34 lines
1.2 KiB
Text
34 lines
1.2 KiB
Text
|
import macros
|
||
|
|
|
||
|
|
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 subset_trans {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3
|
||
|
|
:= take x : A,
|
||
|
|
assume Hin : x ∈ s1,
|
||
|
|
have x ∈ s3 :
|
||
|
|
let L1 : x ∈ s2 := H1 x Hin
|
||
|
|
in H2 x L1
|
||
|
|
|
||
|
|
theorem subset_ext {A : Type} {s1 s2 : Set A} (H : ∀ x, x ∈ s1 = x ∈ s2) : s1 = s2
|
||
|
|
:= funext H
|
||
|
|
|
||
|
|
theorem subset_antisym {A : Type} {s1 s2 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s1) : s1 = s2
|
||
|
|
:= subset_ext (have (∀ x, x ∈ s1 = x ∈ s2) :
|
||
|
|
take x, have x ∈ s1 = x ∈ s2 :
|
||
|
|
iff_intro (have x ∈ s1 → x ∈ s2 : H1 x)
|
||
|
|
(have x ∈ s2 → x ∈ s1 : H2 x))
|
||
|
|
|
||
|
|
exit
|
||
|
|
|
||
|
|
theorem subset_trans2 {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3
|
||
|
|
:= λ x Hin, H2 x (H1 x Hin)
|
||
|
|
|
||
|
|
theorem subset_antisym2 {A : Type} {s1 s2 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s1) : s1 = s2
|
||
|
|
:= funext (λ x, iff_intro (H1 x) (H2 x))
|