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Step * 1 1 of Lemma FormSafe-iff-FormSafe1'


1. C : Type
2. left : Form(C)
3. right : Form(C)
4. ∀vs:Atom List. (FormSafe1(left) vs ⇐⇒ FormSafe1'(left) vs)
5. ∀vs:Atom List. (FormSafe1(right) vs ⇐⇒ FormSafe1'(right) vs)
6. vs : Atom List
7. ∃as,bs:Atom List
    (set-equal(Atom;vs;as @ bs)
    ∧ (FormSafe1(left) as)
    ∧ (FormSafe1(right) bs)
    ∧ (l_disjoint(Atom;as;FormFvs(right)) ∨ l_disjoint(Atom;bs;FormFvs(left))))
⊢ (∃theta:Atom List. (l_subset(Atom;theta;vs-FormFvs(right)) ∧ (FormSafe1'(left) theta) ∧ (FormSafe1'(right) vs-theta)))
∨ (∃phi:Atom List. (l_subset(Atom;phi;vs-FormFvs(left)) ∧ (FormSafe1'(left) vs-phi) ∧ (FormSafe1'(right) phi)))
BY
{ ((RWO "4< 5<" 0 THENA Auto) THEN RepeatFor 2 (Thin 4) THEN ExRepD) }

1
1. C : Type
2. left : Form(C)
3. right : Form(C)
4. vs : Atom List
5. as : Atom List
6. bs : Atom List
7. set-equal(Atom;vs;as @ bs)
8. FormSafe1(left) as
9. FormSafe1(right) bs
10. l_disjoint(Atom;as;FormFvs(right)) ∨ l_disjoint(Atom;bs;FormFvs(left))
⊢ (∃theta:Atom List. (l_subset(Atom;theta;vs-FormFvs(right)) ∧ (FormSafe1(left) theta) ∧ (FormSafe1(right) vs-theta)))
∨ (∃phi:Atom List. (l_subset(Atom;phi;vs-FormFvs(left)) ∧ (FormSafe1(left) vs-phi) ∧ (FormSafe1(right) phi)))

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Latex:

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Latex:

1.  C  :  Type
2.  left  :  Form(C)
3.  right  :  Form(C)
4.  \mforall{}vs:Atom  List.  (FormSafe1(left)  vs  \mLeftarrow{}{}\mRightarrow{}  FormSafe1'(left)  vs)
5.  \mforall{}vs:Atom  List.  (FormSafe1(right)  vs  \mLeftarrow{}{}\mRightarrow{}  FormSafe1'(right)  vs)
6.  vs  :  Atom  List
7.  \mexists{}as,bs:Atom  List
        (set-equal(Atom;vs;as  @  bs)
        \mwedge{}  (FormSafe1(left)  as)
        \mwedge{}  (FormSafe1(right)  bs)
        \mwedge{}  (l\_disjoint(Atom;as;FormFvs(right))  \mvee{}  l\_disjoint(Atom;bs;FormFvs(left))))
\mvdash{}  (\mexists{}theta:Atom  List
        (l\_subset(Atom;theta;vs-FormFvs(right))
        \mwedge{}  (FormSafe1'(left)  theta)
        \mwedge{}  (FormSafe1'(right)  vs-theta)))
\mvee{}  (\mexists{}phi:Atom  List
        (l\_subset(Atom;phi;vs-FormFvs(left))  \mwedge{}  (FormSafe1'(left)  vs-phi)  \mwedge{}  (FormSafe1'(right)  phi)))


By

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Latex:
((RWO  "4<  5<"  0  THENA  Auto)  THEN  RepeatFor  2  (Thin  4)  THEN  ExRepD)

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