lean2/library/logic/classes/decidable.lean

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-- 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.core.connectives
namespace decidable
inductive decidable (p : Prop) : Type :=
inl : p → decidable p,
inr : ¬p → decidable p
theorem true_decidable [instance] : decidable true :=
inl trivial
theorem false_decidable [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, congr_arg inl (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 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_cases {a b : Prop} {C : decidable a} (Hab : a → b) (Hnab : ¬a → b) : b :=
or_elim (em a) (assume Ha, Hab Ha) (assume Hna, Hnab Hna)
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 (H H1))
theorem and_decidable [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 or_decidable [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 not_decidable [instance] {a : Prop} (Ha : decidable a) : decidable (¬a) :=
rec_on Ha
(assume Ha, inr (not_not_intro Ha))
(assume Hna, inl Hna)
theorem iff_decidable [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 Ha Hna) (assume Hb, absurd Hb Hnb))))
theorem implies_decidable [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 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