Step
*
1
of Lemma
cubical-interval-non-trivial-box
1. I : Cname List
2. J : nameset(I) List
3. x : nameset(I)
4. i : ℕ2
5. bx : open_box(cubical-interval();I;J;x;i)
6. h : name-morph(I;[])
7. ¬(J = [] ∈ (nameset(I) List))
⊢ ¬(filter(λf.(h (fst(f)) =z fst(snd(f)));bx)
= []
∈ ({x:{f:I-face(cubical-interval();I)| (f ∈ bx)} | ↑(h (fst(x)) =z fst(snd(x)))} List))
BY
{ (DVar `bx' THENA Auto) }
1
1. I : Cname List
2. J : nameset(I) List
3. x : nameset(I)
4. i : ℕ2
5. bx : I-face(cubical-interval();I) List
6. adjacent-compatible(cubical-interval();I;bx)
∧ (¬(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)))
7. h : name-morph(I;[])
8. ¬(J = [] ∈ (nameset(I) List))
⊢ ¬(filter(λf.(h (fst(f)) =z fst(snd(f)));bx)
= []
∈ ({x:{f:I-face(cubical-interval();I)| (f ∈ bx)} | ↑(h (fst(x)) =z fst(snd(x)))} List))
Latex:
Latex:
1. I : Cname List
2. J : nameset(I) List
3. x : nameset(I)
4. i : \mBbbN{}2
5. bx : open\_box(cubical-interval();I;J;x;i)
6. h : name-morph(I;[])
7. \mneg{}(J = [])
\mvdash{} \mneg{}(filter(\mlambda{}f.(h (fst(f)) =\msubz{} fst(snd(f)));bx) = [])
By
Latex:
(DVar `bx' THENA Auto)
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