Nuprl Lemma : satisfies-negate-poly-constraint
∀X:polynomial-constraints(). ∀f:ℤ ⟶ ℤ.
((∃Z∈negate-poly-constraint(X). satisfies-poly-constraints(f;Z)) ⇐⇒ ¬satisfies-poly-constraints(f;X))
Proof
Definitions occuring in Statement :
negate-poly-constraint: negate-poly-constraint(X),
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
polynomial-constraints: polynomial-constraints(),
l_exists: (∃x∈L. P[x]),
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],
polynomial-constraints: polynomial-constraints(),
member: t ∈ T,
satisfies-poly-constraints: satisfies-poly-constraints(f;X),
l_all: (∀x∈L.P[x]),
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
iPolynomial: iPolynomial(),
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
cand: A c∧ B,
exists: ∃x:A. B[x],
int_term_value: int_term_value(f;t),
itermMinus: "-"num,
int_term_ind: int_term_ind,
itermAdd: left (+) right,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
l_exists: (∃x∈L. P[x]),
negate-poly-constraint: negate-poly-constraint(X),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
polynomial-constraints_wf,
decidable__all_int_seg,
length_wf,
iPolynomial_wf,
equal-wf-T-base,
int_term_value_wf,
ipolynomial-term_wf,
select_wf,
sq_stable__le,
int_seg_wf,
decidable__equal_int,
le_wf,
decidable__le,
not_wf,
l_all_wf,
l_member_wf,
or_wf,
l_exists_wf,
minus-poly_wf,
const-poly_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
add-ipoly_wf1,
l_exists_iff,
exists_wf,
l_all_iff,
all_wf,
itermMinus_wf,
itermAdd_wf,
squash_wf,
int_term_value_functionality,
equiv_int_terms_transitivity,
minus-poly-equiv,
itermMinus_functionality,
add-ipoly-equiv,
iff_weakening_equal,
const-poly-value,
equal_wf,
minus-is-int-iff,
minus-add,
minus-one-mul,
add-commutes,
add-is-int-iff,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul-top,
zero-add,
one-mul,
add-mul-special,
add-associates,
two-mul,
mul-distributes-right,
zero-mul,
minus-minus,
mul-associates,
add-swap,
add-zero,
omega-shadow,
less_than_wf,
decidable__exists_int_seg,
not-le-2,
minus-zero,
false_wf,
decidable__or,
condition-implies-le,
le-add-cancel-alt,
not-equal-implies-less,
subtype_rel_self,
less-iff-le,
satisfies-poly-constraints_wf,
list_wf,
iff_wf,
negate-poly-constraint_wf,
nil_wf,
list_induction,
list_accum_wf,
cons_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
l_exists_nil,
l_exists_wf_nil,
l_exists_cons,
l_all_nil_iff,
l_all_single,
l_all_wf_nil,
le-add-cancel,
map_wf,
subtype_rel_product,
subtype_rel_list,
l_exists_map,
l_exists_functionality,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
functionEquality,
intEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
instantiate,
dependent_functionElimination,
natural_numberEquality,
isectElimination,
hypothesisEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
setElimination,
because_Cache,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_pairFormation,
productEquality,
setEquality,
dependent_set_memberEquality,
addLevel,
cumulativity,
equalityTransitivity,
equalitySymmetry,
voidElimination,
minusEquality,
orFunctionality,
levelHypothesis,
promote_hyp,
andLevelFunctionality,
orLevelFunctionality,
isect_memberEquality,
voidEquality,
universeEquality,
addEquality,
baseApply,
closedConclusion,
multiplyEquality,
inrFormation,
dependent_pairFormation,
inlFormation,
impliesFunctionality,
independent_pairEquality,
allFunctionality
Latex:
\mforall{}X:polynomial-constraints(). \mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}.
((\mexists{}Z\mmember{}negate-poly-constraint(X). satisfies-poly-constraints(f;Z))
\mLeftarrow{}{}\mRightarrow{} \mneg{}satisfies-poly-constraints(f;X))
Date html generated:
2017_04_14-AM-09_02_54
Last ObjectModification:
2017_02_27-PM-03_48_20
Theory : omega
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