37 lines
1.5 KiB
Text
37 lines
1.5 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, 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 := H1 ↓ x ◂ Hin
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in H2 ↓ x ◂ L1
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theorem subset::ext (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 (λ x, H ↓ x)
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theorem subset::antisym (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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(have (∀ x, x ∈ s1 = x ∈ s2) ⇒ s1 = s2 :
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(subset::ext 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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iff::intro (have x ∈ s1 ⇒ x ∈ s2 : H1 ↓ x)
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(have x ∈ s2 ⇒ x ∈ s1 : H2 ↓ x))
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-- Compact (but less readable) version of the previous theorem
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theorem subset::antisym2 (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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(subset::ext A) ↓ s1 ↓ s2 ◂ (take x, iff::intro (H1 ↓ x) (H2 ↓ x))
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