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Step * 6 1 1 1 of Lemma satisfies_int_formula_dnf

.....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)))
BY
{ (Unfold `l_exists` 0 THEN Auto) }

1
.....decidable?..... 
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(∃i:ℕ||v||. satisfies-poly-constraints(f;v[i]))

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Latex:

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Latex:
.....assertion..... 
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{}  Dec((\mexists{}X\mmember{}v.  satisfies-poly-constraints(f;X)))


By

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Latex:
(Unfold  `l\_exists`  0  THEN  Auto)

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