d4e141e233
Split from #4583 There are two open questions, opinions appreciated: - Should this material be part of `Init` or `Std`? - Should the typeclasses be in the `Std` namespace?
60 lines
2.4 KiB
Text
60 lines
2.4 KiB
Text
/-
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Copyright (c) 2022 Mario Carneiro. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Mario Carneiro, Markus Himmel
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-/
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prelude
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import Init.Data.Bool
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set_option linter.missingDocs true
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/-- `PartialEquivBEq α` says that the `BEq` implementation is a
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partial equivalence relation, that is:
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* it is symmetric: `a == b → b == a`
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* it is transitive: `a == b → b == c → a == c`.
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-/
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class PartialEquivBEq (α) [BEq α] : Prop where
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/-- Symmetry for `BEq`. If `a == b` then `b == a`. -/
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symm : (a : α) == b → b == a
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/-- Transitivity for `BEq`. If `a == b` and `b == c` then `a == c`. -/
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trans : (a : α) == b → b == c → a == c
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/-- `ReflBEq α` says that the `BEq` implementation is reflexive. -/
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class ReflBEq (α) [BEq α] : Prop where
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/-- Reflexivity for `BEq`. -/
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refl : (a : α) == a
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/-- `EquivBEq` says that the `BEq` implementation is an equivalence relation. -/
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class EquivBEq (α) [BEq α] extends PartialEquivBEq α, ReflBEq α : Prop
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@[simp]
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theorem BEq.refl [BEq α] [ReflBEq α] {a : α} : a == a :=
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ReflBEq.refl
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theorem beq_of_eq [BEq α] [ReflBEq α] {a b : α} : a = b → a == b
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| rfl => BEq.refl
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theorem BEq.symm [BEq α] [PartialEquivBEq α] {a b : α} : a == b → b == a :=
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PartialEquivBEq.symm
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theorem BEq.comm [BEq α] [PartialEquivBEq α] {a b : α} : (a == b) = (b == a) :=
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Bool.eq_iff_iff.2 ⟨BEq.symm, BEq.symm⟩
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theorem BEq.symm_false [BEq α] [PartialEquivBEq α] {a b : α} : (a == b) = false → (b == a) = false :=
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BEq.comm (α := α) ▸ id
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theorem BEq.trans [BEq α] [PartialEquivBEq α] {a b c : α} : a == b → b == c → a == c :=
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PartialEquivBEq.trans
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theorem BEq.neq_of_neq_of_beq [BEq α] [PartialEquivBEq α] {a b c : α} :
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(a == b) = false → b == c → (a == c) = false :=
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fun h₁ h₂ => Bool.eq_false_iff.2 fun h₃ => Bool.eq_false_iff.1 h₁ (BEq.trans h₃ (BEq.symm h₂))
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theorem BEq.neq_of_beq_of_neq [BEq α] [PartialEquivBEq α] {a b c : α} :
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a == b → (b == c) = false → (a == c) = false :=
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fun h₁ h₂ => Bool.eq_false_iff.2 fun h₃ => Bool.eq_false_iff.1 h₂ (BEq.trans (BEq.symm h₁) h₃)
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instance (priority := low) [BEq α] [LawfulBEq α] : EquivBEq α where
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refl := LawfulBEq.rfl
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symm h := (beq_iff_eq _ _).2 <| Eq.symm <| (beq_iff_eq _ _).1 h
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trans hab hbc := (beq_iff_eq _ _).2 <| ((beq_iff_eq _ _).1 hab).trans <| (beq_iff_eq _ _).1 hbc
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