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Step * 1 of Lemma extend-face-term_wf


1. I : fset(ℕ)
2. phi : Point(face_lattice(I))
3. u : {I,phi ⊢ _:𝔽}
4. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
5. phi = \/(λpr.irr_face(I;fst(pr);snd(pr))"(v)) ∈ Point(face_lattice(I))
6. fs : fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
⊢ \/(λpr.let as,bs = pr 
         in irr_face(I;as;bs) ∧ u(irr-face-morph(I;as;bs))"(fs)) ∈ Point(face_lattice(I))
BY
{ (Auto THEN RepUR ``cubical-subset rep-sub-sheaf names-cat cat-arrow`` 0 THEN MemTypeCD THEN Auto) }

1
.....set predicate..... 
1. I : fset(ℕ)
2. phi : Point(face_lattice(I))
3. u : {I,phi ⊢ _:𝔽}
4. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
5. phi = \/(λpr.irr_face(I;fst(pr);snd(pr))"(v)) ∈ Point(face_lattice(I))
6. fs : fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
7. ∀z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} . (z ∈ v ∈ ℙ{[1 | i 0]})
8. p1 : fset(names(I))
9. p2 : fset(names(I))
10. ↑fset-disjoint(NamesDeq;p1;p2)
11. <p1, p2> ∈ v
⊢ (phi irr-face-morph(I;p1;p2)) = 1

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Latex:

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Latex:

1.  I  :  fset(\mBbbN{})
2.  phi  :  Point(face\_lattice(I))
3.  u  :  \{I,phi  \mvdash{}  \_:\mBbbF{}\}
4.  v  :  fset(\{p:fset(names(I))  \mtimes{}  fset(names(I))|  \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\}  )
5.  phi  =  \mbackslash{}/(\mlambda{}pr.irr\_face(I;fst(pr);snd(pr))"(v))
6.  fs  :  fset(\{z:\{p:fset(names(I))  \mtimes{}  fset(names(I))|  \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\}  |  z  \mmember{}  v\}\000C  )
\mvdash{}  \mbackslash{}/(\mlambda{}pr.let  as,bs  =  pr 
                  in  irr\_face(I;as;bs)  \mwedge{}  u(irr-face-morph(I;as;bs))"(fs))  \mmember{}  Point(face\_lattice(I))


By

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Latex:
(Auto  THEN  RepUR  ``cubical-subset  rep-sub-sheaf  names-cat  cat-arrow``  0  THEN  MemTypeCD  THEN  Auto)

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