### Nuprl Lemma : fps-deriv-single

`∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[b:bag(X)]. ∀[x:X].`
`  (d<b>/dx = (int-to-ring(r;(#x in b)))*<bag-drop(eq;b;x)> ∈ PowerSeries(X;r))`

Proof

Definitions occuring in Statement :  fps-deriv: `df/dx` fps-scalar-mul: `(c)*f` fps-single: `<c>` power-series: `PowerSeries(X;r)` bag-drop: `bag-drop(eq;bs;a)` bag-count: `(#x in bs)` bag: `bag(T)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T` int-to-ring: `int-to-ring(r;n)` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` crng: `CRng` subtype_rel: `A ⊆r B` nat: `ℕ` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-single: `<c>` fps-scalar-mul: `(c)*f` fps-coeff: `f[b]` fps-deriv: `df/dx` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` 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` infix_ap: `x f y` rng: `Rng` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` top: `Top` rev_uimplies: `rev_uimplies(P;Q)` sq_or: `a ↓∨ b` ringeq_int_terms: `t1 ≡ t2` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)`
Lemmas referenced :  fps-ext fps-deriv_wf fps-single_wf fps-scalar-mul_wf int-to-ring_wf bag-count_wf nat_wf bag-drop_wf bag-drop-property bag-eq_wf cons-bag_wf bool_wf eqtt_to_assert assert-bag-eq eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bag_wf crng_wf deq_wf rng_times_wf squash_wf true_wf rng_wf rng_one_wf subtype_rel_self iff_weakening_equal cons-bag-as-append bag-count-append single-bag_wf add-commutes add_functionality_wrt_eq bag-count-member-no-repeats bag-member-single bag-single-no-repeats bag-append-cancel bag-member_wf bag-member-cons rng_zero_wf itermAdd_wf itermMultiply_wf itermVar_wf itermConstant_wf itermMinus_wf ringeq-iff-rsub-is-0 ring_polynomial_null ring_term_value_add_lemma ring_term_value_mul_lemma ring_term_value_var_lemma ring_term_value_const_lemma int-to-ring-zero ring_term_value_minus_lemma bag-append_wf not_wf or_functionality_wrt_iff set_subtype_base le_wf int_subtype_base decidable__le nat_properties full-omega-unsat intformand_wf intformle_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf bag-count-is-zero rng_car_wf rng_times_zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality hypothesis setElimination rename applyEquality lambdaEquality sqequalRule productElimination independent_isectElimination lambdaFormation dependent_functionElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp instantiate cumulativity independent_functionElimination voidElimination isect_memberEquality axiomEquality universeEquality imageElimination intEquality natural_numberEquality imageMemberEquality baseClosed addEquality voidEquality inlFormation approximateComputation int_eqEquality productEquality independent_pairFormation applyLambdaEquality dependent_set_memberEquality

Latex:
\mforall{}[X:Type].  \mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[b:bag(X)].  \mforall{}[x:X].
(d<b>/dx  =  (int-to-ring(r;(\#x  in  b)))*<bag-drop(eq;b;x)>)

Date html generated: 2018_05_21-PM-10_16_26
Last ObjectModification: 2018_05_19-PM-04_18_06

Theory : power!series

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