Nuprl Lemma : Taylor-theorem-case1

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])
         a ≠ r0
         (∀e:ℝ
              ((r0 < e)
               (∃c:ℝ
                   ((rmin(a;b) ≤ c)
                   ∧ (c ≤ rmax(a;b))
                   ∧ (|Taylor-remainder(I;n;b;a;k,x.F[k;x]) (b c^n (F[n 1;c]/r((n)!))) (b a)| ≤ 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 rdiv: (x/y) rneq: x ≠ y rleq: x ≤ y rless: x < y rabs: |x| rmin: rmin(x;y) rmax: rmax(x;y) rnexp: x^k1 rsub: y req: y rmul: b int-to-real: r(n) real: fact: (n)! int_seg: {i..j-} nat: 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: $n
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T implies:  Q uall: [x:A]. B[x] so_lambda: λ2y.t[x; y] label: ...$L... t rfun: I ⟶ℝ so_apply: x[s1;s2] int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q le: A ≤ B less_than: a < b squash: T rless: x < y sq_exists: x:A [B[x]] nat_plus: + nat: ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: supposing a not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False top: Top prop: subtype_rel: A ⊆B iff: ⇐⇒ Q rev_implies:  Q cand: c∧ B sq_stable: SqStable(P) rneq: x ≠ y guard: {T} uiff: uiff(P;Q) req_int_terms: t1 ≡ t2 so_lambda: λ2x.t[x] so_apply: x[s] subinterval: I ⊆  rev_uimplies: rev_uimplies(P;Q) rat_term_to_real: rat_term_to_real(f;t) rtermSubtract: left "-" right rat_term_ind: rat_term_ind rtermDivide: num "/" denom rtermMultiply: left "*" right rtermVar: rtermVar(var) pi1: fst(t) true: True rtermConstant: "const" rtermAdd: left "+" right pi2: snd(t) rfun-eq: rfun-eq(I;f;g) r-ap: f(x) rdiv: (x/y) Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]) i-member: r ∈ I rccint: [l, u]
Lemmas referenced :  derivative-Taylor-approx Taylor-remainder_wf int_seg_properties nat_plus_properties nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma istype-le istype-less_than subtype_rel_self real_wf i-member_wf int_seg_wf rless_wf int-to-real_wf rneq_wf rsub_wf finite-deriv-seq_wf req_wf rfun_wf istype-nat iproper_wf interval_wf rcc-subinterval rmin_wf rmax_wf rmin-i-member sq_stable__i-member rmax-i-member rleq_wf rmax_strict_ub rless-implies-rless itermSubtract_wf req-iff-rsub-is-0 rmin_strict_lb real_polynomial_null real_term_value_sub_lemma real_term_value_const_lemma real_term_value_var_lemma Rolles-theorem radd_wf Taylor-approx_wf subtype_rel_sets_simple rccint_wf rdiv_wf rmul_wf rnexp_wf fact_wf rless-int function-is-continuous req_functionality rsub_functionality req_weakening rmul_functionality rnexp_functionality rdiv_functionality derivative-sub derivative-const derivative-add derivative_functionality_wrt_subinterval istype-top member_rccint_lemma subtype_rel_dep_function top_wf derivative-rdiv-const derivative-const-mul derivative-id assert-rat-term-eq2 rtermSubtract_wf rtermConstant_wf rtermAdd_wf rtermMultiply_wf rtermVar_wf rtermDivide_wf derivative_functionality rmin-max-cases radd-preserves-rless rless_functionality real_term_value_add_lemma rleq_weakening_equal rmin-rleq-rmax radd_functionality Taylor-approx_functionality trivial-Taylor-approx rmul_preserves_req rinv_wf2 itermMultiply_wf req_transitivity rmul-rinv real_term_value_mul_lemma squash_wf true_wf iff_weakening_equal rabs-neq-zero rabs_wf rmul_preserves_rless rmul-rinv3 rneq-int fact-non-zero rmul_preserves_rleq2 zero-rleq-rabs rminus_wf itermMinus_wf rleq_functionality rabs_functionality real_term_value_minus_lemma req_inversion rabs-rmul
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination isectElimination sqequalRule lambdaEquality_alt applyEquality dependent_set_memberEquality_alt setElimination rename productElimination imageElimination independent_pairFormation natural_numberEquality unionElimination independent_isectElimination approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination universeIsType addEquality productIsType functionEquality setEquality inhabitedIsType equalityIstype equalityTransitivity equalitySymmetry because_Cache functionIsType setIsType imageMemberEquality baseClosed inlFormation_alt inrFormation_alt applyLambdaEquality closedConclusion productEquality instantiate universeEquality

Latex:
\mforall{}I:Interval
    (iproper(I)
    {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \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{}  b  -  a  \mneq{}  r0
                {}\mRightarrow{}  (\mforall{}e:\mBbbR{}
                            ((r0  <  e)
                            {}\mRightarrow{}  (\mexists{}c:\mBbbR{}
                                      ((rmin(a;b)  \mleq{}  c)
                                      \mwedge{}  (c  \mleq{}  rmax(a;b))
                                      \mwedge{}  (|Taylor-remainder(I;n;b;a;k,x.F[k;x])  -  (b  -  c\^{}n  *  (F[n  +  1;c]/r((n)!)))
                                          *  (b  -  a)|  \mleq{}  e))))))))



Date html generated: 2019_10_30-AM-10_10_34
Last ObjectModification: 2019_04_02-AM-09_42_21

Theory : reals


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