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Step * 1 2 1 2 1 of Lemma satisfies-negate-poly-constraints


1. f : ℤ ⟶ ℤ
2. u : polynomial-constraints()
3. u1 : polynomial-constraints()
4. v : polynomial-constraints() List
5. ∀v1:polynomial-constraints() List
     ((∃X∈accumulate (with value sofar and list item X):
           and-poly-constraints(sofar;negate-poly-constraint(X))
          over list:
            v
          with starting value:
           v1). satisfies-poly-constraints(f;X))
     ⇐⇒ (∃Z∈v1. satisfies-poly-constraints(f;Z))
         ∧ (∀X∈v.(∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z))))
6. v1 : polynomial-constraints() List
⊢ (∃Z∈and-poly-constraints(v1;negate-poly-constraint(u1)). satisfies-poly-constraints(f;Z))
  ∧ (∀X∈v.(∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z)))
⇐⇒ (∃Z∈v1. satisfies-poly-constraints(f;Z))
    ∧ (∀X∈[u1 / v].(∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z)))
BY
{ (RWO "satisfies-and-poly-constraints< l_all_cons" 0 THEN Auto) }

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Latex:

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Latex:

1.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
2.  u  :  polynomial-constraints()
3.  u1  :  polynomial-constraints()
4.  v  :  polynomial-constraints()  List
5.  \mforall{}v1:polynomial-constraints()  List
          ((\mexists{}X\mmember{}accumulate  (with  value  sofar  and  list  item  X):
                      and-poly-constraints(sofar;negate-poly-constraint(X))
                    over  list:
                        v
                    with  starting  value:
                      v1).  satisfies-poly-constraints(f;X))
          \mLeftarrow{}{}\mRightarrow{}  (\mexists{}Z\mmember{}v1.  satisfies-poly-constraints(f;Z))
                  \mwedge{}  (\mforall{}X\mmember{}v.(\mexists{}Z\mmember{}negate-poly-constraint(X).  satisfies-poly-constraints(f;Z))))
6.  v1  :  polynomial-constraints()  List
\mvdash{}  (\mexists{}Z\mmember{}and-poly-constraints(v1;negate-poly-constraint(u1)).  satisfies-poly-constraints(f;Z))
    \mwedge{}  (\mforall{}X\mmember{}v.(\mexists{}Z\mmember{}negate-poly-constraint(X).  satisfies-poly-constraints(f;Z)))
\mLeftarrow{}{}\mRightarrow{}  (\mexists{}Z\mmember{}v1.  satisfies-poly-constraints(f;Z))
        \mwedge{}  (\mforall{}X\mmember{}[u1  /  v].(\mexists{}Z\mmember{}negate-poly-constraint(X).  satisfies-poly-constraints(f;Z)))


By

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Latex:
(RWO  "satisfies-and-poly-constraints<  l\_all\_cons"  0  THEN  Auto)

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