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Step * 1 1 1 1 1 2 of Lemma Kan_id_filler_wf1


1. X : CubicalSet
2. A : {X ⊢ _(Kan)}
3. a : {X ⊢ _:Kan-type(A)}
4. b : {X ⊢ _:Kan-type(A)}
5. I : Cname List
6. alpha : X(I)
7. J : nameset(I) List
8. x : nameset(I)
9. i : ℕ2
10. bx : A-open-box(X;(Id_Kan-type(A) a b);I;alpha;J;x;i)
11. l_subset(Cname;[fresh-cname(I) / J];[fresh-cname(I) / I])
12. ¬(fresh-cname(I) ∈ I)
13. cube : Kan-type(A)(iota(fresh-cname(I))(alpha))
14. fills-A-open-box(X;Kan-type(A);[fresh-cname(I) / 
                                    I];iota(fresh-cname(I))(alpha);cubical-id-box(X;Kan-type(A);a;b;I;alpha;bx);cube)
15. is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(a;I;alpha;0))
∧ is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(b;I;alpha;1))
⊢ name-path-endpoints(X;Kan-type(A);a;b;I;alpha;fresh-cname(I);cube)
BY
{ xxxAssert ⌜∀i:ℕ2. (Kan-type(A)((fresh-cname(I):=i)(iota(fresh-cname(I))(alpha))) = Kan-type(A)(alpha) ∈ Type)⌝⋅xxx }

1
.....assertion..... 
1. X : CubicalSet
2. A : {X ⊢ _(Kan)}
3. a : {X ⊢ _:Kan-type(A)}
4. b : {X ⊢ _:Kan-type(A)}
5. I : Cname List
6. alpha : X(I)
7. J : nameset(I) List
8. x : nameset(I)
9. i : ℕ2
10. bx : A-open-box(X;(Id_Kan-type(A) a b);I;alpha;J;x;i)
11. l_subset(Cname;[fresh-cname(I) / J];[fresh-cname(I) / I])
12. ¬(fresh-cname(I) ∈ I)
13. cube : Kan-type(A)(iota(fresh-cname(I))(alpha))
14. fills-A-open-box(X;Kan-type(A);[fresh-cname(I) / 
                                    I];iota(fresh-cname(I))(alpha);cubical-id-box(X;Kan-type(A);a;b;I;alpha;bx);cube)
15. is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(a;I;alpha;0))
∧ is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(b;I;alpha;1))
⊢ ∀i:ℕ2. (Kan-type(A)((fresh-cname(I):=i)(iota(fresh-cname(I))(alpha))) = Kan-type(A)(alpha) ∈ Type)

2
1. X : CubicalSet
2. A : {X ⊢ _(Kan)}
3. a : {X ⊢ _:Kan-type(A)}
4. b : {X ⊢ _:Kan-type(A)}
5. I : Cname List
6. alpha : X(I)
7. J : nameset(I) List
8. x : nameset(I)
9. i : ℕ2
10. bx : A-open-box(X;(Id_Kan-type(A) a b);I;alpha;J;x;i)
11. l_subset(Cname;[fresh-cname(I) / J];[fresh-cname(I) / I])
12. ¬(fresh-cname(I) ∈ I)
13. cube : Kan-type(A)(iota(fresh-cname(I))(alpha))
14. fills-A-open-box(X;Kan-type(A);[fresh-cname(I) / 
                                    I];iota(fresh-cname(I))(alpha);cubical-id-box(X;Kan-type(A);a;b;I;alpha;bx);cube)
15. is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(a;I;alpha;0))
∧ is-A-face(X;Kan-type(A);[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);cube;term-A-face(b;I;alpha;1))
16. ∀i:ℕ2. (Kan-type(A)((fresh-cname(I):=i)(iota(fresh-cname(I))(alpha))) = Kan-type(A)(alpha) ∈ Type)
⊢ name-path-endpoints(X;Kan-type(A);a;b;I;alpha;fresh-cname(I);cube)

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Latex:

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Latex:

1.  X  :  CubicalSet
2.  A  :  \{X  \mvdash{}  \_(Kan)\}
3.  a  :  \{X  \mvdash{}  \_:Kan-type(A)\}
4.  b  :  \{X  \mvdash{}  \_:Kan-type(A)\}
5.  I  :  Cname  List
6.  alpha  :  X(I)
7.  J  :  nameset(I)  List
8.  x  :  nameset(I)
9.  i  :  \mBbbN{}2
10.  bx  :  A-open-box(X;(Id\_Kan-type(A)  a  b);I;alpha;J;x;i)
11.  l\_subset(Cname;[fresh-cname(I)  /  J];[fresh-cname(I)  /  I])
12.  \mneg{}(fresh-cname(I)  \mmember{}  I)
13.  cube  :  Kan-type(A)(iota(fresh-cname(I))(alpha))
14.  fills-A-open-box(X;Kan-type(A);[fresh-cname(I)  / 
                                                                        I];iota(fresh-cname(I))(alpha);...;cube)
15.  is-A-face(X;Kan-type(A);[fresh-cname(I)  / 
                                                          I];iota(fresh-cname(I))(alpha);cube;term-A-face(a;I;alpha;0))
\mwedge{}  is-A-face(X;Kan-type(A);[fresh-cname(I)  / 
                                                      I];iota(fresh-cname(I))(alpha);cube;term-A-face(b;I;alpha;1))
\mvdash{}  name-path-endpoints(X;Kan-type(A);a;b;I;alpha;fresh-cname(I);cube)


By

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Latex:
xxxAssert  \mkleeneopen{}\mforall{}i:\mBbbN{}2
                          (Kan-type(A)((fresh-cname(I):=i)(iota(fresh-cname(I))(alpha)))  =  Kan-type(A)(alpha))\mkleeneclose{}
\mcdot{}xxx

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