The Coq library¶
The Coq library has two parts:
The prelude: definitions and theorems for the most commonly used elementary logical notions and data types. Coq normally loads these files automatically when it starts.
The standard library: general-purpose libraries with definitions and theorems for sets, lists, sorting, arithmetic, etc. To use these files, users must load them explicitly with the
Requirecommand (see Compiled files)
There are also many libraries provided by Coq users' community. These libraries and developments are available for download at http://coq.inria.fr (see Users’ contributions).
This chapter briefly reviews the Coq libraries whose contents can also be browsed at http://coq.inria.fr/stdlib/.
The prelude¶
This section lists the basic notions and results which are directly
available in the standard Coq system. Most of these constructions
are defined in the Prelude module in directory theories/Init
in 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.
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 :
Notation |
Precedence |
Associativity |
|---|---|---|
|
99 |
right |
|
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 |
Logic¶
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 constructs
for form are:
True |
True |
False |
False |
|
not |
and |
|
or |
|
primitive implication |
|
iff |
|
primitive for all |
|
ex |
|
ex2 |
|
eq |
|
eq |
Note
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:
Quantifiers¶
Then we find first-order quantifiers:
- Definition all (A:Set) (P:A -> Prop) := forall x:A, P x.
- all is defined
- Inductive ex (A: Set) (P:A -> Prop) : Prop := ex_intro (x:A) (_:P x).
- ex is defined ex_ind is defined ex_sind is defined
- Inductive ex2 (A:Set) (P Q:A -> Prop) : Prop := ex_intro2 (x:A) (_:P x) (_:Q x).
- ex2 is defined ex2_ind is defined ex2_sind is defined
The following abbreviations are allowed:
|
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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.
- Inductive eq (A:Type) (x:A) : A -> Prop := eq_refl : eq A x x.
- eq is defined eq_rect is defined eq_ind is defined eq_rec is defined eq_sind is defined
Lemmas¶
Finally, a few easy lemmas are provided.
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.
- Theorem f_equal3 : forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) (x1 y1:A1) (x2 y2:A2) (x3 y3:A3), x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.
- 1 subgoal ============================ forall (A1 A2 A3 B : Type) (f : A1 -> A2 -> A3 -> B) (x1 y1 : A1) (x2 y2 : A2) (x3 y3 : A3), x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3
Datatypes¶
In the basic library, we find in Datatypes.v the definition
of the basic data-types of programming,
defined as inductive constructions over the sort Set. Some of
them come with a special syntax shown below (this syntax table is common with
the next section Specification). The constructs for specif are:
prod |
|
sum |
|
sumor |
|
sumbool |
|
sig |
|
sig2 |
|
sigT |
|
sigT2 |
The notation for pairs (elements of type prod) is: (term, term)
Programming¶
- Inductive unit : Set := tt.
- unit is defined unit_rect is defined unit_ind is defined unit_rec is defined unit_sind is defined
- Inductive bool : Set := true | false.
- bool is defined bool_rect is defined bool_ind is defined bool_rec is defined bool_sind is defined
- Inductive nat : Set := O | S (n:nat).
- nat is defined nat_rect is defined nat_ind is defined nat_rec is defined nat_sind is defined
- Inductive option (A:Set) : Set := Some (_:A) | None.
- option is defined option_rect is defined option_ind is defined option_rec is defined option_sind is defined
- Inductive identity (A:Type) (a:A) : A -> Type := refl_identity : identity A a a.
- identity is defined identity_rect is defined identity_ind is defined identity_rec is defined identity_sind is defined
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.
- Inductive sum (A B:Set) : Set := inl (_:A) | inr (_:B).
- sum is defined sum_rect is defined sum_ind is defined sum_rec is defined sum_sind is defined
- Inductive prod (A B:Set) : Set := pair (_:A) (_:B).
- prod is defined prod_rect is defined prod_ind is defined prod_rec is defined prod_sind is defined
- Section projections.
- Variables A B : Set.
- A is declared B is declared
- Definition fst (H: prod A B) := match H with | pair _ _ x y => x end.
- fst is defined
- Definition snd (H: prod A B) := match H with | pair _ _ x y => y end.
- snd is defined
- End projections.
Some operations on bool are also provided: andb (with
infix notation &&), orb (with
infix notation ||), xorb, implb and negb.
Specification¶
The following notions defined in module Specif.v allow to build new data-types and specifications.
They are available with the syntax shown in the previous section Datatypes.
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.
- Inductive sig (A:Set) (P:A -> Prop) : Set := exist (x:A) (_:P x).
- sig is defined sig_rect is defined sig_ind is defined sig_rec is defined sig_sind is defined
- Inductive sig2 (A:Set) (P Q:A -> Prop) : Set := exist2 (x:A) (_:P x) (_:Q x).
- sig2 is defined sig2_rect is defined sig2_ind is defined sig2_rec is defined sig2_sind is defined
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.
- Inductive sigT (A:Type) (P:A -> Type) : Type := existT (x:A) (_:P x).
- sigT is defined sigT_rect is defined sigT_ind is defined sigT_rec is defined sigT_sind is defined
- Section Projections2.
- Variable A : Type.
- A is declared
- Variable P : A -> Type.
- P is declared
- Definition projT1 (H:sigT A P) := let (x, h) := H in x.
- projT1 is defined
- Definition projT2 (H:sigT A P) := match H return P (projT1 H) with existT _ _ x h => h end.
- projT2 is defined
- End Projections2.
- Inductive sigT2 (A: Type) (P Q:A -> Type) : Type := existT2 (x:A) (_:P x) (_:Q x).
- sigT2 is defined sigT2_rect is defined sigT2_ind is defined sigT2_rec is defined sigT2_sind is defined
A related non-dependent construct is the constructive sum
{A}+{B} of two propositions A and B.
- Inductive sumbool (A B:Prop) : Set := left (_:A) | right (_:B).
- sumbool is defined sumbool_rect is defined sumbool_ind is defined sumbool_rec is defined sumbool_sind is defined
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 construction A+{B} in Set.
- Inductive sumor (A:Set) (B:Prop) : Set := | inleft (_:A) | inright (_:B).
- sumor is defined sumor_rect is defined sumor_ind is defined sumor_rec is defined sumor_sind is defined
We may define variants of the axiom of choice, like in Martin-Löf's Intuitionistic Type Theory.
The next construct builds a sum between a data-type A:Type and
an exceptional value encoding errors:
- Definition Exc := option.
- Exc is defined
- Definition value := Some.
- value is defined
- Definition error := None.
- error is defined
This module ends with theorems, relating the sorts Set or
Type and Prop in a way which is consistent with the
realizability interpretation.
Basic Arithmetic¶
The basic library includes a few elementary properties of natural
numbers, together with the definitions of predecessor, addition and
multiplication, in module Peano.v. It also
provides a scope nat_scope gathering standard notations for
common operations (+, *) and a decimal notation for
numbers, allowing for instance to write 3 for S (S (S O))). This also works on
the left hand side of a match expression (see for example
section refine). This scope is opened by default.
Example
The following example is not part of the standard library, but it shows the usage of the notations:
- Fixpoint even (n:nat) : bool := match n with | 0 => true | 1 => false | S (S n) => even n end.
- even is defined even is recursively defined (guarded on 1st argument)
Now comes the content of module Peano:
Finally, it gives the definition of the usual orderings le,
lt, ge and gt.
- Inductive le (n:nat) : nat -> Prop := | le_n : le n n | le_S : forall m:nat, n <= m -> n <= (S m) where "n <= m" := (le n m) : nat_scope.
- Toplevel input, characters 0-137: > Inductive le (n:nat) : nat -> Prop := | le_n : le n n | le_S : forall m:nat, n <= m -> n <= (S m) where "n <= m" := (le n m) : nat_scope. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Notation "_ <= _" was already used in scope nat_scope. [notation-overridden,parsing] le is defined le_ind is defined le_sind is defined Toplevel input, characters 0-137: > Inductive le (n:nat) : nat -> Prop := | le_n : le n n | le_S : forall m:nat, n <= m -> n <= (S m) where "n <= m" := (le n m) : nat_scope. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Notation "_ <= _" was already used in scope nat_scope. [notation-overridden,parsing]
- Definition lt (n m:nat) := S n <= m.
- lt is defined
- Definition ge (n m:nat) := m <= n.
- ge is defined
- Definition gt (n m:nat) := m < n.
- gt is defined
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.
Well-founded recursion¶
The basic library contains the basics of well-founded recursion and
well-founded induction, in module Wf.v.
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.
Accessing the Type level¶
The standard library includes Type level definitions of counterparts of some
logic concepts and basic lemmas about them.
The module Datatypes defines identity, which is the Type level counterpart
of equality:
- Inductive identity (A:Type) (a:A) : A -> Type := identity_refl : identity A a a.
- identity is defined identity_rect is defined identity_ind is defined identity_rec is defined identity_sind is defined
Some properties of identity are proved in the module Logic_Type, which also
provides the definition of Type level negation:
- Definition notT (A:Type) := A -> False.
- notT is defined
Tactics¶
A few tactics defined at the user level are provided in the initial
state, in module Tactics.v. They are listed at
http://coq.inria.fr/stdlib, in paragraph Init, link Tactics.
The standard library¶
Survey¶
The rest of the standard library is structured into the following subdirectories:
Logic : Classical logic and dependent equality
Arith : Basic Peano arithmetic
PArith : Basic positive integer arithmetic
NArith : Basic binary natural number arithmetic
ZArith : Basic relative integer arithmetic
Numbers : Various approaches to natural, integer and cyclic numbers (currently axiomatically and on top of 2^31 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,...)
Floats : Machine implementation of floating-point arithmetic (for the binary64 format)
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
Section Compiled files).
The different modules of the Coq standard library are documented online at https://coq.inria.fr/stdlib.
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.
The following table describes the notations available in scope
nat_scope :
Notation |
Interpretation |
|---|---|
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Notations for integer arithmetic¶
The following table describes the syntax of expressions
for integer arithmetic. It is provided by requiring and opening the module ZArith and opening scope Z_scope.
It specifies how notations are interpreted and, when not
already reserved, the precedence and associativity.
Notation |
Interpretation |
Precedence |
Associativity |
|---|---|---|---|
|
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||
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||
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||
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||
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||
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70 |
no |
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40 |
no |
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||
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Example
- Require Import ZArith.
- [Loading ML file ring_plugin.cmxs ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done]
- Check (2 + 3)%Z.
- (2 + 3)%Z : Z
- Open Scope Z_scope.
- Check 2 + 3.
- 2 + 3 : Z
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 arithmetic. The inverse function was added.
Notation |
Interpretation |
|---|---|
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Example
- Require Import Reals.
- Check (2 + 3)%R.
- (2 + 3)%R : R
- Open Scope R_scope.
- Check 2 + 3.
- 2 + 3 : R
Some tactics for real numbers¶
In addition to the powerful ring, field and lra
tactics (see Chapter Tactics), there are also:
-
Tactic
discrR¶ Proves that two real integer constants are different.
Example
- Require Import DiscrR.
- Open Scope R_scope.
- Goal 5 <> 0.
- 1 subgoal ============================ 5 <> 0
- discrR.
-
Tactic
split_Rabs¶ Allows unfolding the
Rabsconstant and splits corresponding conjunctions.
Example
- Require Import Reals.
- Open Scope R_scope.
- Goal forall x:R, x <= Rabs x.
- 1 subgoal ============================ forall x : R, x <= Rabs x
- intro; split_Rabs.
- 2 subgoals x : R Hlt : x < 0 ============================ x <= - x subgoal 2 is: x <= x
-
Tactic
split_Rmult¶ Splits a condition that a product is non null into subgoals corresponding to the condition on each operand of the product.
Example
- Require Import Reals.
- Open Scope R_scope.
- Goal forall x y z:R, x * y * z <> 0.
- 1 subgoal ============================ forall x y z : R, x * y * z <> 0
- intros; split_Rmult.
- 3 subgoals x, y, z : R ============================ x <> 0 subgoal 2 is: y <> 0 subgoal 3 is: z <> 0
These tactics has been written with the tactic language Ltac
described in Chapter Ltac.
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
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)
The following table shows notations available when opening scope list_scope.
Notation |
Interpretation |
Precedence |
Associativity |
|---|---|---|---|
|
|
60 |
right |
|
|
60 |
right |
Floats library¶
The library of primitive floating-point arithmetic can be loaded by
requiring module Floats:
- Require Import Floats.
- [Loading ML file int63_syntax_plugin.cmxs ... done] [Loading ML file float_syntax_plugin.cmxs ... done]
It exports the module PrimFloat that provides a primitive type
named float, defined in the kernel (see section Primitive Floats),
as well as two variant types float_comparison and float_class:
- Print float.
- *** [ float : Set ]
- Print float_comparison.
- Variant float_comparison : Set := FEq : float_comparison | FLt : float_comparison | FGt : float_comparison | FNotComparable : float_comparison
- Print float_class.
- Variant float_class : Set := PNormal : float_class | NNormal : float_class | PSubn : float_class | NSubn : float_class | PZero : float_class | NZero : float_class | PInf : float_class | NInf : float_class | NaN : float_class
It then defines the primitive operators below, using the processor floating-point operators for binary64 in rounding-to-nearest even:
absoppsubaddmuldivsqrtcompare: compare two floats and return afloat_comparisonclassify: analyze a float and return afloat_classof_int63: round a primitive integer and convert it into a floatnormfr_mantissa: take a float in[0.5; 1.0)and return its mantissafrshiftexp: convert a float to fractional part in[0.5; 1.0)and integer partldshiftexp: multiply a float by an integral power of2next_up: return the next float towards positive infinitynext_down: return the next float towards negative infinity
For special floating-point values, the following constants are also defined:
zeroneg_zeroonetwoinfinityneg_infinitynan: Not a Number (assumed to be unique: the "payload" of NaNs is ignored)
The following table shows the notations available when opening scope
float_scope.
Notation |
Interpretation |
|---|---|
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Floating-point constants are parsed and pretty-printed as (17-digit) decimal constants. This ensures that the composition \(\text{parse} \circ \text{print}\) amounts to the identity.
-
Warning
The constant number is not a binary64 floating-point value. A closest value number will be used and unambiguously printed number. [inexact-float,parsing]¶ Not all decimal constants are floating-point values. This warning is generated when parsing such a constant (for instance
0.1).
-
Flag
Printing Float¶ Turn this flag off (it is on by default) to deactivate decimal printing of floating-point constants. They will then be printed with an hexadecimal representation.
Example
- Open Scope float_scope.
- Eval compute in 1 + 0.5.
- = 1.5 : float
- Eval compute in 1 / 0.
- = infinity : float
- Eval compute in 1 / -0.
- = neg_infinity : float
- Eval compute in 0 / 0.
- = nan : float
- Eval compute in 0 ?= -0.
- = FEq : float_comparison
- Eval compute in nan ?= nan.
- = FNotComparable : float_comparison
- Eval compute in next_down (-1).
- = -1.0000000000000002 : float
The primitive operators are specified with respect to their Gallina
counterpart, using the variant type spec_float, and the injection
Prim2SF:
- Print spec_float.
- Variant spec_float : Set := S754_zero : bool -> spec_float | S754_infinity : bool -> spec_float | S754_nan : spec_float | S754_finite : bool -> positive -> Z -> spec_float Arguments S754_zero _%bool_scope Arguments S754_infinity _%bool_scope Arguments S754_finite _%bool_scope _%positive_scope _%Z_scope
- Check Prim2SF.
- Prim2SF : float -> spec_float
- Check mul_spec.
- mul_spec : forall x y : float, Prim2SF (x * y) = SF64mul (Prim2SF x) (Prim2SF y)
For more details on the available definitions and lemmas, see the
online documentation of the Floats library.
Users’ contributions¶
Numerous users' contributions have been collected and are available at URL http://coq.inria.fr/opam/www/. 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.