2015-04-01 02:56:05 +00:00
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/-
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Copyright (c) 2015 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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Authors: Leonardo de Moura
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-/
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prelude
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import init.relation
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structure setoid [class] (A : Type) :=
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(r : A → A → Prop) (iseqv : equivalence r)
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namespace setoid
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2015-09-30 23:52:34 +00:00
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infix ` ≈ ` := setoid.r
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2015-04-01 02:56:05 +00:00
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variable {A : Type}
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variable [s : setoid A]
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include s
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2015-12-05 18:10:59 +00:00
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theorem refl [refl] (a : A) : a ≈ a :=
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2015-04-01 02:56:05 +00:00
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and.elim_left (@setoid.iseqv A s) a
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2015-12-05 18:10:59 +00:00
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theorem symm [symm] {a b : A} : a ≈ b → b ≈ a :=
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2015-04-01 02:56:05 +00:00
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λ H, and.elim_left (and.elim_right (@setoid.iseqv A s)) a b H
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2015-12-05 18:10:59 +00:00
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theorem trans [trans] {a b c : A} : a ≈ b → b ≈ c → a ≈ c :=
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2015-04-01 02:56:05 +00:00
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λ H₁ H₂, and.elim_right (and.elim_right (@setoid.iseqv A s)) a b c H₁ H₂
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end setoid
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