24 lines
906 B
Text
24 lines
906 B
Text
import macros
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definition Set (A : Type) : Type := A → Bool
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definition element {A : Type} (x : A) (s : Set A) := s x
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infix 60 ∈ : element
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definition subset {A : Type} (s1 : Set A) (s2 : Set A) := ∀ x, x ∈ s1 → x ∈ s2
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infix 50 ⊆ : subset
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theorem subset::trans {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3
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:= λ (x : A) (Hin : x ∈ s1),
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have x ∈ s3 :
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let L1 : x ∈ s2 := H1 x Hin
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in H2 x L1
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theorem subset::ext {A : Type} {s1 s2 : Set A} (H : ∀ x, x ∈ s1 = x ∈ s2) : s1 = s2
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:= abst H
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theorem subset::antisym {A : Type} {s1 s2 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s1) : s1 = s2
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:= subset::ext (have (∀ x, x ∈ s1 = x ∈ s2) :
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λ x, have x ∈ s1 = x ∈ s2 :
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iff::intro (have x ∈ s1 → x ∈ s2 : H1 x)
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(have x ∈ s2 → x ∈ s1 : H2 x))
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