lean2/library/standard/logic/classes/decidable.lean
2014-08-07 16:59:08 -07:00

90 lines
No EOL
3.5 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

----------------------------------------------------------------------------------------------------
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
----------------------------------------------------------------------------------------------------
import logic.connectives.basic logic.connectives.eq
namespace decidable
inductive decidable (p : Prop) : Type :=
| inl : p → decidable p
| inr : ¬p → decidable p
theorem decidable_true [instance] : decidable true :=
inl trivial
theorem decidable_false [instance] : decidable false :=
inr not_false_trivial
theorem induction_on {p : Prop} {C : Prop} (H : decidable p) (H1 : p → C) (H2 : ¬p → C) : C :=
decidable_rec H1 H2 H
definition rec_on [inline] {p : Prop} {C : Type} (H : decidable p) (H1 : p → C) (H2 : ¬p → C) : C :=
decidable_rec H1 H2 H
theorem irrelevant {p : Prop} (d1 d2 : decidable p) : d1 = d2 :=
decidable_rec
(assume Hp1 : p, decidable_rec
(assume Hp2 : p, congr2 inl (refl Hp1)) -- using proof irrelevance for Prop
(assume Hnp2 : ¬p, absurd_elim (inl Hp1 = inr Hnp2) Hp1 Hnp2)
d2)
(assume Hnp1 : ¬p, decidable_rec
(assume Hp2 : p, absurd_elim (inr Hnp1 = inl Hp2) Hp2 Hnp1)
(assume Hnp2 : ¬p, congr2 inr (refl Hnp1)) -- using proof irrelevance for Prop
d2)
d1
theorem em (p : Prop) {H : decidable p} : p ¬p :=
induction_on H (λ Hp, or_inl Hp) (λ Hnp, or_inr Hnp)
theorem by_contradiction {p : Prop} {Hp : decidable p} (H : ¬p → false) : p :=
or_elim (em p)
(assume H1 : p, H1)
(assume H1 : ¬p, false_elim p (H H1))
theorem decidable_and [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : decidable (a ∧ b) :=
rec_on Ha
(assume Ha : a, rec_on Hb
(assume Hb : b, inl (and_intro Ha Hb))
(assume Hnb : ¬b, inr (and_not_right a Hnb)))
(assume Hna : ¬a, inr (and_not_left b Hna))
theorem decidable_or [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : decidable (a b) :=
rec_on Ha
(assume Ha : a, inl (or_inl Ha))
(assume Hna : ¬a, rec_on Hb
(assume Hb : b, inl (or_inr Hb))
(assume Hnb : ¬b, inr (or_not_intro Hna Hnb)))
theorem decidable_not [instance] {a : Prop} (Ha : decidable a) : decidable (¬a) :=
rec_on Ha
(assume Ha, inr (not_not_intro Ha))
(assume Hna, inl Hna)
theorem decidable_iff [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : decidable (a ↔ b) :=
rec_on Ha
(assume Ha, rec_on Hb
(assume Hb : b, inl (iff_intro (assume H, Hb) (assume H, Ha)))
(assume Hnb : ¬b, inr (assume H : a ↔ b, absurd (iff_elim_left H Ha) Hnb)))
(assume Hna, rec_on Hb
(assume Hb : b, inr (assume H : a ↔ b, absurd (iff_elim_right H Hb) Hna))
(assume Hnb : ¬b, inl (iff_intro (assume Ha, absurd_elim b Ha Hna) (assume Hb, absurd_elim a Hb Hnb))))
theorem decidable_implies [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : decidable (a → b) :=
rec_on Ha
(assume Ha : a, rec_on Hb
(assume Hb : b, inl (assume H, Hb))
(assume Hnb : ¬b, inr (assume H : a → b, absurd (H Ha) Hnb)))
(assume Hna : ¬a, inl (assume Ha, absurd_elim b Ha Hna))
theorem decidable_iff_equiv {a b : Prop} (Ha : decidable a) (H : a ↔ b) : decidable b :=
rec_on Ha
(assume Ha : a, inl (iff_elim_left H Ha))
(assume Hna : ¬a, inr (iff_elim_left (iff_flip_sign H) Hna))
theorem decidable_eq_equiv {a b : Prop} (Ha : decidable a) (H : a = b) : decidable b :=
decidable_iff_equiv Ha (eq_to_iff H)
end decidable