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Step * 1 of Lemma satisfies-negate-poly-constraint


1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
⊢ (∃Z∈negate-poly-constraint(<eqs, ineqs>). satisfies-poly-constraints(f;Z)) ⇐⇒ ¬satisfies-poly-constraints(f;<eqs, ine\000Cqs>)
BY
{ Assert ⌜¬satisfies-poly-constraints(f;<eqs, ineqs>)
          ⇐⇒ (∃e∈eqs. (0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(e;const-poly(1))))))
              ∨ (0 ≤ int_term_value(f;ipolynomial-term(add-ipoly(e;const-poly(-1))))))
              ∨ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))⌝⋅ }

1
.....assertion..... 
1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
⊢ ¬satisfies-poly-constraints(f;<eqs, ineqs>)
⇐⇒ (∃e∈eqs. (0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(e;const-poly(1))))))
    ∨ (0 ≤ int_term_value(f;ipolynomial-term(add-ipoly(e;const-poly(-1))))))
    ∨ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))

2
1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. ¬satisfies-poly-constraints(f;<eqs, ineqs>)
⇐⇒ (∃e∈eqs. (0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(e;const-poly(1))))))
    ∨ (0 ≤ int_term_value(f;ipolynomial-term(add-ipoly(e;const-poly(-1))))))
    ∨ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))
⊢ (∃Z∈negate-poly-constraint(<eqs, ineqs>). satisfies-poly-constraints(f;Z)) ⇐⇒ ¬satisfies-poly-constraints(f;<eqs, ine\000Cqs>)

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Latex:

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Latex:

1.  eqs  :  iPolynomial()  List
2.  ineqs  :  iPolynomial()  List
3.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
\mvdash{}  (\mexists{}Z\mmember{}negate-poly-constraint(<eqs,  ineqs>).  satisfies-poly-constraints(f;Z))  \mLeftarrow{}{}\mRightarrow{}  \mneg{}satisfies-poly-con\000Cstraints(f;<eqs,  ineqs>)


By

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Latex:
Assert 
\mkleeneopen{}\mneg{}satisfies-poly-constraints(f;<eqs,  ineqs>)
  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}e\mmember{}eqs.  (0  \mleq{}  int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(e;const-poly(1))))))
          \mvee{}  (0  \mleq{}  int\_term\_value(f;ipolynomial-term(add-ipoly(e;const-poly(-1))))))
          \mvee{}  (\mexists{}ineq\mmember{}ineqs.  ...)\mkleeneclose{}\mcdot{}

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