Step
*
1
2
1
1
of Lemma
cubical-pair_wf
1. X : CubicalSet
2. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
3. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
4. ∀I:Cname List. ∀a:X(I). ∀u:A1 I a. ((A2 I I 1 a u) = u ∈ (A1 I a))
5. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X(I). ∀u:A1 I a.
((A2 I K (f o g) a u) = (A2 J K g f(a) (A2 I J f a u)) ∈ (A1 K (f o g)(a)))
6. X ⊢ <A1, A2>
7. A : I:(Cname List) ⟶ X.<A1, A2>(I) ⟶ Type
8. B1 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X.<A1, A2>(I) ⟶ (A I a) ⟶ (A J f(a))
9. (∀I:Cname List. ∀a:X.<A1, A2>(I). ∀u:A I a. ((B1 I I 1 a u) = u ∈ (A I a)))
∧ (∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X.<A1, A2>(I). ∀u:A I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)) ∈ (A K (f o g)(a))))
10. X.<A1, A2> ⊢ <A, B1>
11. u : I:(Cname List) ⟶ a:X(I) ⟶ <A1, A2>(a)
12. u ∈ {X ⊢ _:<A1, A2>}
13. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((A2 I J f a (u I a)) = (u J f(a)) ∈ (A1 J f(a)))
14. v : I:(Cname List) ⟶ a:X(I) ⟶ (<A, B1>)[u](a)
15. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a)))
16. I : Cname List
17. J : Cname List
18. f : name-morph(I;J)
19. a : X(I)
20. (B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a))
⊢ (B1 I J f (a;u I a) (v I a)) = (v J f(a)) ∈ (A J (f(a);A2 I J f a (u I a)))
BY
{ xxxAssert ⌜(<A, B1>)[u](f(a)) = (A J (f(a);A2 I J f a (u I a))) ∈ Type⌝⋅xxx }
1
.....assertion.....
1. X : CubicalSet
2. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
3. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
4. ∀I:Cname List. ∀a:X(I). ∀u:A1 I a. ((A2 I I 1 a u) = u ∈ (A1 I a))
5. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X(I). ∀u:A1 I a.
((A2 I K (f o g) a u) = (A2 J K g f(a) (A2 I J f a u)) ∈ (A1 K (f o g)(a)))
6. X ⊢ <A1, A2>
7. A : I:(Cname List) ⟶ X.<A1, A2>(I) ⟶ Type
8. B1 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X.<A1, A2>(I) ⟶ (A I a) ⟶ (A J f(a))
9. (∀I:Cname List. ∀a:X.<A1, A2>(I). ∀u:A I a. ((B1 I I 1 a u) = u ∈ (A I a)))
∧ (∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X.<A1, A2>(I). ∀u:A I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)) ∈ (A K (f o g)(a))))
10. X.<A1, A2> ⊢ <A, B1>
11. u : I:(Cname List) ⟶ a:X(I) ⟶ <A1, A2>(a)
12. u ∈ {X ⊢ _:<A1, A2>}
13. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((A2 I J f a (u I a)) = (u J f(a)) ∈ (A1 J f(a)))
14. v : I:(Cname List) ⟶ a:X(I) ⟶ (<A, B1>)[u](a)
15. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a)))
16. I : Cname List
17. J : Cname List
18. f : name-morph(I;J)
19. a : X(I)
20. (B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a))
⊢ (<A, B1>)[u](f(a)) = (A J (f(a);A2 I J f a (u I a))) ∈ Type
2
1. X : CubicalSet
2. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
3. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
4. ∀I:Cname List. ∀a:X(I). ∀u:A1 I a. ((A2 I I 1 a u) = u ∈ (A1 I a))
5. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X(I). ∀u:A1 I a.
((A2 I K (f o g) a u) = (A2 J K g f(a) (A2 I J f a u)) ∈ (A1 K (f o g)(a)))
6. X ⊢ <A1, A2>
7. A : I:(Cname List) ⟶ X.<A1, A2>(I) ⟶ Type
8. B1 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X.<A1, A2>(I) ⟶ (A I a) ⟶ (A J f(a))
9. (∀I:Cname List. ∀a:X.<A1, A2>(I). ∀u:A I a. ((B1 I I 1 a u) = u ∈ (A I a)))
∧ (∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X.<A1, A2>(I). ∀u:A I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)) ∈ (A K (f o g)(a))))
10. X.<A1, A2> ⊢ <A, B1>
11. u : I:(Cname List) ⟶ a:X(I) ⟶ <A1, A2>(a)
12. u ∈ {X ⊢ _:<A1, A2>}
13. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((A2 I J f a (u I a)) = (u J f(a)) ∈ (A1 J f(a)))
14. v : I:(Cname List) ⟶ a:X(I) ⟶ (<A, B1>)[u](a)
15. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I). ((B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a)))
16. I : Cname List
17. J : Cname List
18. f : name-morph(I;J)
19. a : X(I)
20. (B1 I J f ([u])a (v I a)) = (v J f(a)) ∈ (A J ([u])f(a))
21. (<A, B1>)[u](f(a)) = (A J (f(a);A2 I J f a (u I a))) ∈ Type
⊢ (B1 I J f (a;u I a) (v I a)) = (v J f(a)) ∈ (A J (f(a);A2 I J f a (u I a)))
Latex:
Latex:
1. X : CubicalSet
2. A1 : I:(Cname List) {}\mrightarrow{} X(I) {}\mrightarrow{} Type
3. A2 : I:(Cname List) {}\mrightarrow{} J:(Cname List) {}\mrightarrow{} f:name-morph(I;J) {}\mrightarrow{} a:X(I) {}\mrightarrow{} (A1 I a) {}\mrightarrow{} (A1 J f(a))
4. \mforall{}I:Cname List. \mforall{}a:X(I). \mforall{}u:A1 I a. ((A2 I I 1 a u) = u)
5. \mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K). \mforall{}a:X(I). \mforall{}u:A1 I a.
((A2 I K (f o g) a u) = (A2 J K g f(a) (A2 I J f a u)))
6. X \mvdash{} <A1, A2>
7. A : I:(Cname List) {}\mrightarrow{} X.<A1, A2>(I) {}\mrightarrow{} Type
8. B1 : I:(Cname List)
{}\mrightarrow{} J:(Cname List)
{}\mrightarrow{} f:name-morph(I;J)
{}\mrightarrow{} a:X.<A1, A2>(I)
{}\mrightarrow{} (A I a)
{}\mrightarrow{} (A J f(a))
9. (\mforall{}I:Cname List. \mforall{}a:X.<A1, A2>(I). \mforall{}u:A I a. ((B1 I I 1 a u) = u))
\mwedge{} (\mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K). \mforall{}a:X.<A1, A2>(I). \mforall{}u:A I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u))))
10. X.<A1, A2> \mvdash{} <A, B1>
11. u : I:(Cname List) {}\mrightarrow{} a:X(I) {}\mrightarrow{} <A1, A2>(a)
12. u \mmember{} \{X \mvdash{} \_:<A1, A2>\}
13. \mforall{}I,J:Cname List. \mforall{}f:name-morph(I;J). \mforall{}a:X(I). ((A2 I J f a (u I a)) = (u J f(a)))
14. v : I:(Cname List) {}\mrightarrow{} a:X(I) {}\mrightarrow{} (<A, B1>)[u](a)
15. \mforall{}I,J:Cname List. \mforall{}f:name-morph(I;J). \mforall{}a:X(I). ((B1 I J f ([u])a (v I a)) = (v J f(a)))
16. I : Cname List
17. J : Cname List
18. f : name-morph(I;J)
19. a : X(I)
20. (B1 I J f ([u])a (v I a)) = (v J f(a))
\mvdash{} (B1 I J f (a;u I a) (v I a)) = (v J f(a))
By
Latex:
xxxAssert \mkleeneopen{}(<A, B1>)[u](f(a)) = (A J (f(a);A2 I J f a (u I a)))\mkleeneclose{}\mcdot{}xxx
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