31 lines
No EOL
1.3 KiB
Text
31 lines
No EOL
1.3 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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-- Author: Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic.connectives.prop
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namespace equivalence
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section
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parameter {A : Type}
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parameter p : A → A → Prop
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infix `∼`:50 := p
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definition reflexive := ∀a, a ∼ a
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definition symmetric := ∀a b, a ∼ b → b ∼ a
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definition transitive := ∀a b c, a ∼ b → b ∼ c → a ∼ c
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end
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inductive equivalence {A : Type} (p : A → A → Prop) : Prop :=
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| equivalence_intro : reflexive p → symmetric p → transitive p → equivalence p
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theorem equivalence_reflexive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : reflexive p :=
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equivalence_rec (λ r s t, r) H
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theorem equivalence_symmetric [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : symmetric p :=
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equivalence_rec (λ r s t, s) H
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theorem equivalence_transitive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : transitive p :=
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equivalence_rec (λ r s t, t) H
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end equivalence |