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


1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. (∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ)
5. Dec((∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p))))
⊢ ¬((∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ) ∧ (∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p))))
⇐⇒ (∃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))))))
BY
{ D -1 }

1
1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. (∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ)
5. (∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p)))
⊢ ¬((∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ) ∧ (∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p))))
⇐⇒ (∃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. (∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ)
5. ¬(∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p)))
⊢ ¬((∀p∈eqs.int_term_value(f;ipolynomial-term(p)) = 0 ∈ ℤ) ∧ (∀p∈ineqs.0 ≤ int_term_value(f;ipolynomial-term(p))))
⇐⇒ (∃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))))))

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Latex:

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Latex:

1.  eqs  :  iPolynomial()  List
2.  ineqs  :  iPolynomial()  List
3.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
4.  (\mforall{}p\mmember{}eqs.int\_term\_value(f;ipolynomial-term(p))  =  0)
5.  Dec((\mforall{}p\mmember{}ineqs.0  \mleq{}  int\_term\_value(f;ipolynomial-term(p))))
\mvdash{}  \mneg{}((\mforall{}p\mmember{}eqs.int\_term\_value(f;ipolynomial-term(p))  =  0)
    \mwedge{}  (\mforall{}p\mmember{}ineqs.0  \mleq{}  int\_term\_value(f;ipolynomial-term(p))))
\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.  0 
                                        \mleq{}  int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))


By

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Latex:
D  -1

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