9e980b2a78
We are getting a few "stuck at universe constraint" errors after the fix. We may need to add some kind of approximation in the future.
26 lines
848 B
Text
26 lines
848 B
Text
def Set α := α → Prop
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class HasMem (α : outParam $ Type u) (β : Type v) where
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mem : α → β → Prop
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infix:50 " ∈ " => HasMem.mem
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instance {α : Type u} : HasMem α (Set α) := ⟨λ a s => s a⟩
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instance {α : Type u} : LE (Set α) := ⟨λ s t => ∀ {x : α}, x ∈ s → x ∈ t⟩
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class HasInf (P : Type u) where
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inf : P → P → P
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infix:70 " ⊓ " => HasInf.inf
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instance {α : Type u} : HasInf (Set α) := ⟨λ s t x => x ∈ s ∧ x ∈ t⟩
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theorem infLeLeft {s t : Set α} : s ⊓ t ≤ s := And.left
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theorem infLeRight {s t : Set α} : s ⊓ t ≤ t := And.right
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theorem inter_mem_sets_iff {α : Type u} (f : Set (Set α)) (hf : ∀ {s t}, s ∈ f → s ≤ t → t ∈ f) {x y : Set α}
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: x ⊓ y ∈ f → x ∈ f ∧ y ∈ f := by
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intro h
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refine ⟨hf h infLeLeft, hf h ?_⟩
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apply infLeRight
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