0c6108ce7a
Remark: The kernel was already using Sort. So, the limitation was artificial. Moreover, it may seem unnecessary to have quotients of proofs in a proof irrelevant system, but this is useful for proving a more general funext lemma. This more general version is needed in the new tested contributed by @digama0.
34 lines
812 B
Text
34 lines
812 B
Text
/-
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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.logic
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universes u
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class setoid (α : Sort u) :=
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(r : α → α → Prop) (iseqv : equivalence r)
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namespace setoid
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infix ` ≈ ` := setoid.r
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variable {α : Sort u}
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variable [s : setoid α]
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include s
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@[refl] lemma refl (a : α) : a ≈ a :=
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match setoid.iseqv α with
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| ⟨h_refl, h_symm, h_trans⟩ := h_refl a
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end
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@[symm] lemma symm {a b : α} (hab : a ≈ b) : b ≈ a :=
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match setoid.iseqv α with
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| ⟨h_refl, h_symm, h_trans⟩ := h_symm hab
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end
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@[trans] lemma trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c :=
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match setoid.iseqv α with
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| ⟨h_refl, h_symm, h_trans⟩ := h_trans hab hbc
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end
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end setoid
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