Step
*
1
1
2
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1
1
of Lemma
A-open-box-image_wf
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. ¬(x ∈ J)
16. l_subset(Cname;J;I)
17. (∀f∈bx.(fst(f) ∈ [x / J])) ∧ (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
18. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2)) ∧ (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
19. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
20. ∀x:nameset(J). (f x ∈ nameset(K))
21. map(f;J) ∈ nameset(K) List
22. f x ∈ nameset(K)
23. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
24. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ ∀y:nameset(map(f;J)). ∀c:ℕ2. (∃f∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c> ∈ (nameset(K) × ℕ2))
BY
{ (Auto
THEN (Assert J ∈ nameset(J) List BY
(Unfold `nameset` 0 THEN Auto))
THEN (Decide (∃j∈J. ↑eq-cname(f j;y)) THENA Auto)) }
1
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. ¬(x ∈ J)
16. l_subset(Cname;J;I)
17. (∀f∈bx.(fst(f) ∈ [x / J]))
18. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
19. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
20. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
21. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
22. ∀x:nameset(J). (f x ∈ nameset(K))
23. map(f;J) ∈ nameset(K) List
24. f x ∈ nameset(K)
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
26. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
27. y : nameset(map(f;J))
28. c : ℕ2
29. J ∈ nameset(J) List
30. (∃j∈J. ↑eq-cname(f j;y))
⊢ (∃f∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c> ∈ (nameset(K) × ℕ2))
2
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. ¬(x ∈ J)
16. l_subset(Cname;J;I)
17. (∀f∈bx.(fst(f) ∈ [x / J]))
18. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
19. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
20. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
21. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
22. ∀x:nameset(J). (f x ∈ nameset(K))
23. map(f;J) ∈ nameset(K) List
24. f x ∈ nameset(K)
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
26. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
27. y : nameset(map(f;J))
28. c : ℕ2
29. J ∈ nameset(J) List
30. ¬(∃j∈J. ↑eq-cname(f j;y))
⊢ (∃f∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c> ∈ (nameset(K) × ℕ2))
Latex:
Latex:
1. X : CubicalSet
2. A : \{X \mvdash{} \_\}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) \msubseteq{}r nameset(K)
9. x : nameset(I)
10. \mforall{}x:nameset([x / J]). (f x \mmember{} nameset(K))
11. i : \mBbbN{}2
12. nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. \mneg{}(x \mmember{} J)
16. l\_subset(Cname;J;I)
17. (\mforall{}f\mmember{}bx.(fst(f) \mmember{} [x / J])) \mwedge{} (\mforall{}f1,f2\mmember{}bx. \mneg{}(A-face-name(f1) = A-face-name(f2)))
18. (\mexists{}f\mmember{}bx. A-face-name(f) = <x, i>) \mwedge{} (\mforall{}f\mmember{}bx.\mneg{}(A-face-name(f) = <x, 1 - i>))
19. \mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}bx. A-face-name(f) = <y, c>)
20. \mforall{}x:nameset(J). (f x \mmember{} nameset(K))
21. map(f;J) \mmember{} nameset(K) List
22. f x \mmember{} nameset(K)
23. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (fst(fc) \mmember{} [x / J]))
24. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc)))))
\mvdash{} \mforall{}y:nameset(map(f;J)). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c>)
By
Latex:
(Auto
THEN (Assert J \mmember{} nameset(J) List BY
(Unfold `nameset` 0 THEN Auto))
THEN (Decide (\mexists{}j\mmember{}J. \muparrow{}eq-cname(f j;y)) THENA Auto))
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