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


1. f : ℤ ⟶ ℤ
2. u : polynomial-constraints()
3. v : polynomial-constraints() List
⊢ ∀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))))
BY
{ (ListInd (-1) THEN Reduce 0) }

1
1. f : ℤ ⟶ ℤ
2. u : polynomial-constraints()
⊢ ∀v1:polynomial-constraints() List
    ((∃X∈v1. satisfies-poly-constraints(f;X))
    ⇐⇒ (∃Z∈v1. satisfies-poly-constraints(f;Z))
        ∧ (∀X∈[].(∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z))))

2
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))))
⊢ ∀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:
          and-poly-constraints(v1;negate-poly-constraint(u1))). satisfies-poly-constraints(f;X))
    ⇐⇒ (∃Z∈v1. satisfies-poly-constraints(f;Z))
        ∧ (∀X∈[u1 / v].(∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z))))

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Latex:

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Latex:

1.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
2.  u  :  polynomial-constraints()
3.  v  :  polynomial-constraints()  List
\mvdash{}  \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))))


By

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Latex:
(ListInd  (-1)  THEN  Reduce  0)

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