The Coq Standard Library

Here is a short description of the Coq standard library, which is distributed with the system. It provides a set of modules directly available through the Require Import command.

The standard library is composed of the following subdirectories:

Init: The core library (automatically loaded when starting Coq)

Ltac Notations Datatypes Logic Byte Nat Decimal Hexadecimal Number Peano Specif Tactics Tauto Wf (Prelude)

Logic: Classical logic, dependent equality, extensionality, choice axioms

SetIsType StrictProp Classical_Pred_Type Classical_Prop (Classical) ClassicalFacts Decidable Eqdep_dec EqdepFacts Eqdep JMeq ChoiceFacts RelationalChoice ClassicalChoice ClassicalDescription ClassicalEpsilon ClassicalUniqueChoice SetoidChoice Berardi Diaconescu Hurkens ProofIrrelevance ProofIrrelevanceFacts ConstructiveEpsilon Description Epsilon IndefiniteDescription PropExtensionality PropExtensionalityFacts FunctionalExtensionality ExtensionalFunctionRepresentative ExtensionalityFacts WeakFan WKL FinFun PropFacts HLevels Adjointification

Structures: Algebraic structures (types with equality, with order, ...). DecidableType* and OrderedType* are there only for compatibility.

Equalities EqualitiesFacts Orders OrdersTac OrdersAlt OrdersEx OrdersFacts OrdersLists GenericMinMax DecidableType DecidableTypeEx OrderedType OrderedTypeAlt OrderedTypeEx

Bool: Booleans (basic functions and results)

Bool BoolEq BoolOrder DecBool IfProp Sumbool Zerob Bvector

Arith: Basic Peano arithmetic

PeanoNat Between Peano_dec Compare_dec (Arith_base) (Arith) Compare EqNat Euclid Factorial Wf_nat Cantor

PArith: Binary positive integers

BinPosDef BinPos Pnat POrderedType (PArith)

NArith: Binary natural numbers

BinNatDef BinNat Nnat Ndec Ndiv_def Ngcd_def Nsqrt_def (NArith)

ZArith: Binary integers

BinIntDef BinInt Zorder Zcompare Znat Zmin Zmax Zminmax Zabs Zeven auxiliary ZArith_dec Zbool Zmisc Wf_Z Zhints (ZArith_base) Zcomplements Zpow_def Zpow_alt Zpower Zdiv Zquot (ZArith) Zgcd_alt Zwf Znumtheory Int Zpow_facts Zdiv_facts Zbitwise

QArith: Rational numbers

QArith_base Qabs Qpower Qreduction Qring Qfield (QArith) Qreals Qcanon Qcabs Qround QOrderedType Qminmax

Numbers: An experimental modular architecture for arithmetic

  Prelude:

BinNums NumPrelude NaryFunctions AltBinNotations DecimalFacts DecimalNat DecimalPos DecimalN DecimalZ DecimalQ DecimalR DecimalString HexadecimalFacts HexadecimalNat HexadecimalPos HexadecimalN HexadecimalZ HexadecimalQ HexadecimalR HexadecimalString

  NatInt: Abstract mixed natural/integer/cyclic arithmetic

NZAdd NZAddOrder NZAxioms NZBase NZMul NZDiv NZMulOrder NZOrder NZParity NZPow NZSqrt NZLog NZGcd NZBits

  Cyclic: Abstract and 63-bits-based cyclic arithmetic

CyclicAxioms NZCyclic CarryType DoubleType Cyclic63 PrimInt63 Uint63 Sint63 Ring63

  Natural: Abstract and 63-bits-words-based natural arithmetic

NAdd NAddOrder NAxioms NBase NMulOrder NOrder NStrongRec NSub NDiv NDiv0 NMaxMin NParity NPow NSqrt NLog NGcd NLcm NLcm0 NBits NProperties

  Integer: Abstract and concrete (especially 63-bits-words-based) integer arithmetic

ZAdd ZAddOrder ZAxioms ZBase ZLt ZMul ZMulOrder ZSgnAbs ZMaxMin ZParity ZPow ZGcd ZLcm ZBits ZProperties ZDivFloor ZDivTrunc

  Floats: Floating-point arithmetic

FloatClass PrimFloat SpecFloat FloatOps FloatAxioms FloatLemmas (Floats)

Relations: Relations (definitions and basic results)

Relation_Definitions Relation_Operators Relations Operators_Properties

Sets: Sets (classical, constructive, finite, infinite, powerset, etc.)

Classical_sets Constructive_sets Cpo Ensembles Finite_sets_facts Finite_sets Image Infinite_sets Integers Multiset Partial_Order Permut Powerset_Classical_facts Powerset_facts Powerset Relations_1_facts Relations_1 Relations_2_facts Relations_2 Relations_3_facts Relations_3 Uniset

Classes:

Init RelationClasses Morphisms Morphisms_Prop Morphisms_Relations Equivalence CRelationClasses CMorphisms CEquivalence EquivDec SetoidTactics SetoidClass SetoidDec RelationPairs DecidableClass

Setoids:

Setoid

Lists: Polymorphic lists, Streams (infinite sequences)

List ListDec ListSet SetoidList SetoidPermutation Streams StreamMemo ListTactics

Vectors: Dependent datastructures storing their length

Fin VectorDef VectorSpec VectorEq (Vector)

Sorting: Axiomatizations of sorts

Heap Permutation Sorting PermutEq PermutSetoid Mergesort Sorted CPermutation

Wellfounded: Well-founded Relations

Disjoint_Union Inclusion Inverse_Image Lexicographic_Exponentiation Lexicographic_Product List_Extension Transitive_Closure Union Wellfounded Well_Ordering

MSets: Modular implementation of finite sets using lists or efficient trees. This is a modernization of FSets.

MSetInterface MSetFacts MSetDecide MSetProperties MSetEqProperties MSetWeakList MSetList MSetGenTree MSetAVL MSetRBT MSetPositive MSetToFiniteSet (MSets)

FSets: Modular implementation of finite sets/maps using lists or efficient trees. For sets, please consider the more modern MSets.

FSetInterface FSetBridge FSetFacts FSetDecide FSetProperties FSetEqProperties FSetList FSetWeakList FSetCompat FSetAVL FSetPositive (FSets) FSetToFiniteSet FMapInterface FMapWeakList FMapList FMapPositive FMapFacts (FMaps) FMapAVL FMapFullAVL

Strings Implementation of string as list of ascii characters

Byte Ascii String BinaryString HexString OctalString

Reals: Formalization of real numbers

Classical Reals: Real numbers with excluded middle, total order and least upper bounds

Rdefinitions ClassicalDedekindReals ClassicalConstructiveReals Raxioms RIneq DiscrR ROrderedType Rminmax (Rbase) RList Ranalysis Rbasic_fun Rderiv Rfunctions Rgeom R_Ifp Rlimit Rseries Rsigma R_sqr Rtrigo_fun Rtrigo1 Rtrigo Rtrigo_facts Ratan Machin SplitAbsolu SplitRmult Alembert AltSeries ArithProp Binomial Cauchy_prod Cos_plus Cos_rel Exp_prop Integration MVT NewtonInt PSeries_reg PartSum R_sqrt Ranalysis1 Ranalysis2 Ranalysis3 Ranalysis4 Ranalysis5 Ranalysis_reg Rcomplete RiemannInt RiemannInt_SF Rpow_def Rpower Rprod Rsqrt_def Rtopology Rtrigo_alt Rtrigo_calc Rtrigo_def Rtrigo_reg SeqProp SeqSeries Sqrt_reg Rlogic Rregisternames (Reals) Runcountable

Abstract Constructive Reals: Interface of constructive reals, proof of equivalence of all implementations. EXPERIMENTAL

ConstructiveReals ConstructiveRealsMorphisms ConstructiveLUB ConstructiveAbs ConstructiveLimits ConstructiveMinMax ConstructivePower ConstructiveSum

Constructive Cauchy Reals: Cauchy sequences of rational numbers, implementation of the interface. EXPERIMENTAL

ConstructiveRcomplete ConstructiveCauchyReals ConstructiveCauchyRealsMult ConstructiveCauchyAbs

Program: Support for dependently-typed programming

Basics Wf Subset Equality Tactics Utils Syntax Program Combinators

SSReflect: Base libraries for the SSReflect proof language and the small scale reflection formalization technique

ssrmatching ssrclasses ssreflect ssrbool ssrfun

Ltac2: The Ltac2 tactic programming language

Ltac2 Array Bool Char Constant Constr Constructor Control Env Evar Float FMap FSet Fresh Ident Ind Init Int Lazy List Ltac1 Message Meta Notations Option Pattern Printf Proj Pstring RedFlags Ref Std String TransparentState Uint63 Unification

Unicode: Unicode-based notations

Utf8_core Utf8

Compat: Compatibility wrappers for previous versions of Coq

AdmitAxiom Coq818 Coq819 Coq820 Coq818 Coq819

Array: Persistent native arrays

PArray

Primitive strings Native string type

PrimString PrimStringAxioms PString