42 lines
1.2 KiB
Text
42 lines
1.2 KiB
Text
class HasMem (α : outParam $ Type u) (β : Type v) where
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mem : α → β → Prop
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class PartialOrder (P : Type u) extends LE P where
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refl (s : P) : s ≤ s
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antisymm (s t : P) : s ≤ t → t ≤ s → s = t
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trans (s t u : P) : s ≤ t → t ≤ u → s ≤ u
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theorem LE.le.trans {P : Type _} [PartialOrder P] {x y z : P} : x ≤ y → y ≤ z → x ≤ z := PartialOrder.trans _ _ _
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infix:50 " ∈ " => HasMem.mem
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def Set (α : Type u) := α → Prop
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namespace Set
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instance : HasMem α (Set α) := ⟨λ a s => s a⟩
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theorem ext {s t : Set α} (h : ∀ x, x ∈ s ↔ x ∈ t) : s = t :=
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funext $ λ x => propext $ h x
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instance : LE (Set α) := ⟨λ s t => ∀ {x : α}, x ∈ s → x ∈ t⟩
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instance : PartialOrder (Set α) where
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refl := λ s x => id
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antisymm := λ s t hst hts => ext $ λ x => ⟨hst, hts⟩
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trans := λ s t u hst htu x hxs => htu $ hst hxs
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variable (x y z : Set α) (hxy : x ≤ y) (hyz : y ≤ z)
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example : x ≤ z := hxy.trans hyz
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example : x ≤ z := by
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refine hxy.trans ?_
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exact hyz
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example : x ≤ z := by
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apply hxy.trans
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assumption
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example : x ≤ y → y ≤ z → x ≤ z :=
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λ h h' => _ -- goal view has the unfolded `x✝ ∈ x → x✝ ∈ z` instead of `x ≤ y`
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