### Nuprl Lemma : Taylor-approx_functionality

`∀[I:Interval]. ∀[n:ℕ]. ∀[F:ℕn + 1 ⟶ I ⟶ℝ].`
`  ∀[a1,b1,a2,b2:{a:ℝ| a ∈ I} ].`
`    (Taylor-approx(n;a1;b1;i,x.F[i;x]) = Taylor-approx(n;a2;b2;i,x.F[i;x])) supposing ((a1 = a2) and (b1 = b2)) `
`  supposing ∀k:ℕn + 1. ∀x,y:{a:ℝ| a ∈ I} .  ((x = y) `` (F[k;x] = F[k;y]))`

Proof

Definitions occuring in Statement :  Taylor-approx: `Taylor-approx(n;a;b;i,x.F[i; x])` rfun: `I ⟶ℝ` i-member: `r ∈ I` interval: `Interval` req: `x = y` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` 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 :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` Taylor-approx: `Taylor-approx(n;a;b;i,x.F[i; x])` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` nat_plus: `ℕ+` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_apply: `x[s]` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` so_lambda: `λ2x y.t[x; y]` label: `...\$L... t` rfun: `I ⟶ℝ`
Lemmas referenced :  rsum_functionality rmul_wf rdiv_wf int-to-real_wf fact_wf int_seg_subtype_nat false_wf rless-int int_seg_properties nat_properties decidable__lt le_wf nat_plus_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf rless_wf rnexp_wf rsub_wf rmul_functionality itermAdd_wf intformle_wf int_term_value_add_lemma int_formula_prop_le_lemma lelt_wf i-member_wf rnexp_functionality rsub_functionality req_witness Taylor-approx_wf int_seg_wf rfun_wf real_wf req_wf set_wf all_wf nat_wf interval_wf req_weakening rdiv_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality because_Cache sqequalRule lambdaEquality applyEquality functionExtensionality hypothesisEquality hypothesis setElimination rename dependent_set_memberEquality addEquality independent_isectElimination independent_pairFormation lambdaFormation inrFormation dependent_functionElimination productElimination independent_functionElimination unionElimination equalityTransitivity equalitySymmetry Error :applyLambdaEquality,  voidElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll setEquality functionEquality

Latex:
\mforall{}[I:Interval].  \mforall{}[n:\mBbbN{}].  \mforall{}[F:\mBbbN{}n  +  1  {}\mrightarrow{}  I  {}\mrightarrow{}\mBbbR{}].
\mforall{}[a1,b1,a2,b2:\{a:\mBbbR{}|  a  \mmember{}  I\}  ].
(Taylor-approx(n;a1;b1;i,x.F[i;x])  =  Taylor-approx(n;a2;b2;i,x.F[i;x]))  supposing
((a1  =  a2)  and
(b1  =  b2))
supposing  \mforall{}k:\mBbbN{}n  +  1.  \mforall{}x,y:\{a:\mBbbR{}|  a  \mmember{}  I\}  .    ((x  =  y)  {}\mRightarrow{}  (F[k;x]  =  F[k;y]))

Date html generated: 2016_10_26-AM-11_44_27
Last ObjectModification: 2016_08_28-PM-10_37_45

Theory : reals

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