Step
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of Lemma
csm-cubical-sigma
1. X : CubicalSet
2. Delta : CubicalSet
3. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
4. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
5. (∀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))))
6. A : 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 I a) ⟶ (A J f(a))
8. (∀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))))
9. s : A:(Cname List) ⟶ Delta(A) ⟶ X(A)
10. ∀A,B:Cname List. ∀g:name-morph(A;B).
((λx.((snd(X)) A B g (s A x))) = (λx.(s B ((snd(Delta)) A B g x))) ∈ (((fst(Delta)) A) ⟶ ((fst(X)) B)))
11. X ⊢ <A1, A2>
12. X.<A1, A2> ⊢ <A, B1>
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : Delta(I)
17. u : A1 I (s)a
18. x1 : A I ((s)a;u)
19. (s I a;u) ∈ X.<A1, A2>(I)
20. (B1 I J f (s I a;u) x1) = (B1 I J f (s I a;u) x1) ∈ (A J f((s I a;u)))
⊢ f(s I a) = (s J f(a)) ∈ X(J)
BY
{ TACTIC:((InstHyp [⌜I⌝;⌜J⌝;⌜f⌝] 10⋅ THENA Auto) THEN Fold `functor-ob` (-1) THEN Fold `I-cube` (-1)) }
1
1. X : CubicalSet
2. Delta : CubicalSet
3. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
4. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
5. (∀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))))
6. A : 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 I a) ⟶ (A J f(a))
8. (∀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))))
9. s : A:(Cname List) ⟶ Delta(A) ⟶ X(A)
10. ∀A,B:Cname List. ∀g:name-morph(A;B).
((λx.((snd(X)) A B g (s A x))) = (λx.(s B ((snd(Delta)) A B g x))) ∈ (((fst(Delta)) A) ⟶ ((fst(X)) B)))
11. X ⊢ <A1, A2>
12. X.<A1, A2> ⊢ <A, B1>
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : Delta(I)
17. u : A1 I (s)a
18. x1 : A I ((s)a;u)
19. (s I a;u) ∈ X.<A1, A2>(I)
20. (B1 I J f (s I a;u) x1) = (B1 I J f (s I a;u) x1) ∈ (A J f((s I a;u)))
21. (λx.((snd(X)) I J f (s I x))) = (λx.(s J ((snd(Delta)) I J f x))) ∈ (Delta(I) ⟶ X(J))
⊢ f(s I a) = (s J f(a)) ∈ X(J)
Latex:
Latex:
1. X : CubicalSet
2. Delta : CubicalSet
3. A1 : I:(Cname List) {}\mrightarrow{} X(I) {}\mrightarrow{} Type
4. 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))
5. (\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))))
6. A : 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 I a)
{}\mrightarrow{} (A J f(a))
8. (\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))))
9. s : A:(Cname List) {}\mrightarrow{} Delta(A) {}\mrightarrow{} X(A)
10. \mforall{}A,B:Cname List. \mforall{}g:name-morph(A;B).
((\mlambda{}x.((snd(X)) A B g (s A x))) = (\mlambda{}x.(s B ((snd(Delta)) A B g x))))
11. X \mvdash{} <A1, A2>
12. X.<A1, A2> \mvdash{} <A, B1>
13. I : Cname List
14. J : Cname List
15. f : name-morph(I;J)
16. a : Delta(I)
17. u : A1 I (s)a
18. x1 : A I ((s)a;u)
19. (s I a;u) \mmember{} X.<A1, A2>(I)
20. (B1 I J f (s I a;u) x1) = (B1 I J f (s I a;u) x1)
\mvdash{} f(s I a) = (s J f(a))
By
Latex:
TACTIC:((InstHyp [\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}J\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{}] 10\mcdot{} THENA Auto) THEN Fold `functor-ob` (-1) THEN Fold `I-cube` (-1))
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