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


1. ineqs : iPolynomial() List
2. f : ℤ ⟶ ℤ
3. [] ∈ iPolynomial() List
4. u : iPolynomial()
5. v : iPolynomial() List
6. ∀L:polynomial-constraints() List
     ((∃Z∈accumulate (with value pcs and list item e):
           [<[], [minus-poly(add-ipoly(e;const-poly(1)))]>; [<[], [add-ipoly(e;const-poly(-1))]> / pcs]]
          over list:
            v
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈v. (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))))))
         ∨ (∃Z∈L. satisfies-poly-constraints(f;Z)))
⊢ ∀L:polynomial-constraints() List
    ((∃e∈v. (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))))))
     ∨ satisfies-poly-constraints(f;<[], [minus-poly(add-ipoly(u;const-poly(1)))]>)
     ∨ satisfies-poly-constraints(f;<[], [add-ipoly(u;const-poly(-1))]>)
     ∨ (∃Z∈L. satisfies-poly-constraints(f;Z))
    ⇐⇒ (((0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1))))))
        ∨ (0 ≤ int_term_value(f;ipolynomial-term(add-ipoly(u;const-poly(-1))))))
        ∨ (∃e∈v. (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)))))))
        ∨ (∃Z∈L. satisfies-poly-constraints(f;Z)))
BY
{ Assert ⌜satisfies-poly-constraints(f;<[], [minus-poly(add-ipoly(u;const-poly(1)))]>)
          ⇐⇒ 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1)))))⌝⋅ }

1
.....assertion..... 
1. ineqs : iPolynomial() List
2. f : ℤ ⟶ ℤ
3. [] ∈ iPolynomial() List
4. u : iPolynomial()
5. v : iPolynomial() List
6. ∀L:polynomial-constraints() List
     ((∃Z∈accumulate (with value pcs and list item e):
           [<[], [minus-poly(add-ipoly(e;const-poly(1)))]>; [<[], [add-ipoly(e;const-poly(-1))]> / pcs]]
          over list:
            v
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈v. (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))))))
         ∨ (∃Z∈L. satisfies-poly-constraints(f;Z)))
⊢ satisfies-poly-constraints(f;<[], [minus-poly(add-ipoly(u;const-poly(1)))]>)
⇐⇒ 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1)))))

2
1. ineqs : iPolynomial() List
2. f : ℤ ⟶ ℤ
3. [] ∈ iPolynomial() List
4. u : iPolynomial()
5. v : iPolynomial() List
6. ∀L:polynomial-constraints() List
     ((∃Z∈accumulate (with value pcs and list item e):
           [<[], [minus-poly(add-ipoly(e;const-poly(1)))]>; [<[], [add-ipoly(e;const-poly(-1))]> / pcs]]
          over list:
            v
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈v. (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))))))
         ∨ (∃Z∈L. satisfies-poly-constraints(f;Z)))
7. satisfies-poly-constraints(f;<[], [minus-poly(add-ipoly(u;const-poly(1)))]>)
⇐⇒ 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1)))))
⊢ ∀L:polynomial-constraints() List
    ((∃e∈v. (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))))))
     ∨ satisfies-poly-constraints(f;<[], [minus-poly(add-ipoly(u;const-poly(1)))]>)
     ∨ satisfies-poly-constraints(f;<[], [add-ipoly(u;const-poly(-1))]>)
     ∨ (∃Z∈L. satisfies-poly-constraints(f;Z))
    ⇐⇒ (((0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1))))))
        ∨ (0 ≤ int_term_value(f;ipolynomial-term(add-ipoly(u;const-poly(-1))))))
        ∨ (∃e∈v. (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)))))))
        ∨ (∃Z∈L. satisfies-poly-constraints(f;Z)))

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Latex:

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Latex:

1.  ineqs  :  iPolynomial()  List
2.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
3.  []  \mmember{}  iPolynomial()  List
4.  u  :  iPolynomial()
5.  v  :  iPolynomial()  List
6.  \mforall{}L:polynomial-constraints()  List
          ((\mexists{}Z\mmember{}accumulate  (with  value  pcs  and  list  item  e):
                      [<[],  [minus-poly(add-ipoly(e;const-poly(1)))]>
                        [<[],  [add-ipoly(e;const-poly(-1))]>  /  pcs]]
                    over  list:
                        v
                    with  starting  value:
                      L).  satisfies-poly-constraints(f;Z))
          \mLeftarrow{}{}\mRightarrow{}  (\mexists{}e\mmember{}v.  (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{}Z\mmember{}L.  satisfies-poly-constraints(f;Z)))
\mvdash{}  \mforall{}L:polynomial-constraints()  List
        ((\mexists{}e\mmember{}v.  (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{}  satisfies-poly-constraints(f;<[],  [minus-poly(add-ipoly(u;const-poly(1)))]>)
          \mvee{}  satisfies-poly-constraints(f;<[],  [add-ipoly(u;const-poly(-1))]>)
          \mvee{}  (\mexists{}Z\mmember{}L.  satisfies-poly-constraints(f;Z))
        \mLeftarrow{}{}\mRightarrow{}  (((0  \mleq{}  int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1))))))
                \mvee{}  (0  \mleq{}  int\_term\_value(f;ipolynomial-term(add-ipoly(u;const-poly(-1))))))
                \mvee{}  (\mexists{}e\mmember{}v.  (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{}Z\mmember{}L.  satisfies-poly-constraints(f;Z)))


By

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Latex:
Assert  \mkleeneopen{}satisfies-poly-constraints(f;<[],  [minus-poly(add-ipoly(u;const-poly(1)))]>)
                \mLeftarrow{}{}\mRightarrow{}  0  \mleq{}  int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(u;const-poly(1)))))\mkleeneclose{}\mcdot{}

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