Nuprl Lemma : rpoly-nth-deriv_functionality

`∀[d,n:ℕ]. ∀[a,b:ℕd + 1 ⟶ ℝ]. ∀[x1,x2:ℝ].`
`  (rpoly-nth-deriv(n;d;a;x1) = rpoly-nth-deriv(n;d;b;x2)) supposing ((x1 = x2) and (∀i:ℕd + 1. ((a i) = (b i))))`

Proof

Definitions occuring in Statement :  rpoly-nth-deriv: `rpoly-nth-deriv(n;d;a;x)` req: `x = y` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀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` uimplies: `b supposing a` 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` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` rpolynomial: `(Σi≤n. a_i * x^i)` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` subtract: `n - m` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` so_apply: `x[s]` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` int_seg: `{i..j-}` lelt: `i ≤ j < k` nat_plus: `ℕ+` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` rev_uimplies: `rev_uimplies(P;Q)` rneq: `x ≠ y` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int req_weakening int-to-real_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf poly-nth-deriv_wf subtract_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf int_subtype_base add-commutes add-associates minus-one-mul add-swap add-mul-special zero-mul add-zero rsum_functionality rmul_wf rnexp_wf int_seg_subtype_nat false_wf int_seg_wf rmul_functionality decidable__lt lelt_wf req_witness rpoly-nth-deriv_wf req_wf all_wf real_wf nat_wf poly-nth-deriv-req int-rdiv_wf fact_wf subtype_rel_sets nequal_wf nat_plus_properties intformeq_wf int_formula_prop_eq_lemma equal-wf-base nat_plus_wf req_functionality rdiv_wf rless-int rless_wf int-rdiv-req rdiv_functionality rnexp_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis hypothesisEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination sqequalRule natural_numberEquality dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination dependent_set_memberEquality addEquality lambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll multiplyEquality applyEquality functionExtensionality functionEquality setEquality applyLambdaEquality baseClosed inrFormation

Latex:
\mforall{}[d,n:\mBbbN{}].  \mforall{}[a,b:\mBbbN{}d  +  1  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[x1,x2:\mBbbR{}].
(rpoly-nth-deriv(n;d;a;x1)  =  rpoly-nth-deriv(n;d;b;x2))  supposing
((x1  =  x2)  and
(\mforall{}i:\mBbbN{}d  +  1.  ((a  i)  =  (b  i))))

Date html generated: 2017_10_03-PM-00_15_38
Last ObjectModification: 2017_07_28-AM-08_37_59

Theory : reals

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