Nuprl Lemma : Taylor-theorem-case2

I:Interval
  (iproper(I)
   (∀n:ℕ+. ∀F:ℕ2 ⟶ I ⟶ℝ. ∀a,b:{a:ℝa ∈ I} .
        ((∀k:ℕ2. ∀x,y:{a:ℝa ∈ I} .  ((x y)  (F[k;x] F[k;y])))
         finite-deriv-seq(I;n 1;i,x.F[i;x])
         (∀e:ℝ((r0 < e)  (∃d:ℝ((r0 < d) ∧ ((|a b| < d)  (|Taylor-remainder(I;n;b;a;k,x.F[k;x])| ≤ e)))))))))


Proof




Definitions occuring in Statement :  Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]) finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]) rfun: I ⟶ℝ i-member: r ∈ I iproper: iproper(I) interval: Interval rleq: x ≤ y rless: x < y rabs: |x| rsub: y req: y int-to-real: r(n) real: int_seg: {i..j-} nat_plus: + so_apply: x[s1;s2] all: x:A. B[x] exists: x:A. B[x] implies:  Q and: P ∧ Q set: {x:A| B[x]}  function: x:A ⟶ B[x] add: m natural_number: $n
Definitions unfolded in proof :  assert: b bnot: ¬bb bfalse: ff ifthenelse: if then else fi  btrue: tt it: unit: Unit bool: 𝔹 ge: i ≥  subtract: m int_upper: {i...} pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m] rge: x ≥ y req_int_terms: t1 ≡ t2 rdiv: (x/y) rev_uimplies: rev_uimplies(P;Q) uiff: uiff(P;Q) int_nzero: -o sq_type: SQType(T) nequal: a ≠ b ∈  primrec: primrec(n;b;c) fact: (n)! Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]) Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]) subinterval: I ⊆  i-member: r ∈ I rnonneg: rnonneg(x) rleq: x ≤ y rneq: x ≠ y true: True less_than: a < b rccint: [l, u] i-approx: i-approx(I;n) continuous: f[x] continuous for x ∈ I cand: c∧ B guard: {T} rev_implies:  Q iff: ⇐⇒ Q less_than': less_than'(a;b) le: A ≤ B squash: T sq_stable: SqStable(P) real: sq_exists: x:A [B[x]] rless: x < y so_apply: x[s] so_lambda: λ2x.t[x] subtype_rel: A ⊆B lelt: i ≤ j < k int_seg: {i..j-} so_apply: x[s1;s2] rfun: I ⟶ℝ label: ...$L... t so_lambda: λ2y.t[x; y] and: P ∧ Q top: Top false: False exists: x:A. B[x] satisfiable_int_formula: satisfiable_int_formula(fmla) not: ¬A uimplies: supposing a or: P ∨ Q decidable: Dec(P) nat_plus: + nat: uall: [x:A]. B[x] prop: member: t ∈ T implies:  Q all: x:A. B[x]
Lemmas referenced :  rinv-as-rdiv rmul-rinv3 rmul-rinv rinv-of-rmul rinv_functionality2 rmul-int rneq_functionality subtract-add-cancel rsum-one bfalse_wf ifthenelse_wf assert-bnot bool_subtype_base bool_cases_sqequal eqff_to_assert assert_of_lt_int eqtt_to_assert bool_wf lt_int_wf rsum-constant mul_preserves_le nat_properties upper_subtype_nat exp_preserves_le nat_wf int_term_value_subtract_lemma subtract_wf exp_step exp-one rleq-int-fractions rdiv_functionality rnexp-rdiv req_inversion rnexp-int exp_wf3 int_nzero-rational equal_functionality_wrt_subtype_rel2 int-subtype-rationals exp_wf_nat_plus rationals_wf not_functionality_wrt_implies exp-positive exp_wf2 rless_transitivity1 le-add-cancel add_functionality_wrt_le zero-add minus-zero minus-add add-commutes condition-implies-le not-lt-2 zero-rleq-rabs rnexp-rless rabs-rnexp rless_functionality int_upper_properties int_upper_wf rleq_weakening rabs-rmul-rleq rsum_functionality_wrt_rleq rabs-difference-symmetry rleq_weakening_rless rleq-rmax rmin-rleq rabs-rsum radd_functionality_wrt_rleq real_term_value_minus_lemma real_term_value_add_lemma r-triangle-inequality rleq_functionality_wrt_implies iff_weakening_equal rabs-rminus rleq_weakening_equal itermMinus_wf rminus_wf equal_wf radd_functionality real_term_value_const_lemma real_term_value_var_lemma real_term_value_mul_lemma real_term_value_sub_lemma real_polynomial_null rmul-identity1 rinv1 rmul_functionality req_transitivity req_functionality rsum-split-first rsub_functionality rabs_functionality rleq_functionality req_weakening req-iff-rsub-is-0 itermSubtract_wf rinv_wf2 rnexp_zero_lemma fact0_redex_lemma equal-wf-T-base int_formula_prop_eq_lemma intformeq_wf nequal_wf subtype_rel_sets mul_nzero true_wf equal-wf-base int_subtype_base subtype_base_sq int_entire_a radd_wf rnexp_wf rmul_wf rsum_wf rless-int-fractions2 rmin_strict_ub Taylor-remainder_wf mul_bounds_1b member_rccint_lemma squash_wf less_than'_wf sq_stable__rleq sq_stable__all sq_stable__rless int_term_value_mul_lemma itermMultiply_wf rless-int rsub_wf i-approx_wf sq_stable__and mul_nat_plus icompact_wf rmin-rleq-rmax rccint-icompact less_than_wf rccint_wf continuous_functionality_wrt_subinterval function-is-continuous rmax-i-member sq_stable__i-member rmin-i-member rmax_wf rmin_wf rcc-subinterval small-reciprocal-real rleq_transitivity r-bound-property int_seg_properties rmaximum_ub rleq_wf fact-non-zero rneq-int false_wf int_seg_subtype_nat fact_wf rdiv_wf rabs_wf sq_stable__less_than rmaximum_wf r-bound_wf interval_wf iproper_wf nat_plus_wf rfun_wf set_wf subtype_rel_self req_wf all_wf i-member_wf lelt_wf decidable__lt int_seg_wf le_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf full-omega-unsat decidable__le nat_plus_properties finite-deriv-seq_wf real_wf int-to-real_wf rless_wf
Rules used in proof :  promote_hyp equalityElimination universeEquality cumulativity instantiate addLevel applyLambdaEquality productEquality equalitySymmetry equalityTransitivity axiomEquality minusEquality independent_pairEquality inrFormation multiplyEquality imageElimination baseClosed imageMemberEquality functionEquality setEquality productElimination because_Cache functionExtensionality applyEquality independent_pairFormation sqequalRule voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation independent_functionElimination approximateComputation independent_isectElimination unionElimination dependent_functionElimination rename setElimination addEquality dependent_set_memberEquality hypothesisEquality hypothesis natural_numberEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}I:Interval
    (iproper(I)
    {}\mRightarrow{}  (\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}F:\mBbbN{}n  +  2  {}\mrightarrow{}  I  {}\mrightarrow{}\mBbbR{}.  \mforall{}a,b:\{a:\mBbbR{}|  a  \mmember{}  I\}  .
                ((\mforall{}k:\mBbbN{}n  +  2.  \mforall{}x,y:\{a:\mBbbR{}|  a  \mmember{}  I\}  .    ((x  =  y)  {}\mRightarrow{}  (F[k;x]  =  F[k;y])))
                {}\mRightarrow{}  finite-deriv-seq(I;n  +  1;i,x.F[i;x])
                {}\mRightarrow{}  (\mforall{}e:\mBbbR{}
                            ((r0  <  e)
                            {}\mRightarrow{}  (\mexists{}d:\mBbbR{}
                                      ((r0  <  d)
                                      \mwedge{}  ((|a  -  b|  <  d)  {}\mRightarrow{}  (|Taylor-remainder(I;n;b;a;k,x.F[k;x])|  \mleq{}  e)))))))))



Date html generated: 2018_05_22-PM-02_48_48
Last ObjectModification: 2018_05_21-AM-01_22_31

Theory : reals


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