The Coq Proof Assistant

# Library Coq.Init.Datatypes

Set Implicit Arguments.
Require Import Notations.
Require Import Ltac.
Require Import Logic.

# Datatypes with zero and one element

Empty_set is a datatype with no inhabitant
Inductive Empty_set : Set :=.
Register Empty_set as core.Empty_set.type.
unit is a singleton datatype with sole inhabitant tt
Inductive unit : Set :=
tt : unit.
Register unit as core.unit.type.
Register tt as core.unit.tt.

# The boolean datatype

bool is the datatype of the boolean values true and false
Inductive bool : Set :=
| true : bool
| false : bool.
Add Printing If bool.
Declare Scope bool_scope.
Delimit Scope bool_scope with bool.
Bind Scope bool_scope with bool.
Register bool as core.bool.type.
Register true as core.bool.true.
Register false as core.bool.false.
Basic boolean operators
Definition andb (b1 b2:bool) : bool := if b1 then b2 else false.
Definition orb (b1 b2:bool) : bool := if b1 then true else b2.
Definition implb (b1 b2:bool) : bool := if b1 then b2 else true.
Definition xorb (b1 b2:bool) : bool :=
if b1 then
if b2 then false else true
else b2.
Definition negb (b:bool) := if b then false else true.
Infix "||" := orb : bool_scope.
Infix "&&" := andb : bool_scope.
Register andb as core.bool.andb.
Register orb as core.bool.orb.
Register implb as core.bool.implb.
Register xorb as core.bool.xorb.
Register negb as core.bool.negb.
Basic properties of andb
Lemma andb_prop (a b:bool) : andb a b = true -> a = true /\ b = true.
#[global]
Hint Resolve andb_prop: bool.
Register andb_prop as core.bool.andb_prop.
Lemma andb_true_intro (b1 b2:bool) :
b1 = true /\ b2 = true -> andb b1 b2 = true.
#[global]
Hint Resolve andb_true_intro: bool.
Register andb_true_intro as core.bool.andb_true_intro.
Interpretation of booleans as propositions
Inductive eq_true : bool -> Prop := is_eq_true : eq_true true.
#[global]
Hint Constructors eq_true : eq_true.
Register eq_true as core.eq_true.type.
Another way of interpreting booleans as propositions
Definition is_true b := b = true.
is_true can be activated as a coercion by (Local) Coercion is_true : bool >-> Sortclass.
Additional rewriting lemmas about eq_true
Lemma eq_true_ind_r :
forall (P : bool -> Prop) (b : bool), P b -> eq_true b -> P true.
Lemma eq_true_rec_r :
forall (P : bool -> Set) (b : bool), P b -> eq_true b -> P true.
Lemma eq_true_rect_r :
forall (P : bool -> Type) (b : bool), P b -> eq_true b -> P true.
The BoolSpec inductive will be used to relate a boolean value and two propositions corresponding respectively to the true case and the false case. Interest: BoolSpec behave nicely with case and destruct. See also Bool.reflect when Q = ~P.
Inductive BoolSpec (P Q : Prop) : bool -> Prop :=
| BoolSpecT : P -> BoolSpec P Q true
| BoolSpecF : Q -> BoolSpec P Q false.
#[global]
Hint Constructors BoolSpec : core.
Register BoolSpec as core.BoolSpec.type.
Register BoolSpecT as core.BoolSpec.BoolSpecT.
Register BoolSpecF as core.BoolSpec.BoolSpecF.

# Peano natural numbers

nat is the datatype of natural numbers built from O and successor S; note that the constructor name is the letter O. Numbers in nat can be denoted using a decimal notation; e.g. 3%nat abbreviates S (S (S O))
Inductive nat : Set :=
| O : nat
| S : nat -> nat.
Declare Scope hex_nat_scope.
Delimit Scope hex_nat_scope with xnat.
Declare Scope nat_scope.
Delimit Scope nat_scope with nat.
Bind Scope nat_scope with nat.
Arguments S _%_nat.
Register nat as num.nat.type.
Register O as num.nat.O.
Register S as num.nat.S.

# Container datatypes

option A is the extension of A with an extra element None
#[universes(template)]
Inductive option (A:Type) : Type :=
| Some : A -> option A
| None : option A.
Arguments Some {A} a.
Arguments None {A}.
Register option as core.option.type.
Register Some as core.option.Some.
Register None as core.option.None.
Definition option_map (A B:Type) (f:A->B) (o : option A) : option B :=
match o with
| Some a => @Some B (f a)
| None => @None B
end.
sum A B, written A + B, is the disjoint sum of A and B
#[universes(template)]
Inductive sum (A B:Type) : Type :=
| inl : A -> sum A B
| inr : B -> sum A B.
Notation "x + y" := (sum x y) : type_scope.
Arguments inl {A B} _ , [A] B _.
Arguments inr {A B} _ , A [B] _.
Register sum as core.sum.type.
Register inl as core.sum.inl.
Register inr as core.sum.inr.
prod A B, written A * B, is the product of A and B; the pair pair A B a b of a and b is abbreviated (a,b)
#[universes(template)]
Inductive prod (A B:Type) : Type :=
pair : A -> B -> A * B
where "x * y" := (prod x y) : type_scope.
Add Printing Let prod.
Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.
Arguments pair {A B} _ _.
Register prod as core.prod.type.
Register pair as core.prod.intro.
Register prod_rect as core.prod.rect.
Section projections.
Context {A : Type} {B : Type}.
Definition fst (p:A * B) := match p with (x, y) => x end.
Definition snd (p:A * B) := match p with (x, y) => y end.
Register fst as core.prod.proj1.
Register snd as core.prod.proj2.
End projections.
#[global]
Hint Resolve pair inl inr: core.
Lemma surjective_pairing (A B:Type) (p:A * B) : p = (fst p, snd p).
Lemma injective_projections (A B:Type) (p1 p2:A * B) :
fst p1 = fst p2 -> snd p1 = snd p2 -> p1 = p2.
Lemma pair_equal_spec (A B : Type) (a1 a2 : A) (b1 b2 : B) :
(a1, b1) = (a2, b2) <-> a1 = a2 /\ b1 = b2.
Definition curry {A B C:Type} (f:A * B -> C)
(x:A) (y:B) : C := f (x,y).
Definition uncurry {A B C:Type} (f:A -> B -> C)
(p:A * B) : C := match p with (x, y) => f x y end.
Import EqNotations.
Lemma rew_pair A (P Q : A->Type) x1 x2 (y1:P x1) (y2:Q x1) (H:x1=x2) :
(rew H in y1, rew H in y2) = rew [fun x => (P x * Q x)%type] H in (y1,y2).
Polymorphic lists and some operations
#[universes(template)]
Inductive list (A : Type) : Type :=
| nil : list A
| cons : A -> list A -> list A.
Arguments nil {A}.
Arguments cons {A} a l.
Declare Scope list_scope.
Delimit Scope list_scope with list.
Bind Scope list_scope with list.
Infix "::" := cons (at level 60, right associativity) : list_scope.
Register list as core.list.type.
Register nil as core.list.nil.
Register cons as core.list.cons.
Local Open Scope list_scope.
Definition length (A : Type) : list A -> nat :=
fix length l :=
match l with
| nil => O
| _ :: l' => S (length l')
end.
Concatenation of two lists
Definition app (A : Type) : list A -> list A -> list A :=
fix app l m :=
match l with
| nil => m
| a :: l1 => a :: app l1 m
end.
Infix "++" := app (right associativity, at level 60) : list_scope.

# The comparison datatype

Inductive comparison : Set :=
| Eq : comparison
| Lt : comparison
| Gt : comparison.
Register comparison as core.comparison.type.
Register Eq as core.comparison.Eq.
Register Lt as core.comparison.Lt.
Register Gt as core.comparison.Gt.
Lemma comparison_eq_stable (c c' : comparison) : ~~ c = c' -> c = c'.
Definition CompOpp (r:comparison) :=
match r with
| Eq => Eq
| Lt => Gt
| Gt => Lt
end.
Lemma CompOpp_involutive c : CompOpp (CompOpp c) = c.
Lemma CompOpp_inj c c' : CompOpp c = CompOpp c' -> c = c'.
Lemma CompOpp_iff : forall c c', CompOpp c = c' <-> c = CompOpp c'.
The CompareSpec inductive relates a comparison value with three propositions, one for each possible case. Typically, it can be used to specify a comparison function via some equality and order predicates. Interest: CompareSpec behave nicely with case and destruct.
Inductive CompareSpec (Peq Plt Pgt : Prop) : comparison -> Prop :=
| CompEq : Peq -> CompareSpec Peq Plt Pgt Eq
| CompLt : Plt -> CompareSpec Peq Plt Pgt Lt
| CompGt : Pgt -> CompareSpec Peq Plt Pgt Gt.
#[global]
Hint Constructors CompareSpec : core.
Register CompareSpec as core.CompareSpec.type.
Register CompEq as core.CompareSpec.CompEq.
Register CompLt as core.CompareSpec.CompLt.
Register CompGt as core.CompareSpec.CompGt.
For having clean interfaces after extraction, CompareSpec is declared in Prop. For some situations, it is nonetheless useful to have a version in Type. Interestingly, these two versions are equivalent.
Inductive CompareSpecT (Peq Plt Pgt : Prop) : comparison -> Type :=
| CompEqT : Peq -> CompareSpecT Peq Plt Pgt Eq
| CompLtT : Plt -> CompareSpecT Peq Plt Pgt Lt
| CompGtT : Pgt -> CompareSpecT Peq Plt Pgt Gt.
#[global]
Hint Constructors CompareSpecT : core.
Register CompareSpecT as core.CompareSpecT.type.
Register CompEqT as core.CompareSpecT.CompEqT.
Register CompLtT as core.CompareSpecT.CompLtT.
Register CompGtT as core.CompareSpecT.CompGtT.
Lemma CompareSpec2Type Peq Plt Pgt c :
CompareSpec Peq Plt Pgt c -> CompareSpecT Peq Plt Pgt c.
As an alternate formulation, one may also directly refer to predicates eq and lt for specifying a comparison, rather that fully-applied propositions. This CompSpec is now a particular case of CompareSpec.
Definition CompSpec {A} (eq lt : A->A->Prop)(x y:A) : comparison -> Prop :=
CompareSpec (eq x y) (lt x y) (lt y x).
Definition CompSpecT {A} (eq lt : A->A->Prop)(x y:A) : comparison -> Type :=
CompareSpecT (eq x y) (lt x y) (lt y x).
#[global]
Hint Unfold CompSpec CompSpecT : core.
Lemma CompSpec2Type : forall A (eq lt:A->A->Prop) x y c,
CompSpec eq lt x y c -> CompSpecT eq lt x y c.

# Misc Other Datatypes

identity A a is the family of datatypes on A whose sole non-empty member is the singleton datatype identity A a a whose sole inhabitant is denoted identity_refl A a
#[deprecated(since="8.16",note="Use eq instead")]
Notation identity := eq (only parsing).
#[deprecated(since="8.16",note="Use eq_refl instead")]
Notation identity_refl := eq_refl (only parsing).
#[deprecated(since="8.16",note="Use eq_ind instead")]
Notation identity_ind := eq_ind (only parsing).
#[deprecated(since="8.16",note="Use eq_rec instead")]
Notation identity_rec := eq_rec (only parsing).
#[deprecated(since="8.16",note="Use eq_rect instead")]
Notation identity_rect := eq_rect (only parsing).
#[deprecated(since="8.16",note="Use eq_sym instead")]
Notation identity_sym := eq_sym (only parsing).
#[deprecated(since="8.16",note="Use eq_trans instead")]
Notation identity_trans := eq_trans (only parsing).
#[deprecated(since="8.16",note="Use f_equal instead")]
Notation identity_congr := f_equal (only parsing).
#[deprecated(since="8.16",note="Use not_eq_sym instead")]
Notation not_identity_sym := not_eq_sym (only parsing).
#[deprecated(since="8.16",note="Use eq_ind_r instead")]
Notation identity_ind_r := eq_ind_r (only parsing).
#[deprecated(since="8.16",note="Use eq_rec_r instead")]
Notation identity_rec_r := eq_rec_r (only parsing).
#[deprecated(since="8.16",note="Use eq_rect_r instead")]
Notation identity_rect_r := eq_rect_r (only parsing).
Register eq as core.identity.type.
Register eq_refl as core.identity.refl.
Register eq_ind as core.identity.ind.
Register eq_sym as core.identity.sym.
Register eq_trans as core.identity.trans.
Register f_equal as core.identity.congr.
#[deprecated(since="8.16",note="Use eq_refl instead")]
Notation refl_id := eq_refl (only parsing).
#[deprecated(since="8.16",note="Use eq_sym instead")]
Notation sym_id := eq_sym (only parsing).
#[deprecated(since="8.16",note="Use eq_trans instead")]
Notation trans_id := eq_trans (only parsing).
#[deprecated(since="8.16",note="Use not_eq_sym instead")]
Notation sym_not_id := not_eq_sym (only parsing).
Identity type
Definition ID := forall A:Type, A -> A.
Definition id : ID := fun A x => x.
Definition IDProp := forall A:Prop, A -> A.
Definition idProp : IDProp := fun A x => x.
Register idProp as core.IDProp.idProp.