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Step * 1 of Lemma extend-face-term-le


1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
5. [%1] : 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. 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)) ≤ phi
BY
{ Assert ⌜∀p1,p2:fset(names(I)).
            ((↑fset-disjoint(NamesDeq;p1;p2)) ⇒ <p1, p2> ∈ v ⇒ (irr-face-morph(I;p1;p2) ∈ I,phi(I)))⌝⋅ }

1
.....assertion..... 
1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
5. [%1] : 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. v = fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
⊢ ∀p1,p2:fset(names(I)).  ((↑fset-disjoint(NamesDeq;p1;p2)) ⇒ <p1, p2> ∈ v ⇒ (irr-face-morph(I;p1;p2) ∈ I,phi(I)))

2
1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
5. [%1] : 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. v = fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
8. ∀p1,p2:fset(names(I)).  ((↑fset-disjoint(NamesDeq;p1;p2)) ⇒ <p1, p2> ∈ v ⇒ (irr-face-morph(I;p1;p2) ∈ I,phi(I)))
⊢ \/(λpr.let as,bs = pr 
         in irr_face(I;as;bs) ∧ u(irr-face-morph(I;as;bs))"(fs)) ≤ phi

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Latex:

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Latex:

1.  I  :  fset(\mBbbN{})
2.  phi  :  \mBbbF{}(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.  [\%1]  :  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  )
7.  v  =  fs
\mvdash{}  \mbackslash{}/(\mlambda{}pr.let  as,bs  =  pr 
                  in  irr\_face(I;as;bs)  \mwedge{}  u(irr-face-morph(I;as;bs))"(fs))  \mleq{}  phi


By

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Latex:
Assert  \mkleeneopen{}\mforall{}p1,p2:fset(names(I)).
                    ((\muparrow{}fset-disjoint(NamesDeq;p1;p2))
                    {}\mRightarrow{}  <p1,  p2>  \mmember{}  v
                    {}\mRightarrow{}  (irr-face-morph(I;p1;p2)  \mmember{}  I,phi(I)))\mkleeneclose{}\mcdot{}

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