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
Requirecommand (see Section 6.4.1);
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.
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 _ - _50 left _ * _40 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 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 < | 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 < 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 < refl_equal : eq A x x.
Lemmas
Finally, a few easy lemmas are provided.
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 < 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.
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 < 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 } (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 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 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 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 < 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.
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}.
We may define variants of the axiom of choice, like in Martin-Löf’s Intuitionistic Type Theory.
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 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 < 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 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 < match n with
Coq < | 0 => true
Coq < | 1 => false
Coq < | S (S n) => even n
Coq < end.
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 < 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 < 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 < 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 < | 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 < forall (n:nat) (P:nat -> Prop), P 0 -> (forall m:nat, P (S m)) -> P n.
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 < 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 < 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 < 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 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 < 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:
| Logic | Classical logic and dependent equality |
| Arith | Basic Peano arithmetic |
| NArith | Basic positive integer arithmetic |
| ZArith | Basic relative integer arithmetic |
| Numbers | Various approaches to natural, integer and cyclic numbers (currently axiomatically and on top of 231 binary words) |
| 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.) |
| FSets | Specification and implementations of finite sets and finite maps (by lists and by AVL trees) |
| Reals | Axiomatization of real numbers (classical, basic functions, integer part, fractional part, limit, derivative, Cauchy series, power series and results,...) |
| Relations | Relations (definitions and basic results) |
| Sorting | Sorted list (basic definitions and heapsort correctness) |
| Strings | 8-bits characters and strings |
| 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 <= yZle _ > _Zgt x >= yZge x < y < zx < y /\y < zx < y <= zx < y /\y <= zx <= y < zx <= y /\y < zx <= y <= zx <= 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 < 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 <= yle _ > _gt x >= yge x < y < zx < y /\y < zx < y <= zx < y /\y <= zx <= y < zx <= y /\y < zx <= 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.
Notation Interpretation _ < _Rlt x <= yRle _ > _Rgt x >= yRge x < y < zx < y /\y < zx < y <= zx < y /\y <= zx <= y < zx <= y /\y < zx <= 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 < 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:
-
discrR
Proves that a real integer constant c1 is different from another real integer constant c2.
Coq < Require Import DiscrR.
Coq < Goal 5 <> 0.
Coq < discrR.
Proof completed.
- split_Rabs allows to unfold Rabs constant and splits
corresponding conjonctions.
Coq < Require Import SplitAbsolu.
Coq < Goal forall x:R, x <= Rabs x.
Coq < intro; split_Rabs.
2 subgoals
x : R
r : x < 0
============================
x <= - x
subgoal 2 is:
x <= x
- split_Rmult allows to split a condition that a product is
non null into subgoals corresponding to the condition on each
operand of the product.
Coq < Require Import SplitRmult.
Coq < Goal forall x y z:R, x * y * z <> 0.
Coq < intros; split_Rmult.
3 subgoals
x : R
y : R
z : R
============================
x <> 0
subgoal 2 is:
y <> 0
subgoal 3 is:
z <> 0
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, 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