38 lines
1.4 KiB
Text
38 lines
1.4 KiB
Text
/-
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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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Module: logic.subsingleton
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Author: Floris van Doorn
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-/
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import logic.eq
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inductive subsingleton [class] (A : Type) : Prop :=
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intro : (∀ a b : A, a = b) → subsingleton A
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namespace subsingleton
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definition elim {A : Type} {H : subsingleton A} : ∀(a b : A), a = b := subsingleton.rec (fun p, p) H
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end subsingleton
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protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P :=
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subsingleton.intro (λa b, !proof_irrel)
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theorem irrelevant [instance] (p : Prop) : subsingleton (decidable p) :=
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subsingleton.intro (fun 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 decidable.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 decidable.inr (eq.refl Hnp1)) -- using proof irrelevance for Prop
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d2)
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d1)
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protected theorem rec_subsingleton [instance] {p : Prop} [H : decidable p]
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{H1 : p → Type} {H2 : ¬p → Type}
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(H3 : Π(h : p), subsingleton (H1 h)) (H4 : Π(h : ¬p), subsingleton (H2 h))
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: subsingleton (decidable.rec_on H H1 H2) :=
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decidable.rec_on H (λh, H3 h) (λh, H4 h) --this can be proven using dependent version of "by_cases"
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