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authorGravatar Pierre-Marie Pédrot <pierre-marie.pedrot@inria.fr>2016-01-21 16:45:23 +0100
committerGravatar Pierre-Marie Pédrot <pierre-marie.pedrot@inria.fr>2016-01-21 16:45:23 +0100
commitbe0eca32fae93ed4793c2f839bb9e725b6a963d1 (patch)
treec2c5dce5ce24f5a2a8cade9e69410599c00e2b55 /theories
parent9c2662eecc398f38be3b6280a8f760cc439bc31c (diff)
parent5e23fb90b39dfa014ae5c4fb46eb713cca09dbff (diff)
Merge branch 'v8.5'
Diffstat (limited to 'theories')
-rw-r--r--theories/Arith/Arith.v2
-rw-r--r--theories/Arith/Arith_base.v2
-rw-r--r--theories/Arith/Between.v2
-rw-r--r--theories/Arith/Bool_nat.v2
-rw-r--r--theories/Arith/Compare.v2
-rw-r--r--theories/Arith/Compare_dec.v2
-rw-r--r--theories/Arith/Div2.v2
-rw-r--r--theories/Arith/EqNat.v2
-rw-r--r--theories/Arith/Euclid.v2
-rw-r--r--theories/Arith/Even.v2
-rw-r--r--theories/Arith/Factorial.v2
-rw-r--r--theories/Arith/Gt.v2
-rw-r--r--theories/Arith/Le.v2
-rw-r--r--theories/Arith/Lt.v2
-rw-r--r--theories/Arith/Max.v2
-rw-r--r--theories/Arith/Min.v2
-rw-r--r--theories/Arith/Minus.v2
-rw-r--r--theories/Arith/Mult.v2
-rw-r--r--theories/Arith/PeanoNat.v2
-rw-r--r--theories/Arith/Peano_dec.v2
-rw-r--r--theories/Arith/Wf_nat.v2
-rw-r--r--theories/Bool/Bool.v2
-rw-r--r--theories/Bool/BoolEq.v2
-rw-r--r--theories/Bool/Bvector.v2
-rw-r--r--theories/Bool/DecBool.v2
-rw-r--r--theories/Bool/IfProp.v2
-rw-r--r--theories/Bool/Sumbool.v2
-rw-r--r--theories/Bool/Zerob.v2
-rw-r--r--theories/Classes/CEquivalence.v2
-rw-r--r--theories/Classes/CMorphisms.v2
-rw-r--r--theories/Classes/CRelationClasses.v2
-rw-r--r--theories/Classes/DecidableClass.v2
-rw-r--r--theories/Classes/EquivDec.v2
-rw-r--r--theories/Classes/Equivalence.v2
-rw-r--r--theories/Classes/Init.v2
-rw-r--r--theories/Classes/Morphisms.v2
-rw-r--r--theories/Classes/Morphisms_Prop.v2
-rw-r--r--theories/Classes/Morphisms_Relations.v2
-rw-r--r--theories/Classes/RelationClasses.v2
-rw-r--r--theories/Classes/SetoidClass.v2
-rw-r--r--theories/Classes/SetoidDec.v2
-rw-r--r--theories/Classes/SetoidTactics.v2
-rw-r--r--theories/Compat/AdmitAxiom.v2
-rw-r--r--theories/Compat/Coq84.v2
-rw-r--r--theories/Compat/Coq85.v2
-rw-r--r--theories/Init/Datatypes.v2
-rw-r--r--theories/Init/Logic.v2
-rw-r--r--theories/Init/Logic_Type.v2
-rw-r--r--theories/Init/Nat.v2
-rw-r--r--theories/Init/Notations.v2
-rw-r--r--theories/Init/Peano.v2
-rw-r--r--theories/Init/Prelude.v2
-rw-r--r--theories/Init/Specif.v2
-rw-r--r--theories/Init/Tactics.v2
-rw-r--r--theories/Init/Wf.v2
-rw-r--r--theories/Lists/List.v2
-rw-r--r--theories/Lists/ListDec.v2
-rw-r--r--theories/Lists/ListSet.v2
-rw-r--r--theories/Lists/ListTactics.v2
-rw-r--r--theories/Lists/StreamMemo.v2
-rw-r--r--theories/Lists/Streams.v2
-rw-r--r--theories/Logic/Berardi.v2
-rw-r--r--theories/Logic/ChoiceFacts.v2
-rw-r--r--theories/Logic/Classical.v2
-rw-r--r--theories/Logic/ClassicalChoice.v2
-rw-r--r--theories/Logic/ClassicalDescription.v2
-rw-r--r--theories/Logic/ClassicalEpsilon.v2
-rw-r--r--theories/Logic/ClassicalFacts.v2
-rw-r--r--theories/Logic/ClassicalUniqueChoice.v2
-rw-r--r--theories/Logic/Classical_Pred_Type.v2
-rw-r--r--theories/Logic/Classical_Prop.v2
-rw-r--r--theories/Logic/ConstructiveEpsilon.v2
-rw-r--r--theories/Logic/Decidable.v2
-rw-r--r--theories/Logic/Description.v2
-rw-r--r--theories/Logic/Diaconescu.v2
-rw-r--r--theories/Logic/Epsilon.v2
-rw-r--r--theories/Logic/Eqdep.v2
-rw-r--r--theories/Logic/EqdepFacts.v2
-rw-r--r--theories/Logic/Eqdep_dec.v2
-rw-r--r--theories/Logic/ExtensionalityFacts.v2
-rw-r--r--theories/Logic/FinFun.v2
-rw-r--r--theories/Logic/FunctionalExtensionality.v2
-rw-r--r--theories/Logic/Hurkens.v2
-rw-r--r--theories/Logic/IndefiniteDescription.v2
-rw-r--r--theories/Logic/JMeq.v2
-rw-r--r--theories/Logic/ProofIrrelevance.v2
-rw-r--r--theories/Logic/ProofIrrelevanceFacts.v2
-rw-r--r--theories/Logic/RelationalChoice.v2
-rw-r--r--theories/Logic/SetIsType.v2
-rw-r--r--theories/Logic/WKL.v2
-rw-r--r--theories/Logic/WeakFan.v2
-rw-r--r--theories/MMaps/MMapAVL.v2158
-rw-r--r--theories/MMaps/MMapFacts.v2434
-rw-r--r--theories/MMaps/MMapInterface.v292
-rw-r--r--theories/MMaps/MMapList.v1144
-rw-r--r--theories/MMaps/MMapPositive.v698
-rw-r--r--theories/MMaps/MMapWeakList.v687
-rw-r--r--theories/MMaps/MMaps.v16
-rw-r--r--theories/MMaps/vo.itarget7
-rw-r--r--theories/NArith/BinNat.v2
-rw-r--r--theories/NArith/BinNatDef.v2
-rw-r--r--theories/NArith/NArith.v2
-rw-r--r--theories/NArith/Ndec.v2
-rw-r--r--theories/NArith/Ndigits.v2
-rw-r--r--theories/NArith/Ndist.v2
-rw-r--r--theories/NArith/Ndiv_def.v2
-rw-r--r--theories/NArith/Ngcd_def.v2
-rw-r--r--theories/NArith/Nnat.v2
-rw-r--r--theories/NArith/Nsqrt_def.v2
-rw-r--r--theories/Numbers/BigNumPrelude.v2
-rw-r--r--theories/Numbers/BinNums.v2
-rw-r--r--theories/Numbers/Cyclic/Abstract/CyclicAxioms.v2
-rw-r--r--theories/Numbers/Cyclic/Abstract/NZCyclic.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v2
-rw-r--r--theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v2
-rw-r--r--theories/Numbers/Cyclic/Int31/Cyclic31.v2
-rw-r--r--theories/Numbers/Cyclic/Int31/Int31.v2
-rw-r--r--theories/Numbers/Cyclic/Int31/Ring31.v2
-rw-r--r--theories/Numbers/Cyclic/ZModulo/ZModulo.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZAdd.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZAddOrder.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZAxioms.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZBase.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZBits.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivEucl.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivFloor.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZDivTrunc.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZGcd.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZLcm.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZLt.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMaxMin.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMul.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZMulOrder.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZParity.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZPow.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZProperties.v2
-rw-r--r--theories/Numbers/Integer/Abstract/ZSgnAbs.v2
-rw-r--r--theories/Numbers/Integer/BigZ/BigZ.v2
-rw-r--r--theories/Numbers/Integer/BigZ/ZMake.v2
-rw-r--r--theories/Numbers/Integer/Binary/ZBinary.v2
-rw-r--r--theories/Numbers/Integer/NatPairs/ZNatPairs.v2
-rw-r--r--theories/Numbers/Integer/SpecViaZ/ZSig.v2
-rw-r--r--theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v2
-rw-r--r--theories/Numbers/NaryFunctions.v2
-rw-r--r--theories/Numbers/NatInt/NZAdd.v2
-rw-r--r--theories/Numbers/NatInt/NZAddOrder.v2
-rw-r--r--theories/Numbers/NatInt/NZAxioms.v2
-rw-r--r--theories/Numbers/NatInt/NZBase.v2
-rw-r--r--theories/Numbers/NatInt/NZBits.v2
-rw-r--r--theories/Numbers/NatInt/NZDiv.v2
-rw-r--r--theories/Numbers/NatInt/NZDomain.v2
-rw-r--r--theories/Numbers/NatInt/NZGcd.v2
-rw-r--r--theories/Numbers/NatInt/NZLog.v2
-rw-r--r--theories/Numbers/NatInt/NZMul.v2
-rw-r--r--theories/Numbers/NatInt/NZMulOrder.v2
-rw-r--r--theories/Numbers/NatInt/NZOrder.v2
-rw-r--r--theories/Numbers/NatInt/NZParity.v2
-rw-r--r--theories/Numbers/NatInt/NZPow.v2
-rw-r--r--theories/Numbers/NatInt/NZProperties.v2
-rw-r--r--theories/Numbers/NatInt/NZSqrt.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NAdd.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NAddOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NAxioms.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NBase.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NBits.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NDefOps.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NDiv.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NGcd.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NIso.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NLcm.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NLog.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NMaxMin.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NMulOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NOrder.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NParity.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NPow.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NProperties.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NSqrt.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NStrongRec.v2
-rw-r--r--theories/Numbers/Natural/Abstract/NSub.v2
-rw-r--r--theories/Numbers/Natural/BigN/BigN.v2
-rw-r--r--theories/Numbers/Natural/BigN/NMake.v2
-rw-r--r--theories/Numbers/Natural/BigN/NMake_gen.ml2
-rw-r--r--theories/Numbers/Natural/BigN/Nbasic.v2
-rw-r--r--theories/Numbers/Natural/Binary/NBinary.v2
-rw-r--r--theories/Numbers/Natural/Peano/NPeano.v2
-rw-r--r--theories/Numbers/Natural/SpecViaZ/NSig.v2
-rw-r--r--theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v2
-rw-r--r--theories/Numbers/NumPrelude.v2
-rw-r--r--theories/Numbers/Rational/BigQ/BigQ.v2
-rw-r--r--theories/Numbers/Rational/BigQ/QMake.v2
-rw-r--r--theories/Numbers/Rational/SpecViaQ/QSig.v2
-rw-r--r--theories/PArith/BinPos.v2
-rw-r--r--theories/PArith/BinPosDef.v2
-rw-r--r--theories/PArith/PArith.v2
-rw-r--r--theories/PArith/POrderedType.v2
-rw-r--r--theories/PArith/Pnat.v2
-rw-r--r--theories/Program/Basics.v2
-rw-r--r--theories/Program/Combinators.v2
-rw-r--r--theories/Program/Equality.v2
-rw-r--r--theories/Program/Program.v2
-rw-r--r--theories/Program/Subset.v2
-rw-r--r--theories/Program/Syntax.v2
-rw-r--r--theories/Program/Tactics.v2
-rw-r--r--theories/Program/Utils.v2
-rw-r--r--theories/Program/Wf.v2
-rw-r--r--theories/QArith/QArith.v2
-rw-r--r--theories/QArith/QArith_base.v2
-rw-r--r--theories/QArith/QOrderedType.v2
-rw-r--r--theories/QArith/Qabs.v2
-rw-r--r--theories/QArith/Qcanon.v2
-rw-r--r--theories/QArith/Qfield.v2
-rw-r--r--theories/QArith/Qminmax.v2
-rw-r--r--theories/QArith/Qpower.v2
-rw-r--r--theories/QArith/Qreals.v2
-rw-r--r--theories/QArith/Qreduction.v2
-rw-r--r--theories/QArith/Qring.v2
-rw-r--r--theories/QArith/Qround.v2
-rw-r--r--theories/Reals/Alembert.v2
-rw-r--r--theories/Reals/AltSeries.v2
-rw-r--r--theories/Reals/ArithProp.v2
-rw-r--r--theories/Reals/Binomial.v2
-rw-r--r--theories/Reals/Cauchy_prod.v2
-rw-r--r--theories/Reals/Cos_plus.v2
-rw-r--r--theories/Reals/Cos_rel.v2
-rw-r--r--theories/Reals/DiscrR.v2
-rw-r--r--theories/Reals/Exp_prop.v2
-rw-r--r--theories/Reals/Integration.v2
-rw-r--r--theories/Reals/MVT.v2
-rw-r--r--theories/Reals/Machin.v2
-rw-r--r--theories/Reals/NewtonInt.v2
-rw-r--r--theories/Reals/PSeries_reg.v2
-rw-r--r--theories/Reals/PartSum.v2
-rw-r--r--theories/Reals/RIneq.v2
-rw-r--r--theories/Reals/RList.v2
-rw-r--r--theories/Reals/ROrderedType.v2
-rw-r--r--theories/Reals/R_Ifp.v2
-rw-r--r--theories/Reals/R_sqr.v2
-rw-r--r--theories/Reals/R_sqrt.v2
-rw-r--r--theories/Reals/Ranalysis.v2
-rw-r--r--theories/Reals/Ranalysis1.v2
-rw-r--r--theories/Reals/Ranalysis2.v2
-rw-r--r--theories/Reals/Ranalysis3.v2
-rw-r--r--theories/Reals/Ranalysis4.v2
-rw-r--r--theories/Reals/Ranalysis5.v2
-rw-r--r--theories/Reals/Ranalysis_reg.v2
-rw-r--r--theories/Reals/Ratan.v2
-rw-r--r--theories/Reals/Raxioms.v2
-rw-r--r--theories/Reals/Rbase.v2
-rw-r--r--theories/Reals/Rbasic_fun.v2
-rw-r--r--theories/Reals/Rcomplete.v2
-rw-r--r--theories/Reals/Rdefinitions.v2
-rw-r--r--theories/Reals/Rderiv.v2
-rw-r--r--theories/Reals/Reals.v2
-rw-r--r--theories/Reals/Rfunctions.v2
-rw-r--r--theories/Reals/Rgeom.v2
-rw-r--r--theories/Reals/RiemannInt.v2
-rw-r--r--theories/Reals/RiemannInt_SF.v2
-rw-r--r--theories/Reals/Rlimit.v2
-rw-r--r--theories/Reals/Rlogic.v2
-rw-r--r--theories/Reals/Rminmax.v2
-rw-r--r--theories/Reals/Rpow_def.v2
-rw-r--r--theories/Reals/Rpower.v2
-rw-r--r--theories/Reals/Rprod.v2
-rw-r--r--theories/Reals/Rseries.v2
-rw-r--r--theories/Reals/Rsigma.v2
-rw-r--r--theories/Reals/Rsqrt_def.v2
-rw-r--r--theories/Reals/Rtopology.v2
-rw-r--r--theories/Reals/Rtrigo.v2
-rw-r--r--theories/Reals/Rtrigo1.v2
-rw-r--r--theories/Reals/Rtrigo_alt.v2
-rw-r--r--theories/Reals/Rtrigo_calc.v2
-rw-r--r--theories/Reals/Rtrigo_def.v2
-rw-r--r--theories/Reals/Rtrigo_fun.v2
-rw-r--r--theories/Reals/Rtrigo_reg.v2
-rw-r--r--theories/Reals/SeqProp.v2
-rw-r--r--theories/Reals/SeqSeries.v2
-rw-r--r--theories/Reals/SplitAbsolu.v2
-rw-r--r--theories/Reals/SplitRmult.v2
-rw-r--r--theories/Reals/Sqrt_reg.v2
-rw-r--r--theories/Relations/Operators_Properties.v2
-rw-r--r--theories/Relations/Relation_Definitions.v2
-rw-r--r--theories/Relations/Relation_Operators.v2
-rw-r--r--theories/Relations/Relations.v2
-rw-r--r--theories/Setoids/Setoid.v2
-rw-r--r--theories/Sets/Classical_sets.v2
-rw-r--r--theories/Sets/Constructive_sets.v2
-rw-r--r--theories/Sets/Cpo.v2
-rw-r--r--theories/Sets/Ensembles.v2
-rw-r--r--theories/Sets/Finite_sets.v2
-rw-r--r--theories/Sets/Finite_sets_facts.v2
-rw-r--r--theories/Sets/Image.v2
-rw-r--r--theories/Sets/Infinite_sets.v2
-rw-r--r--theories/Sets/Integers.v2
-rw-r--r--theories/Sets/Multiset.v2
-rw-r--r--theories/Sets/Partial_Order.v2
-rw-r--r--theories/Sets/Permut.v2
-rw-r--r--theories/Sets/Powerset.v2
-rw-r--r--theories/Sets/Powerset_Classical_facts.v2
-rw-r--r--theories/Sets/Powerset_facts.v2
-rw-r--r--theories/Sets/Relations_1.v2
-rw-r--r--theories/Sets/Relations_1_facts.v2
-rw-r--r--theories/Sets/Relations_2.v2
-rw-r--r--theories/Sets/Relations_2_facts.v2
-rw-r--r--theories/Sets/Relations_3.v2
-rw-r--r--theories/Sets/Relations_3_facts.v2
-rw-r--r--theories/Sets/Uniset.v2
-rw-r--r--theories/Sorting/Heap.v2
-rw-r--r--theories/Sorting/Mergesort.v2
-rw-r--r--theories/Sorting/PermutEq.v2
-rw-r--r--theories/Sorting/PermutSetoid.v2
-rw-r--r--theories/Sorting/Permutation.v2
-rw-r--r--theories/Sorting/Sorted.v2
-rw-r--r--theories/Sorting/Sorting.v2
-rw-r--r--theories/Strings/Ascii.v2
-rw-r--r--theories/Strings/String.v2
-rw-r--r--theories/Structures/EqualitiesFacts.v2
-rw-r--r--theories/Structures/OrdersEx.v2
-rw-r--r--theories/Structures/OrdersLists.v2
-rw-r--r--theories/Unicode/Utf8.v2
-rw-r--r--theories/Unicode/Utf8_core.v2
-rw-r--r--theories/Wellfounded/Disjoint_Union.v2
-rw-r--r--theories/Wellfounded/Inclusion.v2
-rw-r--r--theories/Wellfounded/Inverse_Image.v2
-rw-r--r--theories/Wellfounded/Lexicographic_Exponentiation.v2
-rw-r--r--theories/Wellfounded/Lexicographic_Product.v2
-rw-r--r--theories/Wellfounded/Transitive_Closure.v2
-rw-r--r--theories/Wellfounded/Union.v2
-rw-r--r--theories/Wellfounded/Well_Ordering.v2
-rw-r--r--theories/Wellfounded/Wellfounded.v2
-rw-r--r--theories/ZArith/BinInt.v2
-rw-r--r--theories/ZArith/BinIntDef.v2
-rw-r--r--theories/ZArith/Wf_Z.v2
-rw-r--r--theories/ZArith/ZArith.v2
-rw-r--r--theories/ZArith/ZArith_base.v2
-rw-r--r--theories/ZArith/ZArith_dec.v2
-rw-r--r--theories/ZArith/Zabs.v2
-rw-r--r--theories/ZArith/Zbool.v2
-rw-r--r--theories/ZArith/Zcompare.v2
-rw-r--r--theories/ZArith/Zcomplements.v2
-rw-r--r--theories/ZArith/Zdigits.v2
-rw-r--r--theories/ZArith/Zdiv.v2
-rw-r--r--theories/ZArith/Zeuclid.v2
-rw-r--r--theories/ZArith/Zeven.v2
-rw-r--r--theories/ZArith/Zgcd_alt.v2
-rw-r--r--theories/ZArith/Zhints.v2
-rw-r--r--theories/ZArith/Zlogarithm.v2
-rw-r--r--theories/ZArith/Zmax.v2
-rw-r--r--theories/ZArith/Zmin.v2
-rw-r--r--theories/ZArith/Zminmax.v2
-rw-r--r--theories/ZArith/Zmisc.v2
-rw-r--r--theories/ZArith/Znat.v2
-rw-r--r--theories/ZArith/Znumtheory.v2
-rw-r--r--theories/ZArith/Zorder.v2
-rw-r--r--theories/ZArith/Zpow_alt.v2
-rw-r--r--theories/ZArith/Zpow_def.v2
-rw-r--r--theories/ZArith/Zpow_facts.v2
-rw-r--r--theories/ZArith/Zpower.v2
-rw-r--r--theories/ZArith/Zquot.v2
-rw-r--r--theories/ZArith/Zsqrt_compat.v2
-rw-r--r--theories/ZArith/Zwf.v2
-rw-r--r--theories/ZArith/auxiliary.v2
-rw-r--r--theories/theories.itarget1
370 files changed, 361 insertions, 7798 deletions
diff --git a/theories/Arith/Arith.v b/theories/Arith/Arith.v
index 620a42014..953f7e4db 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 a99c4113d..b378828ef 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 067235413..f998e8619 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 f91f3340a..602555b60 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 400f2d81a..610e9a697 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 a97cf6dce..976507b56 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 1c65a192a..016cb85e9 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 2771670ef..206fc0ab5 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 eaacab02d..fbe98d172 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 0f94a8ed6..3c8c250ab 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 7d29f23c6..b119bb000 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 e406ff0dc..dfd576946 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 875863e4a..ceb91187b 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 b783ca33c..f824ee6fb 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 26875373b..65534b2e3 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 f2fa3aec6..38e59b7b8 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 6e312e4fd..bc3a318cf 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 2d82920b3..965812432 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 799031a20..e3240bb78 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 a7ede3fc5..340a79689 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 647648300..94bbd50a9 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 cc12cf47f..721ab6932 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 11af2fd1b..aec4f0bbf 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 7c63f0698..09f643c8e 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 e0b8ec9bf..501366ce2 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 a0acbe8c1..11f3d1d6f 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 c2e9183b6..fd7f42e2b 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 e146f25f0..16e47ac5a 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 65353ed20..bd01de47e 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 b13671cec..c41eb2fa2 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 *)
diff --git a/theories/Classes/CRelationClasses.v b/theories/Classes/CRelationClasses.v
index 35b2b8a3d..3d7ef01fb 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 9fe3d0fe8..a3b7e3115 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 59e800c27..52313735e 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 c281af800..c45889479 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 9574cf857..c13b36fd1 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 1bdce6546..8d942d908 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 096c96e5d..159001771 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 68a8c06a3..6048fe067 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 6248babd0..11c204dae 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 *)
diff --git a/theories/Classes/SetoidClass.v b/theories/Classes/SetoidClass.v
index f20100fe9..4b133a4d1 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 bf05934e9..7201c0b16 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 8ca93341c..145d451f0 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
index 68607f6b2..4d9f55cfe 100644
--- a/theories/Compat/AdmitAxiom.v
+++ b/theories/Compat/AdmitAxiom.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 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/Coq84.v b/theories/Compat/Coq84.v
index 1c70a894a..90083b00d 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 *)
diff --git a/theories/Compat/Coq85.v b/theories/Compat/Coq85.v
index 7ce04a662..af2e04f88 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/Init/Datatypes.v b/theories/Init/Datatypes.v
index aaffc0978..ddaf08bf7 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 50f853f0e..85123cc44 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 1e1264639..4a5f2ad69 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 afb46436c..b8920586f 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 2f6d824d2..e1ddaeaef 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 7a14ab39b..3749baf61 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 0efb8744d..04a263ad9 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 1384901b7..6c0221856 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 a7d3f8062..59fdbb42f 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 6501b1e18..985ecaf27 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 aa9df50a7..62b01eb7d 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 *)
diff --git a/theories/Lists/ListDec.v b/theories/Lists/ListDec.v
index 8bd2daaff..3e2eeac04 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 c8ed95cd4..655d3940c 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 f19d95a9e..537d5f684 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 74d464c5d..5a16cc439 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 cc4fb1793..7ec3d2503 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 d72f4072b..1e0bd0fed 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 d23274981..1420a000b 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 600db472e..14d83501f 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 07153b35f..7041ee40b 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 bdad50e27..9e6d07b2f 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 2d9a1ffdf..0e91613d9 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 2ae0ade80..afd64efdf 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 *)
diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v
index 4b0ec15eb..57f367e54 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 8468ced35..6665798dd 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 be75c4e96..2c69d4f0e 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 6f5bfae4d..a304dd24e 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 545f92bd9..2ba7253c4 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 70cc0787c..0239222e3 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 645173541..23af5afc6 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 fe17cde47..ffbb5758f 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 d9ffe68d8..f3a2783e1 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 34aba4864..30e26c7c6 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 65011e8e3..b7b4dec22 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 61ee9eb9a..0e34e7e97 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 670aa2199..064668015 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 eb50a3aae..04d9a6704 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 5c87011e5..8ded74763 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 *)
diff --git a/theories/Logic/IndefiniteDescription.v b/theories/Logic/IndefiniteDescription.v
index 9875710e6..21be50323 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 98cddf0ad..2f95856b4 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 eb00dedd2..305839cd0 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 6ab6abcf4..19b3e9e6f 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 615981308..d16835f81 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 f110237e7..33fce6cc7 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 abe6a8d99..95f3e83f1 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 *)
diff --git a/theories/Logic/WeakFan.v b/theories/Logic/WeakFan.v
index 365661be0..4416d38db 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 *)
diff --git a/theories/MMaps/MMapAVL.v b/theories/MMaps/MMapAVL.v
deleted file mode 100644
index d840f1f32..000000000
--- 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 8b356d750..000000000
--- 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 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),
- 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 05c5e5d8f..000000000
--- 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 c521178cb..000000000
--- 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 adbec7057..000000000
--- 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 additional 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 656c61e11..000000000
--- 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 054d07225..000000000
--- 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 a7bbd266e..000000000
--- 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 641ec02fa..4c3985737 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 9de2e7e10..4df4242e8 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 436145430..12d3ad2fd 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 5b1815bd4..37fd9dded 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 55ef451ee..a1e51c9d3 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 5467f9cb0..0ab8be587 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 5ae388e31..4bdf6529f 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 1750ffeb9..277fd26f0 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 0dcaa71de..4e007878f 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 da7829a9c..dd44b6e2d 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 6817947c8..45a7527c9 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 1dd5d82ab..e4f3cd6cf 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 8b84a484b..3312161ae 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 8adeda37d..df9b83392 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 a7c288624..407bcca4b 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 e68cd0336..e94a891dd 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 e137349ef..4ebe8fac1 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 cd55f9d8c..09d7329b6 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 6a1d741ea..195527dd5 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 ff9f50a55..f65b47c8c 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 537f557db..b99013900 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 ab8c8617a..d07ce3018 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 c1f314e9f..abd567a85 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 aca572166..0e58b8155 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 f5e936cf0..ff4b998e7 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 4fde3f53e..215b8bd58 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 b93b4eb39..c115a831d 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 ec8801c4e..f7fdc1795 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 e341ea8a5..6bf5e9d4c 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 2a9fa539a..ad10e65f0 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 6634eab1d..9b1b30f87 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 9dd0ec0ee..c919e121d 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 ab73ebfe1..278e1bcff 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 *)
diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v
index c8260e516..310748ddb 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 *)
diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v
index 464fe354b..043010779 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 *)
diff --git a/theories/Numbers/Integer/Abstract/ZGcd.v b/theories/Numbers/Integer/Abstract/ZGcd.v
index cf6ff79ec..30adaeb4b 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 9a1768eb1..ef33308c2 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 6d0cdb019..0c92918d1 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 07c78ead3..24a47f002 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 2d78d8f30..830c0a7da 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 487aaae1c..320c8f355 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 195a0277e..952d8e9c1 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 87de2c782..02b8501c7 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 5cfeeb216..1dec3c586 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 b379853ed..a10552ab3 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 b28bc40f8..ec495d094 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 af4f1d937..8673b8a58 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 042081066..eae8204d4 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 02f02fbc5..0aaf33657 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 30ac32b53..a360327a4 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 c9dc687cb..32410d1d0 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 376620ddc..eaac7d690 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 501583ae9..8cc52940a 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 619a66348..705f29102 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 c88341fa9..5a7f20728 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 c0afa0982..cf4ad3544 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 1c1185972..209bd9471 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 e0dfdedbd..b2c0be6f5 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 *)
diff --git a/theories/Numbers/NatInt/NZDomain.v b/theories/Numbers/NatInt/NZDomain.v
index ffb04f080..3881a27ff 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 42bee3151..1d3672943 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 9cd1f8776..3dfa2eef2 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 89ace4deb..36cd08271 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 e79e50a9f..8569fc56b 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 c1e83529f..0b9d65983 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 6b9a680a8..4a1352ccd 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 384521192..261febbcf 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 0f3a5caf7..79a9d7335 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 894c08063..7f9bb9c25 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 638cfc7ec..eecec3ac1 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 144bce72b..9d68a006e 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 d300f857d..25e285d59 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 40453214c..f949a0f6f 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 6f8a8fe52..3dd603e26 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 892b9266b..b3a53617b 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 d3d3eb0fb..84e1219ef 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 *)
diff --git a/theories/Numbers/Natural/Abstract/NGcd.v b/theories/Numbers/Natural/Abstract/NGcd.v
index a1f4ddf88..1eac134d5 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 c296315d5..e6cac0ba9 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 0fe8e1055..3d7497997 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 605c0aadf..0c9da29bb 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 e0710561e..b4c9db916 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 c41275d21..a4b523963 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 90053a736..60e955f5f 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 80a579f19..e1f573f65 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 *)
diff --git a/theories/Numbers/Natural/Abstract/NPow.v b/theories/Numbers/Natural/Abstract/NPow.v
index 9cc23004a..c9b2177f9 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 cb3501d41..819a5be93 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 8dc66884f..4ae414071 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 896ffc182..fa3a5351e 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 18ebe77b7..21f12037c 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 f7f4347b0..29a1145e0 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 bdcdd5ca3..98949736c 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 6de77e0a9..601fa108f 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 8fe9ea924..18d0262c9 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 d54bedd1a..037b10d99 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 96eb7b352..58b1b0180 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 1049c1567..258e03159 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 11569b3f9..355da4cc6 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 f67e0e966..f323aaeb1 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 b64cfb643..fe38ea4f2 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 c11e07fa5..4ac36425b 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 5f831bfc2..a40d94059 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 921e2d67e..0ccfad7b2 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 fefd1d76f..74a292c63 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 93352c6ba..6ee8d6d7b 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 92483ac83..7619f6395 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 e529a8c41..9c2608f4a 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 e5be0ca92..644b9b5a7 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 e246041b6..772018aa0 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 ae6fe7dd0..27e1ca844 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 5af6f4d74..4a6f2786f 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 ce1f7768d..c8f37318d 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 *)
diff --git a/theories/Program/Syntax.v b/theories/Program/Syntax.v
index 892305b49..2fccf6249 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 c1d958b9d..66ca3e577 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 65fe87801..396c96181 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 6e5919b34..a2fd05cd9 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 c32fb9509..5ad08b650 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 7f19b4ce5..62304876e 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 fa0b92091..25e98f0b0 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 f77f409e7..c60d04517 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 f7f83bf03..d966b050c 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 083e40aed..bbaf6027e 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 0fd053252..86584d9ee 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 8bd643aa7..af89d3005 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 add43b960..048e409cd 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 1d304964b..c50c38b28 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 78c464ae7..da11c2b1d 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 964a4bae0..0ed6d557c 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 011328ec0..a98d529fa 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 3e99c9893..c3ab8edc5 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 c4e410ed5..6fca9c8ad 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 d48f42fc3..f878abfab 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 28de11863..5cf6f17d4 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 49ba9a6ed..b14d807d2 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 6d30319c9..f5fcac47d 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 75fd4c0aa..4e2a7c3c6 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 be96b94e9..569518f7b 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 222d106fb..e3760e01e 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 599769578..26c51583b 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 1a94f6a8f..19db476fd 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 832e7adc4..ed5ae90c2 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 94b881ccc..03ac6582b 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 b710c75cc..37d54a6df 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 8dca0197e..f26bac2bb 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 abf8a99d2..924d5117b 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 0531bd0a1..f2dc7fd09 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 57ee1d9a6..b6d072837 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 f1e2d6fa4..445ffcb21 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 20319a2b1..a6b1a26e0 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 3cda675a4..88ebb88b2 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 875eebbbd..9e3abd815 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 eb646913c..0254218c4 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 407f64100..4e88714d6 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 ae2013f01..661bc8c76 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 27615c597..d172139f5 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 4cf908865..2465f0399 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 cc45139d2..e13ef1f2c 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 f545d3a08..9d55e4e63 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 7a879f458..e56ce28d6 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 bb30c0ef3..c889d7347 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 1766f3773..3520c26c3 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 50eb59b22..f3f8f7409 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 3a332d21e..bd330ac9b 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 9cb8a10bc..8265f65ad 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 1c353803f..0a49d4983 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 d930c5aaf..7423ffce7 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 856fff80a..4c0466ac0 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 1484ab2a8..7885d697f 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 c8887dfb5..e424a732a 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 07792942e..b9a9458c8 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 ba1fe90f4..152988dcc 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 1d697f3c9..791718a45 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 e30ea334b..b3ce6fa33 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 1ee9410fb..883e4e1a9 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 fd16ea610..744fd6641 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 458d1f8c3..ced2b3dac 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 b8ec8d3ca..b3c9c7449 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 72e4142b8..df3b95be4 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 440583582..ecef0d682 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 9e485ec5e..4d2418639 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 3d36cb341..a5092d22d 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 281c152ba..9ba14ee73 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 ef3e31f12..0d2a9a8ba 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 b921ee7b4..f395f9ae3 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 7845e6c46..eed612d94 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 9a6fb945e..5a2a07c42 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 25fe4848b..1123e7ee7 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 64f4f1c9e..a78a6e198 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 fec285188..074a76315 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 dd8738e18..10527442e 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 95d9cfa98..220cebeac 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 a187f955f..b6005b9d1 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 4e52017ec..ffd682d62 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 ce849a167..ce6bdbc60 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 75cffa7f6..55b301c37 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 aa2c144bf..8a4bb9f42 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 193bec780..8d2344f93 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 f2fac097a..b5db90306 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 c0cddbe18..8f579214a 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 22cb3dae5..f38dd6fdf 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 b1c12c7f9..31cc11e1a 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 6cf4d2507..34ea857d1 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 5860f960b..98eeb7d74 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 944e0dd16..057fc9b66 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 46dbe9942..ec38b8923 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 35d5f91af..3610ebce6 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 c9c1e5b71..adbde18e6 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 587d48ab1..d636e0468 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 40fd5e67c..09c90506b 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 e9347ce31..63e84199d 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 c9148e007..de96fa560 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 f650a50cd..bf8a42612 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 ea48fd918..f1026e31a 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 e05435019..da93e922d 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 de6770ee0..92b299885 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 0180c7d47..89fb900ce 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 86ba903fd..e297d97e3 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 6313dbf66..20c6feb9a 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 593b2e9be..4b967c163 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 9bae43c26..45d27e35e 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 64dda4487..0697a5e47 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 ba7da256d..8470b7955 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 dc4a1e0a7..df03ff1c6 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 712b8fd69..6e9702f29 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 3dbd9cb4e..97cb746f3 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 ac1f158a3..943bb48e9 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 d5827d87a..cee3d63f0 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).
diff --git a/theories/Structures/OrdersEx.v b/theories/Structures/OrdersEx.v
index b484257b9..89c563882 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/OrdersLists.v b/theories/Structures/OrdersLists.v
index 41e65d728..bf8529bc7 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 *)
diff --git a/theories/Unicode/Utf8.v b/theories/Unicode/Utf8.v
index 3b4beda96..d5c2fa739 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 fe13f0ef0..44a0700af 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 ee4329bd7..ebfc27b38 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 d09c4112a..1ff9b0055 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 aa6fa6ee6..7786c8b3b 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 e05dab7a4..d90f9956b 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 *)
diff --git a/theories/Wellfounded/Lexicographic_Product.v b/theories/Wellfounded/Lexicographic_Product.v
index 0d8ed8dd1..4b8447f49 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 b76e9661c..eb12d5d7b 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 b2e8ea928..61355c8d1 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 f8a17b56f..b5acc2877 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 a76d5e951..397f35aab 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 cb0c6880c..5aa397f8a 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 9bb86fd53..8c2e7d941 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 09909bdb6..4fbbac268 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 04cccd048..f86a7e52a 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 4c93173bf..a1da544de 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 ac69cebd0..8947295e0 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 146009bcc..df9db5b22 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 61eb2a340..41d1b2b50 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 d4ac72e94..2627f1743 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 9604a06e4..5bdb32fcf 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 b5d04719c..bc3694bc1 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 363b4fd03..2ba865bd0 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 *)
diff --git a/theories/ZArith/Zeuclid.v b/theories/ZArith/Zeuclid.v
index f5cacc7ec..38a824cd4 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 d88bf7a98..d4051ef71 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 14286bde1..3977ca9d3 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 1942c2ab0..c4e201e32 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 6e3495690..fdfd71e12 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 c436b3ad8..529e9f1d4 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 1cfa2e035..782a51586 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 05a94a8e9..ea9a5f861 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 b401e6b66..65831c783 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 20e7c2e88..06428a7c4 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 f69cf3156..ee6efb3cb 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 e090302ed..73dee0cf2 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 c8627477b..79a5a555e 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 *)
diff --git a/theories/ZArith/Zpow_def.v b/theories/ZArith/Zpow_def.v
index 740c45fd1..9eafa0769 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 ac41a98f6..2930e677d 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 747bd4fd2..6f3a89f18 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 6db92edb7..0d92f1d54 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 *)
diff --git a/theories/ZArith/Zsqrt_compat.v b/theories/ZArith/Zsqrt_compat.v
index 65959a69b..b80eb4451 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 cba709e86..1ac00bddd 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 25ef852a5..c6c389760 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 b7de41641..aacab2d97 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