Step
*
1
of Lemma
csm-cubical-sigma
.....assertion.....
1. X : CubicalSet
2. Delta : CubicalSet
3. A : {X ⊢ _}
4. B : {X.A ⊢ _}
5. s : Delta ⟶ X
⊢ (Σ A B)s
= Σ (A)s (B)(s o p;q)
∈ (A:I:(Cname List) ⟶ Delta(I) ⟶ Type × (I:(Cname List)
⟶ J:(Cname List)
⟶ f:name-morph(I;J)
⟶ a:Delta(I)
⟶ (A I a)
⟶ (A J f(a))))
BY
{ ((Assert X ⊢ A BY
Auto)
THEN (Assert X.A ⊢ B BY
Auto)
THEN RepeatFor 2 (DVar `A')
THEN RepeatFor 2 (DVar `B')
THEN (Assert s ∈ Delta ⟶ X BY
Auto)
THEN DVar `s'⋅
THEN All Reduce
THEN RepUR ``cubical-sigma csm-ap-type cubical-type-at`` 0
THEN MemCD
THEN (RepeatFor 2 ((EqCD THENA Auto)) ORELSE Auto)
THEN ((RepeatFor 3 (At ⌜Type⌝ ((FunExt THENW Auto))⋅)
THEN Reduce 0
THEN Reduce (-1)
THEN D -1
THEN RepUR ``cubical-type-ap-morph`` 0)
ORELSE RepeatFor 2 ((EqCD THEN Auto))
)) }
1
.....subterm..... T:t
2:n
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))
6. ∀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)))
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))
10. ∀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)))
11. s : A:cat-ob(NameCat) ⟶ (cat-arrow(TypeCat) (Delta A) (X A))
12. ∀A,B:cat-ob(NameCat). ∀g:cat-arrow(NameCat) A B.
((cat-comp(TypeCat) (Delta A) (X A) (X B) (s A) (X A B g))
= (cat-comp(TypeCat) (Delta A) (Delta B) (X B) (Delta A B g) (s B))
∈ (cat-arrow(TypeCat) (Delta A) (X B)))
13. X ⊢ <A1, A2>
14. X.<A1, A2> ⊢ <A, B1>
15. s ∈ Delta ⟶ X
16. I : Cname List
17. a : Delta(I)
18. u : A1 I (s)a
⊢ ((s)a;u) = <(s o p)(a;u), (q)(a;u)> ∈ X.<A1, A2>(I)
2
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:cat-ob(NameCat) ⟶ (cat-arrow(TypeCat) (Delta A) (X A))
10. ∀A,B:cat-ob(NameCat). ∀g:cat-arrow(NameCat) A B.
((cat-comp(TypeCat) (Delta A) (X A) (X B) (s A) (X A B g))
= (cat-comp(TypeCat) (Delta A) (Delta B) (X B) (Delta A B g) (s B))
∈ (cat-arrow(TypeCat) (Delta A) (X B)))
11. X ⊢ <A1, A2>
12. X.<A1, A2> ⊢ <A, B1>
13. s ∈ Delta ⟶ X
14. I : Cname List
15. J : Cname List
16. f : name-morph(I;J)
17. a : Delta(I)
18. u : A1 I (s)a
19. x1 : A I ((s)a;u)
⊢ <A2 I J f (s)a u, B1 I J f ((s)a;u) x1>
= <A2 I J f (s)a u, B1 I J f ((s o p;q))(a;u) x1>
∈ (u:A1 J (s)f(a) × (A J ((s)f(a);u)))
Latex:
Latex:
.....assertion.....
1. X : CubicalSet
2. Delta : CubicalSet
3. A : \{X \mvdash{} \_\}
4. B : \{X.A \mvdash{} \_\}
5. s : Delta {}\mrightarrow{} X
\mvdash{} (\mSigma{} A B)s = \mSigma{} (A)s (B)(s o p;q)
By
Latex:
((Assert X \mvdash{} A BY
Auto)
THEN (Assert X.A \mvdash{} B BY
Auto)
THEN RepeatFor 2 (DVar `A')
THEN RepeatFor 2 (DVar `B')
THEN (Assert s \mmember{} Delta {}\mrightarrow{} X BY
Auto)
THEN DVar `s'\mcdot{}
THEN All Reduce
THEN RepUR ``cubical-sigma csm-ap-type cubical-type-at`` 0
THEN MemCD
THEN (RepeatFor 2 ((EqCD THENA Auto)) ORELSE Auto)
THEN ((RepeatFor 3 (At \mkleeneopen{}Type\mkleeneclose{} ((FunExt THENW Auto))\mcdot{})
THEN Reduce 0
THEN Reduce (-1)
THEN D -1
THEN RepUR ``cubical-type-ap-morph`` 0)
ORELSE RepeatFor 2 ((EqCD THEN Auto))
))
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