Step
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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))
7. ∀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)))
8. a : {X ⊢ _:<A1, A2>}
9. b : {X ⊢ _:<A1, A2>}
10. X ⊢ <A1, A2>
11. Delta ⊢ (<A1, A2>)s
12. I : Cname List
13. a@0 : Delta(I)
14. 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
15. z : {z:Cname| ¬(z ∈ I)}
16. p1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z)
17. z1 : {z:Cname| ¬(z ∈ I)}
18. q1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z1)
19. ∀[I,J,f,a,u,s:Top]. ((u a f) ~ (u (s)a f))
20. ∀[I,J:Cname List]. ∀[f:name-morph(I;J)]. ∀[a:X(I)]. ∀[u:<A1, A2>(a)]. ((u a f) ∈ <A1, A2>(f(a)))
⊢ (p1 iota(z)((s)a@0) rename-one-name(z;z1)) = (p1 (s)iota(z)(a@0) rename-one-name(z;z1)) ∈ <A1, A2>(iota(z1)((s)a@0))
BY
{ TACTIC:Assert ⌜rename-one-name(z;z1)((s)iota(z)(a@0)) = iota(z1)((s)a@0) ∈ X([z1 / I])⌝⋅ }
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))
7. ∀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)))
8. a : {X ⊢ _:<A1, A2>}
9. b : {X ⊢ _:<A1, A2>}
10. X ⊢ <A1, A2>
11. Delta ⊢ (<A1, A2>)s
12. I : Cname List
13. a@0 : Delta(I)
14. 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
15. z : {z:Cname| ¬(z ∈ I)}
16. p1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z)
17. z1 : {z:Cname| ¬(z ∈ I)}
18. q1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z1)
19. ∀[I,J,f,a,u,s:Top]. ((u a f) ~ (u (s)a f))
20. ∀[I,J:Cname List]. ∀[f:name-morph(I;J)]. ∀[a:X(I)]. ∀[u:<A1, A2>(a)]. ((u a f) ∈ <A1, A2>(f(a)))
⊢ rename-one-name(z;z1)((s)iota(z)(a@0)) = iota(z1)((s)a@0) ∈ X([z1 / I])
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))
7. ∀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)))
8. a : {X ⊢ _:<A1, A2>}
9. b : {X ⊢ _:<A1, A2>}
10. X ⊢ <A1, A2>
11. Delta ⊢ (<A1, A2>)s
12. I : Cname List
13. a@0 : Delta(I)
14. 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
15. z : {z:Cname| ¬(z ∈ I)}
16. p1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z)
17. z1 : {z:Cname| ¬(z ∈ I)}
18. q1 : named-path(X;<A1, A2>;a;b;I;(s)a@0;z1)
19. ∀[I,J,f,a,u,s:Top]. ((u a f) ~ (u (s)a f))
20. ∀[I,J:Cname List]. ∀[f:name-morph(I;J)]. ∀[a:X(I)]. ∀[u:<A1, A2>(a)]. ((u a f) ∈ <A1, A2>(f(a)))
21. rename-one-name(z;z1)((s)iota(z)(a@0)) = iota(z1)((s)a@0) ∈ X([z1 / I])
⊢ (p1 iota(z)((s)a@0) rename-one-name(z;z1)) = (p1 (s)iota(z)(a@0) rename-one-name(z;z1)) ∈ <A1, A2>(iota(z1)((s)a@0))
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)
7. \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)))
8. a : \{X \mvdash{} \_:<A1, A2>\}
9. b : \{X \mvdash{} \_:<A1, A2>\}
10. X \mvdash{} <A1, A2>
11. Delta \mvdash{} (<A1, A2>)s
12. I : Cname List
13. a@0 : Delta(I)
14. 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 J f (s)a u)\000C>(a)s;(b)s;I;a@0)
15. z : \{z:Cname| \mneg{}(z \mmember{} I)\}
16. p1 : named-path(X;<A1, A2>a;b;I;(s)a@0;z)
17. z1 : \{z:Cname| \mneg{}(z \mmember{} I)\}
18. q1 : named-path(X;<A1, A2>a;b;I;(s)a@0;z1)
19. \mforall{}[I,J,f,a,u,s:Top]. ((u a f) \msim{} (u (s)a f))
20. \mforall{}[I,J:Cname List]. \mforall{}[f:name-morph(I;J)]. \mforall{}[a:X(I)]. \mforall{}[u:<A1, A2>(a)]. ((u a f) \mmember{} <A1, A2>(f(a))\000C)
\mvdash{} (p1 iota(z)((s)a@0) rename-one-name(z;z1)) = (p1 (s)iota(z)(a@0) rename-one-name(z;z1))
By
Latex:
TACTIC:Assert \mkleeneopen{}rename-one-name(z;z1)((s)iota(z)(a@0)) = iota(z1)((s)a@0)\mkleeneclose{}\mcdot{}
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