Commit 5dc6c4ff by Sven Wille

### mehr zftrash

parent b8a58821
 ... ... @@ -7,9 +7,11 @@ import fnd http://mathhub.info/MitM/Foundation ❚ theory Set_Operations : fnd:?Logic = include http://mathhub.info/MitM/smglom/other_foundations/mmt_tg?ZF_Base ❙ bigintersection : set ⟶ set ❘ = [S] ⟪ (⋃ S) | ([x] ∀ [Z] (Z ∈ S ⇒ x ∈ Z)) ⟫ ❘ # ⋂ 1 ❙ bigintersection : {S : set} ⊦ (∃ [x : set] (x ∈ S)) ⟶ set ❘ = [S,h] ⟪ (⋃ S) | ([x] ∀ [Z] (Z ∈ S ⇒ x ∈ Z)) ⟫ ❘ # ⋂ 1 2 ❙ // the intersection of the empty set is undefined ❙ binaryIntersection : set ⟶ set ⟶ set ❘ = [A , B] ⋂ upair A B ❘ # 1 ∩ 2 prec 10 ❙ upairNotEmpty : ⊦ ∀[a] ∀ [b] ∃[x] x ∈ upair a b ❙ binaryIntersection : set ⟶ set ⟶ set ❘ = [A , B] ⋂ (upair A B) (forallE (forallE upairNotEmpty A) B) ❘ # 1 ∩ 2 prec 10 ❙ set_complement : set ⟶ set ⟶ set ❘= [A,B] ⟪ A | ([x] ¬ (x ∈ B)) ⟫ ❘# 1 \ 2 ❙ ... ... @@ -21,6 +23,7 @@ theory Set_Operations : fnd:?Logic = succ : set ⟶ set ❘ = [a] cons a a ❙ ❚ theory Set_Properties : fnd:?Logic = ... ... @@ -43,7 +46,7 @@ theory Set_Properties : fnd:?Logic = ball : set ⟶ (set ⟶ prop) ⟶ prop ❘ = [s , p ] ∀ [x] x ∈ s ⇒ p s ❙ bex : set ⟶ (set ⟶ prop) ⟶ prop ❘ = [s , p] ∃[x] x ∈ s ⇒ p s ❙ bex : set ⟶ (set ⟶ prop) ⟶ prop ❘ = [s , p] ∃[x] x ∈ s ∧ p s ❙ ❚ ... ...
 namespace http://mathhub.info/MitM/smglom/other_foundations/mmt_tg ❚ import fnd http://mathhub.info/MitM/Foundation ❚ theory BoolZF : fnd:?Logic = include http://mathhub.info/MitM/smglom/other_foundations/mmt_tg?Set_Operations❙ zero = ∅ ❘# ff ❙ one = singleton ∅ ❘# tt❙ two = succ one ❙ boolSet : set ❘ = two ❙ cond = [c,a,b : set] if (c ≐ tt) then a else b ❙ notzf = [b] cond b ff tt ❙ andzf = [a,b] cond a b ff❙ orzf = [a,b] cond a tt b❙ xor = [a,b] cond (orzf (andzf a b) (andzf (notzf a) (notzf b))) ff tt❙ boolToboolzf = [b] if b then tt else ff ❙ ❚ \ No newline at end of file
 ... ... @@ -35,6 +35,9 @@ theory PairsTG : fnd:?Logic = lefts : set ⟶ set ❘ = [s] ⟪ ⋃ ⋃ s | ([x] ∃[y] tuple x y ∈ s) ⟫ ❙ rights : set ⟶ set ❘ = [s] ⟪ ⋃ ⋃ s | ([x] ∃ [y] tuple y x ∈ s) ⟫ ❙ split : set ⟶ (set ⟶ set ⟶set) ⟶ set ❘ = [p,f] f (projl p) (projr p) ❙ ❚ theory Paris_theorems : fnd:?Logic = ... ...
 ... ... @@ -39,11 +39,20 @@ theory Relations : fnd:?Logic = ❚ theory Abstractions : fnd:?Logic = // include ?Relatioins ❙ // Lambda = [A,b] ❙ include ?Relations ❙ include http://mathhub.info/MitM/smglom/other_foundations/mmt_tg?ReplacementFunction ❙ Lambda = [A,b] replaceFn A ([x] opair x (b x)) ❙ apply = [f,xs] ⋃ (image f xs)❙ Pii = [X,Y] ⟪ ℘ (Sigma X Y) | ([f] X ⊆‍ (domain f) ∧ is_function f) ⟫ ❙ // the name Pi is already used by lf ❙ function_space = [A,B] Pii A ([x : set] B ) ❙ ❚ theory Relations_theorems : fnd:?Logic = // include ?Relations ❙ include ?Relations ❙ theorem_cantor : ⊦ ∀ [A] ∀ [b] ∃ [S] S ∈ ℘ A ⇒ ∀ [x] x ∈ A ⇒ b x ≠ S❙ ❚ \ No newline at end of file
 namespace http://mathhub.info/MitM/smglom/other_foundations/mmt_tg ❚ import fnd http://mathhub.info/MitM/Foundation ❚ // an interpretation of the sum type in zfc set theory ❚ theory SumType : fnd:?Logic = include http://mathhub.info/MitM/smglom/other_foundations/mmt_tg?PairsTG ❙ include http://mathhub.info/MitM/smglom/other_foundations/mmt_tg?BoolZF ❙ sumType = [a,b] (ff × a) ∪ (tt × b) ❘ # sumSet ❙ Inl = [v] opair ff v ❙ Inr = [v] opair tt v ❙ case = [a,b] ❙ ❚
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