Step
*
2
1
of Lemma
poset-functor-extends-box-faces
1. G : Groupoid
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(cubical-nerve(cat(G));I;J;x;i)
7. ¬↑null(J)
8. (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
9. i1 : ℕ||bx||
10. ∀f:name-morph(I-[dimension(bx[i1])];[]). (((dimension(bx[i1]):=direction(bx[i1])) o f) ∈ name-morph(I;[]))
11. i2 : nameset(I-[dimension(bx[i1])])
12. c : {c:name-morph(I-[dimension(bx[i1])];[])| (c i2) = 0 ∈ ℕ2}
13. ((dimension(bx[i1]):=direction(bx[i1])) o c) ∈ name-morph(I;[])
14. (((dimension(bx[i1]):=direction(bx[i1])) o c) i2) = 0 ∈ ℕ2
⊢ (cube(bx[i1]) c flip(c;i2) (λx.Ax))
= nerve_box_edge(bx;((dimension(bx[i1]):=direction(bx[i1])) o c);i2)
∈ (cat-arrow(cat(G)) nerve_box_label(bx;((dimension(bx[i1]):=direction(bx[i1])) o c))
nerve_box_label(bx;((dimension(bx[i1]):=direction(bx[i1])) o flip(c;i2))))
BY
{ (InstLemma `nerve_box_edge_same` [⌜cat(G)⌝;⌜I⌝;⌜J⌝;⌜x⌝;⌜i⌝;⌜bx⌝;⌜i2⌝;⌜((dimension(bx[i1]):=direction(bx[i1])) o c)⌝;
⌜bx[i1]⌝]⋅
THENA Auto
) }
1
1. G : Groupoid
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(cubical-nerve(cat(G));I;J;x;i)
7. ¬↑null(J)
8. (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
9. i1 : ℕ||bx||
10. ∀f:name-morph(I-[dimension(bx[i1])];[]). (((dimension(bx[i1]):=direction(bx[i1])) o f) ∈ name-morph(I;[]))
11. i2 : nameset(I-[dimension(bx[i1])])
12. c : {c:name-morph(I-[dimension(bx[i1])];[])| (c i2) = 0 ∈ ℕ2}
13. ((dimension(bx[i1]):=direction(bx[i1])) o c) ∈ name-morph(I;[])
14. (((dimension(bx[i1]):=direction(bx[i1])) o c) i2) = 0 ∈ ℕ2
15. (bx[i1] ∈ bx)
⊢ direction(bx[i1]) = (((dimension(bx[i1]):=direction(bx[i1])) o c) dimension(bx[i1])) ∈ ℕ2
2
1. G : Groupoid
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(cubical-nerve(cat(G));I;J;x;i)
7. ¬↑null(J)
8. (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
9. i1 : ℕ||bx||
10. ∀f:name-morph(I-[dimension(bx[i1])];[]). (((dimension(bx[i1]):=direction(bx[i1])) o f) ∈ name-morph(I;[]))
11. i2 : nameset(I-[dimension(bx[i1])])
12. c : {c:name-morph(I-[dimension(bx[i1])];[])| (c i2) = 0 ∈ ℕ2}
13. ((dimension(bx[i1]):=direction(bx[i1])) o c) ∈ name-morph(I;[])
14. (((dimension(bx[i1]):=direction(bx[i1])) o c) i2) = 0 ∈ ℕ2
15. (bx[i1] ∈ bx)
16. direction(bx[i1]) = (((dimension(bx[i1]):=direction(bx[i1])) o c) dimension(bx[i1])) ∈ ℕ2
⊢ direction(bx[i1]) = (flip(((dimension(bx[i1]):=direction(bx[i1])) o c);i2) dimension(bx[i1])) ∈ ℕ2
3
1. G : Groupoid
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(cubical-nerve(cat(G));I;J;x;i)
7. ¬↑null(J)
8. (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
9. i1 : ℕ||bx||
10. ∀f:name-morph(I-[dimension(bx[i1])];[]). (((dimension(bx[i1]):=direction(bx[i1])) o f) ∈ name-morph(I;[]))
11. i2 : nameset(I-[dimension(bx[i1])])
12. c : {c:name-morph(I-[dimension(bx[i1])];[])| (c i2) = 0 ∈ ℕ2}
13. ((dimension(bx[i1]):=direction(bx[i1])) o c) ∈ name-morph(I;[])
14. (((dimension(bx[i1]):=direction(bx[i1])) o c) i2) = 0 ∈ ℕ2
15. (cube(bx[i1]) ((dimension(bx[i1]):=direction(bx[i1])) o c) flip(((dimension(bx[i1]):=direction(bx[i1])) o c);i2)
(λx.Ax))
= nerve_box_edge(bx;((dimension(bx[i1]):=direction(bx[i1])) o c);i2)
∈ (cat-arrow(cat(G)) nerve_box_label(bx;((dimension(bx[i1]):=direction(bx[i1])) o c))
nerve_box_label(bx;flip(((dimension(bx[i1]):=direction(bx[i1])) o c);i2)))
⊢ (cube(bx[i1]) c flip(c;i2) (λx.Ax))
= nerve_box_edge(bx;((dimension(bx[i1]):=direction(bx[i1])) o c);i2)
∈ (cat-arrow(cat(G)) nerve_box_label(bx;((dimension(bx[i1]):=direction(bx[i1])) o c))
nerve_box_label(bx;((dimension(bx[i1]):=direction(bx[i1])) o flip(c;i2))))
Latex:
Latex:
1. G : Groupoid
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : \mBbbN{}2
6. bx : open\_box(cubical-nerve(cat(G));I;J;x;i)
7. \mneg{}\muparrow{}null(J)
8. (\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2)))
9. i1 : \mBbbN{}||bx||
10. \mforall{}f:name-morph(I-[dimension(bx[i1])];[])
(((dimension(bx[i1]):=direction(bx[i1])) o f) \mmember{} name-morph(I;[]))
11. i2 : nameset(I-[dimension(bx[i1])])
12. c : \{c:name-morph(I-[dimension(bx[i1])];[])| (c i2) = 0\}
13. ((dimension(bx[i1]):=direction(bx[i1])) o c) \mmember{} name-morph(I;[])
14. (((dimension(bx[i1]):=direction(bx[i1])) o c) i2) = 0
\mvdash{} (cube(bx[i1]) c flip(c;i2) (\mlambda{}x.Ax))
= nerve\_box\_edge(bx;((dimension(bx[i1]):=direction(bx[i1])) o c);i2)
By
Latex:
(InstLemma `nerve\_box\_edge\_same` [\mkleeneopen{}cat(G)\mkleeneclose{};\mkleeneopen{}I\mkleeneclose{};\mkleeneopen{}J\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}i\mkleeneclose{};\mkleeneopen{}bx\mkleeneclose{};\mkleeneopen{}i2\mkleeneclose{};
\mkleeneopen{}((dimension(bx[i1]):=direction(bx[i1])) o c)\mkleeneclose{};\mkleeneopen{}bx[i1]\mkleeneclose{}]\mcdot{}
THENA Auto
)
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