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Step * 1 1 1 1 of Lemma boundary-singleton-complex


1. k : ℕ
2. n : ℕ
3. c : {c:ℚCube(k)| dim(c) = n ∈ ℤ} 
4. ↑Inhabited(c)
5. v : {f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)}  List
6. remove-repeats(rc-deq(k);rat-cube-faces(k;c)) = v ∈ ({f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)}  List)
7. v ∈ ℚCube(k) List
8. x : ℚCube(k)
9. (x ∈ remove-repeats(rc-deq(k);rat-cube-faces(k;c)))
⊢ ↑in-complex-boundary(k;x;[c])
BY
{ (GenListD (-1)
   THEN (RWO "member-rat-cube-faces" (-1) THEN Auto)
   THEN RepUR ``in-complex-boundary`` 0
   THEN AutoSplit) }

1
1. k : ℕ
2. n : ℕ
3. c : {c:ℚCube(k)| dim(c) = n ∈ ℤ} 
4. ↑Inhabited(c)
5. v : {f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)}  List
6. remove-repeats(rc-deq(k);rat-cube-faces(k;c)) = v ∈ ({f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)}  List)
7. v ∈ ℚCube(k) List
8. x : ℚCube(k)
9. ¬↑is-rat-cube-face(k;x;c)
10. x ≤ c
11. dim(x) = (dim(c) - 1) ∈ ℤ
⊢ ↑isOdd(0)

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Latex:

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Latex:

1.  k  :  \mBbbN{}
2.  n  :  \mBbbN{}
3.  c  :  \{c:\mBbbQ{}Cube(k)|  dim(c)  =  n\} 
4.  \muparrow{}Inhabited(c)
5.  v  :  \{f:\mBbbQ{}Cube(k)|  f  \mleq{}  c  \mwedge{}  (dim(f)  =  (dim(c)  -  1))\}    List
6.  remove-repeats(rc-deq(k);rat-cube-faces(k;c))  =  v
7.  v  \mmember{}  \mBbbQ{}Cube(k)  List
8.  x  :  \mBbbQ{}Cube(k)
9.  (x  \mmember{}  remove-repeats(rc-deq(k);rat-cube-faces(k;c)))
\mvdash{}  \muparrow{}in-complex-boundary(k;x;[c])


By

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Latex:
(GenListD  (-1)
  THEN  (RWO  "member-rat-cube-faces"  (-1)  THEN  Auto)
  THEN  RepUR  ``in-complex-boundary``  0
  THEN  AutoSplit)

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