Proof of Nuprl Lemma : open_box_image

Step * 1 1 1 2 2 of Lemma open_box_image_wf

1. X : CubicalSet
2. I : Cname List
3. J : Cname List
4. K : Cname List
5. f : name-morph(I;K)
6. nameset(map(f;J)) ⊆r nameset(K)
7. x : nameset(I)
8. ∀x:nameset([x / J]). (f x ∈ nameset(K))
9. i : ℕ2
10. nameset([x / J]) ⊆r name-morph-domain(f;I)
11. bx : I-face(X;I) List
12. adjacent-compatible(X;I;bx)
13. (¬(x ∈ J))
∧ l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈bx. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
  ∧ (∃f∈bx. face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
  ∧ (∀f∈bx.¬(face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
14. ∀x:nameset(J). (f x ∈ nameset(K))
15. map(f;J) ∈ nameset(K) List
16. f x ∈ nameset(K)
17. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
18. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ (¬(f x ∈ map(f;J)))
∧ l_subset(Cname;map(f;J);K)

∧ ((∀y:nameset(map(f;J)). ∀c:ℕ2.  (∃f∈open_box_image(X;I;K;f;bx). face-name(f) = <y, c> ∈ (nameset(K) × ℕ2)))

  ∧ (∃f@0∈open_box_image(X;I;K;f;bx). face-name(f@0) = <f x, i> ∈ (nameset(K) × ℕ2))
  ∧ (∀f@0∈open_box_image(X;I;K;f;bx).¬(face-name(f@0) = <f x, 1 - i> ∈ (nameset(K) × ℕ2))))
∧ (∀f@0∈open_box_image(X;I;K;f;bx).(fst(f@0) ∈ [f x / map(f;J)]))
∧ (∀f1,f2∈open_box_image(X;I;K;f;bx).  ¬(face-name(f1) = face-name(f2) ∈ (nameset(K) × ℕ2)))
BY
{ RepeatFor 2 (ParallelOp -6) }

1
1. X : CubicalSet
2. I : Cname List
3. J : Cname List
4. K : Cname List
5. f : name-morph(I;K)
6. nameset(map(f;J)) ⊆r nameset(K)
7. x : nameset(I)
8. ∀x:nameset([x / J]). (f x ∈ nameset(K))
9. i : ℕ2
10. nameset([x / J]) ⊆r name-morph-domain(f;I)
11. bx : I-face(X;I) List
12. adjacent-compatible(X;I;bx)
13. l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈bx. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
  ∧ (∃f∈bx. face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
  ∧ (∀f∈bx.¬(face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
14. ∀x:nameset(J). (f x ∈ nameset(K))
15. map(f;J) ∈ nameset(K) List
16. f x ∈ nameset(K)
17. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
18. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
19. (f x ∈ map(f;J))
⊢ (x ∈ J)

2
1. X : CubicalSet
2. I : Cname List
3. J : Cname List
4. K : Cname List
5. f : name-morph(I;K)
6. nameset(map(f;J)) ⊆r nameset(K)
7. x : nameset(I)
8. ∀x:nameset([x / J]). (f x ∈ nameset(K))
9. i : ℕ2
10. nameset([x / J]) ⊆r name-morph-domain(f;I)
11. bx : I-face(X;I) List
12. adjacent-compatible(X;I;bx)
13. ¬(x ∈ J)
14. ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈bx. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
    ∧ (∃f∈bx. face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
    ∧ (∀f∈bx.¬(face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
15. l_subset(Cname;J;I)
16. ∀x:nameset(J). (f x ∈ nameset(K))
17. map(f;J) ∈ nameset(K) List
18. f x ∈ nameset(K)
19. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
20. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ l_subset(Cname;map(f;J);K)

3
1. X : CubicalSet
2. I : Cname List
3. J : Cname List
4. K : Cname List
5. f : name-morph(I;K)
6. nameset(map(f;J)) ⊆r nameset(K)
7. x : nameset(I)
8. ∀x:nameset([x / J]). (f x ∈ nameset(K))
9. i : ℕ2
10. nameset([x / J]) ⊆r name-morph-domain(f;I)
11. bx : I-face(X;I) List
12. adjacent-compatible(X;I;bx)
13. ¬(x ∈ J)
14. l_subset(Cname;J;I)
15. ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈bx. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
    ∧ (∃f∈bx. face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
    ∧ (∀f∈bx.¬(face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
16. ∀x:nameset(J). (f x ∈ nameset(K))
17. map(f;J) ∈ nameset(K) List
18. f x ∈ nameset(K)
19. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
20. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))

⊢ ((∀y:nameset(map(f;J)). ∀c:ℕ2.  (∃f∈open_box_image(X;I;K;f;bx). face-name(f) = <y, c> ∈ (nameset(K) × ℕ2)))

Latex:

Latex:
1. X : CubicalSet 2. I : Cname List 3. J : Cname List 4. K : Cname List 5. f : name-morph(I;K) 6. nameset(map(f;J)) \msubseteq{}r nameset(K) 7. x : nameset(I) 8. \mforall{}x:nameset([x / J]). (f x \mmember{} nameset(K)) 9. i : \mBbbN{}2 10. nameset([x / J]) \msubseteq{}r name-morph-domain(f;I) 11. bx : I-face(X;I) List 12. adjacent-compatible(X;I;bx) 13. (\mneg{}(x \mmember{} J)) \mwedge{} l\_subset(Cname;J;I) \mwedge{} ((\mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}bx. face-name(f) = <y, c>)) \mwedge{} (\mexists{}f\mmember{}bx. face-name(f) = <x, i>) \mwedge{} (\mforall{}f\mmember{}bx.\mneg{}(face-name(f) = <x, 1 - i>))) \mwedge{} (\mforall{}f\mmember{}bx.(fst(f) \mmember{} [x / J])) \mwedge{} (\mforall{}f1,f2\mmember{}bx. \mneg{}(face-name(f1) = face-name(f2))) 14. \mforall{}x:nameset(J). (f x \mmember{} nameset(K)) 15. map(f;J) \mmember{} nameset(K) List 16. f x \mmember{} nameset(K) 17. \mforall{}fc:I-face(X;I). ((fc \mmember{} bx) {}\mRightarrow{} (fst(fc) \mmember{} [x / J])) 18. \mforall{}fc:I-face(X;I). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc))))) \mvdash{} (\mneg{}(f x \mmember{} map(f;J))) \mwedge{} l\_subset(Cname;map(f;J);K) \mwedge{} ((\mforall{}y:nameset(map(f;J)). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}open\_box\_image(X;I;K;f;bx). face-name(f) = <y, c>)) \mwedge{} (\mexists{}f@0\mmember{}open\_box\_image(X;I;K;f;bx). face-name(f@0) = <f x, i>) \mwedge{} (\mforall{}f@0\mmember{}open\_box\_image(X;I;K;f;bx).\mneg{}(face-name(f@0) = <f x, 1 - i>))) \mwedge{} (\mforall{}f@0\mmember{}open\_box\_image(X;I;K;f;bx).(fst(f@0) \mmember{} [f x / map(f;J)])) \mwedge{} (\mforall{}f1,f2\mmember{}open\_box\_image(X;I;K;f;bx). \mneg{}(face-name(f1) = face-name(f2))) By Latex: RepeatFor 2 (ParallelOp -6) Home Index