38 lines
1.4 KiB
Text
38 lines
1.4 KiB
Text
/-
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Module: logic.subsingleton
|
|
Author: Floris van Doorn
|
|
-/
|
|
|
|
import logic.eq
|
|
|
|
inductive subsingleton [class] (A : Type) : Prop :=
|
|
intro : (∀ a b : A, a = b) → subsingleton A
|
|
|
|
namespace subsingleton
|
|
definition elim {A : Type} {H : subsingleton A} : ∀(a b : A), a = b := subsingleton.rec (fun p, p) H
|
|
end subsingleton
|
|
|
|
protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P :=
|
|
subsingleton.intro (λa b, !proof_irrel)
|
|
|
|
theorem irrelevant [instance] (p : Prop) : subsingleton (decidable p) :=
|
|
subsingleton.intro (fun d1 d2,
|
|
decidable.rec
|
|
(assume Hp1 : p, decidable.rec
|
|
(assume Hp2 : p, congr_arg decidable.inl (eq.refl Hp1)) -- using proof irrelevance for Prop
|
|
(assume Hnp2 : ¬p, absurd Hp1 Hnp2)
|
|
d2)
|
|
(assume Hnp1 : ¬p, decidable.rec
|
|
(assume Hp2 : p, absurd Hp2 Hnp1)
|
|
(assume Hnp2 : ¬p, congr_arg decidable.inr (eq.refl Hnp1)) -- using proof irrelevance for Prop
|
|
d2)
|
|
d1)
|
|
|
|
protected theorem rec_subsingleton [instance] {p : Prop} [H : decidable p]
|
|
{H1 : p → Type} {H2 : ¬p → Type}
|
|
[H3 : Π(h : p), subsingleton (H1 h)] [H4 : Π(h : ¬p), subsingleton (H2 h)]
|
|
: subsingleton (decidable.rec_on H H1 H2) :=
|
|
decidable.rec_on H (λh, H3 h) (λh, H4 h) --this can be proven using dependent version of "by_cases"
|