The Coq Proof Assistant

# Library Coq.Init.Datatypes

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

# Datatypes with zero and one element

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

# 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.
Delimit Scope bool_scope with bool.
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 :=
match b1, b2 with
| true, true => false
| true, false => true
| false, true => true
| false, false => false
end.
Definition negb (b:bool) := if b then false else true.
Infix "||" := orb : bool_scope.
Infix "&&" := andb : bool_scope.
Basic properties of andb
Lemma andb_prop : forall a b:bool, andb a b = true -> a = true /\ b = true.
Hint Resolve andb_prop: bool.
Lemma andb_true_intro :
forall b1 b2:bool, b1 = true /\ b2 = true -> andb b1 b2 = true.
Hint Resolve andb_true_intro: bool.
Interpretation of booleans as propositions
Inductive eq_true : bool -> Prop := is_eq_true : eq_true true.
Hint Constructors eq_true : eq_true.
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.
Hint Constructors BoolSpec.

# 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.
Delimit Scope nat_scope with nat.

# Container datatypes

option A is the extension of A with an extra element None
Inductive option (A:Type) : Type :=
| Some : A -> option A
| None : option A.
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
Inductive sum (A B:Type) : Type :=
| inl : A -> sum A B
| inr : B -> sum A B.
Notation "x + y" := (sum x y) : type_scope.
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)
Inductive prod (A B:Type) : Type :=
pair : A -> B -> prod A B.
Add Printing Let prod.
Notation "x * y" := (prod x y) : type_scope.
Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.
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.
End projections.
Hint Resolve pair inl inr: core.
Lemma surjective_pairing :
forall (A B:Type) (p:A * B), p = pair (fst p) (snd p).
Lemma injective_projections :
forall (A B:Type) (p1 p2:A * B),
fst p1 = fst p2 -> snd p1 = snd p2 -> p1 = p2.
Definition prod_uncurry (A B C:Type) (f:prod A B -> C)
(x:A) (y:B) : C := f (pair x y).
Definition prod_curry (A B C:Type) (f:A -> B -> C)
(p:prod A B) : C := match p with
| pair x y => f x y
end.
Polymorphic lists and some operations
Inductive list (A : Type) : Type :=
| nil : list A
| cons : A -> list A -> list A.
Infix "::" := cons (at level 60, right associativity) : list_scope.
Delimit Scope list_scope with list.
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.
Lemma comparison_eq_stable : forall c c' : comparison, ~~ c = c' -> c = c'.
Definition CompOpp (r:comparison) :=
match r with
| Eq => Eq
| Lt => Gt
| Gt => Lt
end.
Lemma CompOpp_involutive : forall c, CompOpp (CompOpp c) = c.
Lemma CompOpp_inj : forall 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.
Hint Constructors CompareSpec.
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.
Hint Constructors CompareSpecT.
Lemma CompareSpec2Type : forall 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).
Hint Unfold CompSpec CompSpecT.
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
Inductive identity (A:Type) (a:A) : A -> Type :=
identity_refl : identity a a.
Hint Resolve identity_refl: core.
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.