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Step * of Lemma cu-box-in-box_wf

∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[d:ℕ2]. ∀[box:open_box(c𝕌;I;J;x;d)].
  (cu-box-in-box(I;box) ∈ Type)
BY
{ TACTIC:((UnivCD THENA Auto) THEN DVar `box' THEN Unfold `cu-box-in-box` 0) }

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)))
⊢ {u:i:ℕ||box|| ⟶ cu-cube-family(cube(box[i]);I-[dimension(box[i])];1)| 
   ∀i,j:ℕ||box||.
     ((¬(dimension(box[i]) = dimension(box[j]) ∈ Cname))
     ⇒ (cu-cube-restriction(cube(box[i]);I-[dimension(box[i])];I-[dimension(box[i]); dimension(box[j])];
                             (dimension(box[j]):=direction(box[j]));1;u i)
        = 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]))))))}  ∈ Type

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Latex:

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Latex:
\mforall{}[I:Cname  List].  \mforall{}[J:nameset(I)  List].  \mforall{}[x:nameset(I)].  \mforall{}[d:\mBbbN{}2].  \mforall{}[box:open\_box(c\mBbbU{};I;J;x;d)].
    (cu-box-in-box(I;box)  \mmember{}  Type)


By

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Latex:
TACTIC:((UnivCD  THENA  Auto)  THEN  DVar  `box'  THEN  Unfold  `cu-box-in-box`  0)

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