_spec0 m (Hm : Sort m) m' (Hm' : Sort m') (x:key) : - exists y, X.eq y x /\ - find x (merge m m') = at_least_one_then_f y (find x m) (find x m'). -Proof. - intros. - rewrite <- merge_equiv. - unfold merge'. - assert (H:=combine_spec Hm Hm' x). - assert (H2:=combine_sorted Hm Hm'). - set (f':= fun k p => f k (fst p) (snd p)). - set (m0 := combine m m') in *; clearbody m0. - set (o:=find x m) in *; clearbody o. - set (o':=find x m') in *; clearbody o'. - clear Hm Hm' m m'. revert H. - match goal with |- ?G => - assert (G/\(find x m0 = None -> - find x (fold_right_pair option_cons (mapi f' m0) nil) = None)); - [|intuition] end. - induction m0; simpl in *; intuition. - - exists x; split; [easy|]. - destruct o; destruct o'; simpl in *; try discriminate; auto. - - destruct a as (k,(oo,oo')); simpl in *. - inversion_clear H2. - destruct (X.compare_spec x k) as [Heq|Hlt|Hlt]; simpl in *. - + (* x = k *) - exists k; split; [easy|]. - assert (at_least_one_then_f k o o' = f k oo oo'). - { destruct o; destruct o'; simpl in *; inversion_clear H; auto. } - rewrite H2. - unfold f'; simpl. - destruct (f k oo oo'); simpl. - * elim X.compare_spec; trivial; try MX.order. - * destruct (IHm0 H0) as (_,H4); apply H4; auto. - case_eq (find x m0); intros; auto. - assert (eqk (elt:=oee') (k,(oo,oo')) (x,(oo,oo'))). - now compute. - symmetry in H5. - destruct (Sort_Inf_NotIn H0 (Inf_eq H5 H1)). - exists p; apply find_spec; auto. - + (* x < k *) - destruct (f' k (oo,oo')); simpl. - * elim X.compare_spec; trivial; try MX.order. - destruct o; destruct o'; simpl in *; try discriminate; auto. - now exists x. - * apply IHm0; trivial. - rewrite <- H. - case_eq (find x m0); intros; auto. - assert (ltk (elt:=oee') (x,(oo,oo')) (k,(oo,oo'))). - red; auto. - destruct (Sort_Inf_NotIn H0 (Inf_lt H3 H1)). - exists p; apply find_spec; auto. - + (* k < x *) - unfold f'; simpl. - destruct (f k oo oo'); simpl. - * elim X.compare_spec; trivial; try MX.order. - intros. apply IHm0; auto. - * apply IHm0; auto. - - - (* None -> None *) - destruct a as (k,(oo,oo')). - simpl. - inversion_clear H2. - destruct (X.compare_spec x k) as [Hlt|Heq|Hlt]; try discriminate. - + (* x < k *) - unfold f'; simpl. - destruct (f k oo oo'); simpl. - elim X.compare_spec; trivial; try MX.order. intros. - apply IHm0; auto. - case_eq (find x m0); intros; auto. - assert (ltk (elt:=oee') (x,(oo,oo')) (k,(oo,oo'))). - now compute. - destruct (Sort_Inf_NotIn H0 (Inf_lt H3 H1)). - exists p; apply find_spec; auto. - + (* k < x *) - unfold f'; simpl. - destruct (f k oo oo'); simpl. - elim X.compare_spec; trivial; try MX.order. intros. - apply IHm0; auto. - apply IHm0; auto. -Qed. - -(** Specification of [merge] *) - -Lemma merge_spec1 m (Hm : Sort m) m' (Hm' : Sort m')(x:key) : - In x m \/ In x m' -> - exists y, X.eq y x /\ - find x (merge m m') = f y (find x m) (find x m'). -Proof. - intros. - destruct (merge_spec0 Hm Hm' x) as (y,(Hy,H')). - exists y; split; [easy|]. rewrite H'. - destruct H as [(e,H)|(e,H)]; - apply find_spec in H; trivial; rewrite H; simpl; auto. - now destruct (find x m). -Qed. - -Lemma merge_spec2 m (Hm : Sort m) m' (Hm' : Sort m')(x:key) : - In x (merge m m') -> In x m \/ In x m'. -Proof. - intros. - destruct H as (e,H). - apply find_spec in H; auto using merge_sorted. - destruct (merge_spec0 Hm Hm' x) as (y,(Hy,H')). - rewrite H in H'. - destruct (find x m) eqn:F. - - apply find_spec in F; eauto. - - destruct (find x m') eqn:F'. - + apply find_spec in F'; eauto. - + simpl in H'. discriminate. -Qed. - -End Elt3. -End Raw. - -Module Make (X: OrderedType) <: S with Module E := X. -Module Raw := Raw X. -Module E := X. - -Definition key := E.t. -Definition eq_key {elt} := @Raw.PX.eqk elt. -Definition eq_key_elt {elt} := @Raw.PX.eqke elt. -Definition lt_key {elt} := @Raw.PX.ltk elt. - -Record t_ (elt:Type) := Mk - {this :> Raw.t elt; - sorted : sort Raw.PX.ltk this}. -Definition t := t_. - -Definition empty {elt} := Mk (Raw.empty_sorted elt). - -Section Elt. - Variable elt elt' elt'':Type. - - Implicit Types m : t elt. - Implicit Types x y : key. - Implicit Types e : elt. - - Definition is_empty m : bool := Raw.is_empty m.(this). - Definition add x e m : t elt := Mk (Raw.add_sorted m.(sorted) x e). - Definition find x m : option elt := Raw.find x m.(this). - Definition remove x m : t elt := Mk (Raw.remove_sorted m.(sorted) x). - Definition mem x m : bool := Raw.mem x m.(this). - Definition map f m : t elt' := Mk (Raw.map_sorted f m.(sorted)). - Definition mapi (f:key->elt->elt') m : t elt' := - Mk (Raw.mapi_sorted f m.(sorted)). - Definition merge f m (m':t elt') : t elt'' := - Mk (Raw.merge_sorted f m.(sorted) m'.(sorted)). - Definition bindings m : list (key*elt) := Raw.bindings m.(this). - Definition cardinal m := length m.(this). - Definition fold {A:Type}(f:key->elt->A->A) m (i:A) : A := - Raw.fold f m.(this) i. - Definition equal cmp m m' : bool := Raw.equal cmp m.(this) m'.(this). - - Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e m.(this). - Definition In x m : Prop := Raw.PX.In x m.(this). - - Definition Equal m m' := forall y, find y m = find y m'. - Definition Equiv (eq_elt:elt->elt->Prop) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). - Definition Equivb cmp m m' : Prop := @Raw.Equivb elt cmp m.(this) m'.(this). - - Instance MapsTo_compat : - Proper (E.eq==>Logic.eq==>Logic.eq==>iff) MapsTo. - Proof. - intros x x' Hx e e' <- m m' <-. unfold MapsTo. now rewrite Hx. - Qed. - - Lemma find_spec m : forall x e, find x m = Some e <-> MapsTo x e m. - Proof. exact (Raw.find_spec m.(sorted)). Qed. - - Lemma mem_spec m : forall x, mem x m = true <-> In x m. - Proof. exact (Raw.mem_spec m.(sorted)). Qed. - - Lemma empty_spec : forall x, find x empty = None. - Proof. exact (Raw.empty_spec _). Qed. - - Lemma is_empty_spec m : is_empty m = true <-> (forall x, find x m = None). - Proof. exact (Raw.is_empty_spec m.(this)). Qed. - - Lemma add_spec1 m : forall x e, find x (add x e m) = Some e. - Proof. exact (Raw.add_spec1 m.(this)). Qed. - Lemma add_spec2 m : forall x y e, ~E.eq x y -> find y (add x e m) = find y m. - Proof. exact (Raw.add_spec2 m.(this)). Qed. - - Lemma remove_spec1 m : forall x, find x (remove x m) = None. - Proof. exact (Raw.remove_spec1 m.(sorted)). Qed. - Lemma remove_spec2 m : forall x y, ~E.eq x y -> find y (remove x m) = find y m. - Proof. exact (Raw.remove_spec2 m.(sorted)). Qed. - - Lemma bindings_spec1 m : forall x e, - InA eq_key_elt (x,e) (bindings m) <-> MapsTo x e m. - Proof. exact (Raw.bindings_spec1 m.(this)). Qed. - Lemma bindings_spec2w m : NoDupA eq_key (bindings m). - Proof. exact (Raw.bindings_spec2w m.(sorted)). Qed. - Lemma bindings_spec2 m : sort lt_key (bindings m). - Proof. exact (Raw.bindings_spec2 m.(sorted)). Qed. - - Lemma cardinal_spec m : cardinal m = length (bindings m). - Proof. reflexivity. Qed. - - Lemma fold_spec m : forall (A : Type) (i : A) (f : key -> elt -> A -> A), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. - Proof. exact (Raw.fold_spec m.(this)). Qed. - - Lemma equal_spec m m' : forall cmp, equal cmp m m' = true <-> Equivb cmp m m'. - Proof. exact (Raw.equal_spec m.(sorted) m'.(sorted)). Qed. - -End Elt. - - Lemma map_spec {elt elt'} (f:elt->elt') m : - forall x, find x (map f m) = option_map f (find x m). - Proof. exact (Raw.map_spec f m.(this)). Qed. - - Lemma mapi_spec {elt elt'} (f:key->elt->elt') m : - forall x, exists y, - E.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). - Proof. exact (Raw.mapi_spec f m.(this)). Qed. - - Lemma merge_spec1 {elt elt' elt''} - (f:key->option elt->option elt'->option elt'') m m' : - forall x, - In x m \/ In x m' -> - exists y, E.eq y x /\ find x (merge f m m') = f y (find x m) (find x m'). - Proof. exact (Raw.merge_spec1 f m.(sorted) m'.(sorted)). Qed. - - Lemma merge_spec2 {elt elt' elt''} - (f:key->option elt->option elt'->option elt'') m m' : - forall x, - In x (merge f m m') -> In x m \/ In x m'. - Proof. exact (Raw.merge_spec2 m.(sorted) m'.(sorted)). Qed. - -End Make. - -Module Make_ord (X: OrderedType)(D : OrderedType) <: -Sord with Module Data := D - with Module MapS.E := X. - -Module Data := D. -Module MapS := Make(X). -Import MapS. - -Module MD := OrderedTypeFacts(D). -Import MD. - -Definition t := MapS.t D.t. - -Definition cmp e e' := - match D.compare e e' with Eq => true | _ => false end. - -Fixpoint eq_list (m m' : list (X.t * D.t)) : Prop := - match m, m' with - | nil, nil => True - | (x,e)::l, (x',e')::l' => - match X.compare x x' with - | Eq => D.eq e e' /\ eq_list l l' - | _ => False - end - | _, _ => False - end. - -Definition eq m m' := eq_list m.(this) m'.(this). - -Fixpoint lt_list (m m' : list (X.t * D.t)) : Prop := - match m, m' with - | nil, nil => False - | nil, _ => True - | _, nil => False - | (x,e)::l, (x',e')::l' => - match X.compare x x' with - | Lt => True - | Gt => False - | Eq => D.lt e e' \/ (D.eq e e' /\ lt_list l l') - end - end. - -Definition lt m m' := lt_list m.(this) m'.(this). - -Lemma eq_equal : forall m m', eq m m' <-> equal cmp m m' = true. -Proof. - intros (l,Hl); induction l. - intros (l',Hl'); unfold eq; simpl. - destruct l'; unfold equal; simpl; intuition. - intros (l',Hl'); unfold eq. - destruct l'. - destruct a; unfold equal; simpl; intuition. - destruct a as (x,e). - destruct p as (x',e'). - unfold equal; simpl. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; simpl; intuition. - unfold cmp at 1. - elim D.compare_spec; try MD.order; simpl. - inversion_clear Hl. - inversion_clear Hl'. - destruct (IHl H (Mk H3)). - unfold equal, eq in H5; simpl in H5; auto. - destruct (andb_prop _ _ H); clear H. - generalize H0; unfold cmp. - elim D.compare_spec; try MD.order; simpl; try discriminate. - destruct (andb_prop _ _ H); clear H. - inversion_clear Hl. - inversion_clear Hl'. - destruct (IHl H (Mk H3)). - unfold equal, eq in H6; simpl in H6; auto. -Qed. - -Lemma eq_spec m m' : eq m m' <-> Equivb cmp m m'. -Proof. - now rewrite eq_equal, equal_spec. -Qed. - -Lemma eq_refl : forall m : t, eq m m. -Proof. - intros (m,Hm); induction m; unfold eq; simpl; auto. - destruct a. - destruct (X.compare_spec t0 t0) as [Hlt|Heq|Hlt]; auto. - - split. reflexivity. inversion_clear Hm. apply (IHm H). - - MapS.Raw.MX.order. - - MapS.Raw.MX.order. -Qed. - -Lemma eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1. -Proof. - intros (m,Hm); induction m; - intros (m', Hm'); destruct m'; unfold eq; simpl; - try destruct a as (x,e); try destruct p as (x',e'); auto. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; - elim X.compare_spec; try MapS.Raw.MX.order; intuition. - inversion_clear Hm; inversion_clear Hm'. - apply (IHm H0 (Mk H4)); auto. -Qed. - -Lemma eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3. -Proof. - intros (m1,Hm1); induction m1; - intros (m2, Hm2); destruct m2; - intros (m3, Hm3); destruct m3; unfold eq; simpl; - try destruct a as (x,e); - try destruct p as (x',e'); - try destruct p0 as (x'',e''); try contradiction; auto. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; - destruct (X.compare_spec x' x'') as [Hlt'|Heq'|Hlt']; - elim X.compare_spec; try MapS.Raw.MX.order; intuition. - now transitivity e'. - inversion_clear Hm1; inversion_clear Hm2; inversion_clear Hm3. - apply (IHm1 H1 (Mk H6) (Mk H8)); intuition. -Qed. - -Instance eq_equiv : Equivalence eq. -Proof. split; [exact eq_refl|exact eq_sym|exact eq_trans]. Qed. - -Lemma lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3. -Proof. - intros (m1,Hm1); induction m1; - intros (m2, Hm2); destruct m2; - intros (m3, Hm3); destruct m3; unfold lt; simpl; - try destruct a as (x,e); - try destruct p as (x',e'); - try destruct p0 as (x'',e''); try contradiction; auto. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; - destruct (X.compare_spec x' x'') as [Hlt'|Heq'|Hlt']; - elim X.compare_spec; try MapS.Raw.MX.order; intuition. - left; transitivity e'; auto. - left; MD.order. - left; MD.order. - right. - split. - transitivity e'; auto. - inversion_clear Hm1; inversion_clear Hm2; inversion_clear Hm3. - apply (IHm1 H2 (Mk H6) (Mk H8)); intuition. -Qed. - -Lemma lt_irrefl : forall m, ~ lt m m. -Proof. - intros (m,Hm); induction m; unfold lt; simpl; auto. - destruct a. - destruct (X.compare_spec t0 t0) as [Hlt|Heq|Hlt]; auto. - - intuition. MD.order. inversion_clear Hm. now apply (IHm H0). - - MapS.Raw.MX.order. -Qed. - -Instance lt_strorder : StrictOrder lt. -Proof. split; [exact lt_irrefl|exact lt_trans]. Qed. - -Lemma lt_compat1 : forall m1 m1' m2, eq m1 m1' -> lt m1 m2 -> lt m1' m2. -Proof. - intros (m1,Hm1); induction m1; - intros (m1',Hm1'); destruct m1'; - intros (m2,Hm2); destruct m2; unfold eq, lt; - try destruct a as (x,e); - try destruct p as (x',e'); - try destruct p0 as (x'',e''); try contradiction; simpl; auto. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; - destruct (X.compare_spec x' x'') as [Hlt'|Heq'|Hlt']; - elim X.compare_spec; try MapS.Raw.MX.order; intuition. - left; MD.order. - right. - split. - MD.order. - inversion_clear Hm1; inversion_clear Hm1'; inversion_clear Hm2. - apply (IHm1 H0 (Mk H6) (Mk H8)); intuition. -Qed. - -Lemma lt_compat2 : forall m1 m2 m2', eq m2 m2' -> lt m1 m2 -> lt m1 m2'. -Proof. - intros (m1,Hm1); induction m1; - intros (m2,Hm2); destruct m2; - intros (m2',Hm2'); destruct m2'; unfold eq, lt; - try destruct a as (x,e); - try destruct p as (x',e'); - try destruct p0 as (x'',e''); try contradiction; simpl; auto. - destruct (X.compare_spec x x') as [Hlt|Heq|Hlt]; - destruct (X.compare_spec x' x'') as [Hlt'|Heq'|Hlt']; - elim X.compare_spec; try MapS.Raw.MX.order; intuition. - left; MD.order. - right. - split. - MD.order. - inversion_clear Hm1; inversion_clear Hm2; inversion_clear Hm2'. - apply (IHm1 H0 (Mk H6) (Mk H8)); intuition. -Qed. - -Instance lt_compat : Proper (eq==>eq==>iff) lt. -Proof. - intros m1 m1' H1 m2 m2' H2. split; intros. - now apply (lt_compat2 H2), (lt_compat1 H1). - symmetry in H1, H2. - now apply (lt_compat2 H2), (lt_compat1 H1). -Qed. - -Ltac cmp_solve := - unfold eq, lt; simpl; elim X.compare_spec; try Raw.MX.order; auto. - -Fixpoint compare_list m1 m2 := match m1, m2 with -| nil, nil => Eq -| nil, _ => Lt -| _, nil => Gt -| (k1,e1)::m1, (k2,e2)::m2 => - match X.compare k1 k2 with - | Lt => Lt - | Gt => Gt - | Eq => match D.compare e1 e2 with - | Lt => Lt - | Gt => Gt - | Eq => compare_list m1 m2 - end - end -end. - -Definition compare m1 m2 := compare_list m1.(this) m2.(this). - -Lemma compare_spec : forall m1 m2, CompSpec eq lt m1 m2 (compare m1 m2). -Proof. - unfold CompSpec. - intros (m1,Hm1)(m2,Hm2). unfold compare, eq, lt; simpl. - revert m2 Hm2. - induction m1 as [|(k1,e1) m1 IH1]; destruct m2 as [|(k2,e2) m2]; - try constructor; simpl; intros; auto. - elim X.compare_spec; simpl; try constructor; auto; intros. - elim D.compare_spec; simpl; try constructor; auto; intros. - inversion_clear Hm1; inversion_clear Hm2. - destruct (IH1 H1 _ H3); simpl; try constructor; auto. - elim X.compare_spec; try Raw.MX.order. right. now split. - elim X.compare_spec; try Raw.MX.order. now left. - elim X.compare_spec; try Raw.MX.order; auto. -Qed. - -End Make_ord. diff --git a/theories/MMaps/MMapPositive.v b/theories/MMaps/MMapPositive.v deleted file mode 100644 index d3aab238..00000000 --- a/theories/MMaps/MMapPositive.v +++ /dev/null @@ -1,698 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(** * MMapPositive : an implementation of MMapInterface for [positive] keys. *) - -Require Import Bool PeanoNat BinPos Orders OrdersEx OrdersLists MMapInterface. - -Set Implicit Arguments. -Local Open Scope lazy_bool_scope. -Local Open Scope positive_scope. -Local Unset Elimination Schemes. - -(** This file is an adaptation to the [MMap] framework of a work by - Xavier Leroy and Sandrine Blazy (used for building certified compilers). - Keys are of type [positive], and maps are binary trees: the sequence - of binary digits of a positive number corresponds to a path in such a tree. - This is quite similar to the [IntMap] library, except that no path - compression is implemented, and that the current file is simple enough to be - self-contained. *) - -(** Reverses the positive [y] and concatenate it with [x] *) - -Fixpoint rev_append (y x : positive) : positive := - match y with - | 1 => x - | y~1 => rev_append y x~1 - | y~0 => rev_append y x~0 - end. -Local Infix "@" := rev_append (at level 60). -Definition rev x := x@1. - -(** The module of maps over positive keys *) - -Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. - - Module E:=PositiveOrderedTypeBits. - Module ME:=KeyOrderedType E. - - Definition key := positive : Type. - - Definition eq_key {A} (p p':key*A) := E.eq (fst p) (fst p'). - - Definition eq_key_elt {A} (p p':key*A) := - E.eq (fst p) (fst p') /\ (snd p) = (snd p'). - - Definition lt_key {A} (p p':key*A) := E.lt (fst p) (fst p'). - - Instance eqk_equiv {A} : Equivalence (@eq_key A) := _. - Instance eqke_equiv {A} : Equivalence (@eq_key_elt A) := _. - Instance ltk_strorder {A} : StrictOrder (@lt_key A) := _. - - Inductive tree (A : Type) := - | Leaf : tree A - | Node : tree A -> option A -> tree A -> tree A. - - Arguments Leaf {A}. - - Scheme tree_ind := Induction for tree Sort Prop. - - Definition t := tree. - - Definition empty {A} : t A := Leaf. - - Section A. - Variable A:Type. - - Fixpoint is_empty (m : t A) : bool := - match m with - | Leaf => true - | Node l None r => (is_empty l) &&& (is_empty r) - | _ => false - end. - - Fixpoint find (i : key) (m : t A) : option A := - match m with - | Leaf => None - | Node l o r => - match i with - | xH => o - | xO ii => find ii l - | xI ii => find ii r - end - end. - - Fixpoint mem (i : key) (m : t A) : bool := - match m with - | Leaf => false - | Node l o r => - match i with - | xH => match o with None => false | _ => true end - | xO ii => mem ii l - | xI ii => mem ii r - end - end. - - Fixpoint add (i : key) (v : A) (m : t A) : t A := - match m with - | Leaf => - match i with - | xH => Node Leaf (Some v) Leaf - | xO ii => Node (add ii v Leaf) None Leaf - | xI ii => Node Leaf None (add ii v Leaf) - end - | Node l o r => - match i with - | xH => Node l (Some v) r - | xO ii => Node (add ii v l) o r - | xI ii => Node l o (add ii v r) - end - end. - - (** helper function to avoid creating empty trees that are not leaves *) - - Definition node (l : t A) (o: option A) (r : t A) : t A := - match o,l,r with - | None,Leaf,Leaf => Leaf - | _,_,_ => Node l o r - end. - - Fixpoint remove (i : key) (m : t A) : t A := - match m with - | Leaf => Leaf - | Node l o r => - match i with - | xH => node l None r - | xO ii => node (remove ii l) o r - | xI ii => node l o (remove ii r) - end - end. - - (** [bindings] *) - - Fixpoint xbindings (m : t A) (i : positive) (a: list (key*A)) := - match m with - | Leaf => a - | Node l None r => xbindings l i~0 (xbindings r i~1 a) - | Node l (Some e) r => xbindings l i~0 ((rev i,e) :: xbindings r i~1 a) - end. - - Definition bindings (m : t A) := xbindings m 1 nil. - - (** [cardinal] *) - - Fixpoint cardinal (m : t A) : nat := - match m with - | Leaf => 0%nat - | Node l None r => (cardinal l + cardinal r)%nat - | Node l (Some _) r => S (cardinal l + cardinal r) - end. - - (** Specification proofs *) - - Definition MapsTo (i:key)(v:A)(m:t A) := find i m = Some v. - Definition In (i:key)(m:t A) := exists e:A, MapsTo i e m. - - Lemma MapsTo_compat : Proper (E.eq==>eq==>eq==>iff) MapsTo. - Proof. - intros k k' Hk e e' He m m' Hm. red in Hk. now subst. - Qed. - - Lemma find_spec m x e : find x m = Some e <-> MapsTo x e m. - Proof. reflexivity. Qed. - - Lemma mem_find : - forall m x, mem x m = match find x m with None => false | _ => true end. - Proof. - induction m; destruct x; simpl; auto. - Qed. - - Lemma mem_spec : forall m x, mem x m = true <-> In x m. - Proof. - unfold In, MapsTo; intros m x; rewrite mem_find. - split. - - destruct (find x m). - exists a; auto. - intros; discriminate. - - destruct 1 as (e0,H0); rewrite H0; auto. - Qed. - - Lemma gleaf : forall (i : key), find i Leaf = None. - Proof. destruct i; simpl; auto. Qed. - - Theorem empty_spec: - forall (i: key), find i empty = None. - Proof. exact gleaf. Qed. - - Lemma is_empty_spec m : - is_empty m = true <-> forall k, find k m = None. - Proof. - induction m; simpl. - - intuition. apply empty_spec. - - destruct o. split; try discriminate. - intros H. now specialize (H xH). - rewrite <- andb_lazy_alt, andb_true_iff, IHm1, IHm2. - clear IHm1 IHm2. - split. - + intros (H1,H2) k. destruct k; simpl; auto. - + intros H; split; intros k. apply (H (xO k)). apply (H (xI k)). - Qed. - - Theorem add_spec1: - forall (m: t A) (i: key) (x: A), find i (add i x m) = Some x. - Proof. - intros m i; revert m. - induction i; destruct m; simpl; auto. - Qed. - - Theorem add_spec2: - forall (m: t A) (i j: key) (x: A), - i <> j -> find j (add i x m) = find j m. - Proof. - intros m i j; revert m i. - induction j; destruct i, m; simpl; intros; - rewrite ?IHj, ?gleaf; auto; try congruence. - Qed. - - Lemma rleaf : forall (i : key), remove i Leaf = Leaf. - Proof. destruct i; simpl; auto. Qed. - - Lemma gnode l o r i : find i (node l o r) = find i (Node l o r). - Proof. - destruct o,l,r; simpl; trivial. - destruct i; simpl; now rewrite ?gleaf. - Qed. - - Opaque node. - - Theorem remove_spec1: - forall (m: t A)(i: key), find i (remove i m) = None. - Proof. - induction m; simpl. - - intros; rewrite rleaf. apply gleaf. - - destruct i; simpl remove; rewrite gnode; simpl; auto. - Qed. - - Theorem remove_spec2: - forall (m: t A)(i j: key), - i <> j -> find j (remove i m) = find j m. - Proof. - induction m; simpl; intros. - - now rewrite rleaf. - - destruct i; simpl; rewrite gnode; destruct j; simpl; trivial; - try apply IHm1; try apply IHm2; congruence. - Qed. - - Local Notation InL := (InA eq_key_elt). - - Lemma xbindings_spec: forall m j acc k e, - InL (k,e) (xbindings m j acc) <-> - InL (k,e) acc \/ exists x, k=(j@x) /\ find x m = Some e. - Proof. - induction m as [|l IHl o r IHr]; simpl. - - intros. split; intro H. - + now left. - + destruct H as [H|[x [_ H]]]. assumption. - now rewrite gleaf in H. - - intros j acc k e. case o as [e'|]; - rewrite IHl, ?InA_cons, IHr; clear IHl IHr; split. - + intros [[H|[H|H]]|H]; auto. - * unfold eq_key_elt, E.eq, fst, snd in H. destruct H as (->,<-). - right. now exists 1. - * destruct H as (x,(->,H)). right. now exists x~1. - * destruct H as (x,(->,H)). right. now exists x~0. - + intros [H|H]; auto. - destruct H as (x,(->,H)). - destruct x; simpl in *. - * left. right. right. now exists x. - * right. now exists x. - * left. left. injection H as ->. reflexivity. - + intros [[H|H]|H]; auto. - * destruct H as (x,(->,H)). right. now exists x~1. - * destruct H as (x,(->,H)). right. now exists x~0. - + intros [H|H]; auto. - destruct H as (x,(->,H)). - destruct x; simpl in *. - * left. right. now exists x. - * right. now exists x. - * discriminate. - Qed. - - Lemma lt_rev_append: forall j x y, E.lt x y -> E.lt (j@x) (j@y). - Proof. induction j; intros; simpl; auto. Qed. - - Lemma xbindings_sort m j acc : - sort lt_key acc -> - (forall x p, In x m -> InL p acc -> E.lt (j@x) (fst p)) -> - sort lt_key (xbindings m j acc). - Proof. - revert j acc. - induction m as [|l IHl o r IHr]; simpl; trivial. - intros j acc Hacc Hsacc. destruct o as [e|]. - - apply IHl;[constructor;[apply IHr; [apply Hacc|]|]|]. - + intros. now apply Hsacc. - + case_eq (xbindings r j~1 acc); [constructor|]. - intros (z,e') q H. constructor. - assert (H': InL (z,e') (xbindings r j~1 acc)). - { rewrite H. now constructor. } - clear H q. rewrite xbindings_spec in H'. - destruct H' as [H'|H']. - * apply (Hsacc 1 (z,e')); trivial. now exists e. - * destruct H' as (x,(->,H)). - red. simpl. now apply lt_rev_append. - + intros x (y,e') Hx Hy. inversion_clear Hy. - rewrite H. simpl. now apply lt_rev_append. - rewrite xbindings_spec in H. - destruct H as [H|H]. - * now apply Hsacc. - * destruct H as (z,(->,H)). simpl. - now apply lt_rev_append. - - apply IHl; [apply IHr; [apply Hacc|]|]. - + intros. now apply Hsacc. - + intros x (y,e') Hx H. rewrite xbindings_spec in H. - destruct H as [H|H]. - * now apply Hsacc. - * destruct H as (z,(->,H)). simpl. - now apply lt_rev_append. - Qed. - - Lemma bindings_spec1 m k e : - InA eq_key_elt (k,e) (bindings m) <-> MapsTo k e m. - Proof. - unfold bindings, MapsTo. rewrite xbindings_spec. - split; [ intros [H|(y & H & H')] | intros IN ]. - - inversion H. - - simpl in *. now subst. - - right. now exists k. - Qed. - - Lemma bindings_spec2 m : sort lt_key (bindings m). - Proof. - unfold bindings. - apply xbindings_sort. constructor. inversion 2. - Qed. - - Lemma bindings_spec2w m : NoDupA eq_key (bindings m). - Proof. - apply ME.Sort_NoDupA. - apply bindings_spec2. - Qed. - - Lemma xbindings_length m j acc : - length (xbindings m j acc) = (cardinal m + length acc)%nat. - Proof. - revert j acc. - induction m; simpl; trivial; intros. - destruct o; simpl; rewrite IHm1; simpl; rewrite IHm2; - now rewrite ?Nat.add_succ_r, Nat.add_assoc. - Qed. - - Lemma cardinal_spec m : cardinal m = length (bindings m). - Proof. - unfold bindings. rewrite xbindings_length. simpl. - symmetry. apply Nat.add_0_r. - Qed. - - (** [map] and [mapi] *) - - Variable B : Type. - - Section Mapi. - - Variable f : key -> option A -> option B. - - Fixpoint xmapi (m : t A) (i : key) : t B := - match m with - | Leaf => Leaf - | Node l o r => Node (xmapi l (i~0)) - (f (rev i) o) - (xmapi r (i~1)) - end. - - End Mapi. - - Definition mapi (f : key -> A -> B) m := - xmapi (fun k => option_map (f k)) m 1. - - Definition map (f : A -> B) m := mapi (fun _ => f) m. - - End A. - - Lemma xgmapi: - forall (A B: Type) (f: key -> option A -> option B) (i j : key) (m: t A), - (forall k, f k None = None) -> - find i (xmapi f m j) = f (j@i) (find i m). - Proof. - induction i; intros; destruct m; simpl; rewrite ?IHi; auto. - Qed. - - Theorem mapi_spec0 : - forall (A B: Type) (f: key -> A -> B) (i: key) (m: t A), - find i (mapi f m) = option_map (f i) (find i m). - Proof. - intros. unfold mapi. rewrite xgmapi; simpl; auto. - Qed. - - Lemma mapi_spec : - forall (A B: Type) (f: key -> A -> B) (m: t A) (i:key), - exists j, E.eq j i /\ - find i (mapi f m) = option_map (f j) (find i m). - Proof. - intros. - exists i. split. reflexivity. apply mapi_spec0. - Qed. - - Lemma map_spec : - forall (elt elt':Type)(f:elt->elt')(m: t elt)(x:key), - find x (map f m) = option_map f (find x m). - Proof. - intros; unfold map. apply mapi_spec0. - Qed. - - Section merge. - Variable A B C : Type. - Variable f : key -> option A -> option B -> option C. - - Fixpoint xmerge (m1 : t A)(m2 : t B)(i:positive) : t C := - match m1 with - | Leaf => xmapi (fun k => f k None) m2 i - | Node l1 o1 r1 => - match m2 with - | Leaf => xmapi (fun k o => f k o None) m1 i - | Node l2 o2 r2 => - Node (xmerge l1 l2 (i~0)) - (f (rev i) o1 o2) - (xmerge r1 r2 (i~1)) - end - end. - - Lemma xgmerge: forall (i j: key)(m1:t A)(m2: t B), - (forall i, f i None None = None) -> - find i (xmerge m1 m2 j) = f (j@i) (find i m1) (find i m2). - Proof. - induction i; intros; destruct m1; destruct m2; simpl; auto; - rewrite ?xgmapi, ?IHi; simpl; auto. - Qed. - - End merge. - - Definition merge {A B C}(f:key->option A->option B->option C) m1 m2 := - xmerge - (fun k o1 o2 => match o1,o2 with - | None,None => None - | _, _ => f k o1 o2 - end) - m1 m2 xH. - - Lemma merge_spec1 {A B C}(f:key->option A->option B->option C) : - forall m m' x, - In x m \/ In x m' -> - exists y, E.eq y x /\ - find x (merge f m m') = f y (find x m) (find x m'). - Proof. - intros. exists x. split. reflexivity. - unfold merge. - rewrite xgmerge; simpl; auto. - rewrite <- 2 mem_spec, 2 mem_find in H. - destruct (find x m); simpl; auto. - destruct (find x m'); simpl; auto. intuition discriminate. - Qed. - - Lemma merge_spec2 {A B C}(f:key->option A->option B->option C) : - forall m m' x, In x (merge f m m') -> In x m \/ In x m'. - Proof. - intros. - rewrite <-mem_spec, mem_find in H. - unfold merge in H. - rewrite xgmerge in H; simpl; auto. - rewrite <- 2 mem_spec, 2 mem_find. - destruct (find x m); simpl in *; auto. - destruct (find x m'); simpl in *; auto. - Qed. - - Section Fold. - - Variables A B : Type. - Variable f : key -> A -> B -> B. - - (** the additional argument, [i], records the current path, in - reverse order (this should be more efficient: we reverse this argument - only at present nodes only, rather than at each node of the tree). - we also use this convention in all functions below - *) - - Fixpoint xfold (m : t A) (v : B) (i : key) := - match m with - | Leaf => v - | Node l (Some x) r => - xfold r (f (rev i) x (xfold l v i~0)) i~1 - | Node l None r => - xfold r (xfold l v i~0) i~1 - end. - Definition fold m i := xfold m i 1. - - End Fold. - - Lemma fold_spec : - forall {A}(m:t A){B}(i : B) (f : key -> A -> B -> B), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. - Proof. - unfold fold, bindings. intros A m B i f. revert m i. - set (f' := fun a p => f (fst p) (snd p) a). - assert (H: forall m i j acc, - fold_left f' acc (xfold f m i j) = - fold_left f' (xbindings m j acc) i). - { induction m as [|l IHl o r IHr]; intros; trivial. - destruct o; simpl; now rewrite IHr, <- IHl. } - intros. exact (H m i 1 nil). - Qed. - - Fixpoint equal (A:Type)(cmp : A -> A -> bool)(m1 m2 : t A) : bool := - match m1, m2 with - | Leaf, _ => is_empty m2 - | _, Leaf => is_empty m1 - | Node l1 o1 r1, Node l2 o2 r2 => - (match o1, o2 with - | None, None => true - | Some v1, Some v2 => cmp v1 v2 - | _, _ => false - end) - &&& equal cmp l1 l2 &&& equal cmp r1 r2 - end. - - Definition Equal (A:Type)(m m':t A) := - forall y, find y m = find y m'. - Definition Equiv (A:Type)(eq_elt:A->A->Prop) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). - Definition Equivb (A:Type)(cmp: A->A->bool) := Equiv (Cmp cmp). - - Lemma equal_1 : forall (A:Type)(m m':t A)(cmp:A->A->bool), - Equivb cmp m m' -> equal cmp m m' = true. - Proof. - induction m. - - (* m = Leaf *) - destruct 1 as (E,_); simpl. - apply is_empty_spec; intros k. - destruct (find k m') eqn:F; trivial. - assert (H : In k m') by now exists a. - rewrite <- E in H. - destruct H as (x,H). red in H. now rewrite gleaf in H. - - (* m = Node *) - destruct m'. - + (* m' = Leaf *) - destruct 1 as (E,_); simpl. - destruct o. - * assert (H : In xH (@Leaf A)). - { rewrite <- E. now exists a. } - destruct H as (e,H). now red in H. - * apply andb_true_intro; split; apply is_empty_spec; intros k. - destruct (find k m1) eqn:F; trivial. - assert (H : In (xO k) (@Leaf A)). - { rewrite <- E. exists a; auto. } - destruct H as (x,H). red in H. now rewrite gleaf in H. - destruct (find k m2) eqn:F; trivial. - assert (H : In (xI k) (@Leaf A)). - { rewrite <- E. exists a; auto. } - destruct H as (x,H). red in H. now rewrite gleaf in H. - + (* m' = Node *) - destruct 1. - assert (Equivb cmp m1 m'1). - { split. - intros k; generalize (H (xO k)); unfold In, MapsTo; simpl; auto. - intros k e e'; generalize (H0 (xO k) e e'); unfold In, MapsTo; simpl; auto. } - assert (Equivb cmp m2 m'2). - { split. - intros k; generalize (H (xI k)); unfold In, MapsTo; simpl; auto. - intros k e e'; generalize (H0 (xI k) e e'); unfold In, MapsTo; simpl; auto. } - simpl. - destruct o; destruct o0; simpl. - repeat (apply andb_true_intro; split); auto. - apply (H0 xH); red; auto. - generalize (H xH); unfold In, MapsTo; simpl; intuition. - destruct H4; try discriminate; eauto. - generalize (H xH); unfold In, MapsTo; simpl; intuition. - destruct H5; try discriminate; eauto. - apply andb_true_intro; split; auto. - Qed. - - Lemma equal_2 : forall (A:Type)(m m':t A)(cmp:A->A->bool), - equal cmp m m' = true -> Equivb cmp m m'. - Proof. - induction m. - (* m = Leaf *) - simpl. - split; intros. - split. - destruct 1; red in H0; destruct k; discriminate. - rewrite is_empty_spec in H. - intros (e,H'). red in H'. now rewrite H in H'. - red in H0; destruct k; discriminate. - (* m = Node *) - destruct m'. - (* m' = Leaf *) - simpl. - destruct o; intros; try discriminate. - destruct (andb_prop _ _ H); clear H. - split; intros. - split; unfold In, MapsTo; destruct 1. - destruct k; simpl in *; try discriminate. - rewrite is_empty_spec in H1. - now rewrite H1 in H. - rewrite is_empty_spec in H0. - now rewrite H0 in H. - destruct k; simpl in *; discriminate. - unfold In, MapsTo; destruct k; simpl in *; discriminate. - (* m' = Node *) - destruct o; destruct o0; simpl; intros; try discriminate. - destruct (andb_prop _ _ H); clear H. - destruct (andb_prop _ _ H0); clear H0. - destruct (IHm1 _ _ H2); clear H2 IHm1. - destruct (IHm2 _ _ H1); clear H1 IHm2. - split; intros. - destruct k; unfold In, MapsTo in *; simpl; auto. - split; eauto. - destruct k; unfold In, MapsTo in *; simpl in *. - eapply H4; eauto. - eapply H3; eauto. - congruence. - destruct (andb_prop _ _ H); clear H. - destruct (IHm1 _ _ H0); clear H0 IHm1. - destruct (IHm2 _ _ H1); clear H1 IHm2. - split; intros. - destruct k; unfold In, MapsTo in *; simpl; auto. - split; eauto. - destruct k; unfold In, MapsTo in *; simpl in *. - eapply H3; eauto. - eapply H2; eauto. - try discriminate. - Qed. - - Lemma equal_spec : forall (A:Type)(m m':t A)(cmp:A->A->bool), - equal cmp m m' = true <-> Equivb cmp m m'. - Proof. - split. apply equal_2. apply equal_1. - Qed. - -End PositiveMap. - -(** Here come some additionnal facts about this implementation. - Most are facts that cannot be derivable from the general interface. *) - -Module PositiveMapAdditionalFacts. - Import PositiveMap. - - (* Derivable from the Map interface *) - Theorem gsspec {A} i j x (m: t A) : - find i (add j x m) = if E.eq_dec i j then Some x else find i m. - Proof. - destruct (E.eq_dec i j) as [->|]; - [ apply add_spec1 | apply add_spec2; auto ]. - Qed. - - (* Not derivable from the Map interface *) - Theorem gsident {A} i (m:t A) v : - find i m = Some v -> add i v m = m. - Proof. - revert m. - induction i; destruct m; simpl in *; try congruence. - - intro H; now rewrite (IHi m2 H). - - intro H; now rewrite (IHi m1 H). - Qed. - - Lemma xmapi_ext {A B}(f g: key -> option A -> option B) : - (forall k (o : option A), f k o = g k o) -> - forall m i, xmapi f m i = xmapi g m i. - Proof. - induction m; intros; simpl; auto. now f_equal. - Qed. - - Theorem xmerge_commut{A B C} - (f: key -> option A -> option B -> option C) - (g: key -> option B -> option A -> option C) : - (forall k o1 o2, f k o1 o2 = g k o2 o1) -> - forall m1 m2 i, xmerge f m1 m2 i = xmerge g m2 m1 i. - Proof. - intros E. - induction m1; destruct m2; intros i; simpl; trivial; f_equal; - try apply IHm1_1; try apply IHm1_2; try apply xmapi_ext; - intros; apply E. - Qed. - - Theorem merge_commut{A B C} - (f: key -> option A -> option B -> option C) - (g: key -> option B -> option A -> option C) : - (forall k o1 o2, f k o1 o2 = g k o2 o1) -> - forall m1 m2, merge f m1 m2 = merge g m2 m1. - Proof. - intros E m1 m2. - unfold merge. apply xmerge_commut. - intros k [x1|] [x2|]; trivial. - Qed. - -End PositiveMapAdditionalFacts. diff --git a/theories/MMaps/MMapWeakList.v b/theories/MMaps/MMapWeakList.v deleted file mode 100644 index 656c61e1..00000000 --- a/theories/MMaps/MMapWeakList.v +++ /dev/null @@ -1,687 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - -(** * Finite map library *) - -(** This file proposes an implementation of the non-dependant interface - [MMapInterface.WS] using lists of pairs, unordered but without redundancy. *) - -Require Import MMapInterface EqualitiesFacts. - -Set Implicit Arguments. -Unset Strict Implicit. - -Lemma Some_iff {A} (a a' : A) : Some a = Some a' <-> a = a'. -Proof. split; congruence. Qed. - -Module Raw (X:DecidableType). - -Module Import PX := KeyDecidableType X. - -Definition key := X.t. -Definition t (elt:Type) := list (X.t * elt). - -Ltac dec := match goal with - | |- context [ X.eq_dec ?x ?x ] => - let E := fresh "E" in destruct (X.eq_dec x x) as [E|E]; [ | now elim E] - | H : X.eq ?x ?y |- context [ X.eq_dec ?x ?y ] => - let E := fresh "E" in destruct (X.eq_dec x y) as [_|E]; [ | now elim E] - | H : ~X.eq ?x ?y |- context [ X.eq_dec ?x ?y ] => - let E := fresh "E" in destruct (X.eq_dec x y) as [E|_]; [ now elim H | ] - | |- context [ X.eq_dec ?x ?y ] => - let E := fresh "E" in destruct (X.eq_dec x y) as [E|E] -end. - -Section Elt. - -Variable elt : Type. -Notation NoDupA := (@NoDupA _ eqk). - -(** * [find] *) - -Fixpoint find (k:key) (s: t elt) : option elt := - match s with - | nil => None - | (k',x)::s' => if X.eq_dec k k' then Some x else find k s' - end. - -Lemma find_spec : forall m (Hm:NoDupA m) x e, - find x m = Some e <-> MapsTo x e m. -Proof. - unfold PX.MapsTo. - induction m as [ | (k,e) m IH]; simpl. - - split; inversion 1. - - intros Hm k' e'. rewrite InA_cons. - change (eqke (k',e') (k,e)) with (X.eq k' k /\ e' = e). - inversion_clear Hm. dec. - + rewrite Some_iff; intuition. - elim H. apply InA_eqk with (k',e'); auto. - + rewrite IH; intuition. -Qed. - -(** * [mem] *) - -Fixpoint mem (k : key) (s : t elt) : bool := - match s with - | nil => false - | (k',_) :: l => if X.eq_dec k k' then true else mem k l - end. - -Lemma mem_spec : forall m (Hm:NoDupA m) x, mem x m = true <-> In x m. -Proof. - induction m as [ | (k,e) m IH]; simpl; intros Hm x. - - split. discriminate. inversion_clear 1. inversion H0. - - inversion_clear Hm. rewrite PX.In_cons; simpl. - rewrite <- IH by trivial. - dec; intuition. -Qed. - -(** * [empty] *) - -Definition empty : t elt := nil. - -Lemma empty_spec x : find x empty = None. -Proof. - reflexivity. -Qed. - -Lemma empty_NoDup : NoDupA empty. -Proof. - unfold empty; auto. -Qed. - -(** * [is_empty] *) - -Definition is_empty (l : t elt) : bool := if l then true else false. - -Lemma is_empty_spec m : is_empty m = true <-> forall x, find x m = None. -Proof. - destruct m; simpl; intuition; try discriminate. - specialize (H a). - revert H. now dec. -Qed. - -(* Not part of the exported specifications, used later for [merge]. *) - -Lemma find_eq : forall m (Hm:NoDupA m) x x', - X.eq x x' -> find x m = find x' m. -Proof. - induction m; simpl; auto; destruct a; intros. - inversion_clear Hm. - rewrite (IHm H1 x x'); auto. - dec; dec; trivial. - elim E0. now transitivity x. - elim E. now transitivity x'. -Qed. - -(** * [add] *) - -Fixpoint add (k : key) (x : elt) (s : t elt) : t elt := - match s with - | nil => (k,x) :: nil - | (k',y) :: l => if X.eq_dec k k' then (k,x)::l else (k',y)::add k x l - end. - -Lemma add_spec1 m x e : find x (add x e m) = Some e. -Proof. - induction m as [ | (k,e') m IH]; simpl. - - now dec. - - dec; simpl; now dec. -Qed. - -Lemma add_spec2 m x y e : ~X.eq x y -> find y (add x e m) = find y m. -Proof. - intros N. - assert (N' : ~X.eq y x) by now contradict N. - induction m as [ | (k,e') m IH]; simpl. - - dec; trivial. - - repeat (dec; simpl); trivial. elim N. now transitivity k. -Qed. - -Lemma add_InA : forall m x y e e', - ~ X.eq x y -> InA eqk (y,e) (add x e' m) -> InA eqk (y,e) m. -Proof. - induction m as [ | (k,e') m IH]; simpl; intros. - - inversion_clear H0. elim H. symmetry; apply H1. inversion_clear H1. - - revert H0; dec; rewrite !InA_cons. - + rewrite E. intuition. - + intuition. right; eapply IH; eauto. -Qed. - -Lemma add_NoDup : forall m (Hm:NoDupA m) x e, NoDupA (add x e m). -Proof. - induction m as [ | (k,e') m IH]; simpl; intros Hm x e. - - constructor; auto. now inversion 1. - - inversion_clear Hm. dec; constructor; auto. - + contradict H. apply InA_eqk with (x,e); auto. - + contradict H; apply add_InA with x e; auto. -Qed. - -(** * [remove] *) - -Fixpoint remove (k : key) (s : t elt) : t elt := - match s with - | nil => nil - | (k',x) :: l => if X.eq_dec k k' then l else (k',x) :: remove k l - end. - -Lemma remove_spec1 m (Hm: NoDupA m) x : find x (remove x m) = None. -Proof. - induction m as [ | (k,e') m IH]; simpl; trivial. - inversion_clear Hm. - repeat (dec; simpl); auto. - destruct (find x m) eqn:F; trivial. - apply find_spec in F; trivial. - elim H. apply InA_eqk with (x,e); auto. -Qed. - -Lemma remove_spec2 m (Hm: NoDupA m) x y : ~X.eq x y -> - find y (remove x m) = find y m. -Proof. - induction m as [ | (k,e') m IH]; simpl; trivial; intros E. - inversion_clear Hm. - repeat (dec; simpl); auto. - elim E. now transitivity k. -Qed. - -Lemma remove_InA : forall m (Hm:NoDupA m) x y e, - InA eqk (y,e) (remove x m) -> InA eqk (y,e) m. -Proof. - induction m as [ | (k,e') m IH]; simpl; trivial; intros. - inversion_clear Hm. - revert H; dec; rewrite !InA_cons; intuition. - right; eapply H; eauto. -Qed. - -Lemma remove_NoDup : forall m (Hm:NoDupA m) x, NoDupA (remove x m). -Proof. - induction m. - simpl; intuition. - intros. - inversion_clear Hm. - destruct a as (x',e'). - simpl; case (X.eq_dec x x'); auto. - constructor; auto. - contradict H; apply remove_InA with x; auto. -Qed. - -(** * [bindings] *) - -Definition bindings (m: t elt) := m. - -Lemma bindings_spec1 m x e : InA eqke (x,e) (bindings m) <-> MapsTo x e m. -Proof. - reflexivity. -Qed. - -Lemma bindings_spec2w m (Hm:NoDupA m) : NoDupA (bindings m). -Proof. - trivial. -Qed. - -(** * [fold] *) - -Fixpoint fold (A:Type)(f:key->elt->A->A)(m:t elt) (acc : A) : A := - match m with - | nil => acc - | (k,e)::m' => fold f m' (f k e acc) - end. - -Lemma fold_spec : forall m (A:Type)(i:A)(f:key->elt->A->A), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. -Proof. - induction m as [ | (k,e) m IH]; simpl; auto. -Qed. - -(** * [equal] *) - -Definition check (cmp : elt -> elt -> bool)(k:key)(e:elt)(m': t elt) := - match find k m' with - | None => false - | Some e' => cmp e e' - end. - -Definition submap (cmp : elt -> elt -> bool)(m m' : t elt) : bool := - fold (fun k e b => andb (check cmp k e m') b) m true. - -Definition equal (cmp : elt -> elt -> bool)(m m' : t elt) : bool := - andb (submap cmp m m') (submap (fun e' e => cmp e e') m' m). - -Definition Submap (cmp:elt->elt->bool) m m' := - (forall k, In k m -> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true). - -Definition Equivb (cmp:elt->elt->bool) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true). - -Lemma submap_1 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, - Submap cmp m m' -> submap cmp m m' = true. -Proof. - unfold Submap, submap. - induction m. - simpl; auto. - destruct a; simpl; intros. - destruct H. - inversion_clear Hm. - assert (H3 : In t0 m'). - { apply H; exists e; auto with *. } - destruct H3 as (e', H3). - assert (H4 : find t0 m' = Some e') by now apply find_spec. - unfold check at 2. rewrite H4. - rewrite (H0 t0); simpl; auto with *. - eapply IHm; auto. - split; intuition. - apply H. - destruct H6 as (e'',H6); exists e''; auto. - apply H0 with k; auto. -Qed. - -Lemma submap_2 : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, - submap cmp m m' = true -> Submap cmp m m'. -Proof. - unfold Submap, submap. - induction m. - simpl; auto. - intuition. - destruct H0; inversion H0. - inversion H0. - - destruct a; simpl; intros. - inversion_clear Hm. - rewrite andb_b_true in H. - assert (check cmp t0 e m' = true). - clear H1 H0 Hm' IHm. - set (b:=check cmp t0 e m') in *. - generalize H; clear H; generalize b; clear b. - induction m; simpl; auto; intros. - destruct a; simpl in *. - destruct (andb_prop _ _ (IHm _ H)); auto. - rewrite H2 in H. - destruct (IHm H1 m' Hm' cmp H); auto. - unfold check in H2. - case_eq (find t0 m'); [intros e' H5 | intros H5]; - rewrite H5 in H2; try discriminate. - split; intros. - destruct H6 as (e0,H6); inversion_clear H6. - compute in H7; destruct H7; subst. - exists e'. - apply PX.MapsTo_eq with t0; auto with *. - apply find_spec; auto. - apply H3. - exists e0; auto. - inversion_clear H6. - compute in H8; destruct H8; subst. - assert (H8 : MapsTo t0 e'0 m'). { eapply PX.MapsTo_eq; eauto. } - apply find_spec in H8; trivial. congruence. - apply H4 with k; auto. -Qed. - -(** Specification of [equal] *) - -Lemma equal_spec : forall m (Hm:NoDupA m) m' (Hm': NoDupA m') cmp, - equal cmp m m' = true <-> Equivb cmp m m'. -Proof. - unfold Equivb, equal. - split. - - intros. - destruct (andb_prop _ _ H); clear H. - generalize (submap_2 Hm Hm' H0). - generalize (submap_2 Hm' Hm H1). - firstorder. - - intuition. - apply andb_true_intro; split; apply submap_1; unfold Submap; firstorder. -Qed. -End Elt. -Section Elt2. -Variable elt elt' : Type. - -(** * [map] and [mapi] *) - -Fixpoint map (f:elt -> elt') (m:t elt) : t elt' := - match m with - | nil => nil - | (k,e)::m' => (k,f e) :: map f m' - end. - -Fixpoint mapi (f: key -> elt -> elt') (m:t elt) : t elt' := - match m with - | nil => nil - | (k,e)::m' => (k,f k e) :: mapi f m' - end. - -(** Specification of [map] *) - -Lemma map_spec (f:elt->elt')(m:t elt)(x:key) : - find x (map f m) = option_map f (find x m). -Proof. - induction m as [ | (k,e) m IH]; simpl; trivial. - dec; simpl; trivial. -Qed. - -Lemma map_NoDup m (Hm : NoDupA (@eqk elt) m)(f:elt->elt') : - NoDupA (@eqk elt') (map f m). -Proof. - induction m; simpl; auto. - intros. - destruct a as (x',e'). - inversion_clear Hm. - constructor; auto. - contradict H. - clear IHm H0. - induction m; simpl in *; auto. - inversion H. - destruct a; inversion H; auto. -Qed. - -(** Specification of [mapi] *) - -Lemma mapi_spec (f:key->elt->elt')(m:t elt)(x:key) : - exists y, X.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). -Proof. - induction m as [ | (k,e) m IH]; simpl; trivial. - - now exists x. - - dec; simpl. - + now exists k. - + destruct IH as (y,(Hy,H)). now exists y. -Qed. - -Lemma mapi_NoDup : forall m (Hm : NoDupA (@eqk elt) m)(f: key->elt->elt'), - NoDupA (@eqk elt') (mapi f m). -Proof. - induction m; simpl; auto. - intros. - destruct a as (x',e'). - inversion_clear Hm; auto. - constructor; auto. - contradict H. - clear IHm H0. - induction m; simpl in *; auto. - inversion_clear H. - destruct a; inversion_clear H; auto. -Qed. - -End Elt2. - -Lemma mapfst_InA {elt}(m:t elt) x : - InA X.eq x (List.map fst m) <-> In x m. -Proof. - induction m as [| (k,e) m IH]; simpl; auto. - - split; inversion 1. inversion H0. - - rewrite InA_cons, In_cons. simpl. now rewrite IH. -Qed. - -Lemma mapfst_NoDup {elt}(m:t elt) : - NoDupA X.eq (List.map fst m) <-> NoDupA eqk m. -Proof. - induction m as [| (k,e) m IH]; simpl. - - split; constructor. - - split; inversion_clear 1; constructor; try apply IH; trivial. - + contradict H0. rewrite mapfst_InA. eapply In_alt'; eauto. - + rewrite mapfst_InA. contradict H0. now apply In_alt'. -Qed. - -Lemma filter_NoDup f (m:list key) : - NoDupA X.eq m -> NoDupA X.eq (List.filter f m). -Proof. - induction 1; simpl. - - constructor. - - destruct (f x); trivial. constructor; trivial. - contradict H. rewrite InA_alt in *. destruct H as (y,(Hy,H)). - exists y; split; trivial. now rewrite filter_In in H. -Qed. - -Lemma NoDupA_unique_repr (l:list key) x y : - NoDupA X.eq l -> X.eq x y -> List.In x l -> List.In y l -> x = y. -Proof. - intros H E Hx Hy. - induction H; simpl in *. - - inversion Hx. - - intuition; subst; trivial. - elim H. apply InA_alt. now exists y. - elim H. apply InA_alt. now exists x. -Qed. - -Section Elt3. - -Variable elt elt' elt'' : Type. - -Definition restrict (m:t elt)(k:key) := - match find k m with - | None => true - | Some _ => false - end. - -Definition domains (m:t elt)(m':t elt') := - List.map fst m ++ List.filter (restrict m) (List.map fst m'). - -Lemma domains_InA m m' (Hm : NoDupA eqk m) x : - InA X.eq x (domains m m') <-> In x m \/ In x m'. -Proof. - unfold domains. - assert (Proper (X.eq==>eq) (restrict m)). - { intros k k' Hk. unfold restrict. now rewrite (find_eq Hm Hk). } - rewrite InA_app_iff, filter_InA, !mapfst_InA; intuition. - unfold restrict. - destruct (find x m) eqn:F. - - left. apply find_spec in F; trivial. now exists e. - - now right. -Qed. - -Lemma domains_NoDup m m' : NoDupA eqk m -> NoDupA eqk m' -> - NoDupA X.eq (domains m m'). -Proof. - intros Hm Hm'. unfold domains. - apply NoDupA_app; auto with *. - - now apply mapfst_NoDup. - - now apply filter_NoDup, mapfst_NoDup. - - intros x. - rewrite mapfst_InA. intros (e,H). - apply find_spec in H; trivial. - rewrite InA_alt. intros (y,(Hy,H')). - rewrite (find_eq Hm Hy) in H. - rewrite filter_In in H'. destruct H' as (_,H'). - unfold restrict in H'. now rewrite H in H'. -Qed. - -Fixpoint fold_keys (f:key->option elt'') l := - match l with - | nil => nil - | k::l => - match f k with - | Some e => (k,e)::fold_keys f l - | None => fold_keys f l - end - end. - -Lemma fold_keys_In f l x e : - List.In (x,e) (fold_keys f l) <-> List.In x l /\ f x = Some e. -Proof. - induction l as [|k l IH]; simpl. - - intuition. - - destruct (f k) eqn:F; simpl; rewrite IH; clear IH; intuition; - try left; congruence. -Qed. - -Lemma fold_keys_NoDup f l : - NoDupA X.eq l -> NoDupA eqk (fold_keys f l). -Proof. - induction 1; simpl. - - constructor. - - destruct (f x); trivial. - constructor; trivial. contradict H. - apply InA_alt in H. destruct H as ((k,e'),(E,H)). - rewrite fold_keys_In in H. - apply InA_alt. exists k. now split. -Qed. - -Variable f : key -> option elt -> option elt' -> option elt''. - -Definition merge m m' : t elt'' := - fold_keys (fun k => f k (find k m) (find k m')) (domains m m'). - -Lemma merge_NoDup m (Hm:NoDupA (@eqk elt) m) m' (Hm':NoDupA (@eqk elt') m') : - NoDupA (@eqk elt'') (merge m m'). -Proof. - now apply fold_keys_NoDup, domains_NoDup. -Qed. - -Lemma merge_spec1 m (Hm:NoDupA eqk m) m' (Hm':NoDupA eqk m') x : - In x m \/ In x m' -> - exists y:key, X.eq y x /\ - find x (merge m m') = f y (find x m) (find x m'). -Proof. - assert (Hmm' : NoDupA eqk (merge m m')) by now apply merge_NoDup. - rewrite <- domains_InA; trivial. - rewrite InA_alt. intros (y,(Hy,H)). - exists y; split; [easy|]. - rewrite (find_eq Hm Hy), (find_eq Hm' Hy). - destruct (f y (find y m) (find y m')) eqn:F. - - apply find_spec; trivial. - red. apply InA_alt. exists (y,e). split. now split. - unfold merge. apply fold_keys_In. now split. - - destruct (find x (merge m m')) eqn:F'; trivial. - rewrite <- F; clear F. symmetry. - apply find_spec in F'; trivial. - red in F'. rewrite InA_alt in F'. - destruct F' as ((y',e'),(E,F')). - unfold merge in F'; rewrite fold_keys_In in F'. - destruct F' as (H',F'). - compute in E; destruct E as (Hy',<-). - replace y with y'; trivial. - apply (@NoDupA_unique_repr (domains m m')); auto. - now apply domains_NoDup. - now transitivity x. -Qed. - -Lemma merge_spec2 m (Hm:NoDupA eqk m) m' (Hm':NoDupA eqk m') x : - In x (merge m m') -> In x m \/ In x m'. -Proof. - rewrite <- domains_InA; trivial. - intros (e,H). red in H. rewrite InA_alt in H. destruct H as ((k,e'),(E,H)). - unfold merge in H; rewrite fold_keys_In in H. destruct H as (H,_). - apply InA_alt. exists k. split; trivial. now destruct E. -Qed. - -End Elt3. -End Raw. - - -Module Make (X: DecidableType) <: WS with Module E:=X. - Module Raw := Raw X. - - Module E := X. - Definition key := E.t. - Definition eq_key {elt} := @Raw.PX.eqk elt. - Definition eq_key_elt {elt} := @Raw.PX.eqke elt. - - Record t_ (elt:Type) := Mk - {this :> Raw.t elt; - nodup : NoDupA Raw.PX.eqk this}. - Definition t := t_. - - Definition empty {elt} : t elt := Mk (Raw.empty_NoDup elt). - -Section Elt. - Variable elt elt' elt'':Type. - Implicit Types m : t elt. - Implicit Types x y : key. - Implicit Types e : elt. - - Definition find x m : option elt := Raw.find x m.(this). - Definition mem x m : bool := Raw.mem x m.(this). - Definition is_empty m : bool := Raw.is_empty m.(this). - Definition add x e m : t elt := Mk (Raw.add_NoDup m.(nodup) x e). - Definition remove x m : t elt := Mk (Raw.remove_NoDup m.(nodup) x). - Definition map f m : t elt' := Mk (Raw.map_NoDup m.(nodup) f). - Definition mapi (f:key->elt->elt') m : t elt' := - Mk (Raw.mapi_NoDup m.(nodup) f). - Definition merge f m (m':t elt') : t elt'' := - Mk (Raw.merge_NoDup f m.(nodup) m'.(nodup)). - Definition bindings m : list (key*elt) := Raw.bindings m.(this). - Definition cardinal m := length m.(this). - Definition fold {A}(f:key->elt->A->A) m (i:A) : A := Raw.fold f m.(this) i. - Definition equal cmp m m' : bool := Raw.equal cmp m.(this) m'.(this). - Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e m.(this). - Definition In x m : Prop := Raw.PX.In x m.(this). - - Definition Equal m m' := forall y, find y m = find y m'. - Definition Equiv (eq_elt:elt->elt->Prop) m m' := - (forall k, In k m <-> In k m') /\ - (forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e'). - Definition Equivb cmp m m' : Prop := Raw.Equivb cmp m.(this) m'.(this). - - Instance MapsTo_compat : - Proper (E.eq==>Logic.eq==>Logic.eq==>iff) MapsTo. - Proof. - intros x x' Hx e e' <- m m' <-. unfold MapsTo. now rewrite Hx. - Qed. - - Lemma find_spec m : forall x e, find x m = Some e <-> MapsTo x e m. - Proof. exact (Raw.find_spec m.(nodup)). Qed. - - Lemma mem_spec m : forall x, mem x m = true <-> In x m. - Proof. exact (Raw.mem_spec m.(nodup)). Qed. - - Lemma empty_spec : forall x, find x empty = None. - Proof. exact (Raw.empty_spec _). Qed. - - Lemma is_empty_spec m : is_empty m = true <-> (forall x, find x m = None). - Proof. exact (Raw.is_empty_spec m.(this)). Qed. - - Lemma add_spec1 m : forall x e, find x (add x e m) = Some e. - Proof. exact (Raw.add_spec1 m.(this)). Qed. - Lemma add_spec2 m : forall x y e, ~E.eq x y -> find y (add x e m) = find y m. - Proof. exact (Raw.add_spec2 m.(this)). Qed. - - Lemma remove_spec1 m : forall x, find x (remove x m) = None. - Proof. exact (Raw.remove_spec1 m.(nodup)). Qed. - Lemma remove_spec2 m : forall x y, ~E.eq x y -> find y (remove x m) = find y m. - Proof. exact (Raw.remove_spec2 m.(nodup)). Qed. - - Lemma bindings_spec1 m : forall x e, - InA eq_key_elt (x,e) (bindings m) <-> MapsTo x e m. - Proof. exact (Raw.bindings_spec1 m.(this)). Qed. - Lemma bindings_spec2w m : NoDupA eq_key (bindings m). - Proof. exact (Raw.bindings_spec2w m.(nodup)). Qed. - - Lemma cardinal_spec m : cardinal m = length (bindings m). - Proof. reflexivity. Qed. - - Lemma fold_spec m : forall (A : Type) (i : A) (f : key -> elt -> A -> A), - fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (bindings m) i. - Proof. exact (Raw.fold_spec m.(this)). Qed. - - Lemma equal_spec m m' : forall cmp, equal cmp m m' = true <-> Equivb cmp m m'. - Proof. exact (Raw.equal_spec m.(nodup) m'.(nodup)). Qed. - -End Elt. - - Lemma map_spec {elt elt'} (f:elt->elt') m : - forall x, find x (map f m) = option_map f (find x m). - Proof. exact (Raw.map_spec f m.(this)). Qed. - - Lemma mapi_spec {elt elt'} (f:key->elt->elt') m : - forall x, exists y, - E.eq y x /\ find x (mapi f m) = option_map (f y) (find x m). - Proof. exact (Raw.mapi_spec f m.(this)). Qed. - - Lemma merge_spec1 {elt elt' elt''} - (f:key->option elt->option elt'->option elt'') m m' : - forall x, - In x m \/ In x m' -> - exists y, E.eq y x /\ find x (merge f m m') = f y (find x m) (find x m'). - Proof. exact (Raw.merge_spec1 f m.(nodup) m'.(nodup)). Qed. - - Lemma merge_spec2 {elt elt' elt''} - (f:key->option elt->option elt'->option elt'') m m' : - forall x, - In x (merge f m m') -> In x m \/ In x m'. - Proof. exact (Raw.merge_spec2 m.(nodup) m'.(nodup)). Qed. - -End Make. diff --git a/theories/MMaps/MMaps.v b/theories/MMaps/MMaps.v deleted file mode 100644 index 054d0722..00000000 --- a/theories/MMaps/MMaps.v +++ /dev/null @@ -1,16 +0,0 @@ -(***********************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) -(* \VV/ *************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(***********************************************************************) - - -Require Export Orders OrdersEx OrdersAlt. -Require Export Equalities. -Require Export MMapInterface. -Require Export MMapFacts. -Require Export MMapWeakList. -Require Export MMapList. -Require Export MMapPositive. diff --git a/theories/MMaps/vo.itarget b/theories/MMaps/vo.itarget deleted file mode 100644 index a7bbd266..00000000 --- a/theories/MMaps/vo.itarget +++ /dev/null @@ -1,7 +0,0 @@ -MMapInterface.vo -MMapFacts.vo -MMapWeakList.vo -MMapList.vo -MMapPositive.vo -MMaps.vo -MMapAVL.vo
\ No newline at end of file diff --git a/theories/NArith/BinNat.v b/theories/NArith/BinNat.v index 641ec02f..4c398573 100644 --- a/theories/NArith/BinNat.v +++ b/theories/NArith/BinNat.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/NArith/BinNatDef.v b/theories/NArith/BinNatDef.v index 9de2e7e1..4df4242e 100644 --- a/theories/NArith/BinNatDef.v +++ b/theories/NArith/BinNatDef.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/NArith/NArith.v b/theories/NArith/NArith.v index 43614543..12d3ad2f 100644 --- a/theories/NArith/NArith.v +++ b/theories/NArith/NArith.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/NArith/Ndec.v b/theories/NArith/Ndec.v index 5b1815bd..37fd9dde 100644 --- a/theories/NArith/Ndec.v +++ b/theories/NArith/Ndec.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/NArith/Ndigits.v b/theories/NArith/Ndigits.v index 55ef451e..a1e51c9d 100644 --- a/theories/NArith/Ndigits.v +++ b/theories/NArith/Ndigits.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/NArith/Ndist.v b/theories/NArith/Ndist.v index 5467f9cb..0ab8be58 100644 --- a/theories/NArith/Ndist.v +++ b/theories/NArith/Ndist.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/NArith/Ndiv_def.v b/theories/NArith/Ndiv_def.v index 5ae388e3..4bdf6529 100644 --- a/theories/NArith/Ndiv_def.v +++ b/theories/NArith/Ndiv_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/NArith/Ngcd_def.v b/theories/NArith/Ngcd_def.v index 1750ffeb..277fd26f 100644 --- a/theories/NArith/Ngcd_def.v +++ b/theories/NArith/Ngcd_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/NArith/Nnat.v b/theories/NArith/Nnat.v index 0dcaa71d..4e007878 100644 --- a/theories/NArith/Nnat.v +++ b/theories/NArith/Nnat.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/NArith/Nsqrt_def.v b/theories/NArith/Nsqrt_def.v index da7829a9..dd44b6e2 100644 --- a/theories/NArith/Nsqrt_def.v +++ b/theories/NArith/Nsqrt_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/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 *) -(* <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/Numbers/BinNums.v b/theories/Numbers/BinNums.v index 1dd5d82a..e4f3cd6c 100644 --- a/theories/Numbers/BinNums.v +++ b/theories/Numbers/BinNums.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/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v index 8b84a484..3312161a 100644 --- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v +++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.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/Numbers/Cyclic/Abstract/NZCyclic.v b/theories/Numbers/Cyclic/Abstract/NZCyclic.v index 8adeda37..df9b8339 100644 --- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v +++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.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/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v index a7c28862..407bcca4 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.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/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v index e68cd033..e94a891d 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.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/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v index e137349e..4ebe8fac 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.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/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v index cd55f9d8..09d7329b 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.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/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v index 6a1d741e..195527dd 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.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/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v index ff9f50a5..f65b47c8 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.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/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v index 537f557d..b9901390 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.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/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v index ab8c8617..d07ce301 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.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/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v index c1f314e9..abd567a8 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.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/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index aca57216..0e58b815 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.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/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v index f5e936cf..ff4b998e 100644 --- a/theories/Numbers/Cyclic/Int31/Int31.v +++ b/theories/Numbers/Cyclic/Int31/Int31.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/Numbers/Cyclic/Int31/Ring31.v b/theories/Numbers/Cyclic/Int31/Ring31.v index 4fde3f53..215b8bd5 100644 --- a/theories/Numbers/Cyclic/Int31/Ring31.v +++ b/theories/Numbers/Cyclic/Int31/Ring31.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/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v index b93b4eb3..c115a831 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.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/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v index ec8801c4..f7fdc179 100644 --- a/theories/Numbers/Integer/Abstract/ZAdd.v +++ b/theories/Numbers/Integer/Abstract/ZAdd.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/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v index e341ea8a..6bf5e9d4 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 1999-2015 *) +(* <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/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v index 2a9fa539..ad10e65f 100644 --- a/theories/Numbers/Integer/Abstract/ZAxioms.v +++ b/theories/Numbers/Integer/Abstract/ZAxioms.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/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v index 6634eab1..9b1b30f8 100644 --- a/theories/Numbers/Integer/Abstract/ZBase.v +++ b/theories/Numbers/Integer/Abstract/ZBase.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/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v index 9dd0ec0e..c919e121 100644 --- a/theories/Numbers/Integer/Abstract/ZBits.v +++ b/theories/Numbers/Integer/Abstract/ZBits.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/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v index d0df8fb4..278e1bcf 100644 --- a/theories/Numbers/Integer/Abstract/ZDivEucl.v +++ b/theories/Numbers/Integer/Abstract/ZDivEucl.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 *) @@ -391,7 +391,7 @@ rewrite <- (add_0_r (b*(a/b))) at 2. apply add_cancel_l. Qed. -(** Some additionnal inequalities about div. *) +(** Some additional inequalities about div. *) Theorem div_lt_upper_bound: forall a b q, 0<b -> 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 *) -(* <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 *) @@ -436,7 +436,7 @@ rewrite <- (add_0_r (b*(a/b))) at 2. apply add_cancel_l. Qed. -(** Some additionnal inequalities about div. *) +(** Some additional inequalities about div. *) Theorem div_lt_upper_bound: forall a b q, 0<b -> 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 *) -(* <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 *) @@ -404,7 +404,7 @@ Proof. intros. rewrite rem_eq by order. rewrite sub_move_r; nzsimpl; tauto. Qed. -(** Some additionnal inequalities about quot. *) +(** Some additional inequalities about quot. *) Theorem quot_lt_upper_bound: forall a b q, 0<=a -> 0<b -> 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 *) -(* <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/Numbers/Integer/Abstract/ZLcm.v b/theories/Numbers/Integer/Abstract/ZLcm.v index 9a1768eb..ef33308c 100644 --- a/theories/Numbers/Integer/Abstract/ZLcm.v +++ b/theories/Numbers/Integer/Abstract/ZLcm.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/Numbers/Integer/Abstract/ZLt.v b/theories/Numbers/Integer/Abstract/ZLt.v index 6d0cdb01..0c92918d 100644 --- a/theories/Numbers/Integer/Abstract/ZLt.v +++ b/theories/Numbers/Integer/Abstract/ZLt.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/Numbers/Integer/Abstract/ZMaxMin.v b/theories/Numbers/Integer/Abstract/ZMaxMin.v index 07c78ead..24a47f00 100644 --- a/theories/Numbers/Integer/Abstract/ZMaxMin.v +++ b/theories/Numbers/Integer/Abstract/ZMaxMin.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/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v index 2d78d8f3..830c0a7d 100644 --- a/theories/Numbers/Integer/Abstract/ZMul.v +++ b/theories/Numbers/Integer/Abstract/ZMul.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/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v index 487aaae1..320c8f35 100644 --- a/theories/Numbers/Integer/Abstract/ZMulOrder.v +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.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/Numbers/Integer/Abstract/ZParity.v b/theories/Numbers/Integer/Abstract/ZParity.v index 195a0277..952d8e9c 100644 --- a/theories/Numbers/Integer/Abstract/ZParity.v +++ b/theories/Numbers/Integer/Abstract/ZParity.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/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v index 87de2c78..02b8501c 100644 --- a/theories/Numbers/Integer/Abstract/ZPow.v +++ b/theories/Numbers/Integer/Abstract/ZPow.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/Numbers/Integer/Abstract/ZProperties.v b/theories/Numbers/Integer/Abstract/ZProperties.v index 5cfeeb21..1dec3c58 100644 --- a/theories/Numbers/Integer/Abstract/ZProperties.v +++ b/theories/Numbers/Integer/Abstract/ZProperties.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/Numbers/Integer/Abstract/ZSgnAbs.v b/theories/Numbers/Integer/Abstract/ZSgnAbs.v index b379853e..a10552ab 100644 --- a/theories/Numbers/Integer/Abstract/ZSgnAbs.v +++ b/theories/Numbers/Integer/Abstract/ZSgnAbs.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/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v index b28bc40f..ec495d09 100644 --- a/theories/Numbers/Integer/BigZ/BigZ.v +++ b/theories/Numbers/Integer/BigZ/BigZ.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/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index af4f1d93..8673b8a5 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.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/Numbers/Integer/Binary/ZBinary.v b/theories/Numbers/Integer/Binary/ZBinary.v index 04208106..eae8204d 100644 --- a/theories/Numbers/Integer/Binary/ZBinary.v +++ b/theories/Numbers/Integer/Binary/ZBinary.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/Numbers/Integer/NatPairs/ZNatPairs.v b/theories/Numbers/Integer/NatPairs/ZNatPairs.v index 02f02fbc..0aaf3365 100644 --- a/theories/Numbers/Integer/NatPairs/ZNatPairs.v +++ b/theories/Numbers/Integer/NatPairs/ZNatPairs.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/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v index 30ac32b5..a360327a 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSig.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSig.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/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v index c9dc687c..32410d1d 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.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/Numbers/NaryFunctions.v b/theories/Numbers/NaryFunctions.v index 376620dd..eaac7d69 100644 --- a/theories/Numbers/NaryFunctions.v +++ b/theories/Numbers/NaryFunctions.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/Numbers/NatInt/NZAdd.v b/theories/Numbers/NatInt/NZAdd.v index 501583ae..8cc52940 100644 --- a/theories/Numbers/NatInt/NZAdd.v +++ b/theories/Numbers/NatInt/NZAdd.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/Numbers/NatInt/NZAddOrder.v b/theories/Numbers/NatInt/NZAddOrder.v index 619a6634..705f2910 100644 --- a/theories/Numbers/NatInt/NZAddOrder.v +++ b/theories/Numbers/NatInt/NZAddOrder.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/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index c88341fa..5a7f2072 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.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/Numbers/NatInt/NZBase.v b/theories/Numbers/NatInt/NZBase.v index c0afa098..cf4ad354 100644 --- a/theories/Numbers/NatInt/NZBase.v +++ b/theories/Numbers/NatInt/NZBase.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/Numbers/NatInt/NZBits.v b/theories/Numbers/NatInt/NZBits.v index 1c118597..209bd947 100644 --- a/theories/Numbers/NatInt/NZBits.v +++ b/theories/Numbers/NatInt/NZBits.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/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v index 4a127216..b2c0be6f 100644 --- a/theories/Numbers/NatInt/NZDiv.v +++ b/theories/Numbers/NatInt/NZDiv.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 *) @@ -307,7 +307,7 @@ rewrite <- (add_0_r (b*(a/b))) at 2. apply add_cancel_l. Qed. -(** Some additionnal inequalities about div. *) +(** Some additional inequalities about div. *) Theorem div_lt_upper_bound: forall a b q, 0<=a -> 0<b -> 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 *) -(* <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/Numbers/NatInt/NZGcd.v b/theories/Numbers/NatInt/NZGcd.v index 42bee315..1d367294 100644 --- a/theories/Numbers/NatInt/NZGcd.v +++ b/theories/Numbers/NatInt/NZGcd.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/Numbers/NatInt/NZLog.v b/theories/Numbers/NatInt/NZLog.v index 9cd1f877..3dfa2eef 100644 --- a/theories/Numbers/NatInt/NZLog.v +++ b/theories/Numbers/NatInt/NZLog.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/Numbers/NatInt/NZMul.v b/theories/Numbers/NatInt/NZMul.v index 89ace4de..36cd0827 100644 --- a/theories/Numbers/NatInt/NZMul.v +++ b/theories/Numbers/NatInt/NZMul.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/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v index e79e50a9..8569fc56 100644 --- a/theories/Numbers/NatInt/NZMulOrder.v +++ b/theories/Numbers/NatInt/NZMulOrder.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/Numbers/NatInt/NZOrder.v b/theories/Numbers/NatInt/NZOrder.v index c1e83529..0b9d6598 100644 --- a/theories/Numbers/NatInt/NZOrder.v +++ b/theories/Numbers/NatInt/NZOrder.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/Numbers/NatInt/NZParity.v b/theories/Numbers/NatInt/NZParity.v index 6b9a680a..4a1352cc 100644 --- a/theories/Numbers/NatInt/NZParity.v +++ b/theories/Numbers/NatInt/NZParity.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/Numbers/NatInt/NZPow.v b/theories/Numbers/NatInt/NZPow.v index 38452119..261febbc 100644 --- a/theories/Numbers/NatInt/NZPow.v +++ b/theories/Numbers/NatInt/NZPow.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/Numbers/NatInt/NZProperties.v b/theories/Numbers/NatInt/NZProperties.v index 0f3a5caf..79a9d733 100644 --- a/theories/Numbers/NatInt/NZProperties.v +++ b/theories/Numbers/NatInt/NZProperties.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/Numbers/NatInt/NZSqrt.v b/theories/Numbers/NatInt/NZSqrt.v index 894c0806..7f9bb9c2 100644 --- a/theories/Numbers/NatInt/NZSqrt.v +++ b/theories/Numbers/NatInt/NZSqrt.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/Numbers/Natural/Abstract/NAdd.v b/theories/Numbers/Natural/Abstract/NAdd.v index 638cfc7e..eecec3ac 100644 --- a/theories/Numbers/Natural/Abstract/NAdd.v +++ b/theories/Numbers/Natural/Abstract/NAdd.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/Numbers/Natural/Abstract/NAddOrder.v b/theories/Numbers/Natural/Abstract/NAddOrder.v index 144bce72..9d68a006 100644 --- a/theories/Numbers/Natural/Abstract/NAddOrder.v +++ b/theories/Numbers/Natural/Abstract/NAddOrder.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/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v index d300f857..25e285d5 100644 --- a/theories/Numbers/Natural/Abstract/NAxioms.v +++ b/theories/Numbers/Natural/Abstract/NAxioms.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/Numbers/Natural/Abstract/NBase.v b/theories/Numbers/Natural/Abstract/NBase.v index 40453214..f949a0f6 100644 --- a/theories/Numbers/Natural/Abstract/NBase.v +++ b/theories/Numbers/Natural/Abstract/NBase.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/Numbers/Natural/Abstract/NBits.v b/theories/Numbers/Natural/Abstract/NBits.v index 6f8a8fe5..3dd603e2 100644 --- a/theories/Numbers/Natural/Abstract/NBits.v +++ b/theories/Numbers/Natural/Abstract/NBits.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/Numbers/Natural/Abstract/NDefOps.v b/theories/Numbers/Natural/Abstract/NDefOps.v index 892b9266..b3a53617 100644 --- a/theories/Numbers/Natural/Abstract/NDefOps.v +++ b/theories/Numbers/Natural/Abstract/NDefOps.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/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v index fb68c139..84e1219e 100644 --- a/theories/Numbers/Natural/Abstract/NDiv.v +++ b/theories/Numbers/Natural/Abstract/NDiv.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 *) @@ -137,7 +137,7 @@ Proof. intros; apply mul_succ_div_gt; auto'. Qed. Lemma div_exact : forall a b, b~=0 -> (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 *) -(* <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/Numbers/Natural/Abstract/NIso.v b/theories/Numbers/Natural/Abstract/NIso.v index c296315d..e6cac0ba 100644 --- a/theories/Numbers/Natural/Abstract/NIso.v +++ b/theories/Numbers/Natural/Abstract/NIso.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/Numbers/Natural/Abstract/NLcm.v b/theories/Numbers/Natural/Abstract/NLcm.v index 0fe8e105..3d749799 100644 --- a/theories/Numbers/Natural/Abstract/NLcm.v +++ b/theories/Numbers/Natural/Abstract/NLcm.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/Numbers/Natural/Abstract/NLog.v b/theories/Numbers/Natural/Abstract/NLog.v index 605c0aad..0c9da29b 100644 --- a/theories/Numbers/Natural/Abstract/NLog.v +++ b/theories/Numbers/Natural/Abstract/NLog.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/Numbers/Natural/Abstract/NMaxMin.v b/theories/Numbers/Natural/Abstract/NMaxMin.v index e0710561..b4c9db91 100644 --- a/theories/Numbers/Natural/Abstract/NMaxMin.v +++ b/theories/Numbers/Natural/Abstract/NMaxMin.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/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v index c41275d2..a4b52396 100644 --- a/theories/Numbers/Natural/Abstract/NMulOrder.v +++ b/theories/Numbers/Natural/Abstract/NMulOrder.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/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v index 90053a73..60e955f5 100644 --- a/theories/Numbers/Natural/Abstract/NOrder.v +++ b/theories/Numbers/Natural/Abstract/NOrder.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/Numbers/Natural/Abstract/NParity.v b/theories/Numbers/Natural/Abstract/NParity.v index b3526c9a..e1f573f6 100644 --- a/theories/Numbers/Natural/Abstract/NParity.v +++ b/theories/Numbers/Natural/Abstract/NParity.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 *) @@ -8,7 +8,7 @@ Require Import Bool NSub NZParity. -(** Some additionnal properties of [even], [odd]. *) +(** Some additional properties of [even], [odd]. *) Module Type NParityProp (Import N : NAxiomsSig')(Import NP : NSubProp N). diff --git a/theories/Numbers/Natural/Abstract/NPow.v b/theories/Numbers/Natural/Abstract/NPow.v index 9cc23004..c9b2177f 100644 --- a/theories/Numbers/Natural/Abstract/NPow.v +++ b/theories/Numbers/Natural/Abstract/NPow.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/Numbers/Natural/Abstract/NProperties.v b/theories/Numbers/Natural/Abstract/NProperties.v index cb3501d4..819a5be9 100644 --- a/theories/Numbers/Natural/Abstract/NProperties.v +++ b/theories/Numbers/Natural/Abstract/NProperties.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/Numbers/Natural/Abstract/NSqrt.v b/theories/Numbers/Natural/Abstract/NSqrt.v index 8dc66884..4ae41407 100644 --- a/theories/Numbers/Natural/Abstract/NSqrt.v +++ b/theories/Numbers/Natural/Abstract/NSqrt.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/Numbers/Natural/Abstract/NStrongRec.v b/theories/Numbers/Natural/Abstract/NStrongRec.v index 896ffc18..fa3a5351 100644 --- a/theories/Numbers/Natural/Abstract/NStrongRec.v +++ b/theories/Numbers/Natural/Abstract/NStrongRec.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/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v index 18ebe77b..21f12037 100644 --- a/theories/Numbers/Natural/Abstract/NSub.v +++ b/theories/Numbers/Natural/Abstract/NSub.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/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v index f7f4347b..29a1145e 100644 --- a/theories/Numbers/Natural/BigN/BigN.v +++ b/theories/Numbers/Natural/BigN/BigN.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/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v index bdcdd5ca..98949736 100644 --- a/theories/Numbers/Natural/BigN/NMake.v +++ b/theories/Numbers/Natural/BigN/NMake.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/Numbers/Natural/BigN/NMake_gen.ml b/theories/Numbers/Natural/BigN/NMake_gen.ml index 6de77e0a..601fa108 100644 --- a/theories/Numbers/Natural/BigN/NMake_gen.ml +++ b/theories/Numbers/Natural/BigN/NMake_gen.ml @@ -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/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index 8fe9ea92..18d0262c 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.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/Numbers/Natural/Binary/NBinary.v b/theories/Numbers/Natural/Binary/NBinary.v index d54bedd1..037b10d9 100644 --- a/theories/Numbers/Natural/Binary/NBinary.v +++ b/theories/Numbers/Natural/Binary/NBinary.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/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 96eb7b35..58b1b018 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.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/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v index 1049c156..258e0315 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSig.v +++ b/theories/Numbers/Natural/SpecViaZ/NSig.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/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 11569b3f..355da4cc 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.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/Numbers/NumPrelude.v b/theories/Numbers/NumPrelude.v index f67e0e96..f323aaeb 100644 --- a/theories/Numbers/NumPrelude.v +++ b/theories/Numbers/NumPrelude.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/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v index b64cfb64..fe38ea4f 100644 --- a/theories/Numbers/Rational/BigQ/BigQ.v +++ b/theories/Numbers/Rational/BigQ/BigQ.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/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v index c11e07fa..4ac36425 100644 --- a/theories/Numbers/Rational/BigQ/QMake.v +++ b/theories/Numbers/Rational/BigQ/QMake.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/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v index 5f831bfc..a40d9405 100644 --- a/theories/Numbers/Rational/SpecViaQ/QSig.v +++ b/theories/Numbers/Rational/SpecViaQ/QSig.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/PArith/BinPos.v b/theories/PArith/BinPos.v index 921e2d67..0ccfad7b 100644 --- a/theories/PArith/BinPos.v +++ b/theories/PArith/BinPos.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/PArith/BinPosDef.v b/theories/PArith/BinPosDef.v index fefd1d76..74a292c6 100644 --- a/theories/PArith/BinPosDef.v +++ b/theories/PArith/BinPosDef.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/PArith/PArith.v b/theories/PArith/PArith.v index 93352c6b..6ee8d6d7 100644 --- a/theories/PArith/PArith.v +++ b/theories/PArith/PArith.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/PArith/POrderedType.v b/theories/PArith/POrderedType.v index 92483ac8..7619f639 100644 --- a/theories/PArith/POrderedType.v +++ b/theories/PArith/POrderedType.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/PArith/Pnat.v b/theories/PArith/Pnat.v index e529a8c4..9c2608f4 100644 --- a/theories/PArith/Pnat.v +++ b/theories/PArith/Pnat.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/Program/Basics.v b/theories/Program/Basics.v index e5be0ca9..644b9b5a 100644 --- a/theories/Program/Basics.v +++ b/theories/Program/Basics.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/Program/Combinators.v b/theories/Program/Combinators.v index e246041b..772018aa 100644 --- a/theories/Program/Combinators.v +++ b/theories/Program/Combinators.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/Program/Equality.v b/theories/Program/Equality.v index ae6fe7dd..27e1ca84 100644 --- a/theories/Program/Equality.v +++ b/theories/Program/Equality.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/Program/Program.v b/theories/Program/Program.v index 5af6f4d7..4a6f2786 100644 --- a/theories/Program/Program.v +++ b/theories/Program/Program.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/Program/Subset.v b/theories/Program/Subset.v index 50b89b5c..c8f37318 100644 --- a/theories/Program/Subset.v +++ b/theories/Program/Subset.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 *) @@ -82,7 +82,7 @@ Qed. Definition match_eq (A B : Type) (x : A) (fn : {y : A | y = x} -> B) : B := fn (exist _ x eq_refl). -(* This is what we want to be able to do: replace the originaly matched object by a new, +(* This is what we want to be able to do: replace the originally matched object by a new, propositionally equal one. If [fn] works on [x] it should work on any [y | y = x]. *) Lemma match_eq_rewrite : forall (A B : Type) (x : A) (fn : {y : A | y = x} -> B) diff --git a/theories/Program/Syntax.v b/theories/Program/Syntax.v index 892305b4..2fccf624 100644 --- a/theories/Program/Syntax.v +++ b/theories/Program/Syntax.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/Program/Tactics.v b/theories/Program/Tactics.v index c1d958b9..66ca3e57 100644 --- a/theories/Program/Tactics.v +++ b/theories/Program/Tactics.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/Program/Utils.v b/theories/Program/Utils.v index 65fe8780..396c9618 100644 --- a/theories/Program/Utils.v +++ b/theories/Program/Utils.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/Program/Wf.v b/theories/Program/Wf.v index d89919b0..154200d7 100644 --- a/theories/Program/Wf.v +++ b/theories/Program/Wf.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/QArith/QArith.v b/theories/QArith/QArith.v index c32fb950..5ad08b65 100644 --- a/theories/QArith/QArith.v +++ b/theories/QArith/QArith.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/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 7f19b4ce..62304876 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_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/QArith/QOrderedType.v b/theories/QArith/QOrderedType.v index fa0b9209..25e98f0b 100644 --- a/theories/QArith/QOrderedType.v +++ b/theories/QArith/QOrderedType.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/QArith/Qabs.v b/theories/QArith/Qabs.v index f77f409e..c60d0451 100644 --- a/theories/QArith/Qabs.v +++ b/theories/QArith/Qabs.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/QArith/Qcanon.v b/theories/QArith/Qcanon.v index f7f83bf0..d966b050 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.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/QArith/Qfield.v b/theories/QArith/Qfield.v index 083e40ae..bbaf6027 100644 --- a/theories/QArith/Qfield.v +++ b/theories/QArith/Qfield.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/QArith/Qminmax.v b/theories/QArith/Qminmax.v index 0fd05325..86584d9e 100644 --- a/theories/QArith/Qminmax.v +++ b/theories/QArith/Qminmax.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/QArith/Qpower.v b/theories/QArith/Qpower.v index 8bd643aa..af89d300 100644 --- a/theories/QArith/Qpower.v +++ b/theories/QArith/Qpower.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/QArith/Qreals.v b/theories/QArith/Qreals.v index add43b96..048e409c 100644 --- a/theories/QArith/Qreals.v +++ b/theories/QArith/Qreals.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/QArith/Qreduction.v b/theories/QArith/Qreduction.v index 1d304964..c50c38b2 100644 --- a/theories/QArith/Qreduction.v +++ b/theories/QArith/Qreduction.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/QArith/Qring.v b/theories/QArith/Qring.v index 78c464ae..da11c2b1 100644 --- a/theories/QArith/Qring.v +++ b/theories/QArith/Qring.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/QArith/Qround.v b/theories/QArith/Qround.v index 964a4bae..0ed6d557 100644 --- a/theories/QArith/Qround.v +++ b/theories/QArith/Qround.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/Reals/Alembert.v b/theories/Reals/Alembert.v index 011328ec..a98d529f 100644 --- a/theories/Reals/Alembert.v +++ b/theories/Reals/Alembert.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/Reals/AltSeries.v b/theories/Reals/AltSeries.v index 3e99c989..c3ab8edc 100644 --- a/theories/Reals/AltSeries.v +++ b/theories/Reals/AltSeries.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/Reals/ArithProp.v b/theories/Reals/ArithProp.v index c4e410ed..6fca9c8a 100644 --- a/theories/Reals/ArithProp.v +++ b/theories/Reals/ArithProp.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/Reals/Binomial.v b/theories/Reals/Binomial.v index d48f42fc..f878abfa 100644 --- a/theories/Reals/Binomial.v +++ b/theories/Reals/Binomial.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/Reals/Cauchy_prod.v b/theories/Reals/Cauchy_prod.v index 28de1186..5cf6f17d 100644 --- a/theories/Reals/Cauchy_prod.v +++ b/theories/Reals/Cauchy_prod.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/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v index 49ba9a6e..b14d807d 100644 --- a/theories/Reals/Cos_plus.v +++ b/theories/Reals/Cos_plus.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/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v index 6d30319c..f5fcac47 100644 --- a/theories/Reals/Cos_rel.v +++ b/theories/Reals/Cos_rel.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/Reals/DiscrR.v b/theories/Reals/DiscrR.v index 75fd4c0a..4e2a7c3c 100644 --- a/theories/Reals/DiscrR.v +++ b/theories/Reals/DiscrR.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/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index be96b94e..569518f7 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.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/Reals/Integration.v b/theories/Reals/Integration.v index 222d106f..e3760e01 100644 --- a/theories/Reals/Integration.v +++ b/theories/Reals/Integration.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/Reals/MVT.v b/theories/Reals/MVT.v index 59976957..26c51583 100644 --- a/theories/Reals/MVT.v +++ b/theories/Reals/MVT.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/Reals/Machin.v b/theories/Reals/Machin.v index 1a94f6a8..19db476f 100644 --- a/theories/Reals/Machin.v +++ b/theories/Reals/Machin.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/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v index 832e7adc..ed5ae90c 100644 --- a/theories/Reals/NewtonInt.v +++ b/theories/Reals/NewtonInt.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/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v index 94b881cc..03ac6582 100644 --- a/theories/Reals/PSeries_reg.v +++ b/theories/Reals/PSeries_reg.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/Reals/PartSum.v b/theories/Reals/PartSum.v index b710c75c..37d54a6d 100644 --- a/theories/Reals/PartSum.v +++ b/theories/Reals/PartSum.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/Reals/RIneq.v b/theories/Reals/RIneq.v index 8dca0197..f26bac2b 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.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/Reals/RList.v b/theories/Reals/RList.v index abf8a99d..924d5117 100644 --- a/theories/Reals/RList.v +++ b/theories/Reals/RList.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/Reals/ROrderedType.v b/theories/Reals/ROrderedType.v index 0531bd0a..f2dc7fd0 100644 --- a/theories/Reals/ROrderedType.v +++ b/theories/Reals/ROrderedType.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/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v index 57ee1d9a..b6d07283 100644 --- a/theories/Reals/R_Ifp.v +++ b/theories/Reals/R_Ifp.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/Reals/R_sqr.v b/theories/Reals/R_sqr.v index f1e2d6fa..445ffcb2 100644 --- a/theories/Reals/R_sqr.v +++ b/theories/Reals/R_sqr.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/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index 20319a2b..a6b1a26e 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.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/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v index 3cda675a..88ebb88b 100644 --- a/theories/Reals/Ranalysis.v +++ b/theories/Reals/Ranalysis.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/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v index 875eebbb..9e3abd81 100644 --- a/theories/Reals/Ranalysis1.v +++ b/theories/Reals/Ranalysis1.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/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v index eb646913..0254218c 100644 --- a/theories/Reals/Ranalysis2.v +++ b/theories/Reals/Ranalysis2.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/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v index 407f6410..4e88714d 100644 --- a/theories/Reals/Ranalysis3.v +++ b/theories/Reals/Ranalysis3.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/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v index ae2013f0..661bc8c7 100644 --- a/theories/Reals/Ranalysis4.v +++ b/theories/Reals/Ranalysis4.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/Reals/Ranalysis5.v b/theories/Reals/Ranalysis5.v index 27615c59..d172139f 100644 --- a/theories/Reals/Ranalysis5.v +++ b/theories/Reals/Ranalysis5.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/Reals/Ranalysis_reg.v b/theories/Reals/Ranalysis_reg.v index 4cf90886..2465f039 100644 --- a/theories/Reals/Ranalysis_reg.v +++ b/theories/Reals/Ranalysis_reg.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/Reals/Ratan.v b/theories/Reals/Ratan.v index cc45139d..e13ef1f2 100644 --- a/theories/Reals/Ratan.v +++ b/theories/Reals/Ratan.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/Reals/Raxioms.v b/theories/Reals/Raxioms.v index f545d3a0..9d55e4e6 100644 --- a/theories/Reals/Raxioms.v +++ b/theories/Reals/Raxioms.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/Reals/Rbase.v b/theories/Reals/Rbase.v index 7a879f45..e56ce28d 100644 --- a/theories/Reals/Rbase.v +++ b/theories/Reals/Rbase.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/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v index bb30c0ef..c889d734 100644 --- a/theories/Reals/Rbasic_fun.v +++ b/theories/Reals/Rbasic_fun.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/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v index 1766f377..3520c26c 100644 --- a/theories/Reals/Rcomplete.v +++ b/theories/Reals/Rcomplete.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/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v index 50eb59b2..f3f8f740 100644 --- a/theories/Reals/Rdefinitions.v +++ b/theories/Reals/Rdefinitions.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/Reals/Rderiv.v b/theories/Reals/Rderiv.v index 3a332d21..bd330ac9 100644 --- a/theories/Reals/Rderiv.v +++ b/theories/Reals/Rderiv.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/Reals/Reals.v b/theories/Reals/Reals.v index 9cb8a10b..8265f65a 100644 --- a/theories/Reals/Reals.v +++ b/theories/Reals/Reals.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/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v index 1c353803..0a49d498 100644 --- a/theories/Reals/Rfunctions.v +++ b/theories/Reals/Rfunctions.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/Reals/Rgeom.v b/theories/Reals/Rgeom.v index d930c5aa..7423ffce 100644 --- a/theories/Reals/Rgeom.v +++ b/theories/Reals/Rgeom.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/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index 856fff80..4c0466ac 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.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/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v index 1484ab2a..7885d697 100644 --- a/theories/Reals/RiemannInt_SF.v +++ b/theories/Reals/RiemannInt_SF.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/Reals/Rlimit.v b/theories/Reals/Rlimit.v index c8887dfb..e424a732 100644 --- a/theories/Reals/Rlimit.v +++ b/theories/Reals/Rlimit.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/Reals/Rlogic.v b/theories/Reals/Rlogic.v index 07792942..b9a9458c 100644 --- a/theories/Reals/Rlogic.v +++ b/theories/Reals/Rlogic.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/Reals/Rminmax.v b/theories/Reals/Rminmax.v index ba1fe90f..152988dc 100644 --- a/theories/Reals/Rminmax.v +++ b/theories/Reals/Rminmax.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/Reals/Rpow_def.v b/theories/Reals/Rpow_def.v index 1d697f3c..791718a4 100644 --- a/theories/Reals/Rpow_def.v +++ b/theories/Reals/Rpow_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/Reals/Rpower.v b/theories/Reals/Rpower.v index e30ea334..b3ce6fa3 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.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/Reals/Rprod.v b/theories/Reals/Rprod.v index 1ee9410f..883e4e1a 100644 --- a/theories/Reals/Rprod.v +++ b/theories/Reals/Rprod.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/Reals/Rseries.v b/theories/Reals/Rseries.v index fd16ea61..744fd664 100644 --- a/theories/Reals/Rseries.v +++ b/theories/Reals/Rseries.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/Reals/Rsigma.v b/theories/Reals/Rsigma.v index 458d1f8c..ced2b3da 100644 --- a/theories/Reals/Rsigma.v +++ b/theories/Reals/Rsigma.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/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v index b8ec8d3c..b3c9c744 100644 --- a/theories/Reals/Rsqrt_def.v +++ b/theories/Reals/Rsqrt_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/Reals/Rtopology.v b/theories/Reals/Rtopology.v index 72e4142b..df3b95be 100644 --- a/theories/Reals/Rtopology.v +++ b/theories/Reals/Rtopology.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/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index 44058358..ecef0d68 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.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/Reals/Rtrigo1.v b/theories/Reals/Rtrigo1.v index 9e485ec5..4d241863 100644 --- a/theories/Reals/Rtrigo1.v +++ b/theories/Reals/Rtrigo1.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/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v index 3d36cb34..a5092d22 100644 --- a/theories/Reals/Rtrigo_alt.v +++ b/theories/Reals/Rtrigo_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/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v index 281c152b..9ba14ee7 100644 --- a/theories/Reals/Rtrigo_calc.v +++ b/theories/Reals/Rtrigo_calc.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/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v index ef3e31f1..0d2a9a8b 100644 --- a/theories/Reals/Rtrigo_def.v +++ b/theories/Reals/Rtrigo_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/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index b921ee7b..f395f9ae 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.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/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v index 7845e6c4..eed612d9 100644 --- a/theories/Reals/Rtrigo_reg.v +++ b/theories/Reals/Rtrigo_reg.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/Reals/SeqProp.v b/theories/Reals/SeqProp.v index 9a6fb945..5a2a07c4 100644 --- a/theories/Reals/SeqProp.v +++ b/theories/Reals/SeqProp.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/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v index 25fe4848..1123e7ee 100644 --- a/theories/Reals/SeqSeries.v +++ b/theories/Reals/SeqSeries.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/Reals/SplitAbsolu.v b/theories/Reals/SplitAbsolu.v index 64f4f1c9..a78a6e19 100644 --- a/theories/Reals/SplitAbsolu.v +++ b/theories/Reals/SplitAbsolu.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/Reals/SplitRmult.v b/theories/Reals/SplitRmult.v index fec28518..074a7631 100644 --- a/theories/Reals/SplitRmult.v +++ b/theories/Reals/SplitRmult.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/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v index dd8738e1..10527442 100644 --- a/theories/Reals/Sqrt_reg.v +++ b/theories/Reals/Sqrt_reg.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/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v index 95d9cfa9..220cebea 100644 --- a/theories/Relations/Operators_Properties.v +++ b/theories/Relations/Operators_Properties.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/Relations/Relation_Definitions.v b/theories/Relations/Relation_Definitions.v index a187f955..b6005b9d 100644 --- a/theories/Relations/Relation_Definitions.v +++ b/theories/Relations/Relation_Definitions.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/Relations/Relation_Operators.v b/theories/Relations/Relation_Operators.v index 4e52017e..ffd682d6 100644 --- a/theories/Relations/Relation_Operators.v +++ b/theories/Relations/Relation_Operators.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/Relations/Relations.v b/theories/Relations/Relations.v index ce849a16..ce6bdbc6 100644 --- a/theories/Relations/Relations.v +++ b/theories/Relations/Relations.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/Setoids/Setoid.v b/theories/Setoids/Setoid.v index 75cffa7f..55b301c3 100644 --- a/theories/Setoids/Setoid.v +++ b/theories/Setoids/Setoid.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/Sets/Classical_sets.v b/theories/Sets/Classical_sets.v index aa2c144b..8a4bb9f4 100644 --- a/theories/Sets/Classical_sets.v +++ b/theories/Sets/Classical_sets.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/Sets/Constructive_sets.v b/theories/Sets/Constructive_sets.v index 193bec78..8d2344f9 100644 --- a/theories/Sets/Constructive_sets.v +++ b/theories/Sets/Constructive_sets.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/Sets/Cpo.v b/theories/Sets/Cpo.v index f2fac097..b5db9030 100644 --- a/theories/Sets/Cpo.v +++ b/theories/Sets/Cpo.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/Sets/Ensembles.v b/theories/Sets/Ensembles.v index c0cddbe1..8f579214 100644 --- a/theories/Sets/Ensembles.v +++ b/theories/Sets/Ensembles.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/Sets/Finite_sets.v b/theories/Sets/Finite_sets.v index 22cb3dae..f38dd6fd 100644 --- a/theories/Sets/Finite_sets.v +++ b/theories/Sets/Finite_sets.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/Sets/Finite_sets_facts.v b/theories/Sets/Finite_sets_facts.v index b1c12c7f..31cc11e1 100644 --- a/theories/Sets/Finite_sets_facts.v +++ b/theories/Sets/Finite_sets_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/Sets/Image.v b/theories/Sets/Image.v index 6cf4d250..34ea857d 100644 --- a/theories/Sets/Image.v +++ b/theories/Sets/Image.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/Sets/Infinite_sets.v b/theories/Sets/Infinite_sets.v index 5860f960..98eeb7d7 100644 --- a/theories/Sets/Infinite_sets.v +++ b/theories/Sets/Infinite_sets.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/Sets/Integers.v b/theories/Sets/Integers.v index 944e0dd1..057fc9b6 100644 --- a/theories/Sets/Integers.v +++ b/theories/Sets/Integers.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/Sets/Multiset.v b/theories/Sets/Multiset.v index 46dbe994..ec38b892 100644 --- a/theories/Sets/Multiset.v +++ b/theories/Sets/Multiset.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/Sets/Partial_Order.v b/theories/Sets/Partial_Order.v index 35d5f91a..3610ebce 100644 --- a/theories/Sets/Partial_Order.v +++ b/theories/Sets/Partial_Order.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/Sets/Permut.v b/theories/Sets/Permut.v index c9c1e5b7..adbde18e 100644 --- a/theories/Sets/Permut.v +++ b/theories/Sets/Permut.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/Sets/Powerset.v b/theories/Sets/Powerset.v index 587d48ab..d636e046 100644 --- a/theories/Sets/Powerset.v +++ b/theories/Sets/Powerset.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/Sets/Powerset_Classical_facts.v b/theories/Sets/Powerset_Classical_facts.v index 40fd5e67..09c90506 100644 --- a/theories/Sets/Powerset_Classical_facts.v +++ b/theories/Sets/Powerset_Classical_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/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v index e9347ce3..63e84199 100644 --- a/theories/Sets/Powerset_facts.v +++ b/theories/Sets/Powerset_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/Sets/Relations_1.v b/theories/Sets/Relations_1.v index c9148e00..de96fa56 100644 --- a/theories/Sets/Relations_1.v +++ b/theories/Sets/Relations_1.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/Sets/Relations_1_facts.v b/theories/Sets/Relations_1_facts.v index f650a50c..bf8a4261 100644 --- a/theories/Sets/Relations_1_facts.v +++ b/theories/Sets/Relations_1_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/Sets/Relations_2.v b/theories/Sets/Relations_2.v index ea48fd91..f1026e31 100644 --- a/theories/Sets/Relations_2.v +++ b/theories/Sets/Relations_2.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/Sets/Relations_2_facts.v b/theories/Sets/Relations_2_facts.v index e0543501..da93e922 100644 --- a/theories/Sets/Relations_2_facts.v +++ b/theories/Sets/Relations_2_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/Sets/Relations_3.v b/theories/Sets/Relations_3.v index de6770ee..92b29988 100644 --- a/theories/Sets/Relations_3.v +++ b/theories/Sets/Relations_3.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/Sets/Relations_3_facts.v b/theories/Sets/Relations_3_facts.v index 0180c7d4..89fb900c 100644 --- a/theories/Sets/Relations_3_facts.v +++ b/theories/Sets/Relations_3_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/Sets/Uniset.v b/theories/Sets/Uniset.v index 86ba903f..e297d97e 100644 --- a/theories/Sets/Uniset.v +++ b/theories/Sets/Uniset.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/Sorting/Heap.v b/theories/Sorting/Heap.v index 6313dbf6..20c6feb9 100644 --- a/theories/Sorting/Heap.v +++ b/theories/Sorting/Heap.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/Sorting/Mergesort.v b/theories/Sorting/Mergesort.v index 593b2e9b..4b967c16 100644 --- a/theories/Sorting/Mergesort.v +++ b/theories/Sorting/Mergesort.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/Sorting/PermutEq.v b/theories/Sorting/PermutEq.v index 9bae43c2..45d27e35 100644 --- a/theories/Sorting/PermutEq.v +++ b/theories/Sorting/PermutEq.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/Sorting/PermutSetoid.v b/theories/Sorting/PermutSetoid.v index 64dda448..0697a5e4 100644 --- a/theories/Sorting/PermutSetoid.v +++ b/theories/Sorting/PermutSetoid.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/Sorting/Permutation.v b/theories/Sorting/Permutation.v index fcb4e787..e159efa8 100644 --- a/theories/Sorting/Permutation.v +++ b/theories/Sorting/Permutation.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/Sorting/Sorted.v b/theories/Sorting/Sorted.v index dc4a1e0a..df03ff1c 100644 --- a/theories/Sorting/Sorted.v +++ b/theories/Sorting/Sorted.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/Sorting/Sorting.v b/theories/Sorting/Sorting.v index 712b8fd6..6e9702f2 100644 --- a/theories/Sorting/Sorting.v +++ b/theories/Sorting/Sorting.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/Strings/Ascii.v b/theories/Strings/Ascii.v index 3dbd9cb4..97cb746f 100644 --- a/theories/Strings/Ascii.v +++ b/theories/Strings/Ascii.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/Strings/String.v b/theories/Strings/String.v index ac1f158a..943bb48e 100644 --- a/theories/Strings/String.v +++ b/theories/Strings/String.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/Structures/EqualitiesFacts.v b/theories/Structures/EqualitiesFacts.v index 8e2b2d08..cee3d63f 100644 --- a/theories/Structures/EqualitiesFacts.v +++ b/theories/Structures/EqualitiesFacts.v @@ -10,7 +10,7 @@ Require Import Equalities Bool SetoidList RelationPairs. Set Implicit Arguments. -(** * Keys and datas used in MMap *) +(** * Keys and datas used in the future MMaps *) Module KeyDecidableType(D:DecidableType). @@ -60,7 +60,7 @@ Module KeyDecidableType(D:DecidableType). Hint Resolve eqke_1 eqke_2 eqk_1. - (* Additionnal facts *) + (* Additional facts *) Lemma InA_eqke_eqk {elt} p (m:list (key*elt)) : InA eqke p m -> InA eqk p m. diff --git a/theories/Structures/OrderedType.v b/theories/Structures/OrderedType.v index cc8c2261..93ca383b 100644 --- a/theories/Structures/OrderedType.v +++ b/theories/Structures/OrderedType.v @@ -342,7 +342,7 @@ Module KeyOrderedType(O:OrderedType). compute in Hxx'; compute in Hyy'. rewrite Hxx', Hyy'; auto. Qed. - (* Additionnal facts *) + (* Additional facts *) Lemma eqk_not_ltk : forall x x', eqk x x' -> ~ltk x x'. Proof. diff --git a/theories/Structures/OrdersEx.v b/theories/Structures/OrdersEx.v index b484257b..89c56388 100644 --- a/theories/Structures/OrdersEx.v +++ b/theories/Structures/OrdersEx.v @@ -87,7 +87,7 @@ End PairOrderedType. (** Even if [positive] can be seen as an ordered type with respect to the usual order (see above), we can also use a lexicographic order over bits (lower bits are considered first). This is more natural when using - [positive] as indexes for sets or maps (see MSetPositive and MMapPositive. *) + [positive] as indexes for sets or maps (see MSetPositive). *) Local Open Scope positive. diff --git a/theories/Structures/OrdersFacts.v b/theories/Structures/OrdersFacts.v index 88fbd8c1..954d3df2 100644 --- a/theories/Structures/OrdersFacts.v +++ b/theories/Structures/OrdersFacts.v @@ -90,7 +90,7 @@ Module Type OrderedTypeFullFacts (Import O:OrderedTypeFull'). Instance le_order : PartialOrder eq le. Proof. compute; iorder. Qed. - Instance le_antisym : Antisymmetric eq le. + Instance le_antisym : Antisymmetric _ eq le. Proof. apply partial_order_antisym; auto with *. Qed. Lemma le_not_gt_iff : forall x y, x<=y <-> ~y<x. diff --git a/theories/Structures/OrdersLists.v b/theories/Structures/OrdersLists.v index 4d49ac84..bf8529bc 100644 --- a/theories/Structures/OrdersLists.v +++ b/theories/Structures/OrdersLists.v @@ -54,7 +54,7 @@ Hint Immediate In_eq Inf_lt. End OrderedTypeLists. -(** * Results about keys and data as manipulated in MMaps. *) +(** * Results about keys and data as manipulated in the future MMaps. *) Module KeyOrderedType(O:OrderedType). Include KeyDecidableType(O). (* provides eqk, eqke *) @@ -76,7 +76,7 @@ Module KeyOrderedType(O:OrderedType). Instance ltk_compat' {elt} : Proper (eqke==>eqke==>iff) (@ltk elt). Proof. eapply subrelation_proper; eauto with *. Qed. - (* Additionnal facts *) + (* Additional facts *) Instance pair_compat {elt} : Proper (O.eq==>Logic.eq==>eqke) (@pair key elt). Proof. apply pair_compat. Qed. diff --git a/theories/Unicode/Utf8.v b/theories/Unicode/Utf8.v index 3b4beda9..d5c2fa73 100644 --- a/theories/Unicode/Utf8.v +++ b/theories/Unicode/Utf8.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/Unicode/Utf8_core.v b/theories/Unicode/Utf8_core.v index fe13f0ef..44a0700a 100644 --- a/theories/Unicode/Utf8_core.v +++ b/theories/Unicode/Utf8_core.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/Wellfounded/Disjoint_Union.v b/theories/Wellfounded/Disjoint_Union.v index ee4329bd..ebfc27b3 100644 --- a/theories/Wellfounded/Disjoint_Union.v +++ b/theories/Wellfounded/Disjoint_Union.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/Wellfounded/Inclusion.v b/theories/Wellfounded/Inclusion.v index d09c4112..1ff9b005 100644 --- a/theories/Wellfounded/Inclusion.v +++ b/theories/Wellfounded/Inclusion.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/Wellfounded/Inverse_Image.v b/theories/Wellfounded/Inverse_Image.v index aa6fa6ee..7786c8b3 100644 --- a/theories/Wellfounded/Inverse_Image.v +++ b/theories/Wellfounded/Inverse_Image.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/Wellfounded/Lexicographic_Exponentiation.v b/theories/Wellfounded/Lexicographic_Exponentiation.v index dd9e4c98..992263cb 100644 --- a/theories/Wellfounded/Lexicographic_Exponentiation.v +++ b/theories/Wellfounded/Lexicographic_Exponentiation.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 *) @@ -95,7 +95,7 @@ Section Wf_Lexicographic_Exponentiation. intros. - inversion H. assert ([b; a] = ([] ++ [b]) ++ [a]) by auto with sets. - destruct (app_inj_tail (l ++ [y]) ([] ++ [b]) _ _ H0) as ((?, <-)/app_inj_tail, <-). + destruct (app_inj_tail (l ++ [y]) ([] ++ [b]) _ _ H0) as ((?, <-)%app_inj_tail, <-). inversion H1; subst; [ apply rt_step; assumption | apply rt_refl ]. - inversion H0. + apply app_cons_not_nil in H3 as (). diff --git a/theories/Wellfounded/Lexicographic_Product.v b/theories/Wellfounded/Lexicographic_Product.v index 0d8ed8dd..4b8447f4 100644 --- a/theories/Wellfounded/Lexicographic_Product.v +++ b/theories/Wellfounded/Lexicographic_Product.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/Wellfounded/Transitive_Closure.v b/theories/Wellfounded/Transitive_Closure.v index b76e9661..eb12d5d7 100644 --- a/theories/Wellfounded/Transitive_Closure.v +++ b/theories/Wellfounded/Transitive_Closure.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/Wellfounded/Union.v b/theories/Wellfounded/Union.v index b2e8ea92..61355c8d 100644 --- a/theories/Wellfounded/Union.v +++ b/theories/Wellfounded/Union.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/Wellfounded/Well_Ordering.v b/theories/Wellfounded/Well_Ordering.v index f8a17b56..b5acc287 100644 --- a/theories/Wellfounded/Well_Ordering.v +++ b/theories/Wellfounded/Well_Ordering.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/Wellfounded/Wellfounded.v b/theories/Wellfounded/Wellfounded.v index a76d5e95..397f35aa 100644 --- a/theories/Wellfounded/Wellfounded.v +++ b/theories/Wellfounded/Wellfounded.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/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 *) diff --git a/theories/theories.itarget b/theories/theories.itarget index b7de4164..aacab2d9 100644 --- a/theories/theories.itarget +++ b/theories/theories.itarget @@ -4,7 +4,6 @@ Classes/vo.otarget Compat/vo.otarget FSets/vo.otarget MSets/vo.otarget -MMaps/vo.otarget Structures/vo.otarget Init/vo.otarget Lists/vo.otarget |