Nuprl Lemma : rpolynomial-locally-non-zero
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ.
(((Σi≤n. a_i * r0^i) < r0) ⇒ (r0 < (Σi≤n. a_i * r1^i)) ⇒ locally-non-constant(λx.(Σi≤n. a_i * x^i);r0;r1;r0))
Proof
Definitions occuring in Statement :
locally-non-constant: locally-non-constant(f;a;b;c),
rpolynomial: (Σi≤n. a_i * x^i),
rless: x < y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
locally-non-constant: locally-non-constant(f;a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
rsub: x - y,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
real: ℝ,
r-ap: f(x),
cand: A c∧ B,
rneq: x ≠ y,
int-to-real: r(n),
le: A ≤ B,
uiff: uiff(P;Q),
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff
Lemmas referenced :
rleq_wf,
int-to-real_wf,
rless_wf,
real_wf,
rpolynomial_wf,
int_seg_wf,
nat_wf,
small-reciprocal-real-ext,
rsub_wf,
radd-preserves-rless,
radd_wf,
rminus_wf,
rless_functionality,
radd-zero-both,
req_weakening,
radd-rminus-assoc,
radd_comm,
radd_functionality,
exp-fastexp,
imax_wf,
exp_wf2,
imax_nat_plus,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
absval_wf,
rleq_weakening_equal,
rleq_weakening_rless,
absval_ifthenelse,
rneq_wf,
lt_int_wf,
assert_wf,
bnot_wf,
not_wf,
le_wf,
minus-is-int-iff,
itermAdd_wf,
itermMinus_wf,
int_term_value_add_lemma,
int_term_value_minus_lemma,
false_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
mul-commutes,
mul-swap,
mul-associates,
zero-mul,
zero-add,
add-commutes,
rpolynomial-locally-non-zero-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
functionExtensionality,
applyEquality,
addEquality,
setElimination,
rename,
functionEquality,
dependent_functionElimination,
dependent_set_memberEquality,
because_Cache,
productElimination,
independent_functionElimination,
addLevel,
independent_isectElimination,
levelHypothesis,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productEquality,
inrFormation,
dependent_set_memberFormation,
inlFormation,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}.
(((\mSigma{}i\mleq{}n. a\_i * r0\^{}i) < r0)
{}\mRightarrow{} (r0 < (\mSigma{}i\mleq{}n. a\_i * r1\^{}i))
{}\mRightarrow{} locally-non-constant(\mlambda{}x.(\mSigma{}i\mleq{}n. a\_i * x\^{}i);r0;r1;r0))
Date html generated:
2017_10_03-PM-00_37_02
Last ObjectModification:
2017_07_28-AM-08_44_10
Theory : reals
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