lean2/hott/init/datatypes.hlean
Floris van Doorn fa1979c128 feat(datatypes): let the type of unit be the lowest non-Prop universe
The definitional package (brec_on and cases_on) now use poly_unit instead of unit

closes #698
2015-06-25 17:33:46 -07:00

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