(** 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.(** 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.(** 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.(********************************************************************)
(** * 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.(********************************************************************)
(** * 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.(********************************************************************)
(** * 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.(********************************************************************)
(** * Container datatypes *)

(* Set Universe Polymorphism. *)

(** [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.(********************************************************************)
(** * Container datatypes *)

(* Set Universe Polymorphism. *)

(** [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.(********************************************************************)
(** * Container datatypes *)

(* Set Universe Polymorphism. *)

(** [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.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.(** [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.(** [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.(** [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.(** [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.(** [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.Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.Definition fst (p:A * B) := match p with (x, y) => x end.Definition snd (p:A * B) := match p with (x, y) => y end.

(* Specifications: *)
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.