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Chapter 3  The Coq library

The Coq library is structured into three parts:
The initial library:
it contains elementary logical notions and datatypes. 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 6.4.1);

User contributions:
Other specification and proof developments coming from the Coq users' community. These libraries are no longer distributed with the system. They are available by anonymous FTP (see section 3.3).
This chapter briefly reviews these libraries.

3.1  The basic library

This section lists the basic notions and results which are directly available in the standard Coq system 1.

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 precedence and associativity of very common notations, and avoid users to use them with other precedence, which may be confusing.
Notation Precedence Associativity
_ <-> _ 95 no
_ \/ _ 85 right
_ /\ _ 80 right
~ _ 75 right
_ = _ 70 no
_ = _ = _ 70 no
_ = _ :> _ 70 no
_ <> _ 70 no
_ <> _ :> _ 70 no
_ < _ 70 no
_ > _ 70 no
_ <= _ 70 no
_ >= _ 70 no
_ < _ < _ 70 no
_ < _ <= _ 70 no
_ <= _ < _ 70 no
_ <= _ <= _ 70 no
_ + _ 50 left
_ - _ 50 left
_ * _ 40 left
_ / _ 40 left
- _ 35 right
/ _ 35 right
_ ^ _ 30 right

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, P ex A (fun x:A => P)
exists x, P ex _ (fun x => P)
exists2 x:A, P & Q ex2 A (fun x:A => P) (fun x:A => Q)
exists2 x, P & Q ex2 _ (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 <     refl_equal : 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 sym_eq : x = y -> y = x.

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

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

Coq < Theorem sym_not_eq : 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 sym_eq sym_not_eq : 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) (x1 y1:A1) (x2 y2:A2)
Coq <    (x3 y3:A3), 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 } (sigS)
  | { ident : specif & specif & specif } (sigS2)
       
term ::= ( term , term ) (pair)

Figure 3.3: Syntax of datatypes 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.

identity is logically equivalent to equality but it lives in sort Set. Computationaly, it behaves like unit.

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.

3.1.4  Specification

The following notions3 allows to build new datatypes and specifications. They are available with the syntax shown on Figure 3.34.

For instance, given A:Set and P:A->Prop, the construct {x:A | P x} (in abstract syntax (sig A P)) is a Set. 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 Set constructor.



Coq < Inductive sigS (A:Set) (P:A -> Set) : Set := existS (x:A) (_:P x).

Coq < Section sigSprojections.

Coq < Variable A : Set.

Coq < Variable P : A -> Set.

Coq < Definition projS1 (H:sigS A P) := let (x, h) := H in x.

Coq < Definition projS2 (H:sigS A P) :=
Coq <   match H return P (projS1 H) with
Coq <     existS x h => h
Coq <   end.

Coq < End sigSprojections.

Coq < Inductive sigS2 (A: Set) (P Q:A -> Set) : Set :=
Coq <     existS2 (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 := inleft (_:A) | 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 constructs builds a sum between a data type A:Set 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 and Prop in a way which is consistent with the realizability interpretation.

Coq < Definition except := False_rec.

Coq < Notation Except := (except _).

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

Coq < Theorem and_rec :
Coq <  forall (A B:Prop) (P:Set), (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 multiplication5. 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 10.1). 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 (plus p m)
Coq <   end.

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

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

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

Coq < Lemma mult_n_Sm : forall n m:nat, plus (mult n m) n = mult 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, le n m -> le n (S m).

Coq < Infix "+" := plus : 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), 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 induction6.

Coq < Section Well_founded.

Coq < Variable A : Set.

Coq < Variable R : A -> A -> Prop.

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

Coq < Lemma Acc_inv : forall x:A, Acc x -> forall y:A, R y x -> Acc y.
Coq < Section AccRec.

Coq < Variable P : A -> Set.

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

Coq < Fixpoint Acc_rec (x:A) (a:Acc x) {struct a} : P x :=
Coq <   F x (Acc_inv x a)
Coq <     (fun (y:A) (h:R y x) => Acc_rec y (Acc_inv x a y h)).

Coq < End AccRec.

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.
Acc_rec 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 -> Set.

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 definitions7 of the counterparts of some datatypes 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 datatypes at the Type level.

3.2  The standard library

3.2.1  Survey

The rest of the standard library is structured into the following subdirectories:
Logic Classical logic and dependent equality
Arith Basic Peano arithmetic
ZArith Basic integer arithmetic
Bool Booleans (basic functions and results)
Lists Monomorphic and polymorphic lists (basic functions and results), Streams (infinite sequences defined with co-inductive types)
Sets Sets (classical, constructive, finite, infinite, power set, etc.)
IntMap Representation of finite sets by an efficient structure of map (trees indexed by binary integers).
Reals Axiomatization of Real Numbers (classical, basic functions, integer part, fractional part, limit, derivative, Cauchy series, power series and results,... Requires the ZArith library).
Relations Relations (definitions and basic results).
Sorting Sorted list (basic definitions and heapsort correctness).
Wellfounded Well-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 3.2.2 is described the syntax of expressions for integer arithmetics. It is provided by requiring and opening the module ZArith and opening scope Z_scope.

Notation Interpretation Precedence Associativity
_ < _ Zlt    
x <= y Zle    
_ > _ Zgt    
x >= y Zge    
x < y < z x < y /\ y < z    
x < y <= z x < y /\ y <= z    
x <= y < z x <= y /\ y < z    
x <= y <= z x <= y /\ y <= z    
_ ?= _ Zcompare 70 no
_ + _ Zplus    
_ - _ Zminus    
_ * _ Zmult    
_ / _ Zdiv    
_ mod _ Zmod 40 no
- _ Zopp    
_ ^ _ Zpower    

Figure 3.4: Definition of the scope for integer arithmetics (Z_scope)

Figure 3.2.2 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.

Coq < Check  (2 + 3)%Z.
(2 + 3)%Z
     : 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 3.2.3 describes notation available in scope nat_scope.
Notation Interpretation
_ < _ lt
x <= y le
_ > _ gt
x >= y ge
x < y < z x < y /\ y < z
x < y <= z x < y /\ y <= z
x <= y < z x <= y /\ y < z
x <= y <= z x <= 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.
Notation Interpretation
_ < _ Rlt
x <= y Rle
_ > _ Rgt
x >= y Rge
x < y < z x < y /\ y < z
x < y <= z x < y /\ y <= z
x <= y < z x <= y /\ y < z
x <= y <= z x <= 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.

Coq < Check  (2 + 3)%R.
(2 + 3)%R
     : 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. More details are available in document http://coq.inria.fr/~desmettr/Reals.ps.

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:
length length
head first element (with default)
tail all but first element
app concatenation
rev reverse
nth accessing n-th element (with default)
map applying a function
flat_map applying a function returning lists
fold_left iterator (from head to tail)
fold_right iterator (from tail to head)

Table show notations available when opening scope list_scope.
Notation Interpretation Precedence Associativity
_ ++ _ app 60 right
_ :: _ cons 60 right

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 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. There is a small search engine to look for keywords in all contributions. You will also find informations on how to submit a new contribution.

The users' contributions may also be obtained by anonymous FTP from site ftp.inria.fr, in directory INRIA/coq/ and searchable on-line at http://coq.inria.fr/contribs-eng.html

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, and Wf. 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 syntax can be found in the module SpecifSyntax.v
5
This is in module Peano.v
6
This is defined in module Wf.v
7
This is in module Logic_Type.v
8
http://coq.inria.fr
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