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-- 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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----------------------------------------------------------------------------------------------------
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import logic.connectives.basic logic.connectives.eq
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namespace decidable
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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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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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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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definition rec_on [inline] {p : Prop} {C : Type} (H : decidable p) (H1 : p → C) (H2 : ¬p → C) : 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, congr2 inl (refl Hp1)) -- using proof irrelevance for Prop
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(assume Hnp2 : ¬p, absurd_elim (inl Hp1 = inr Hnp2) Hp1 Hnp2)
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d2)
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(assume Hnp1 : ¬p, decidable_rec
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(assume Hp2 : p, absurd_elim (inr Hnp1 = inl Hp2) Hp2 Hnp1)
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(assume Hnp2 : ¬p, congr2 inr (refl Hnp1)) -- using proof irrelevance for Prop
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d2)
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d1
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theorem decidable_true [instance] : decidable true :=
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inl trivial
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theorem decidable_false [instance] : decidable false :=
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inr not_false_trivial
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theorem decidable_and [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : 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 decidable_or [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : 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 decidable_not [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 decidable_iff [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : 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 (iff_intro (assume Ha, absurd_elim b Ha Hna) (assume Hb, absurd_elim a Hb Hnb))))
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theorem decidable_implies [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) : 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_elim b Ha Hna))
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end |