Step
*
5
1
1
1
of Lemma
lift-id-faces-wf
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : nameset(I) List
7. x : nameset(I)
8. i : ℕ2
9. alpha : X(I)
10. box : A-face(X;(Id_A a b);I;alpha) List
11. A-adjacent-compatible(X;(Id_A a b);I;alpha;box)
12. ¬(x ∈ J)
13. l_subset(Cname;J;I)
14. ∀y:nameset(J). ∀c:ℕ2. (∃f∈box. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
15. (∃f∈box. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
16. ∀i@0:ℕ||box||. (¬(A-face-name(box[i@0]) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
17. (∀f∈box.(fst(f) ∈ [x / J]))
18. (∀f1,f2∈box. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
19. A-adjacent-compatible(X;A;I+;iota'(I)(alpha);lift-id-faces(X;A;I;alpha;box))
20. ¬(x ∈ J)
21. l_subset(Cname;J;I+)
22. ∀y:nameset(J). ∀c:ℕ2. (∃f∈lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <y, c> ∈ (nameset(I+) × ℕ2))
23. (∃f∈lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <x, i> ∈ (nameset(I+) × ℕ2))
24. i@0 : ℕ||box||@i
25. x1 : nameset(I)@i
26. i1 : ℕ2@i
27. v2 : (Id_A a b)((x1:=i1)(alpha))@i
28. box[i@0] = <x1, i1, v2> ∈ A-face(X;(Id_A a b);I;alpha)
29. ¬(<x1, i1> = <x, 1 - i> ∈ (nameset(I) × ℕ2))
30. <x1, i1> = <x, 1 - i> ∈ (nameset(I+) × ℕ2)
⊢ False
BY
{ TACTIC:(D -2 THEN HypSubst' (-1)0) }
1
1. X : CubicalSet
2. A : {X ⊢ _}
3. a : {X ⊢ _:A}
4. b : {X ⊢ _:A}
5. I : Cname List
6. J : nameset(I) List
7. x : nameset(I)
8. i : ℕ2
9. alpha : X(I)
10. box : A-face(X;(Id_A a b);I;alpha) List
11. A-adjacent-compatible(X;(Id_A a b);I;alpha;box)
12. ¬(x ∈ J)
13. l_subset(Cname;J;I)
14. ∀y:nameset(J). ∀c:ℕ2. (∃f∈box. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
15. (∃f∈box. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
16. ∀i@0:ℕ||box||. (¬(A-face-name(box[i@0]) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
17. (∀f∈box.(fst(f) ∈ [x / J]))
18. (∀f1,f2∈box. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
19. A-adjacent-compatible(X;A;I+;iota'(I)(alpha);lift-id-faces(X;A;I;alpha;box))
20. ¬(x ∈ J)
21. l_subset(Cname;J;I+)
22. ∀y:nameset(J). ∀c:ℕ2. (∃f∈lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <y, c> ∈ (nameset(I+) × ℕ2))
23. (∃f∈lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <x, i> ∈ (nameset(I+) × ℕ2))
24. i@0 : ℕ||box||@i
25. x1 : nameset(I)@i
26. i1 : ℕ2@i
27. v2 : (Id_A a b)((x1:=i1)(alpha))@i
28. box[i@0] = <x1, i1, v2> ∈ A-face(X;(Id_A a b);I;alpha)
29. <x1, i1> = <x, 1 - i> ∈ (nameset(I+) × ℕ2)
⊢ <x, 1 - i> = <x, 1 - i> ∈ (nameset(I) × ℕ2)
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 : nameset(I) List
7. x : nameset(I)
8. i : \mBbbN{}2
9. alpha : X(I)
10. box : A-face(X;(Id\_A a b);I;alpha) List
11. A-adjacent-compatible(X;(Id\_A a b);I;alpha;box)
12. \mneg{}(x \mmember{} J)
13. l\_subset(Cname;J;I)
14. \mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}box. A-face-name(f) = <y, c>)
15. (\mexists{}f\mmember{}box. A-face-name(f) = <x, i>)
16. \mforall{}i@0:\mBbbN{}||box||. (\mneg{}(A-face-name(box[i@0]) = <x, 1 - i>))
17. (\mforall{}f\mmember{}box.(fst(f) \mmember{} [x / J]))
18. (\mforall{}f1,f2\mmember{}box. \mneg{}(A-face-name(f1) = A-face-name(f2)))
19. A-adjacent-compatible(X;A;I+;iota'(I)(alpha);lift-id-faces(X;A;I;alpha;box))
20. \mneg{}(x \mmember{} J)
21. l\_subset(Cname;J;I+)
22. \mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <y, c>)
23. (\mexists{}f\mmember{}lift-id-faces(X;A;I;alpha;box). A-face-name(f) = <x, i>)
24. i@0 : \mBbbN{}||box||@i
25. x1 : nameset(I)@i
26. i1 : \mBbbN{}2@i
27. v2 : (Id\_A a b)((x1:=i1)(alpha))@i
28. box[i@0] = <x1, i1, v2>
29. \mneg{}(<x1, i1> = <x, 1 - i>)
30. <x1, i1> = <x, 1 - i>
\mvdash{} False
By
Latex:
TACTIC:(D -2 THEN HypSubst' (-1)0)
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