/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Authors: Leonardo de Moura

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

set_option structure.eta_thm     true
set_option structure.proj_mk_thm true

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

inductive true : Prop :=
intro : true

inductive false : Prop

inductive empty : Type

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

inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
refl : heq a a

structure prod (A B : Type) :=
mk :: (pr1 : A) (pr2 : B)

inductive and (a b : Prop) : Prop :=
intro : a → b → and a b

definition and.elim_left {a b : Prop} (H : and a b) : a  :=
and.rec (λa b, a) H

definition and.left := @and.elim_left

definition and.elim_right {a b : Prop} (H : and a b) : b :=
and.rec (λa b, b) H

definition and.right := @and.elim_right

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 or (a b : Prop) : Prop :=
inl {} : a → or a b,
inr {} : b → or a b

definition or.intro_left {a : Prop} (b : Prop) (Ha : a) : or a b :=
or.inl Ha

definition or.intro_right (a : Prop) {b : Prop} (Hb : b) : or a b :=
or.inr Hb

structure sigma {A : Type} (B : A → Type) :=
mk :: (pr1 : A) (pr2 : B pr1)

-- 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 :=
  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 :=
  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