Step
*
7
of Lemma
satisfies_int_formula_dnf
1. form : int_formula()
2. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;form) 
⇐⇒ (∃X∈int_formula_dnf(form). satisfies-poly-constraints(f;X)))
3. f : ℤ ⟶ ℤ
⊢ ¬int_formula_prop(f;form) 
⇐⇒ (∃X∈negate-poly-constraints(int_formula_dnf(form)). satisfies-poly-constraints(f;X))
BY
{ ((RWO "2" 0 THENA Auto) THEN GenConclTerms Auto [⌜int_formula_dnf(form)⌝]⋅) }
1
1. form : int_formula()
2. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;form) 
⇐⇒ (∃X∈int_formula_dnf(form). satisfies-poly-constraints(f;X)))
3. f : ℤ ⟶ ℤ
4. v : polynomial-constraints() List
5. int_formula_dnf(form) = v ∈ (polynomial-constraints() List)
⊢ ¬(∃X∈v. satisfies-poly-constraints(f;X)) 
⇐⇒ (∃X∈negate-poly-constraints(v). satisfies-poly-constraints(f;X))
Latex:
Latex:
1.  form  :  int\_formula()
2.  \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
          (int\_formula\_prop(f;form)  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}X\mmember{}int\_formula\_dnf(form).  satisfies-poly-constraints(f;X)))
3.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
\mvdash{}  \mneg{}int\_formula\_prop(f;form)
\mLeftarrow{}{}\mRightarrow{}  (\mexists{}X\mmember{}negate-poly-constraints(int\_formula\_dnf(form)).  satisfies-poly-constraints(f;X))
By
Latex:
((RWO  "2"  0  THENA  Auto)  THEN  GenConclTerms  Auto  [\mkleeneopen{}int\_formula\_dnf(form)\mkleeneclose{}]\mcdot{})
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