Nuprl Lemma : satisfiable_int_formula_dnf
∀fmla:int_formula()
(satisfiable_int_formula(fmla) ⇐⇒ (∃X∈int_formula_dnf(fmla). satisfiable_polynomial_constraints(X)))
Proof
Definitions occuring in Statement :
int_formula_dnf: int_formula_dnf(fmla),
satisfiable_polynomial_constraints: satisfiable_polynomial_constraints(X),
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_formula: int_formula(),
l_exists: (∃x∈L. P[x]),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
l_exists: (∃x∈L. P[x]),
satisfiable_polynomial_constraints: satisfiable_polynomial_constraints(X),
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T
Lemmas referenced :
int_formula_prop_wf,
and_wf,
l_exists_iff,
sq_stable__le,
select_wf,
satisfies-poly-constraints_wf,
satisfies_int_formula_dnf,
int_formula_wf,
l_member_wf,
satisfiable_polynomial_constraints_wf,
int_formula_dnf_wf,
polynomial-constraints_wf,
l_exists_wf,
satisfiable_int_formula_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache
Latex:
\mforall{}fmla:int\_formula()
(satisfiable\_int\_formula(fmla)
\mLeftarrow{}{}\mRightarrow{} (\mexists{}X\mmember{}int\_formula\_dnf(fmla). satisfiable\_polynomial\_constraints(X)))
Date html generated:
2016_05_14-AM-07_09_50
Last ObjectModification:
2016_01_14-PM-08_41_03
Theory : omega
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