Proof of Nuprl Lemma : satisfies-negate-poly-constraint

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

1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. ∀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:
            eqs
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈eqs. (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)))
⊢ (∃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:
        eqs
      with starting value:
       map(λineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]>;ineqs)). satisfies-poly-constraints(f;Z))
⇐⇒ (∃e∈eqs. (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))))))
    ∨ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))
BY
{ (((Assert [] ∈ iPolynomial() List BY Auto) THEN (RWO "-2" 0 THENA Auto))

   THEN Assert ⌜(∃Z∈map(λineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]>;ineqs). satisfies-poly-constraints(f;Z))

                ⇐⇒ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))⌝⋅
   ) }

1
.....assertion.....
1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. ∀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:
            eqs
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈eqs. (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)))
5. [] ∈ iPolynomial() List
⊢ (∃Z∈map(λineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]>;ineqs). satisfies-poly-constraints(f;Z))
⇐⇒ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))

2
1. eqs : iPolynomial() List
2. ineqs : iPolynomial() List
3. f : ℤ ⟶ ℤ
4. ∀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:
            eqs
          with starting value:
           L). satisfies-poly-constraints(f;Z))
     ⇐⇒ (∃e∈eqs. (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)))
5. [] ∈ iPolynomial() List
6. (∃Z∈map(λineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]>;ineqs). satisfies-poly-constraints(f;Z))
⇐⇒ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))
⊢ (∃e∈eqs. (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∈map(λineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]>;ineqs). satisfies-poly-constraints(f;Z))
⇐⇒ (∃e∈eqs. (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))))))
    ∨ (∃ineq∈ineqs. 0 ≤ int_term_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1))))))

Latex:

Latex:
1. eqs : iPolynomial() List 2. ineqs : iPolynomial() List 3. f : \mBbbZ{} {}\mrightarrow{} \mBbbZ{} 4. \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: eqs with starting value: L). satisfies-poly-constraints(f;Z)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}e\mmember{}eqs. (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{} (\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: eqs with starting value: map(\mlambda{}ineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]> ineqs)). satisfies-poly-constraints(f;Z)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}e\mmember{}eqs. (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{}ineq\mmember{}ineqs. 0 \mleq{} int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;const-poly(1)))))) By Latex: (((Assert [] \mmember{} iPolynomial() List BY Auto) THEN (RWO "-2" 0 THENA Auto)) THEN Assert \mkleeneopen{}(\mexists{}Z\mmember{}map(\mlambda{}ineq.<[], [minus-poly(add-ipoly(ineq;const-poly(1)))]> ineqs). satisfies-poly-constraints(f;Z)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}ineq\mmember{}ineqs. 0 \mleq{} int\_term\_value(f;ipolynomial-term(minus-poly(add-ipoly(ineq;...)))))\mkleeneclose{}\mcdot{} ) Home Index