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Step * 6 of Lemma satisfies_int_formula_dnf


1. left : int_formula()
2. right : int_formula()
3. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;left) ⇐⇒ (∃X∈int_formula_dnf(left). satisfies-poly-constraints(f;X)))
4. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;right) ⇐⇒ (∃X∈int_formula_dnf(right). satisfies-poly-constraints(f;X)))
5. f : ℤ ⟶ ℤ
⊢ int_formula_prop(f;left) ⇒ int_formula_prop(f;right)
⇐⇒ (∃X∈negate-poly-constraints(int_formula_dnf(left)) @ int_formula_dnf(right). satisfies-poly-constraints(f;X))
BY
{ ((RWO "3 4" 0 THENA Auto)
   THEN GenConclTerms Auto [⌜int_formula_dnf(left)⌝;⌜int_formula_dnf(right)⌝]⋅
   THEN (RWO "l_exists_append" 0 THENA Auto)) }

1
1. left : int_formula()
2. right : int_formula()
3. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;left) ⇐⇒ (∃X∈int_formula_dnf(left). satisfies-poly-constraints(f;X)))
4. ∀f:ℤ ⟶ ℤ. (int_formula_prop(f;right) ⇐⇒ (∃X∈int_formula_dnf(right). satisfies-poly-constraints(f;X)))
5. f : ℤ ⟶ ℤ
6. v : polynomial-constraints() List
7. int_formula_dnf(left) = v ∈ (polynomial-constraints() List)
8. v1 : polynomial-constraints() List
9. int_formula_dnf(right) = v1 ∈ (polynomial-constraints() List)
⊢ (∃X∈v. satisfies-poly-constraints(f;X)) ⇒ (∃X∈v1. satisfies-poly-constraints(f;X))
⇐⇒ (∃X∈negate-poly-constraints(v). satisfies-poly-constraints(f;X)) ∨ (∃X∈v1. satisfies-poly-constraints(f;X))

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Latex:

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Latex:

1.  left  :  int\_formula()
2.  right  :  int\_formula()
3.  \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
          (int\_formula\_prop(f;left)  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}X\mmember{}int\_formula\_dnf(left).  satisfies-poly-constraints(f;X)))
4.  \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
          (int\_formula\_prop(f;right)  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}X\mmember{}int\_formula\_dnf(right).  satisfies-poly-constraints(f;X)))
5.  f  :  \mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}
\mvdash{}  int\_formula\_prop(f;left)  {}\mRightarrow{}  int\_formula\_prop(f;right)
\mLeftarrow{}{}\mRightarrow{}  (\mexists{}X\mmember{}negate-poly-constraints(int\_formula\_dnf(left))
        @  int\_formula\_dnf(right).  satisfies-poly-constraints(f;X))


By

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Latex:
((RWO  "3  4"  0  THENA  Auto)
  THEN  GenConclTerms  Auto  [\mkleeneopen{}int\_formula\_dnf(left)\mkleeneclose{};\mkleeneopen{}int\_formula\_dnf(right)\mkleeneclose{}]\mcdot{}
  THEN  (RWO  "l\_exists\_append"  0  THENA  Auto))

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