b54a9ec9b9
We swap the arguments for `Membership.mem` so that when proceeded by a `SetLike` coercion, as is often the case in Mathlib, the resulting expression is recognized as eta expanded and reduce for many computations. The most beneficial outcome is that the discrimination tree keys for instances and simp lemmas concerning subsets become more robust resulting in more efficient searches. Closes `RFC` #4932 --------- Co-authored-by: Kim Morrison <kim@tqft.net> Co-authored-by: Henrik Böving <hargonix@gmail.com>
39 lines
879 B
Text
39 lines
879 B
Text
def Set (α : Type u) := α → Prop
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def setOf {α : Type u} (p : α → Prop) : Set α :=
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p
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namespace Set
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protected def mem (s : Set α) (a : α) :=
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s a
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instance : Membership α (Set α) :=
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⟨Set.mem⟩
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theorem ext {a b : Set α} (h : ∀ (x : α), x ∈ a ↔ x ∈ b) : a = b :=
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funext (fun x => propext (h x))
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protected def subset (s₁ s₂ : Set α) :=
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∀ {a}, a ∈ s₁ → a ∈ s₂
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class Subset (α : Type u) where
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/-- Subset relation: `a ⊆ b` -/
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subset : α → α → Prop
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/-- Subset relation: `a ⊆ b` -/
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infix:50 " ⊆ " => Subset.subset
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instance : Subset (Set α) :=
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⟨Set.subset⟩
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instance : EmptyCollection (Set α) :=
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⟨λ _ => False⟩
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example (U : Type) (A B : Set U) : A ⊆ B → A = A :=
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fun h =>
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match @h with | (h : A ⊆ B) => sorry
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example (U : Type) (A B : Set U) : A ⊆ B → A = A := by
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intro (h : A ⊆ B)
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sorry
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