### Nuprl Lemma : rpolynomial_unroll

`∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ].`
`  ((Σi≤n. a_i * x^i) = if (n =z 0) then a 0 else ((a n) * x^n) + (Σi≤n - 1. a_i * x^i) fi )`

Proof

Definitions occuring in Statement :  rpolynomial: `(Σi≤n. a_i * x^i)` rnexp: `x^k1` req: `x = y` rmul: `a * b` radd: `a + b` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rpolynomial: `(Σi≤n. a_i * x^i)` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` int_upper: `{i...}` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  req_witness rpolynomial_wf int_seg_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int false_wf nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf lelt_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_upper_subtype_nat le_wf nequal-le-implies zero-add radd_wf rmul_wf decidable__le intformle_wf int_formula_prop_le_lemma rnexp_wf subtract_wf int_upper_properties itermSubtract_wf int_term_value_subtract_lemma subtype_rel_dep_function real_wf int_seg_subtype subtract-add-cancel subtype_rel_self nat_wf rsum_wf int_seg_subtype_nat lt_int_wf assert_of_lt_int int-to-real_wf less_than_wf rnexp_zero_lemma rmul-one req_weakening req_functionality rsum_unroll radd_comm
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality functionExtensionality applyEquality natural_numberEquality addEquality setElimination rename hypothesis because_Cache lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination dependent_set_memberEquality equalityTransitivity equalitySymmetry independent_pairFormation dependent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp instantiate cumulativity independent_functionElimination hypothesis_subsumption functionEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[a:\mBbbN{}n  +  1  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[x:\mBbbR{}].
((\mSigma{}i\mleq{}n.  a\_i  *  x\^{}i)  =  if  (n  =\msubz{}  0)  then  a  0  else  ((a  n)  *  x\^{}n)  +  (\mSigma{}i\mleq{}n  -  1.  a\_i  *  x\^{}i)  fi  )

Date html generated: 2017_10_03-AM-08_58_11
Last ObjectModification: 2017_07_28-AM-07_37_59

Theory : reals

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