33 lines
1.2 KiB
Text
33 lines
1.2 KiB
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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:= take x : A,
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assume 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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:= funext 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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take 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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exit
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theorem subset_trans2 {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3
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:= λ x Hin, H2 x (H1 x Hin)
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theorem subset_antisym2 {A : Type} {s1 s2 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s1) : s1 = s2
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:= funext (λ x, iff_intro (H1 x) (H2 x))
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