Step
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of Lemma
cubical-snd_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)))
∧ (∀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))))
5. X ⊢ <A1, A2>
6. A@0 : I:(Cname List) ⟶ X.<A1, A2>(I) ⟶ Type
7. B1 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X.<A1, A2>(I) ⟶ (A@0 I a) ⟶ (A@0 J f(a))
8. ∀I:Cname List. ∀a:X.<A1, A2>(I). ∀u:A@0 I a. ((B1 I I 1 a u) = u ∈ (A@0 I a))
9. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X.<A1, A2>(I). ∀u:A@0 I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)) ∈ (A@0 K (f o g)(a)))
10. p : I:(Cname List) ⟶ a:X(I) ⟶ (u:<A1, A2>(a) × <A@0, B1>((a;u)))
11. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I).
(<A2 I J f a (fst((p I a))), B1 I J f (a;fst((p I a))) (snd((p I a)))>
= (p J f(a))
∈ (u:<A1, A2>(f(a)) × <A@0, B1>((f(a);u))))
12. p.1 ∈ {X ⊢ _:<A1, A2>}
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : X(I)
17. <A2 I J f a (fst((p I a))), B1 I J f ([p.1])a (snd((p I a)))> = (p J f(a)) ∈ (u:<A1, A2>(f(a)) × <A@0, B1>((f(a);u))\000C)
⊢ (B1 I J f ([p.1])a (snd((p I a)))) = (snd((p J f(a)))) ∈ (A@0 J ([p.1])f(a))
BY
{ xxxRepUR ``cubical-type-at`` -1xxx }
1
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)))
∧ (∀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))))
5. X ⊢ <A1, A2>
6. A@0 : I:(Cname List) ⟶ X.<A1, A2>(I) ⟶ Type
7. B1 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X.<A1, A2>(I) ⟶ (A@0 I a) ⟶ (A@0 J f(a))
8. ∀I:Cname List. ∀a:X.<A1, A2>(I). ∀u:A@0 I a. ((B1 I I 1 a u) = u ∈ (A@0 I a))
9. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X.<A1, A2>(I). ∀u:A@0 I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)) ∈ (A@0 K (f o g)(a)))
10. p : I:(Cname List) ⟶ a:X(I) ⟶ (u:<A1, A2>(a) × <A@0, B1>((a;u)))
11. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:X(I).
(<A2 I J f a (fst((p I a))), B1 I J f (a;fst((p I a))) (snd((p I a)))>
= (p J f(a))
∈ (u:<A1, A2>(f(a)) × <A@0, B1>((f(a);u))))
12. p.1 ∈ {X ⊢ _:<A1, A2>}
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : X(I)
17. <A2 I J f a (fst((p I a))), B1 I J f ([p.1])a (snd((p I a)))> = (p J f(a)) ∈ (u:A1 J f(a) × (A@0 J (f(a);u)))
⊢ (B1 I J f ([p.1])a (snd((p I a)))) = (snd((p J f(a)))) ∈ (A@0 J ([p.1])f(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))
\mwedge{} (\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))))
5. X \mvdash{} <A1, A2>
6. A@0 : I:(Cname List) {}\mrightarrow{} X.<A1, A2>(I) {}\mrightarrow{} Type
7. B1 : I:(Cname List)
{}\mrightarrow{} J:(Cname List)
{}\mrightarrow{} f:name-morph(I;J)
{}\mrightarrow{} a:X.<A1, A2>(I)
{}\mrightarrow{} (A@0 I a)
{}\mrightarrow{} (A@0 J f(a))
8. \mforall{}I:Cname List. \mforall{}a:X.<A1, A2>(I). \mforall{}u:A@0 I a. ((B1 I I 1 a u) = u)
9. \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@0 I a.
((B1 I K (f o g) a u) = (B1 J K g f(a) (B1 I J f a u)))
10. p : I:(Cname List) {}\mrightarrow{} a:X(I) {}\mrightarrow{} (u:<A1, A2>(a) \mtimes{} <A@0, B1>((a;u)))
11. \mforall{}I,J:Cname List. \mforall{}f:name-morph(I;J). \mforall{}a:X(I).
(<A2 I J f a (fst((p I a))), B1 I J f (a;fst((p I a))) (snd((p I a)))> = (p J f(a)))
12. p.1 \mmember{} \{X \mvdash{} \_:<A1, A2>\}
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : X(I)
17. <A2 I J f a (fst((p I a))), B1 I J f ([p.1])a (snd((p I a)))> = (p J f(a))
\mvdash{} (B1 I J f ([p.1])a (snd((p I a)))) = (snd((p J f(a))))
By
Latex:
xxxRepUR ``cubical-type-at`` -1xxx
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