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

.....subterm..... T:t
3:n
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)
∧ (¬(x ∈ J))
∧ l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈box. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
  ∧ (∃f∈box. face-name(f) = <x, d> ∈ (nameset(I) × ℕ2))
  ∧ (∀f∈box.¬(face-name(f) = <x, 1 - d> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈box.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈box.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
7. u : i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
8. i : ℕ||box||
9. j : ℕ||box||
10. ¬(dimension(box[i]) = dimension(box[j]) ∈ Cname)
⊢ 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[i]);I-[dimension(box[i]); dimension(box[j])];(1 o (dimension(box[j]):=direction(box[j]))))
BY
{ TACTIC:DoSubsume }

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. adjacent-compatible(c𝕌;I;box)
∧ (¬(x ∈ J))
∧ l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈box. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
  ∧ (∃f∈box. face-name(f) = <x, d> ∈ (nameset(I) × ℕ2))
  ∧ (∀f∈box.¬(face-name(f) = <x, 1 - d> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈box.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈box.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
7. u : i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
8. i : ℕ||box||
9. j : ℕ||box||
10. ¬(dimension(box[i]) = dimension(box[j]) ∈ Cname)
⊢ 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]))))

2
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)
∧ (¬(x ∈ J))
∧ l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2.  (∃f∈box. face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
  ∧ (∃f∈box. face-name(f) = <x, d> ∈ (nameset(I) × ℕ2))
  ∧ (∀f∈box.¬(face-name(f) = <x, 1 - d> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈box.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈box.  ¬(face-name(f1) = face-name(f2) ∈ (nameset(I) × ℕ2)))
7. u : i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
8. i : ℕ||box||
9. j : ℕ||box||
10. ¬(dimension(box[i]) = dimension(box[j]) ∈ Cname)
11. 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]))))
    ⊆r cu-cube-family(cube(box[i]);I-[dimension(box[i]);
                                      dimension(box[j])];(1 o (dimension(box[j]):=direction(box[j]))))

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Latex:

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Latex:
.....subterm.....  T:t
3:n
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)
\mwedge{}  (\mneg{}(x  \mmember{}  J))
\mwedge{}  l\_subset(Cname;J;I)
\mwedge{}  ((\mforall{}y:nameset(J).  \mforall{}c:\mBbbN{}2.    (\mexists{}f\mmember{}box.  face-name(f)  =  <y,  c>))
    \mwedge{}  (\mexists{}f\mmember{}box.  face-name(f)  =  <x,  d>)
    \mwedge{}  (\mforall{}f\mmember{}box.\mneg{}(face-name(f)  =  <x,  1  -  d>)))
\mwedge{}  (\mforall{}f\mmember{}box.(fst(f)  \mmember{}  [x  /  J]))
\mwedge{}  (\mforall{}f1,f2\mmember{}box.    \mneg{}(face-name(f1)  =  face-name(f2)))
7.  u  :  i:\mBbbN{}||box||  {}\mrightarrow{}  cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)
8.  i  :  \mBbbN{}||box||
9.  j  :  \mBbbN{}||box||
10.  \mneg{}(dimension(box[i])  =  dimension(box[j]))
\mvdash{}  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)
    \mmember{}  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:DoSubsume

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