99 lines
3.5 KiB
Text
99 lines
3.5 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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import logic.core.connectives
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inductive decidable (p : Prop) : Type :=
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inl : p → decidable p,
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inr : ¬p → decidable p
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namespace decidable
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theorem true_decidable [instance] : decidable true :=
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inl trivial
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theorem false_decidable [instance] : decidable false :=
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inr not_false_trivial
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protected theorem induction_on {p : Prop} {C : Prop} (H : decidable p) (H1 : p → C) (H2 : ¬p → C) : C :=
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decidable.rec H1 H2 H
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protected definition rec_on {p : Prop} {C : Type} (H : decidable p) (H1 : p → C) (H2 : ¬p → C) :
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C :=
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decidable.rec H1 H2 H
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theorem irrelevant {p : Prop} (d1 d2 : decidable p) : d1 = d2 :=
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decidable.rec
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(assume Hp1 : p, decidable.rec
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(assume Hp2 : p, congr_arg inl (eq.refl Hp1)) -- using proof irrelevance for Prop
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(assume Hnp2 : ¬p, absurd Hp1 Hnp2)
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d2)
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(assume Hnp1 : ¬p, decidable.rec
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(assume Hp2 : p, absurd Hp2 Hnp1)
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(assume Hnp2 : ¬p, congr_arg inr (eq.refl Hnp1)) -- using proof irrelevance for Prop
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d2)
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d1
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theorem em (p : Prop) {H : decidable p} : p ∨ ¬p :=
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induction_on H (λ Hp, or.inl Hp) (λ Hnp, or.inr Hnp)
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theorem by_cases {a : Prop} {b : Type} {C : decidable a} (Hab : a → b) (Hnab : ¬a → b) : b :=
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rec_on C (assume Ha, Hab Ha) (assume Hna, Hnab Hna)
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theorem by_contradiction {p : Prop} {Hp : decidable p} (H : ¬p → false) : p :=
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or.elim (em p)
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(assume H1 : p, H1)
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(assume H1 : ¬p, false_elim (H H1))
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theorem and_decidable [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) :
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decidable (a ∧ b) :=
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rec_on Ha
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(assume Ha : a, rec_on Hb
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(assume Hb : b, inl (and.intro Ha Hb))
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(assume Hnb : ¬b, inr (and.not_right a Hnb)))
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(assume Hna : ¬a, inr (and.not_left b Hna))
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theorem or_decidable [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) :
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decidable (a ∨ b) :=
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rec_on Ha
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(assume Ha : a, inl (or.inl Ha))
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(assume Hna : ¬a, rec_on Hb
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(assume Hb : b, inl (or.inr Hb))
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(assume Hnb : ¬b, inr (or.not_intro Hna Hnb)))
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theorem not_decidable [instance] {a : Prop} (Ha : decidable a) : decidable (¬a) :=
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rec_on Ha
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(assume Ha, inr (not_not_intro Ha))
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(assume Hna, inl Hna)
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theorem iff_decidable [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) :
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decidable (a ↔ b) :=
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rec_on Ha
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(assume Ha, rec_on Hb
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(assume Hb : b, inl (iff.intro (assume H, Hb) (assume H, Ha)))
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(assume Hnb : ¬b, inr (assume H : a ↔ b, absurd (iff.elim_left H Ha) Hnb)))
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(assume Hna, rec_on Hb
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(assume Hb : b, inr (assume H : a ↔ b, absurd (iff.elim_right H Hb) Hna))
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(assume Hnb : ¬b, inl
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(iff.intro (assume Ha, absurd Ha Hna) (assume Hb, absurd Hb Hnb))))
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theorem implies_decidable [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) :
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decidable (a → b) :=
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rec_on Ha
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(assume Ha : a, rec_on Hb
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(assume Hb : b, inl (assume H, Hb))
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(assume Hnb : ¬b, inr (assume H : a → b, absurd (H Ha) Hnb)))
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(assume Hna : ¬a, inl (assume Ha, absurd Ha Hna))
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theorem decidable_iff_equiv {a b : Prop} (Ha : decidable a) (H : a ↔ b) : decidable b :=
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rec_on Ha
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(assume Ha : a, inl (iff.elim_left H Ha))
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(assume Hna : ¬a, inr (iff.elim_left (iff.flip_sign H) Hna))
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theorem decidable_eq_equiv {a b : Prop} (Ha : decidable a) (H : a = b) : decidable b :=
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decidable_iff_equiv Ha (eq_to_iff H)
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end decidable
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definition decidable_eq (A : Type) := Π (a b : A), decidable (a = b)
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