lean2/hott/init/datatypes.hlean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jakob von Raumer

Basic datatypes
-/

prelude
notation [parsing-only] `Type'` := Type.{_+1}
notation [parsing-only] `Type₊` := Type.{_+1}
notation `Type₀` := Type.{0}
notation `Type₁` := Type.{1}
notation `Type₂` := Type.{2}
notation `Type₃` := Type.{3}

inductive poly_unit.{l} : Type.{l} :=
star : poly_unit

inductive unit : Type₀ :=
star : unit

inductive empty : Type₀

inductive eq.{l} {A : Type.{l}} (a : A) : A → Type.{l} :=
refl : eq a a

structure lift.{l₁ l₂} (A : Type.{l₁}) : Type.{max l₁ l₂} :=
up :: (down : A)

inductive prod (A B : Type) :=
mk : A → B → prod A B

definition prod.pr1 [reducible] [unfold 3] {A B : Type} (p : prod A B) : A :=
prod.rec (λ a b, a) p

definition prod.pr2 [reducible] [unfold 3] {A B : Type} (p : prod A B) : B :=
prod.rec (λ a b, b) p

definition prod.destruct [reducible] := @prod.cases_on

inductive sum (A B : Type) : Type :=
| inl {} : A → sum A B
| inr {} : B → sum A B

definition sum.intro_left [reducible] {A : Type} (B : Type) (a : A) : sum A B :=
sum.inl a

definition sum.intro_right [reducible] (A : Type) {B : Type} (b : B) : sum A B :=
sum.inr b

inductive sigma {A : Type} (B : A → Type) :=
mk : Π (a : A), B a → sigma B

definition sigma.pr1 [reducible] [unfold 3] {A : Type} {B : A → Type} (p : sigma B) : A :=
sigma.rec (λ a b, a) p

definition sigma.pr2 [reducible] [unfold 3] {A : Type} {B : A → Type} (p : sigma B) : B (sigma.pr1 p) :=
sigma.rec (λ a b, b) p

-- pos_num and num are two auxiliary datatypes used when parsing numerals such as 13, 0, 26.
-- The parser will generate the terms (pos (bit1 (bit1 (bit0 one)))), zero, and (pos (bit0 (bit1 (bit1 one)))).
-- This representation can be coerced in whatever we want (e.g., naturals, integers, reals, etc).
inductive pos_num : Type :=
| one  : pos_num
| bit1 : pos_num → pos_num
| bit0 : pos_num → pos_num

namespace pos_num
  definition succ (a : pos_num) : pos_num :=
  pos_num.rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
end pos_num

inductive num : Type :=
| zero  : num
| pos   : pos_num → num

namespace num
  open pos_num
  definition succ (a : num) : num :=
  num.rec_on a (pos one) (λp, pos (succ p))
end num

inductive bool : Type :=
| ff : bool
| tt : bool

inductive char : Type :=
mk : bool → bool → bool → bool → bool → bool → bool → bool → char

inductive string : Type :=
| empty : string
| str   : char → string → string

inductive nat :=
| zero : nat
| succ : nat → nat

inductive option (A : Type) : Type :=
| none {} : option A
| some    : A → option A