### 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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