/- 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} 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.elim_right {a b : Prop} (H : and a b) : b := and.rec (λa b, b) H inductive sum (A B : Type) : Type := inl : A → sum A B, inr : B → sum A B inductive or (a b : Prop) : Prop := intro_left : a → or a b, intro_right : b → or a b -- 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 inductive num : Type := zero : num, pos : pos_num → 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