lean2/examples/lean/set.lean

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Definition Set (A : Type) : Type := A → Bool
Definition element {A : Type} (x : A) (s : Set A) := s x
Infix 60 ∈ : element
Definition subset {A : Type} (s1 : Set A) (s2 : Set A) := ∀ x, x ∈ s1 ⇒ x ∈ s2
Infix 50 ⊆ : subset
Theorem SubsetProp {A : Type} {s1 s2 : Set A} {x : A} (H1 : s1 ⊆ s2) (H2 : x ∈ s1) : x ∈ s2 :=
MP (ForallElim H1 x) H2
Theorem SubsetTrans {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3 :=
ForallIntro (λ x,
Discharge (λ Hin : x ∈ s1,
let L1 : x ∈ s2 := SubsetProp H1 Hin,
L2 : x ∈ s3 := SubsetProp H2 L1
in L2)).
Definition transitive {A : Type} (R : A → A → Bool) := ∀ x y z, R x y ⇒ R y z ⇒ R x z
Theorem SubsetTrans2 {A : Type} : transitive (subset::explicit A) :=
ForallIntro (λ s1, ForallIntro (λ s2, ForallIntro (λ s3,
Discharge (λ H1, (Discharge (λ H2,
SubsetTrans H1 H2)))))).
Theorem SubsetRefl {A : Type} (s : Set A) : s ⊆ s :=
ForallIntro (λ x, Discharge (λ H : x ∈ s, H))
Definition union {A : Type} (s1 : Set A) (s2 : Set A) := λ x, x ∈ s1 x ∈ s2
Infix 55 : union