Nuprl Lemma : satisfies-negate-poly-constraints
∀f:ℤ ⟶ ℤ. ∀L:polynomial-constraints() List.
((∀X∈L.¬satisfies-poly-constraints(f;X)) ⇐⇒ (∃X∈negate-poly-constraints(L). satisfies-poly-constraints(f;X)))
Proof
Definitions occuring in Statement :
negate-poly-constraints: negate-poly-constraints(Xs),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
polynomial-constraints: polynomial-constraints(),
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
cons: [a / b],
negate-poly-constraints: negate-poly-constraints(Xs),
ifthenelse: if b then t else f fi ,
btrue: tt,
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
polynomial-constraints: polynomial-constraints(),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
polynomial-constraints_wf,
list-cases,
product_subtype_list,
list_wf,
null_nil_lemma,
true_wf,
l_all_nil_iff,
l_all_wf_nil,
le_wf,
int_term_value_wf,
ipolynomial-term_wf,
iff_wf,
l_exists_single,
satisfies-poly-constraints_wf,
nil_wf,
subtype_rel_product,
iPolynomial_wf,
subtype_rel_list,
l_exists_wf,
cons_wf,
l_member_wf,
negate-poly-constraint_wf,
null_cons_lemma,
spread_cons_lemma,
equal_wf,
l_all_cons,
l_all_wf,
list_accum_wf,
and-poly-constraints_wf,
list_induction,
all_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
satisfies-and-poly-constraints,
negate-poly-constraints_wf,
not_wf,
satisfies-negate-poly-constraint,
l_all_iff
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
functionEquality,
intEquality,
independent_pairFormation,
natural_numberEquality,
productEquality,
addLevel,
impliesFunctionality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
andLevelFunctionality,
lambdaEquality,
functionExtensionality,
applyEquality,
independent_pairEquality,
because_Cache,
independent_functionElimination,
setElimination,
rename,
setEquality,
equalityTransitivity,
equalitySymmetry,
allFunctionality,
allLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}. \mforall{}L:polynomial-constraints() List.
((\mforall{}X\mmember{}L.\mneg{}satisfies-poly-constraints(f;X))
\mLeftarrow{}{}\mRightarrow{} (\mexists{}X\mmember{}negate-poly-constraints(L). satisfies-poly-constraints(f;X)))
Date html generated:
2017_04_14-AM-09_03_06
Last ObjectModification:
2017_02_27-PM-03_43_54
Theory : omega
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