Nuprl Lemma : rpoly-nth-deriv_wf

`∀[d,n:ℕ]. ∀[a:ℕd + 1 ⟶ ℝ]. ∀[x:ℝ].  (rpoly-nth-deriv(n;d;a;x) ∈ ℝ)`

Proof

Definitions occuring in Statement :  rpoly-nth-deriv: `rpoly-nth-deriv(n;d;a;x)` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rpoly-nth-deriv: `rpoly-nth-deriv(n;d;a;x)` nat: `ℕ` all: `∀x:A. B[x]` 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 ` bfalse: `ff` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` subtract: `n - m` sq_type: `SQType(T)`
Lemmas referenced :  lt_int_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf less_than_wf eqtt_to_assert assert_of_lt_int int-to-real_wf le_int_wf le_wf bnot_wf eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int rpolynomial_wf subtract_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_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_wf poly-nth-deriv_wf itermAdd_wf int_term_value_add_lemma subtype_base_sq int_subtype_base add-commutes add-associates minus-one-mul add-swap add-mul-special zero-mul add-zero equal_wf real_wf int_seg_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis hypothesisEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry baseClosed independent_functionElimination productElimination independent_isectElimination natural_numberEquality dependent_set_memberEquality dependent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll addEquality instantiate cumulativity multiplyEquality axiomEquality functionEquality

Latex:
\mforall{}[d,n:\mBbbN{}].  \mforall{}[a:\mBbbN{}d  +  1  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[x:\mBbbR{}].    (rpoly-nth-deriv(n;d;a;x)  \mmember{}  \mBbbR{})

Date html generated: 2017_10_03-PM-00_15_14
Last ObjectModification: 2017_07_28-AM-08_37_41

Theory : reals

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