### Nuprl Lemma : derivative-Taylor-approx

`∀I:Interval`
`  (iproper(I)`
`  `` (∀n:ℕ. ∀F:ℕn + 2 ⟶ I ⟶ℝ. ∀b:{a:ℝ| a ∈ I} .`
`        ((∀k:ℕn + 2. ∀x,y:{a:ℝ| a ∈ I} .  ((x = y) `` (F[k;x] = F[k;y])))`
`        `` finite-deriv-seq(I;n + 1;i,x.F[i;x])`
`        `` d(Taylor-approx(n;a;b;i,x.F[i;x]))/da = λx.b - x^n * (F[n + 1;x]/r((n)!)) on I)))`

Proof

Definitions occuring in Statement :  Taylor-approx: `Taylor-approx(n;a;b;i,x.F[i; x])` finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` derivative: `d(f[x])/dx = λz.g[z] on I` rfun: `I ⟶ℝ` i-member: `r ∈ I` iproper: `iproper(I)` interval: `Interval` rdiv: `(x/y)` rnexp: `x^k1` rsub: `x - y` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` fact: `(n)!` int_seg: `{i..j-}` nat: `ℕ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `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` Taylor-approx: `Taylor-approx(n;a;b;i,x.F[i; x])` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` label: `...\$L... t` rfun: `I ⟶ℝ` so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` int_upper: `{i...}` nat_plus: `ℕ+` rneq: `x ≠ y` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` subtract: `n - m` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_uimplies: `rev_uimplies(P;Q)` finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` rfun-eq: `rfun-eq(I;f;g)` r-ap: `f(x)` req_int_terms: `t1 ≡ t2` true: `True` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` less_than: `a < b` squash: `↓T` rdiv: `(x/y)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt sqequalRule universeIsType cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality_alt addEquality setElimination rename hypothesis natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation applyEquality productElimination productIsType because_Cache inhabitedIsType functionIsType functionEquality setEquality setIsType equalityTransitivity equalitySymmetry inrFormation_alt applyLambdaEquality closedConclusion equalityElimination equalityIsType4 baseApply baseClosed promote_hyp instantiate cumulativity equalityIsType1 intEquality minusEquality imageMemberEquality imageElimination universeEquality multiplyEquality

Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}F:\mBbbN{}n  +  2  {}\mrightarrow{}  I  {}\mrightarrow{}\mBbbR{}.  \mforall{}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{}  d(Taylor-approx(n;a;b;i,x.F[i;x]))/da  =  \mlambda{}x.b  -  x\^{}n  *  (F[n  +  1;x]/r((n)!))  on  I)))

Date html generated: 2019_10_30-AM-10_09_49
Last ObjectModification: 2018_11_12-PM-01_59_29

Theory : reals

Home Index