38 lines
1.7 KiB
Text
38 lines
1.7 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 SubsetTrans (A : Type) : ∀ s1 s2 s3 : Set A, s1 ⊆ s2 ⇒ s2 ⊆ s3 ⇒ s1 ⊆ s3 :=
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take s1 s2 s3, Assume (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3),
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have s1 ⊆ s3 :
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take x, Assume Hin : x ∈ s1,
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have x ∈ s3 :
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let L1 : x ∈ s2 := MP (Instantiate H1 x) Hin
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in MP (Instantiate H2 x) L1
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Theorem SubsetExt (A : Type) : ∀ s1 s2 : Set A, (∀ x, x ∈ s1 = x ∈ s2) ⇒ s1 = s2 :=
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take s1 s2, Assume (H : ∀ x, x ∈ s1 = x ∈ s2),
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Abst (fun x, Instantiate H x)
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Theorem SubsetAntiSymm (A : Type) : ∀ s1 s2 : Set A, s1 ⊆ s2 ⇒ s2 ⊆ s1 ⇒ s1 = s2 :=
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take s1 s2, Assume (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s1),
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have s1 = s2 :
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MP (have (∀ x, x ∈ s1 = x ∈ s2) ⇒ s1 = s2 :
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Instantiate (SubsetExt A) s1 s2)
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(have (∀ x, x ∈ s1 = x ∈ s2) :
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take x, have x ∈ s1 = x ∈ s2 :
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let L1 : x ∈ s1 ⇒ x ∈ s2 := Instantiate H1 x,
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L2 : x ∈ s2 ⇒ x ∈ s1 := Instantiate H2 x
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in ImpAntisym L1 L2)
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-- Compact (but less readable) version of the previous theorem
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Theorem SubsetAntiSymm2 (A : Type) : ∀ s1 s2 : Set A, s1 ⊆ s2 ⇒ s2 ⊆ s1 ⇒ s1 = s2 :=
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take s1 s2, Assume H1 H2,
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MP (Instantiate (SubsetExt A) s1 s2)
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(take x, ImpAntisym (Instantiate H1 x) (Instantiate H2 x))
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