Chapter 3  The Coq library

The Coq library is structured into two parts:

The initial library:
it contains elementary logical notions and data-types. It constitutes the basic state of the system directly available when running Coq;
The standard library:
general-purpose libraries containing various developments of Coq axiomatizations about sets, lists, sorting, arithmetic, etc. This library comes with the system and its modules are directly accessible through the Require command (see Section 6.5.1);

In addition, user-provided libraries or developments are provided by Coq users’ community. These libraries and developments are available for download at http://coq.inria.fr (see Section 3.3).

The chapter briefly reviews the Coq libraries.

3.1  The basic library

This section lists the basic notions and results which are directly available in the standard Coq system1.

3.1.1  Notations

This module defines the parsing and pretty-printing of many symbols (infixes, prefixes, etc.). However, it does not assign a meaning to these notations. The purpose of this is to define and fix once for all the precedence and associativity of very common notations. The main notations fixed in the initial state are listed on Figure 3.1.


NotationPrecedenceAssociativity
_ <-> _95no
_ \/ _85right
_ /\ _80right
~ _75right
_ = _70no
_ = _ = _70no
_ = _ :> _70no
_ <> _70no
_ <> _ :> _70no
_ < _70no
_ > _70no
_ <= _70no
_ >= _70no
_ < _ < _70no
_ < _ <= _70no
_ <= _ < _70no
_ <= _ <= _70no
_ + _50left
_ || _50left
_ - _50left
_ * _40left
_ && _40left
_ / _40left
- _35right
/ _35right
_ ^ _30right
Figure 3.1: Notations in the initial state

3.1.2  Logic


form::=True(True)
 |False(False)
 |~ form(not)
 |form /\ form(and)
 |form \/ form(or)
 |form -> form(primitive implication)
 |form <-> form(iff)
 |forall ident : type , form(primitive for all)
 |exists ident [: specif] , form(ex)
 |exists2 ident [: specif] , form & form(ex2)
 |term = term(eq)
 |term = term :> specif(eq)
Figure 3.2: Syntax of formulas

The basic library of Coq comes with the definitions of standard (intuitionistic) logical connectives (they are defined as inductive constructions). They are equipped with an appealing syntax enriching the (subclass form) of the syntactic class term. The syntax extension is shown on Figure 3.2.


Remark: Implication is not defined but primitive (it is a non-dependent product of a proposition over another proposition). There is also a primitive universal quantification (it is a dependent product over a proposition). The primitive universal quantification allows both first-order and higher-order quantification.

Propositional Connectives

First, we find propositional calculus connectives:

Coq < Inductive True : Prop := I.

Coq < Inductive False :  Prop := .

Coq < Definition not (A: Prop) := A -> False.

Coq < Inductive and (A B:Prop) : Prop := conj (_:A) (_:B).

Coq < Section Projections.

Coq < Variables A B : Prop.

Coq < Theorem proj1 : A /\ B -> A.

Coq < Theorem proj2 : A /\ B -> B.

Coq < End Projections.

Coq < Inductive or (A B:Prop) : Prop :=
Coq <   | or_introl (_:A)
Coq <   | or_intror (_:B).

Coq < Definition iff (P Q:Prop) := (P -> Q) /\ (Q -> P).

Coq < Definition IF_then_else (P Q R:Prop) := P /\ Q \~ P /\ R.

Quantifiers

Then we find first-order quantifiers:

Coq < Definition all (A:Set) (P:A -> Prop) := forall x:A, P x.

Coq < Inductive ex (A: Set) (P:A -> Prop) : Prop :=
Coq <     ex_intro (x:A) (_:P x).

Coq < Inductive ex2 (A:Set) (P Q:A -> Prop) : Prop :=
Coq <     ex_intro2 (x:A) (_:P x) (_:Q x).

The following abbreviations are allowed:

exists x:A, Pex A (fun x:A => P)
exists x, Pex _ (fun x => P)
exists2 x:A, P & Qex2 A (fun x:A => P) (fun x:A => Q)
exists2 x, P & Qex2 _ (fun x => P) (fun x => Q)

The type annotation “:A” can be omitted when A can be synthesized by the system.

Equality

Then, we find equality, defined as an inductive relation. That is, given a type A and an x of type A, the predicate (eq A x) is the smallest one which contains x. This definition, due to Christine Paulin-Mohring, is equivalent to define eq as the smallest reflexive relation, and it is also equivalent to Leibniz’ equality.

Coq < Inductive eq (A:Type) (x:A) : A -> Prop :=
Coq <     eq_refl : eq A x x.

Lemmas

Finally, a few easy lemmas are provided.

Coq < Theorem absurd : forall A C:Prop, A -> ~ A -> C.

Coq < Section equality.

Coq < Variables A B : Type.

Coq < Variable f : A -> B.

Coq < Variables x y z : A.

Coq < Theorem eq_sym : x = y -> y = x.

Coq < Theorem eq_trans : x = y -> y = z -> x = z.

Coq < Theorem f_equal : x = y -> f x = f y.

Coq < Theorem not_eq_sym : x <> y -> y <> x.

Coq < End equality.

Coq < Definition eq_ind_r :
Coq <   forall (A:Type) (x:A) (P:A->Prop), P x -> forall y:A, y = x -> P y.

Coq < Definition eq_rec_r :
Coq <   forall (A:Type) (x:A) (P:A->Set), P x -> forall y:A, y = x -> P y.

Coq < Definition eq_rect_r :
Coq <   forall (A:Type) (x:A) (P:A->Type), P x -> forall y:A, y = x -> P y.
Coq < Hint Immediate eq_sym not_eq_sym : core.

The theorem f_equal is extended to functions with two to five arguments. The theorem are names f_equal2, f_equal3, f_equal4 and f_equal5. For instance f_equal3 is defined the following way.

Coq < Theorem f_equal3 :
Coq <  forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B)
Coq <    (x1 y1:A1) (x2 y2:A2) (x3 y3:A3),
Coq <    x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.

3.1.3  Datatypes


specif::=specif * specif(prod)
 |specif + specif(sum)
 |specif + { specif }(sumor)
 |{ specif } + { specif } (sumbool)
 |{ ident : specif | form }(sig)
 |{ ident : specif | form & form }(sig2)
 |{ ident : specif & specif }(sigT)
 |{ ident : specif & specif & specif }(sigT2)
    
term::=( term , term )(pair)
Figure 3.3: Syntax of data-types and specifications

In the basic library, we find the definition2 of the basic data-types of programming, again defined as inductive constructions over the sort Set. Some of them come with a special syntax shown on Figure 3.3.

Programming

Coq < Inductive unit : Set := tt.

Coq < Inductive bool : Set := true | false.

Coq < Inductive nat : Set := O | S (n:nat).

Coq < Inductive option (A:Set) : Set := Some (_:A) | None.

Coq < Inductive identity (A:Type) (a:A) : A -> Type :=
Coq <     refl_identity : identity A a a.

Note that zero is the letter O, and not the numeral 0.

The predicate identity is logically equivalent to equality but it lives in sort Type. It is mainly maintained for compatibility.

We then define the disjoint sum of A+B of two sets A and B, and their product A*B.

Coq < Inductive sum (A B:Set) : Set := inl (_:A) | inr (_:B).

Coq < Inductive prod (A B:Set) : Set := pair (_:A) (_:B).

Coq < Section projections.

Coq < Variables A B : Set.

Coq < Definition fst (H: prod A B) := match H with
Coq <                                 | pair x y => x
Coq <                                 end.

Coq < Definition snd (H: prod A B) := match H with
Coq <                                 | pair x y => y
Coq <                                 end.

Coq < End projections.

Some operations on bool are also provided: andb (with infix notation &&), orb (with infix notation ||), xorb, implb and negb.

3.1.4  Specification

The following notions3 allow to build new data-types and specifications. They are available with the syntax shown on Figure 3.3.

For instance, given A:Type and P:A->Prop, the construct {x:A | P x} (in abstract syntax (sig A P)) is a Type. We may build elements of this set as (exist x p) whenever we have a witness x:A with its justification p:P x.

From such a (exist x p) we may in turn extract its witness x:A (using an elimination construct such as match) but not its justification, which stays hidden, like in an abstract data-type. In technical terms, one says that sig is a “weak (dependent) sum”. A variant sig2 with two predicates is also provided.

Coq < Inductive sig (A:Set) (P:A -> Prop) : Set := exist (x:A) (_:P x).

Coq < Inductive sig2 (A:Set) (P Q:A -> Prop) : Set := 
Coq <   exist2 (x:A) (_:P x) (_:Q x).

A “strong (dependent) sum” {x:A & P x} may be also defined, when the predicate P is now defined as a constructor of types in Type.

Coq < Inductive sigT (A:Type) (P:A -> Type) : Type := existT (x:A) (_:P x).

Coq < Section Projections.

Coq < Variable A : Type.

Coq < Variable P : A -> Type.

Coq < Definition projT1 (H:sigT A P) := let (x, h) := H in x.

Coq < Definition projT2 (H:sigT A P) :=
Coq <   match H return P (projT1 H) with
Coq <     existT x h => h
Coq <   end.

Coq < End Projections.

Coq < Inductive sigT2 (A: Type) (P Q:A -> Type) : Type :=
Coq <     existT2 (x:A) (_:P x) (_:Q x).

A related non-dependent construct is the constructive sum {A}+{B} of two propositions A and B.

Coq < Inductive sumbool (A B:Prop) : Set := left (_:A) | right (_:B).

This sumbool construct may be used as a kind of indexed boolean data-type. An intermediate between sumbool and sum is the mixed sumor which combines A:Set and B:Prop in the Set A+{B}.

Coq < Inductive sumor (A:Set) (B:Prop) : Set :=
Coq < | inleft (_:A)
Coq < | inright (_:B).

We may define variants of the axiom of choice, like in Martin-Löf’s Intuitionistic Type Theory.

Coq < Lemma Choice :
Coq <  forall (S S’:Set) (R:S -> S’ -> Prop),
Coq <    (forall x:S, {y : S’ | R x y}) ->
Coq <    {f : S -> S’ | forall z:S, R z (f z)}.

Coq < Lemma Choice2 :
Coq <  forall (S S’:Set) (R:S -> S’ -> Set),
Coq <    (forall x:S, {y : S’ &  R x y}) ->
Coq <    {f : S -> S’ &  forall z:S, R z (f z)}.

Coq < Lemma bool_choice :
Coq <  forall (S:Set) (R1 R2:S -> Prop),
Coq <    (forall x:S, {R1 x} + {R2 x}) ->
Coq <    {f : S -> bool |
Coq <    forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x}.

The next construct builds a sum between a data-type A:Type and an exceptional value encoding errors:

Coq < Definition Exc := option.

Coq < Definition value := Some.

Coq < Definition error := None.

This module ends with theorems, relating the sorts Set or Type and Prop in a way which is consistent with the realizability interpretation.

Coq < Definition except := False_rec.

Coq < Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C.

Coq < Theorem and_rect :
Coq <  forall (A B:Prop) (P:Type), (A -> B -> P) -> A /\ B -> P.

3.1.5  Basic Arithmetics

The basic library includes a few elementary properties of natural numbers, together with the definitions of predecessor, addition and multiplication4. It also provides a scope nat_scope gathering standard notations for common operations (+, *) and a decimal notation for numbers. That is he can write 3 for (S (S (S O))). This also works on the left hand side of a match expression (see for example section 8.2.3). This scope is opened by default.

The following example is not part of the standard library, but it shows the usage of the notations:

Coq < Fixpoint even (n:nat) : bool :=
Coq <   match n with
Coq <   | 0 => true
Coq <   | 1 => false
Coq <   | S (S n) => even n
Coq <   end.

Coq < Theorem eq_S : forall x y:nat, x = y -> S x = S y.
Coq < Definition pred (n:nat) : nat :=
Coq <   match n with
Coq <   | 0 => 0
Coq <   | S u => u
Coq <   end.

Coq < Theorem pred_Sn : forall m:nat, m = pred (S m).

Coq < Theorem eq_add_S : forall n m:nat, S n = S m -> n = m.

Coq < Hint Immediate eq_add_S : core.

Coq < Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m.
Coq < Definition IsSucc (n:nat) : Prop :=
Coq <   match n with
Coq <   | 0 => False
Coq <   | S p => True
Coq <   end.

Coq < Theorem O_S : forall n:nat, 0 <> S n.

Coq < Theorem n_Sn : forall n:nat, n <> S n.
Coq < Fixpoint plus (n m:nat) {struct n} : nat :=
Coq <   match n with
Coq <   | 0 => m
Coq <   | S p => S (p + m)
Coq <   end.

Coq < where "n + m" := (plus n m) : nat_scope.

Coq < Lemma plus_n_O : forall n:nat, n = n + 0.

Coq < Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
Coq < Fixpoint mult (n m:nat) {struct n} : nat :=
Coq <   match n with
Coq <   | 0 => 0
Coq <   | S p => m + p * m
Coq <   end.

Coq < where "n * m" := (mult n m) : nat_scope.

Coq < Lemma mult_n_O : forall n:nat, 0 = n * 0.

Coq < Lemma mult_n_Sm : forall n m:nat, n * m + n = n * (S m).

Finally, it gives the definition of the usual orderings le, lt, ge, and gt.

Coq < Inductive le (n:nat) : nat -> Prop :=
Coq <   | le_n : le n n
Coq <   | le_S : forall m:nat, n <= m -> n <= (S m).

Coq < where "n <= m" := (le n m) : nat_scope.

Coq < Definition lt (n m:nat) := S n <= m.

Coq < Definition ge (n m:nat) := m <= n.

Coq < Definition gt (n m:nat) := m < n.

Properties of these relations are not initially known, but may be required by the user from modules Le and Lt. Finally, Peano gives some lemmas allowing pattern-matching, and a double induction principle.

Coq < Theorem nat_case :
Coq <  forall (n:nat) (P:nat -> Prop), 
Coq <  P 0 -> (forall m:nat, P (S m)) -> P n.
Coq < Theorem nat_double_ind :
Coq <  forall R:nat -> nat -> Prop,
Coq <    (forall n:nat, R 0 n) ->
Coq <    (forall n:nat, R (S n) 0) ->
Coq <    (forall n m:nat, R n m -> R (S n) (S m)) -> forall n m:nat, R n m.

3.1.6  Well-founded recursion

The basic library contains the basics of well-founded recursion and well-founded induction5.

Coq < Section Well_founded.

Coq < Variable A : Type.

Coq < Variable R : A -> A -> Prop.

Coq < Inductive Acc (x:A) : Prop :=
Coq <     Acc_intro : (forall y:A, R y x -> Acc y) -> Acc x.

Coq < Lemma Acc_inv : Acc x -> forall y:A, R y x -> Acc y.
Coq < Definition well_founded := forall a:A, Acc a.

Coq < Hypothesis Rwf : well_founded.

Coq < Theorem well_founded_induction :
Coq <  forall P:A -> Set,
Coq <    (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a.

Coq < Theorem well_founded_ind :
Coq <  forall P:A -> Prop,
Coq <    (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a.

The automatically generated scheme Acc_rect can be used to define functions by fixpoints using well-founded relations to justify termination. Assuming extensionality of the functional used for the recursive call, the fixpoint equation can be proved.

Coq < Section FixPoint.

Coq < Variable P : A -> Type.

Coq < Variable F : forall x:A, (forall y:A, R y x -> P y) -> P x.

Coq < Fixpoint Fix_F (x:A) (r:Acc x) {struct r} : P x :=
Coq <   F x (fun (y:A) (p:R y x) => Fix_F y (Acc_inv x r y p)).

Coq < Definition Fix (x:A) := Fix_F x (Rwf x).

Coq < Hypothesis F_ext :
Coq <     forall (x:A) (f g:forall y:A, R y x -> P y),
Coq <       (forall (y:A) (p:R y x), f y p = g y p) -> F x f = F x g.

Coq < Lemma Fix_F_eq :
Coq <  forall (x:A) (r:Acc x),
Coq <    F x (fun (y:A) (p:R y x) => Fix_F y (Acc_inv x r y p)) = Fix_F x r.

Coq < Lemma Fix_F_inv : forall (x:A) (r s:Acc x), Fix_F x r = Fix_F x s.

Coq < Lemma fix_eq : forall x:A, Fix x = F x (fun (y:A) (p:R y x) => Fix y).
Coq < End FixPoint.

Coq < End Well_founded.

3.1.7  Accessing the Type level

The basic library includes the definitions6 of the counterparts of some data-types and logical quantifiers at the Type level: negation, pair, and properties of identity.

Coq < Definition notT (A:Type) := A -> False.

Coq < Inductive prodT (A B:Type) : Type := pairT (_:A) (_:B).

At the end, it defines data-types at the Type level.

3.1.8  Tactics

A few tactics defined at the user level are provided in the initial state7.

3.2  The standard library

3.2.1  Survey

The rest of the standard library is structured into the following subdirectories:

LogicClassical logic and dependent equality
ArithBasic Peano arithmetic
PArithBasic positive integer arithmetic
NArithBasic binary natural number arithmetic
ZArithBasic relative integer arithmetic
NumbersVarious approaches to natural, integer and cyclic numbers (currently axiomatically and on top of 231 binary words)
BoolBooleans (basic functions and results)
ListsMonomorphic and polymorphic lists (basic functions and results), Streams (infinite sequences defined with co-inductive types)
SetsSets (classical, constructive, finite, infinite, power set, etc.)
FSetsSpecification and implementations of finite sets and finite maps (by lists and by AVL trees)
RealsAxiomatization of real numbers (classical, basic functions, integer part, fractional part, limit, derivative, Cauchy series, power series and results,...)
RelationsRelations (definitions and basic results)
SortingSorted list (basic definitions and heapsort correctness)
Strings8-bits characters and strings
WellfoundedWell-founded relations (basic results)


These directories belong to the initial load path of the system, and the modules they provide are compiled at installation time. So they are directly accessible with the command Require (see Chapter 6).

The different modules of the Coq standard library are described in the additional document Library.dvi. They are also accessible on the WWW through the Coq homepage 8.

3.2.2  Notations for integer arithmetics

On Figure 3.4 is described the syntax of expressions for integer arithmetics. It is provided by requiring and opening the module ZArith and opening scope Z_scope.


NotationInterpretationPrecedenceAssociativity
_ < _Z.lt  
x <= yZ.le  
_ > _Z.gt  
x >= yZ.ge  
x < y < zx < y /\ y < z  
x < y <= zx < y /\ y <= z  
x <= y < zx <= y /\ y < z  
x <= y <= zx <= y /\ y <= z  
_ ?= _Z.compare70no
_ + _Z.add  
_ - _Z.sub  
_ * _Z.mul  
_ / _Z.div  
_ mod _Z.modulo40no
- _Z.opp  
_ ^ _Z.pow  
Figure 3.4: Definition of the scope for integer arithmetics (Z_scope)

Figure 3.4 shows the notations provided by Z_scope. It specifies how notations are interpreted and, when not already reserved, the precedence and associativity.

Coq < Require Import ZArith.
[Loading ML file z_syntax_plugin.cmxs ... done]
[Loading ML file quote_plugin.cmxs ... done]
[Loading ML file newring_plugin.cmxs ... done]
[Loading ML file omega_plugin.cmxs ... done]

Coq < Check  (2 + 3)\char045Z.
(2 + 3)\char045Z
     : Z

Coq < Open Scope Z_scope.

Coq < Check 2 + 3.
2 + 3
     : Z

3.2.3  Peano’s arithmetic (nat)

While in the initial state, many operations and predicates of Peano’s arithmetic are defined, further operations and results belong to other modules. For instance, the decidability of the basic predicates are defined here. This is provided by requiring the module Arith.

Figure 3.5 describes notation available in scope nat_scope.


NotationInterpretation
_ < _lt
x <= yle
_ > _gt
x >= yge
x < y < zx < y /\ y < z
x < y <= zx < y /\ y <= z
x <= y < zx <= y /\ y < z
x <= y <= zx <= y /\ y <= z
_ + _plus
_ - _minus
_ * _mult
Figure 3.5: Definition of the scope for natural numbers (nat_scope)

3.2.4  Real numbers library

Notations for real numbers

This is provided by requiring and opening the module Reals and opening scope R_scope. This set of notations is very similar to the notation for integer arithmetics. The inverse function was added.


NotationInterpretation
_ < _Rlt
x <= yRle
_ > _Rgt
x >= yRge
x < y < zx < y /\ y < z
x < y <= zx < y /\ y <= z
x <= y < zx <= y /\ y < z
x <= y <= zx <= y /\ y <= z
_ + _Rplus
_ - _Rminus
_ * _Rmult
_ / _Rdiv
- _Ropp
/ _Rinv
_ ^ _pow
Figure 3.6: Definition of the scope for real arithmetics (R_scope)

Coq < Require Import Reals.
[Loading ML file r_syntax_plugin.cmxs ... done]
[Loading ML file ring_plugin.cmxs ... done]
[Loading ML file field_plugin.cmxs ... done]
[Loading ML file fourier_plugin.cmxs ... done]

Coq < Check  (2 + 3)\char045R.
(2 + 3)\char045R
     : R

Coq < Open Scope R_scope.

Coq < Check 2 + 3.
2 + 3
     : R

Some tactics

In addition to the ring, field and fourier tactics (see Chapter 8) there are:

All this tactics has been written with the tactic language Ltac described in Chapter 9.

3.2.5  List library

Some elementary operations on polymorphic lists are defined here. They can be accessed by requiring module List.

It defines the following notions:

lengthlength
headfirst element (with default)
tailall but first element
appconcatenation
revreverse
nthaccessing n-th element (with default)
mapapplying a function
flat_mapapplying a function returning lists
fold_leftiterator (from head to tail)
fold_rightiterator (from tail to head)

Table show notations available when opening scope list_scope.


NotationInterpretationPrecedenceAssociativity
_ ++ _app60right
_ :: _cons60right
Figure 3.7: Definition of the scope for lists (list_scope)

3.3  Users’ contributions

Numerous users’ contributions have been collected and are available at URL http://coq.inria.fr/contribs/. On this web page, you have a list of all contributions with informations (author, institution, quick description, etc.) and the possibility to download them one by one. You will also find informations on how to submit a new contribution.


1
Most of these constructions are defined in the Prelude module in directory theories/Init at the Coq root directory; this includes the modules Notations, Logic, Datatypes, Specif, Peano, Wf and Tactics. Module Logic_Type also makes it in the initial state
2
They are in Datatypes.v
3
They are defined in module Specif.v
4
This is in module Peano.v
5
This is defined in module Wf.v
6
This is in module Logic_Type.v
7
This is in module Tactics.v
8
http://coq.inria.fr