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Step * 6 1 1 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
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)
10. (∃X∈v. satisfies-poly-constraints(f;X)) ⇒ (∃X∈v1. satisfies-poly-constraints(f;X))
⊢ (∀X∈v.¬satisfies-poly-constraints(f;X)) ∨ (∃X∈v1. satisfies-poly-constraints(f;X))
BY
{ Assert ⌜Dec((∃X∈v. satisfies-poly-constraints(f;X)))⌝⋅ }

1
.....assertion..... 
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)
10. (∃X∈v. satisfies-poly-constraints(f;X)) ⇒ (∃X∈v1. satisfies-poly-constraints(f;X))
⊢ Dec((∃X∈v. satisfies-poly-constraints(f;X)))

2
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)
10. (∃X∈v. satisfies-poly-constraints(f;X)) ⇒ (∃X∈v1. satisfies-poly-constraints(f;X))
11. Dec((∃X∈v. satisfies-poly-constraints(f;X)))
⊢ (∀X∈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{}
6.  v  :  polynomial-constraints()  List
7.  int\_formula\_dnf(left)  =  v
8.  v1  :  polynomial-constraints()  List
9.  int\_formula\_dnf(right)  =  v1
10.  (\mexists{}X\mmember{}v.  satisfies-poly-constraints(f;X))  {}\mRightarrow{}  (\mexists{}X\mmember{}v1.  satisfies-poly-constraints(f;X))
\mvdash{}  (\mforall{}X\mmember{}v.\mneg{}satisfies-poly-constraints(f;X))  \mvee{}  (\mexists{}X\mmember{}v1.  satisfies-poly-constraints(f;X))


By

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Latex:
Assert  \mkleeneopen{}Dec((\mexists{}X\mmember{}v.  satisfies-poly-constraints(f;X)))\mkleeneclose{}\mcdot{}

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