# 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)
  Notations Datatypes Logic Logic_Type Peano Specif Tactics Wf (Prelude)
Logic: Classical logic and dependent equality
  SetIsType Classical_Pred_Set Classical_Pred_Type Classical_Prop Classical_Type (Classical) ClassicalFacts Decidable Eqdep_dec EqdepFacts Eqdep JMeq ChoiceFacts RelationalChoice ClassicalChoice ClassicalDescription ClassicalEpsilon ClassicalUniqueChoice Berardi Diaconescu Hurkens ProofIrrelevance ProofIrrelevanceFacts ConstructiveEpsilon Description Epsilon IndefiniteDescription FunctionalExtensionality ExtensionalityFacts
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 DecBool IfProp Sumbool Zerob Bvector
Arith: Basic Peano arithmetic
  Arith_base Le Lt Plus Minus Mult Gt Between Peano_dec Compare_dec (Arith) Min Max Compare Div2 EqNat Euclid Even Bool_nat Factorial Wf_nat
PArith: Binary positive integers
  BinPosDef BinPos Pnat POrderedType (PArith)
NArith: Binary natural numbers
  BinNatDef BinNat Nnat Ndigits Ndist 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 Zsqrt_compat Zpow_def Zpow_alt Zpower ZOdiv_def ZOdiv Zdiv Zquot Zeuclid Zlogarithm (ZArith) Zgcd_alt Zwf Znumtheory Int Zpow_facts Zdigits
QArith: Rational numbers
  QArith_base Qabs Qpower Qreduction Qring Qfield (QArith) Qreals Qcanon Qround QOrderedType Qminmax
Numbers: An experimental modular architecture for arithmetic
Prelude:
  BinNums NumPrelude BigNumPrelude NaryFunctions
NatInt: Abstract mixed natural/integer/cyclic arithmetic
  NZAdd NZAddOrder NZAxioms NZBase NZMul NZDiv NZMulOrder NZOrder NZDomain NZProperties NZParity NZPow NZSqrt NZLog NZGcd NZBits
Cyclic: Abstract and 31-bits-based cyclic arithmetic
  CyclicAxioms NZCyclic DoubleAdd DoubleBase DoubleCyclic DoubleDiv DoubleDivn1 DoubleLift DoubleMul DoubleSqrt DoubleSub DoubleType Cyclic31 Ring31 Int31 ZModulo
Natural: Abstract and 31-bits-words-based natural arithmetic
  NAdd NAddOrder NAxioms NBase NDefOps NIso NMulOrder NOrder NStrongRec NSub NDiv NMaxMin NParity NPow NSqrt NLog NGcd NLcm NBits NProperties NBinary NPeano NSig NSigNAxioms BigN Nbasic NMake NMake_gen
Integer: Abstract and concrete (especially 31-bits-words-based) integer arithmetic
  ZAdd ZAddOrder ZAxioms ZBase ZLt ZMul ZMulOrder ZSgnAbs ZMaxMin ZParity ZPow ZGcd ZLcm ZBits ZProperties ZDivEucl ZDivFloor ZDivTrunc ZBinary ZNatPairs ZSig ZSigZAxioms BigZ ZMake
Rational: Abstract and 31-bits-words-based rational arithmetic
  QSig BigQ QMake
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 EquivDec SetoidTactics SetoidClass SetoidDec RelationPairs
Setoids:
  Setoid
Lists: Polymorphic lists, Streams (infinite sequences)
  List ListSet SetoidList SetoidPermutation Streams StreamMemo ListTactics
Vectors: Dependent datastructures storing their length
  Fin VectorDef VectorSpec (Vector)
Sorting: Axiomatizations of sorts
  Heap Permutation Sorting PermutEq PermutSetoid Mergesort Sorted
Wellfounded: Well-founded Relations
  Disjoint_Union Inclusion Inverse_Image Lexicographic_Exponentiation Lexicographic_Product 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
  Ascii String
Reals: Formalization of real numbers
  Rdefinitions Raxioms RIneq DiscrR ROrderedType Rminmax (Rbase) RList Ranalysis Rbasic_fun Rderiv Rfunctions Rgeom R_Ifp Rlimit Rseries Rsigma R_sqr Rtrigo_fun Rtrigo1 Rtrigo 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 LegacyRfield (Reals)
Program: Support for dependently-typed programming.
  Basics Wf Subset Equality Tactics Utils Syntax Program Combinators
Unicode: Unicode-based notations
  Utf8_core Utf8