Step
*
1
1
of Lemma
csm-cubical-identity
1. X : CubicalSet
2. Delta : CubicalSet
3. s : Delta ⟶ X
4. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
5. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
6. (∀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))))
7. a : {X ⊢ _:<A1, A2>}
8. b : {X ⊢ _:<A1, A2>}
9. X ⊢ <A1, A2>
10. Delta ⊢ (<A1, A2>)s
11. I : Cname List
12. a@0 : Delta(I)
⊢ (p,q:I-path(X;<A1, A2>;a;b;I;(s)a@0)//path-eq(X;<A1, A2>;I;(s)a@0;p;q))
= (p,q:I-path(Delta;<λI,a. (A1 I (s)a), λI,J,f,a,u. (A2 I J f (s)a u)>;(a)s;(b)s;I;a@0)//path-eq(Delta;<λI,a. (A1 I (s)a\000C), λI,J,f,a,u. (A2 I J f (s)a u)>;I;a@0;p;q))
∈ Type
BY
{ TACTIC:Assert ⌜I-path(X;<A1, A2>;a;b;I;(s)a@0) = I-path(Delta;<λI,a. (A1 I (s)a), λI,J,f,a,u. (A2 I J f (s)a u)>;(a)s;\000C(b)s;I;a@0) ∈ Type⌝⋅ }
1
.....assertion.....
1. X : CubicalSet
2. Delta : CubicalSet
3. s : Delta ⟶ X
4. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
5. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
6. (∀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))))
7. a : {X ⊢ _:<A1, A2>}
8. b : {X ⊢ _:<A1, A2>}
9. X ⊢ <A1, A2>
10. Delta ⊢ (<A1, A2>)s
11. I : Cname List
12. a@0 : Delta(I)
⊢ I-path(X;<A1, A2>;a;b;I;(s)a@0) = I-path(Delta;<λI,a. (A1 I (s)a), λI,J,f,a,u. (A2 I J f (s)a u)>;(a)s;(b)s;I;a@0) ∈ T\000Cype
2
1. X : CubicalSet
2. Delta : CubicalSet
3. s : Delta ⟶ X
4. A1 : I:(Cname List) ⟶ X(I) ⟶ Type
5. A2 : I:(Cname List) ⟶ J:(Cname List) ⟶ f:name-morph(I;J) ⟶ a:X(I) ⟶ (A1 I a) ⟶ (A1 J f(a))
6. (∀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))))
7. a : {X ⊢ _:<A1, A2>}
8. b : {X ⊢ _:<A1, A2>}
9. X ⊢ <A1, A2>
10. Delta ⊢ (<A1, A2>)s
11. I : Cname List
12. a@0 : Delta(I)
13. I-path(X;<A1, A2>;a;b;I;(s)a@0) = I-path(Delta;<λI,a. (A1 I (s)a), λI,J,f,a,u. (A2 I J f (s)a u)>;(a)s;(b)s;I;a@0) ∈\000C Type
⊢ (p,q:I-path(X;<A1, A2>;a;b;I;(s)a@0)//path-eq(X;<A1, A2>;I;(s)a@0;p;q))
= (p,q:I-path(Delta;<λI,a. (A1 I (s)a), λI,J,f,a,u. (A2 I J f (s)a u)>;(a)s;(b)s;I;a@0)//path-eq(Delta;<λI,a. (A1 I (s)a\000C), λI,J,f,a,u. (A2 I J f (s)a u)>;I;a@0;p;q))
∈ Type
Latex:
Latex:
1. X : CubicalSet
2. Delta : CubicalSet
3. s : Delta {}\mrightarrow{} X
4. A1 : I:(Cname List) {}\mrightarrow{} X(I) {}\mrightarrow{} Type
5. 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))
6. (\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))))
7. a : \{X \mvdash{} \_:<A1, A2>\}
8. b : \{X \mvdash{} \_:<A1, A2>\}
9. X \mvdash{} <A1, A2>
10. Delta \mvdash{} (<A1, A2>)s
11. I : Cname List
12. a@0 : Delta(I)
\mvdash{} (p,q:I-path(X;<A1, A2>a;b;I;(s)a@0)//path-eq(X;<A1, A2>I;(s)a@0;p;q))
= (p,q:I-path(Delta;<\mlambda{}I,a. (A1 I (s)a), \mlambda{}I,J,f,a,u. (A2 I J f (s)a u)>(a)s;(b)s;I;a@0)//path-eq(Del\000Cta;<\mlambda{}I,a. (A1 I (s)a)
, \mlambda{}I,J,f,a,u. (A2 I J f (s)a
u)
>I;a@0;p;q))
By
Latex:
TACTIC:Assert \mkleeneopen{}I-path(X;<A1, A2>a;b;I;(s)a@0) = I-path(Delta;<\mlambda{}I,a. (A1 I (s)a), \mlambda{}I,J,f,a,u. (A2 I \000CJ f (s)a u)>(a)s;(b)s;I;a@0)\mkleeneclose{}\mcdot{}
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