101 lines
2.7 KiB
Text
101 lines
2.7 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Jakob von Raumer
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Basic datatypes
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-/
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prelude
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notation [parsing-only] `Type'` := Type.{_+1}
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notation [parsing-only] `Type₊` := Type.{_+1}
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notation `Type₀` := Type.{0}
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notation `Type₁` := Type.{1}
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notation `Type₂` := Type.{2}
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notation `Type₃` := Type.{3}
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inductive poly_unit.{l} : Type.{l} :=
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star : poly_unit
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inductive unit : Type₀ :=
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star : unit
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inductive empty : Type₀
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inductive eq.{l} {A : Type.{l}} (a : A) : A → Type.{l} :=
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refl : eq a a
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structure lift.{l₁ l₂} (A : Type.{l₁}) : Type.{max l₁ l₂} :=
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up :: (down : A)
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inductive prod (A B : Type) :=
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mk : A → B → prod A B
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definition prod.pr1 [reducible] [unfold-c 3] {A B : Type} (p : prod A B) : A :=
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prod.rec (λ a b, a) p
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definition prod.pr2 [reducible] [unfold-c 3] {A B : Type} (p : prod A B) : B :=
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prod.rec (λ a b, b) p
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definition prod.destruct [reducible] := @prod.cases_on
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inductive sum (A B : Type) : Type :=
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| inl {} : A → sum A B
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| inr {} : B → sum A B
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definition sum.intro_left [reducible] {A : Type} (B : Type) (a : A) : sum A B :=
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sum.inl a
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definition sum.intro_right [reducible] (A : Type) {B : Type} (b : B) : sum A B :=
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sum.inr b
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inductive sigma {A : Type} (B : A → Type) :=
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mk : Π (a : A), B a → sigma B
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definition sigma.pr1 [reducible] [unfold-c 3] {A : Type} {B : A → Type} (p : sigma B) : A :=
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sigma.rec (λ a b, a) p
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definition sigma.pr2 [reducible] [unfold-c 3] {A : Type} {B : A → Type} (p : sigma B) : B (sigma.pr1 p) :=
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sigma.rec (λ a b, b) p
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-- pos_num and num are two auxiliary datatypes used when parsing numerals such as 13, 0, 26.
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-- The parser will generate the terms (pos (bit1 (bit1 (bit0 one)))), zero, and (pos (bit0 (bit1 (bit1 one)))).
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-- This representation can be coerced in whatever we want (e.g., naturals, integers, reals, etc).
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inductive pos_num : Type :=
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| one : pos_num
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| bit1 : pos_num → pos_num
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| bit0 : pos_num → pos_num
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namespace pos_num
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definition succ (a : pos_num) : pos_num :=
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pos_num.rec_on a (bit0 one) (λn r, bit0 r) (λn r, bit1 n)
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end pos_num
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inductive num : Type :=
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| zero : num
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| pos : pos_num → num
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namespace num
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open pos_num
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definition succ (a : num) : num :=
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num.rec_on a (pos one) (λp, pos (succ p))
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end num
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inductive bool : Type :=
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| ff : bool
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| tt : bool
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inductive char : Type :=
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mk : bool → bool → bool → bool → bool → bool → bool → bool → char
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inductive string : Type :=
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| empty : string
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| str : char → string → string
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inductive nat :=
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| zero : nat
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| succ : nat → nat
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inductive option (A : Type) : Type :=
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| none {} : option A
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| some : A → option A
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