Nuprl Lemma : rpolynomial-locally-non-zero-1
∀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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
rfun: I ⟶ℝ,
r-ap: f(x),
top: Top,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
so_lambda: λ2x.t[x],
i-member: r ∈ I,
rccint: [l, u],
cand: A c∧ B,
so_apply: x[s],
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
poly-nth-deriv: poly-nth-deriv(n;a),
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
real: ℝ,
less_than': less_than'(a;b),
rpolynomial: (Σi≤n. a_i * x^i),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
true: True,
rev_uimplies: rev_uimplies(P;Q),
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rge: x ≥ y,
primrec: primrec(n;b;c),
fact: (n)!,
rneq: x ≠ y,
rdiv: (x/y),
subtract: n - m,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
pi1: fst(t),
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi2: snd(t)
Lemmas referenced :
rless_wf,
int-to-real_wf,
rpolynomial_wf,
int_seg_wf,
real_wf,
istype-nat,
i-member_wf,
rccint_wf,
locally-non-constant-deriv-seq-test,
member_rccint_lemma,
istype-void,
rfun_wf,
finite-deriv-seq_wf,
subtype_rel_self,
rleq_wf,
req_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformand_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
istype-le,
istype-less_than,
rsub_wf,
rsum_wf,
rabs_wf,
rpoly-nth-deriv_wf,
int_seg_properties,
polynomial-deriv-seq,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
bool_subtype_base,
int_subtype_base,
le_wf,
set_subtype_base,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
primrec0_lemma,
req-iff-rsub-is-0,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
itermSubtract_wf,
intformeq_wf,
decidable__equal_int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_weakening_equal,
rleq_weakening_rless,
rnexp_zero_lemma,
sq_stable__less_than,
radd_wf,
istype-false,
int_seg_subtype_nat,
rnexp_wf,
rmul_wf,
rless_functionality,
rsum-split-first,
req_weakening,
rsum_functionality,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
rnexp0,
rmul-zero,
req_functionality,
rmul_functionality,
rsum-zero,
itermMultiply_wf,
radd_functionality,
real_term_value_add_lemma,
real_term_value_mul_lemma,
exp_wf2,
rnexp-int,
rmul-one,
squash_wf,
true_wf,
exp-one,
iff_weakening_equal,
rleq_weakening,
rless_transitivity1,
subtract_wf,
primrec-wf2,
rless_irreflexivity,
rless_transitivity2,
equal-wf-base,
rsum-single,
subtract-add-cancel,
rsum-split-last,
zero-rleq-rabs,
rsum_nonneg,
rless_functionality_wrt_implies,
radd_functionality_wrt_rleq,
poly-nth-deriv_wf,
rabs_functionality,
fact_wf,
int-rdiv_wf,
poly-nth-deriv-req,
rless-int,
fact0_redex_lemma,
rdiv_wf,
int-rdiv-req,
rinv_wf2,
rmul_preserves_req,
req_transitivity,
rinv1,
rabs-rmul,
int_term_value_minus_lemma,
itermMinus_wf,
istype-top,
absval_unfold,
absval_wf,
rabs-int,
rmul-is-positive,
radd-positive-implies,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-is-int-iff,
eq_int_wf,
assert_of_eq_int,
ifthenelse_wf,
neg_assert_of_eq_int,
nequal_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
rtermConstant_wf,
rsum_linearity2,
le-add-cancel2,
nat_plus_inc_int_nzero,
minus-minus,
req_inversion,
rpoly-nth-deriv-linear,
lelt_wf,
rpoly-nth-deriv_functionality,
btrue_neq_bfalse,
assert_elim,
bnot_wf,
bfalse_wf,
btrue_wf,
eq_int_eq_true,
istype-universe,
equal_wf,
absval_pos,
rmul_assoc,
rless-cases,
rmul_preserves_rless,
rmul-rinv,
rsum-triangle-inequality2,
rsub_functionality,
radd-preserves-rless,
rless-implies-rless,
rabs-of-nonneg,
rleq_transitivity,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
functionIsType,
addEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
setIsType,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
productIsType,
productElimination,
applyEquality,
functionEquality,
setEquality,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
imageElimination,
equalityIsType1,
cumulativity,
instantiate,
promote_hyp,
intEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIsType4,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
imageMemberEquality,
universeEquality,
inrFormation_alt,
applyLambdaEquality,
isectIsTypeImplies,
axiomSqEquality,
isect_memberFormation_alt,
lessCases,
minusEquality,
inlFormation_alt,
multiplyEquality,
equalityIstype,
sqequalBase,
hyp_replacement,
int_eqReduceTrueSq,
int_eqReduceFalseSq
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:
2019_10_30-AM-09_12_10
Last ObjectModification:
2019_04_03-AM-00_23_54
Theory : reals
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