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Step * of Lemma extend-face-term-property

∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[J:fset(ℕ)]. ∀[g:I,phi(J)].
  ((extend-face-term(I;phi;u))<g> = u(g) ∈ Point(face_lattice(J)))
BY
{ (Auto
   THEN RepUR ``cubical-subset rep-sub-sheaf cube-cat cat-arrow`` -1
   THEN D -1
   THEN Unfold `extend-face-term` 0
   THEN (GenConclTerm ⌜face_lattice_components(I;phi)⌝⋅ THENA Auto)
   THEN Thin (-1)
   THEN D -1
   THEN (InstLemma `fset-subtype2` [⌜{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} ⌝;
         ⌜product-deq(fset(names(I));fset(names(I));deq-fset(NamesDeq);deq-fset(NamesDeq))⌝;⌜v⌝]⋅
         THENA Auto
         )
   THEN Guard (-1)
   THEN ((GenConcl ⌜v = fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v\000C} )⌝⋅
          THEN Try ((Unfold `guard` -1 THEN Trivial))
          )
         THENW Auto
         )
   THEN Thin (-3)) }

1
1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. J : fset(ℕ)
5. g : J ⟶ I
6. (phi g) = 1
7. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
8. phi = \/(λpr.irr_face(I;fst(pr);snd(pr))"(v)) ∈ Point(face_lattice(I))
9. fs : fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
10. v = 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)))<g> = u(g) ∈ Point(face_lattice(J))

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Latex:

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Latex:
\mforall{}[I:fset(\mBbbN{})].  \mforall{}[phi:\mBbbF{}(I)].  \mforall{}[u:\{I,phi  \mvdash{}  \_:\mBbbF{}\}].  \mforall{}[J:fset(\mBbbN{})].  \mforall{}[g:I,phi(J)].
    ((extend-face-term(I;phi;u))<g>  =  u(g))


By

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Latex:
(Auto
  THEN  RepUR  ``cubical-subset  rep-sub-sheaf  cube-cat  cat-arrow``  -1
  THEN  D  -1
  THEN  Unfold  `extend-face-term`  0
  THEN  (GenConclTerm  \mkleeneopen{}face\_lattice\_components(I;phi)\mkleeneclose{}\mcdot{}  THENA  Auto)
  THEN  Thin  (-1)
  THEN  D  -1
  THEN  (InstLemma  `fset-subtype2`  [\mkleeneopen{}\{p:fset(names(I))  \mtimes{}  fset(names(I))| 
                                                                        \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\}  \mkleeneclose{};
              \mkleeneopen{}product-deq(fset(names(I));fset(names(I));deq-fset(NamesDeq);deq-fset(NamesDeq))\mkleeneclose{};\mkleeneopen{}v\mkleeneclose{}]\mcdot{}
              THENA  Auto
              )
  THEN  Guard  (-1)
  THEN  ((GenConcl  \mkleeneopen{}v  =  fs\mkleeneclose{}\mcdot{}  THEN  Try  ((Unfold  `guard`  -1  THEN  Trivial)))  THENW  Auto)
  THEN  Thin  (-3))

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