### Nuprl Lemma : derivative-rpolynomial

`∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀I:Interval.  d((Σi≤n. a_i * x^i))/dx = λx.rpoly-deriv(n;a;x) on I`

Proof

Definitions occuring in Statement :  rpoly-deriv: `rpoly-deriv(n;a;x)` derivative: `d(f[x])/dx = λz.g[z] on I` interval: `Interval` rpolynomial: `(Σi≤n. a_i * x^i)` 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 :  rpoly-deriv: `rpoly-deriv(n;a;x)` poly-deriv: `poly-deriv(a)` all: `∀x:A. B[x]` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` member: `t ∈ T` uall: `∀[x:A]. B[x]` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` label: `...\$L... t` rfun: `I ⟶ℝ` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` so_apply: `x[s]` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` ge: `i ≥ j ` r-ap: `f(x)` rfun-eq: `rfun-eq(I;f;g)` subtype_rel: `A ⊆r B` true: `True` squash: `↓T` less_than': `less_than'(a;b)` rev_uimplies: `rev_uimplies(P;Q)` nat_plus: `ℕ+` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` real_term_value: `real_term_value(f;t)` int_term_ind: int_term_ind itermSubtract: `left (-) right` itermAdd: `left (+) right` itermMultiply: `left (*) right` itermVar: `vvar`
Lemmas referenced :  interval_wf int_seg_wf real_wf all_wf subtract_wf derivative_wf rpolynomial_wf subtract-add-cancel decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf i-member_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int int-to-real_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma rmul_wf add-member-int_seg2 decidable__lt itermAdd_wf int_term_value_add_lemma lelt_wf set_wf less_than_wf primrec-wf2 nat_properties add-subtract-cancel nat_wf derivative_functionality req_weakening subtype_rel_self subtype_rel_dep_function top_wf false_wf derivative-const rpolynomial_unroll req_functionality rnexp_zero_lemma rnexp_wf radd_wf int_seg_subtype derivative-add derivative-rnexp derivative-const-mul rmul-ac int_subtype_base decidable__equal_int real_term_polynomial itermMultiply_wf req-iff-rsub-is-0
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation cut thin introduction extract_by_obid hypothesis functionEquality sqequalHypSubstitution isectElimination natural_numberEquality addEquality rename setElimination hypothesisEquality because_Cache lambdaEquality dependent_set_memberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality applyEquality setEquality equalityElimination productElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination baseClosed imageMemberEquality

Latex:
\mforall{}n:\mBbbN{}.  \mforall{}a:\mBbbN{}n  +  1  {}\mrightarrow{}  \mBbbR{}.  \mforall{}I:Interval.    d((\mSigma{}i\mleq{}n.  a\_i  *  x\^{}i))/dx  =  \mlambda{}x.rpoly-deriv(n;a;x)  on  I

Date html generated: 2017_10_03-PM-00_16_26
Last ObjectModification: 2017_07_28-AM-08_38_33

Theory : reals

Home Index