Step
*
7
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. (∀f∈box.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
17. (∀f∈box.(fst(f) ∈ [x / J]))
18. ∀i:ℕ||box||. ∀j:ℕi. (¬(A-face-name(box[j]) = A-face-name(box[i]) ∈ (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. (∀f∈lift-id-faces(X;A;I;alpha;box).¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I+) × ℕ2)))
25. (∀f∈lift-id-faces(X;A;I;alpha;box).(fst(f) ∈ [x / J]))
26. i1 : ℕ||box||@i
27. ∀j:ℕi1. (¬(A-face-name(box[j]) = A-face-name(box[i1]) ∈ (nameset(I) × ℕ2)))
28. j : ℕi1@i
29. ¬(A-face-name(box[j]) = A-face-name(box[i1]) ∈ (nameset(I) × ℕ2))
⊢ ¬(A-face-name(lift-id-face(X;A;I;alpha;box[j])) = A-face-name(lift-id-face(X;A;I;alpha;box[i1])) ∈ (nameset(I+) × ℕ2))
BY
{ TACTIC:(MoveToConcl (-1)
THEN (GenConclTerm ⌜box[i1]⌝⋅ THENA Auto)
THEN RepeatFor 2 (D -2)
THEN (GenConclTerm ⌜box[j]⌝⋅ THENA Auto)
THEN RepeatFor 2 (D -2)
THEN RepUR ``lift-id-face A-face-name`` 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. (∀f∈box.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
17. (∀f∈box.(fst(f) ∈ [x / J]))
18. ∀i:ℕ||box||. ∀j:ℕi. (¬(A-face-name(box[j]) = A-face-name(box[i]) ∈ (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. (∀f∈lift-id-faces(X;A;I;alpha;box).¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I+) × ℕ2)))
25. (∀f∈lift-id-faces(X;A;I;alpha;box).(fst(f) ∈ [x / J]))
26. i1 : ℕ||box||@i
27. ∀j:ℕi1. (¬(A-face-name(box[j]) = A-face-name(box[i1]) ∈ (nameset(I) × ℕ2)))
28. j : ℕi1@i
29. x1 : nameset(I)@i
30. i2 : ℕ2@i
31. v2 : (Id_A a b)((x1:=i2)(alpha))@i
32. box[i1] = <x1, i2, v2> ∈ A-face(X;(Id_A a b);I;alpha)
33. x2 : nameset(I)@i
34. i3 : ℕ2@i
35. v4 : (Id_A a b)((x2:=i3)(alpha))@i
36. box[j] = <x2, i3, v4> ∈ A-face(X;(Id_A a b);I;alpha)
⊢ (¬(<x2, i3> = <x1, i2> ∈ (nameset(I) × ℕ2))) ⇒ (¬(<x2, i3> = <x1, i2> ∈ (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{}f\mmember{}box.\mneg{}(A-face-name(f) = <x, 1 - i>))
17. (\mforall{}f\mmember{}box.(fst(f) \mmember{} [x / J]))
18. \mforall{}i:\mBbbN{}||box||. \mforall{}j:\mBbbN{}i. (\mneg{}(A-face-name(box[j]) = A-face-name(box[i])))
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. (\mforall{}f\mmember{}lift-id-faces(X;A;I;alpha;box).\mneg{}(A-face-name(f) = <x, 1 - i>))
25. (\mforall{}f\mmember{}lift-id-faces(X;A;I;alpha;box).(fst(f) \mmember{} [x / J]))
26. i1 : \mBbbN{}||box||@i
27. \mforall{}j:\mBbbN{}i1. (\mneg{}(A-face-name(box[j]) = A-face-name(box[i1])))
28. j : \mBbbN{}i1@i
29. \mneg{}(A-face-name(box[j]) = A-face-name(box[i1]))
\mvdash{} \mneg{}(A-face-name(lift-id-face(X;A;I;alpha;box[j])) = A-face-name(lift-id-face(X;A;I;alpha;box[i1])))
By
Latex:
TACTIC:(MoveToConcl (-1)
THEN (GenConclTerm \mkleeneopen{}box[i1]\mkleeneclose{}\mcdot{} THENA Auto)
THEN RepeatFor 2 (D -2)
THEN (GenConclTerm \mkleeneopen{}box[j]\mkleeneclose{}\mcdot{} THENA Auto)
THEN RepeatFor 2 (D -2)
THEN RepUR ``lift-id-face A-face-name`` 0)
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