Step
*
1
1
1
of Lemma
I-path-morph-comp
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : {z:Cname| ¬(z ∈ I)}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : {z:Cname| ¬(z ∈ I)}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3 ∈ A(iota(z1)(a@0))
16. v : Cname
17. ¬(v ∈ K)
18. v1 : Cname
19. ¬(v1 ∈ J)
⊢ ((u2 iota(z)(a@0) (f o g)[z:=v]) iota(v)((f o g)(a@0)) rename-one-name(v;v))
= (((u2 iota(z)(a@0) rename-one-name(z;z1)) iota(z1)(a@0) f[z1:=v1]) iota(v1)(f(a@0)) g[v1:=v])
∈ A(iota(v)((f o g)(a@0)))
BY
{ (RW (AddrC [2] (LemmaC `cubical-type-ap-morph-id`)) 0 THENA Try (QuickAuto)) }
1
.....wf.....
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : {z:Cname| ¬(z ∈ I)}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : {z:Cname| ¬(z ∈ I)}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3 ∈ A(iota(z1)(a@0))
16. v : Cname
17. ¬(v ∈ K)
18. v1 : Cname
19. ¬(v1 ∈ J)
⊢ (u2 iota(z)(a@0) (f o g)[z:=v]) ∈ A(iota(v)((f o g)(a@0)))
2
.....rewrite subgoal.....
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : {z:Cname| ¬(z ∈ I)}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : {z:Cname| ¬(z ∈ I)}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3 ∈ A(iota(z1)(a@0))
16. v : Cname
17. ¬(v ∈ K)
18. v1 : Cname
19. ¬(v1 ∈ J)
⊢ rename-one-name(v;v) = 1 ∈ name-morph([v / K];[v / K])
3
.....wf.....
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : {z:Cname| ¬(z ∈ I)}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : {z:Cname| ¬(z ∈ I)}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3 ∈ A(iota(z1)(a@0))
16. v : Cname
17. ¬(v ∈ K)
18. v1 : Cname
19. ¬(v1 ∈ J)
⊢ (((u2 iota(z)(a@0) rename-one-name(z;z1)) iota(z1)(a@0) f[z1:=v1]) iota(v1)(f(a@0)) g[v1:=v])
= (((u2 iota(z)(a@0) rename-one-name(z;z1)) iota(z1)(a@0) f[z1:=v1]) iota(v1)(f(a@0)) g[v1:=v])
∈ A(iota(v)((f o g)(a@0)))
4
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : {z:Cname| ¬(z ∈ I)}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : {z:Cname| ¬(z ∈ I)}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3 ∈ A(iota(z1)(a@0))
16. v : Cname
17. ¬(v ∈ K)
18. v1 : Cname
19. ¬(v1 ∈ J)
⊢ (u2 iota(z)(a@0) (f o g)[z:=v])
= (((u2 iota(z)(a@0) rename-one-name(z;z1)) iota(z1)(a@0) f[z1:=v1]) iota(v1)(f(a@0)) g[v1:=v])
∈ A(iota(v)((f o g)(a@0)))
Latex:
Latex:
1. X : CubicalSet
2. A : \{X \mvdash{} \_\}
3. a : \{X \mvdash{} \_:A\}
4. b : \{X \mvdash{} \_:A\}
5. I : Cname List
6. J : Cname List
7. K : Cname List
8. f : name-morph(I;J)
9. g : name-morph(J;K)
10. a@0 : X(I)
11. z : \{z:Cname| \mneg{}(z \mmember{} I)\}
12. u2 : named-path(X;A;a;b;I;a@0;z)
13. z1 : \{z:Cname| \mneg{}(z \mmember{} I)\}
14. u3 : named-path(X;A;a;b;I;a@0;z1)
15. (u2 iota(z)(a@0) rename-one-name(z;z1)) = u3
16. v : Cname
17. \mneg{}(v \mmember{} K)
18. v1 : Cname
19. \mneg{}(v1 \mmember{} J)
\mvdash{} ((u2 iota(z)(a@0) (f o g)[z:=v]) iota(v)((f o g)(a@0)) rename-one-name(v;v))
= (((u2 iota(z)(a@0) rename-one-name(z;z1)) iota(z1)(a@0) f[z1:=v1]) iota(v1)(f(a@0)) g[v1:=v])
By
Latex:
(RW (AddrC [2] (LemmaC `cubical-type-ap-morph-id`)) 0 THENA Try (QuickAuto))
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