62 lines
1.5 KiB
Text
62 lines
1.5 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 (lift empty) _
|
||
|
||
definition false.elim {A : Type} (h : false) : A := lift.rec empty.elim h
|
||
|
||
definition true : Prop := trunctype.mk (lift unit) _
|
||
|
||
definition true.intro : true := lift.up unit.star
|
||
|
||
definition trivial : true := lift.up 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) _
|
||
|
||
definition or.inl := @or.intro_left
|
||
|
||
definition or.inr := @or.intro_right
|
||
|
||
definition or.elim {A B C : Type} [is_prop C] (h₀ : A ∨ B) (h₁ : (A → C)) (h₂ : B → C) : C :=
|
||
begin
|
||
apply trunc.elim_on h₀,
|
||
intro h₃,
|
||
apply sum.elim h₃ h₁ h₂
|
||
end
|
||
|
||
end logic
|