### Nuprl Lemma : locally-non-zero-finite-deriv-seq

`∀a,b:ℝ. ∀f:[a, b] ⟶ℝ.`
`  ((∀u,v:{v:ℝ| v ∈ [a, b]} .`
`      ((u < v)`
`      `` (∃k:ℕ`
`           ∃F:ℕk + 1 ⟶ [a, b] ⟶ℝ`
`            (finite-deriv-seq([a, b];k;i,x.F[i;x])`
`            ∧ (∀x:{x:ℝ| x ∈ [a, b]} . (F[0;x] = f(x)))`
`            ∧ (∃z:{z:ℝ| z ∈ [u, v]} . (r0 < Σ{|F[i;z]| | 0≤i≤k}))))))`
`  `` locally-non-constant(f;a;b;r0))`

Proof

Definitions occuring in Statement :  finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` locally-non-constant: `locally-non-constant(f;a;b;c)` r-ap: `f(x)` rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rsum: `Σ{x[k] | n≤k≤m}` rless: `x < y` rabs: `|x|` req: `x = y` int-to-real: `r(n)` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` set: `{x:A| B[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` top: `Top` and: `P ∧ Q` cand: `A c∧ B` guard: `{T}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` exists: `∃x:A. B[x]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_lambda: `λ2x y.t[x; y]` label: `...\$L... t` rfun: `I ⟶ℝ` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` true: `True` i-member: `r ∈ I` rccint: `[l, u]` sq_stable: `SqStable(P)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rless: `x < y` sq_exists: `∃x:A [B[x]]` real: `ℝ` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` ge: `i ≥ j ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` r-ap: `f(x)` rfun-eq: `rfun-eq(I;f;g)` subinterval: `I ⊆ J `
Lemmas referenced :  member_rccint_lemma istype-void rless_transitivity1 rleq_weakening_rless rleq_wf rless_transitivity2 int_seg_wf finite-deriv-seq_wf rccint_wf istype-false istype-le subtype_rel_self real_wf i-member_wf req_wf istype-less_than r-ap_wf sq_stable__rleq rless_wf int-to-real_wf rsum_wf rabs_wf rleq_transitivity sq_stable__less_than nat_plus_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf decidable__lt itermAdd_wf int_term_value_add_lemma rneq_wf primrec-wf2 all_wf exists_wf istype-nat nat_properties rfun_wf sq_stable__i-member rless_functionality req_weakening rsum_single rabs_functionality rabs-positive-iff radd_wf radd-positive-implies rsum-split-first add-member-int_seg2 subtract-add-cancel subtract_wf itermSubtract_wf int_term_value_subtract_lemma add-subtract-cancel req_witness rsum-shift zero-add derivative_functionality small-reciprocal-rneq-zero non-zero-deriv-non-constant rfun_subtype_3 derivative_functionality_wrt_subinterval subtype_rel_sets_simple
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut hypothesis sqequalHypSubstitution dependent_functionElimination thin sqequalRule introduction extract_by_obid isect_memberEquality_alt voidElimination independent_pairFormation hypothesisEquality independent_functionElimination independent_isectElimination isectElimination dependent_set_memberEquality_alt because_Cache productIsType universeIsType productElimination functionIsType natural_numberEquality addEquality lambdaEquality_alt applyEquality functionEquality setEquality setIsType inhabitedIsType imageMemberEquality baseClosed setElimination rename imageElimination unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality productEquality closedConclusion equalityTransitivity equalitySymmetry

Latex:
\mforall{}a,b:\mBbbR{}.  \mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}.
((\mforall{}u,v:\{v:\mBbbR{}|  v  \mmember{}  [a,  b]\}  .
((u  <  v)
{}\mRightarrow{}  (\mexists{}k:\mBbbN{}
\mexists{}F:\mBbbN{}k  +  1  {}\mrightarrow{}  [a,  b]  {}\mrightarrow{}\mBbbR{}
(finite-deriv-seq([a,  b];k;i,x.F[i;x])
\mwedge{}  (\mforall{}x:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .  (F[0;x]  =  f(x)))
\mwedge{}  (\mexists{}z:\{z:\mBbbR{}|  z  \mmember{}  [u,  v]\}  .  (r0  <  \mSigma{}\{|F[i;z]|  |  0\mleq{}i\mleq{}k\}))))))
{}\mRightarrow{}  locally-non-constant(f;a;b;r0))

Date html generated: 2019_10_30-AM-09_10_28
Last ObjectModification: 2018_11_12-PM-04_18_57

Theory : reals

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