31 lines
1.1 KiB
Text
31 lines
1.1 KiB
Text
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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 SubsetProp {A : Type} {s1 s2 : set A} {x : A} (H1 : s1 ⊆ s2) (H2 : x ∈ s1) : x ∈ s2 :=
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MP (ForallElim H1 x) H2
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Theorem SubsetTrans {A : Type} {s1 s2 s3 : set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3 :=
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ForallIntro (λ x,
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Discharge (λ Hin : x ∈ s1,
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let L1 : x ∈ s2 := SubsetProp H1 Hin,
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L2 : x ∈ s3 := SubsetProp H2 L1
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in L2)).
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Definition transitive {A : Type} (R : A → A → Bool) := ∀ x y z, R x y ⇒ R y z ⇒ R x z
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Theorem SubsetTrans2 {A : Type} : transitive (subset::explicit A) :=
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ForallIntro (λ s1, ForallIntro (λ s2, ForallIntro (λ s3,
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Discharge (λ H1, (Discharge (λ H2,
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SubsetTrans H1 H2)))))).
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Theorem SubsetRefl {A : Type} (s : set A) : s ⊆ s :=
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ForallIntro (λ x, Discharge (λ H : x ∈ s, H))
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Definition union {A : Type} (s1 : set A) (s2 : set A) := λ x, x ∈ s1 ∨ x ∈ s2
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Infix 55 ∪ : union
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