### Nuprl Lemma : poly-nth-deriv-req

`∀[n,d:ℕ]. ∀[a:ℕn + d ⟶ ℝ]. ∀[i:ℕd].  ((poly-nth-deriv(n;a) i) = (r((i + n)!) * (a (i + n)))/(i)!)`

Proof

Definitions occuring in Statement :  poly-nth-deriv: `poly-nth-deriv(n;a)` int-rdiv: `(a)/k1` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` fact: `(n)!` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` poly-nth-deriv: `poly-nth-deriv(n;a)` less_than': `less_than'(a;b)` rneq: `x ≠ y` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rat_term_to_real: `rat_term_to_real(f;t)` rtermDivide: `num "/" denom` rat_term_ind: rat_term_ind rtermMultiply: `left "*" right` rtermVar: `rtermVar(var)` pi1: `fst(t)` true: `True` pi2: `snd(t)` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` poly-deriv: `poly-deriv(a)` assert: `↑b` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` subtract: `n - m` nequal: `a ≠ b ∈ T ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` int_nzero: `ℤ-o` sq_exists: `∃x:A [B[x]]` rless: `x < y` sq_stable: `SqStable(P)` real: `ℝ` rdiv: `(x/y)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than req_witness int_seg_properties poly-nth-deriv_wf decidable__le intformnot_wf int_formula_prop_not_lemma istype-le int-rdiv_wf fact_wf rmul_wf int-to-real_wf itermAdd_wf int_term_value_add_lemma decidable__lt int_seg_wf real_wf istype-nat subtract-1-ge-0 nat_plus_inc_int_nzero primrec0_lemma add-zero rdiv_wf int_seg_subtype_nat istype-false rless-int nat_plus_properties rless_wf assert-rat-term-eq2 rtermVar_wf rtermDivide_wf rtermMultiply_wf req_functionality req_weakening int-rdiv-req neg_assert_of_eq_int assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert int_formula_prop_eq_lemma intformeq_wf satisfiable-full-omega-tt assert_of_eq_int eqtt_to_assert bool_wf eq_int_wf primrec-unroll lelt_wf add-subtract-cancel int_term_value_subtract_lemma itermSubtract_wf subtract_wf add-member-int_seg2 zero-add add-swap add-associates add-commutes int_subtype_base le_wf false_wf nat_plus_wf equal-wf-base nequal_wf less_than_wf subtype_rel_sets rmul_functionality decidable__equal_int fact-non-zero rneq-int equal-wf-T-base fact_unroll_1 req-int rmul-int int_entire_a rinv_wf2 req_wf rneq_functionality sq_stable__less_than rdiv_functionality uiff_transitivity rinv-of-rmul req_transitivity real_term_polynomial itermMultiply_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_var_lemma req-iff-rsub-is-0 rmul-rinv3 rinv-mul-as-rdiv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation_alt natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType productElimination imageElimination isectIsTypeImplies inhabitedIsType functionIsTypeImplies applyEquality dependent_set_memberEquality_alt unionElimination because_Cache addEquality productIsType functionIsType equalityTransitivity equalitySymmetry equalityIstype inrFormation_alt applyLambdaEquality cumulativity instantiate promote_hyp computeAll intEquality lambdaEquality dependent_pairFormation equalityElimination lambdaFormation voidEquality isect_memberEquality dependent_set_memberEquality baseClosed setEquality functionExtensionality inrFormation multiplyEquality closedConclusion baseApply imageMemberEquality

Latex:
\mforall{}[n,d:\mBbbN{}].  \mforall{}[a:\mBbbN{}n  +  d  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[i:\mBbbN{}d].    ((poly-nth-deriv(n;a)  i)  =  (r((i  +  n)!)  *  (a  (i  +  n)))/(i)!)

Date html generated: 2019_10_30-AM-09_02_16
Last ObjectModification: 2019_04_02-AM-09_46_45

Theory : reals

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