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Step * 1 2 3 2 1 1 of Lemma cu-box-in-box_wf


1. I : Cname List
2. J : nameset(I) List
3. x : nameset(I)
4. d : ℕ2
5. box : I-face(c𝕌;I) List
6. adjacent-compatible(c𝕌;I;box)
7. ¬(x ∈ J)
8. l_subset(Cname;J;I)
9. ∀y:nameset(J). ∀c:ℕ2.  (∃f∈box. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
10. (∃f∈box. face-name(f) = <x, d> ∈ (nameset(I) × ℕ2))
11. (∀f∈box.¬(face-name(f) = <x, 1 - d> ∈ (nameset(I) × ℕ2)))
12. (∀f∈box.(fst(f) ∈ [x / J]))
13. (∀f1,f2∈box.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
14. u : i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
15. i : ℕ||box||
16. j : ℕ||box||
17. ¬(dimension(box[i]) = dimension(box[j]) ∈ Cname)
18. cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]); dimension(box[i])];
                        (dimension(box[i]):=direction(box[i]));1;u j)
= cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]); dimension(box[i])];
                      (dimension(box[i]):=direction(box[i]));1;u j)
∈ cu-cube-family(cube(box[j]);I-[dimension(box[j]); dimension(box[i])];(1 o (dimension(box[i]):=direction(box[i]))))
⊢ cu-cube-family(cube(box[j]);I-[dimension(box[j]); dimension(box[i])];(1 o (dimension(box[i]):=direction(box[i]))))
= cu-cube-family(cube(box[i]);I-[dimension(box[i]); dimension(box[j])];(1 o (dimension(box[j]):=direction(box[j]))))
∈ Type
BY
{ TACTIC:((Assert (box[i] ∈ box) BY
                 Auto)
          THEN (Assert (box[j] ∈ box) BY
                      Auto)
          THEN Unfold `adjacent-compatible` 6
          THEN (FLemma `pairwise-implies` [6;-1;-2] THENA Auto)) }

1
1. I : Cname List
2. J : nameset(I) List
3. x : nameset(I)
4. d : ℕ2
5. box : I-face(c𝕌;I) List
6. (∀f1,f2∈box.  face-compatible(c𝕌;I;f1;f2))
7. ¬(x ∈ J)
8. l_subset(Cname;J;I)
9. ∀y:nameset(J). ∀c:ℕ2.  (∃f∈box. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
10. (∃f∈box. face-name(f) = <x, d> ∈ (nameset(I) × ℕ2))
11. (∀f∈box.¬(face-name(f) = <x, 1 - d> ∈ (nameset(I) × ℕ2)))
12. (∀f∈box.(fst(f) ∈ [x / J]))
13. (∀f1,f2∈box.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
14. u : i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
15. i : ℕ||box||
16. j : ℕ||box||
17. ¬(dimension(box[i]) = dimension(box[j]) ∈ Cname)
18. cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]); dimension(box[i])];
                        (dimension(box[i]):=direction(box[i]));1;u j)
= cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]); dimension(box[i])];
                      (dimension(box[i]):=direction(box[i]));1;u j)
∈ cu-cube-family(cube(box[j]);I-[dimension(box[j]); dimension(box[i])];(1 o (dimension(box[i]):=direction(box[i]))))
19. (box[i] ∈ box)
20. (box[j] ∈ box)
21. (box[j] = box[i] ∈ I-face(c𝕌;I)) ∨ face-compatible(c𝕌;I;box[j];box[i]) ∨ face-compatible(c𝕌;I;box[i];box[j])
⊢ cu-cube-family(cube(box[j]);I-[dimension(box[j]); dimension(box[i])];(1 o (dimension(box[i]):=direction(box[i]))))
= cu-cube-family(cube(box[i]);I-[dimension(box[i]); dimension(box[j])];(1 o (dimension(box[j]):=direction(box[j]))))
∈ Type

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Latex:

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Latex:

1.  I  :  Cname  List
2.  J  :  nameset(I)  List
3.  x  :  nameset(I)
4.  d  :  \mBbbN{}2
5.  box  :  I-face(c\mBbbU{};I)  List
6.  adjacent-compatible(c\mBbbU{};I;box)
7.  \mneg{}(x  \mmember{}  J)
8.  l\_subset(Cname;J;I)
9.  \mforall{}y:nameset(J).  \mforall{}c:\mBbbN{}2.    (\mexists{}f\mmember{}box.  face-name(f)  =  <y,  c>)
10.  (\mexists{}f\mmember{}box.  face-name(f)  =  <x,  d>)
11.  (\mforall{}f\mmember{}box.\mneg{}(face-name(f)  =  <x,  1  -  d>))
12.  (\mforall{}f\mmember{}box.(fst(f)  \mmember{}  [x  /  J]))
13.  (\mforall{}f1,f2\mmember{}box.    \mneg{}(face-name(f1)  =  face-name(f2)))
14.  u  :  i:\mBbbN{}||box||  {}\mrightarrow{}  cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
15.  i  :  \mBbbN{}||box||
16.  j  :  \mBbbN{}||box||
17.  \mneg{}(dimension(box[i])  =  dimension(box[j]))
18.  cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]);  dimension(box[i])];
                                                (dimension(box[i]):=direction(box[i]));1;u  j)
=  cu-cube-restriction(cube(box[j]);I-[dimension(box[j])];I-[dimension(box[j]);  dimension(box[i])];
                                            (dimension(box[i]):=direction(box[i]));1;u  j)
\mvdash{}  cu-cube-family(cube(box[j]);I-[dimension(box[j]);
                                                                  dimension(box[i])];(1  o  (dimension(box[i]):=direction(box[i]))))
=  cu-cube-family(cube(box[i]);I-[dimension(box[i]);
                                                                  dimension(box[j])];(1  o  (dimension(box[j]):=direction(box[j]))))


By

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Latex:
TACTIC:((Assert  (box[i]  \mmember{}  box)  BY
                              Auto)
                THEN  (Assert  (box[j]  \mmember{}  box)  BY
                                        Auto)
                THEN  Unfold  `adjacent-compatible`  6
                THEN  (FLemma  `pairwise-implies`  [6;-1;-2]  THENA  Auto))

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