Library Coq.Init.Datatypes
Set Implicit Arguments.
Require Import Notations.
Require Import Logic.
unit is a singleton datatype with sole inhabitant tt
Inductive unit : Set :=
tt : unit.
bool is the datatype of the boolean values true and false
Inductive bool : Set :=
| true : bool
| false : bool.
Add Printing If bool.
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.
Bind Scope nat_scope with nat.
Arguments Scope S [nat_scope].
Empty_set has no inhabitant
Inductive Empty_set : Set :=.
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 refl_identity A a
Inductive identity (A:Type) (a:A) : A -> Type :=
refl_identity : identity (A:=A) a a.
Hint Resolve refl_identity: core v62.
Implicit Arguments identity_ind [A].
Implicit Arguments identity_rec [A].
Implicit Arguments identity_rect [A].
option A is the extension of A with an extra element None
Inductive option (A:Type) : Type :=
| Some : A -> option A
| None : option A.
Implicit Arguments None [A].
Definition option_map (A B:Type) (f:A->B) o :=
match o with
| Some a => Some (f a)
| None => None
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.
Variables A 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 v62.
Lemma surjective_pairing :
forall (A B:Type) (p:A * B), p = pair (fst p) (snd p).
Proof.
destruct p; reflexivity.
Qed.
Lemma injective_projections :
forall (A B:Type) (p1 p2:A * B),
fst p1 = fst p2 -> snd p1 = snd p2 -> p1 = p2.
Proof.
destruct p1; destruct p2; simpl in |- *; intros Hfst Hsnd.
rewrite Hfst; rewrite Hsnd; reflexivity.
Qed.
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.
Comparison
Inductive comparison : Set :=
| Eq : comparison
| Lt : comparison
| Gt : comparison.
Definition CompOpp (r:comparison) :=
match r with
| Eq => Eq
| Lt => Gt
| Gt => Lt
end.
Notation prodT := prod (only parsing).
Notation pairT := pair (only parsing).
Notation prodT_rect := prod_rect (only parsing).
Notation prodT_rec := prod_rec (only parsing).
Notation prodT_ind := prod_ind (only parsing).
Notation fstT := fst (only parsing).
Notation sndT := snd (only parsing).
Notation prodT_uncurry := prod_uncurry (only parsing).
Notation prodT_curry := prod_curry (only parsing).