Step
*
7
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. (∀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. (∀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]))
⊢ (∀f1,f2∈map(λface.lift-id-face(X;A;I;alpha;face);box). ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I+) × ℕ2)))
BY
{ (ParallelOp -8
THEN (RWO "length-map-sq" 0 THENA Auto)
THEN ParallelOp -8
THEN ParallelLast
THEN (RWO "select-map" 0 THENA Auto)
THEN Reduce 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. ¬(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))
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{}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. (\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]))
\mvdash{} (\mforall{}f1,f2\mmember{}map(\mlambda{}face.lift-id-face(X;A;I;alpha;face);box). \mneg{}(A-face-name(f1) = A-face-name(f2)))
By
Latex:
(ParallelOp -8
THEN (RWO "length-map-sq" 0 THENA Auto)
THEN ParallelOp -8
THEN ParallelLast
THEN (RWO "select-map" 0 THENA Auto)
THEN Reduce 0)
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