### Nuprl Lemma : polynomial-deriv-seq

`∀I:Interval. ∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ.  finite-deriv-seq(I;n;i,x.rpoly-nth-deriv(i;n;a;x))`

Proof

Definitions occuring in Statement :  finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` rpoly-nth-deriv: `rpoly-nth-deriv(n;d;a;x)` interval: `Interval` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` finite-deriv-seq: `finite-deriv-seq(I;k;i,x.F[i; x])` rpoly-nth-deriv: `rpoly-nth-deriv(n;d;a;x)` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` int_seg: `{i..j-}` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` guard: `{T}` ge: `i ≥ j ` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` decidable: `Dec(P)` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rpoly-deriv: `rpoly-deriv(n;a;x)` nequal: `a ≠ b ∈ T ` poly-nth-deriv: `poly-nth-deriv(n;a)`
Lemmas referenced :  lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int int_seg_properties nat_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf itermAdd_wf itermConstant_wf int_term_value_add_lemma int_term_value_constant_lemma derivative-rpolynomial subtract_wf decidable__le intformnot_wf intformle_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma le_wf poly-nth-deriv_wf int_seg_subtype_nat false_wf subtype_rel_dep_function int_seg_wf real_wf int_seg_subtype subtype_rel_self nat_wf interval_wf eq_int_wf assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma neg_assert_of_eq_int int_subtype_base decidable__equal_int primrec-unroll add-subtract-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalRule cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis hypothesisEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll promote_hyp instantiate cumulativity independent_functionElimination addEquality dependent_set_memberEquality applyEquality functionEquality

Latex:
\mforall{}I:Interval.  \mforall{}n:\mBbbN{}.  \mforall{}a:\mBbbN{}n  +  1  {}\mrightarrow{}  \mBbbR{}.    finite-deriv-seq(I;n;i,x.rpoly-nth-deriv(i;n;a;x))

Date html generated: 2017_10_03-PM-00_34_09
Last ObjectModification: 2017_07_28-AM-08_43_17

Theory : reals

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