### 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)`
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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