20 lines
649 B
Text
20 lines
649 B
Text
(**
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-- import macros for, assume, mp, ...
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import("macros.lua")
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**)
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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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for s1 s2 s3, assume (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3),
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show s1 ⊆ s3,
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for x, assume Hin : x ∈ s1,
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show 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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