58 lines
1.3 KiB
Text
58 lines
1.3 KiB
Text
|
/-
|
|||
|
|
Copyright (c) 2017 Jeremy Avigad. All rights reserved.
|
|||
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|||
|
|
Authors: Jeremy Avigad
|
|||
|
|
-/
|
|||
|
|
import types.trunc
|
|||
|
|
open funext eq trunc is_trunc prod sum
|
|||
|
|
|
|||
|
|
--reserve prefix `¬`:40
|
|||
|
|
--reserve prefix `~`:40
|
|||
|
|
--reserve infixr ` ∧ `:35
|
|||
|
|
--reserve infixr ` /\ `:35
|
|||
|
|
--reserve infixr ` \/ `:30
|
|||
|
|
--reserve infixr ` ∨ `:30
|
|||
|
|
--reserve infix ` <-> `:20
|
|||
|
|
--reserve infix ` ↔ `:20
|
|||
|
|
|
|||
|
|
namespace logic
|
|||
|
|
|
|||
|
|
section
|
|||
|
|
open trunc_index
|
|||
|
|
|
|||
|
|
definition propext {p q : Prop} (h : p ↔ q) : p = q :=
|
|||
|
|
tua (equiv_of_iff_of_is_prop h)
|
|||
|
|
|
|||
|
|
end
|
|||
|
|
|
|||
|
|
definition false : Prop := trunctype.mk empty _
|
|||
|
|
|
|||
|
|
definition false.elim {A : Type} (h : false) : A := empty.elim h
|
|||
|
|
|
|||
|
|
definition true : Prop := trunctype.mk unit _
|
|||
|
|
|
|||
|
|
definition true.intro : true := unit.star
|
|||
|
|
|
|||
|
|
definition trivial : true := unit.star
|
|||
|
|
|
|||
|
|
definition and (p q : Prop) : Prop := tprod p q
|
|||
|
|
|
|||
|
|
infixr ` ∧ ` := and
|
|||
|
|
infixr ` /\ ` := and
|
|||
|
|
|
|||
|
|
definition and.intro {p q : Prop} (h₁ : p) (h₂ : q) : and p q := prod.mk h₁ h₂
|
|||
|
|
|
|||
|
|
definition and.left {p q : Prop} (h : p ∧ q) : p := prod.pr1 h
|
|||
|
|
|
|||
|
|
definition and.right {p q : Prop} (h : p ∧ q) : q := prod.pr2 h
|
|||
|
|
|
|||
|
|
definition not (p : Prop) : Prop := trunctype.mk (p → empty) _
|
|||
|
|
|
|||
|
|
prefix `~` := not
|
|||
|
|
|
|||
|
|
definition or.inl := @or.intro_left
|
|||
|
|
|
|||
|
|
definition or.inr := @or.intro_right
|
|||
|
|
|
|||
|
|
end logic
|