### Nuprl Lemma : fps-mul-slice

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[n:ℕ]. ∀[f,g:PowerSeries(X;r)].`
`    ([(f*g)]_n = fps-summation(r;upto(n + 1);k.([f]_k*[g]_n - k)) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-slice: `[f]_n` fps-summation: `fps-summation(r;b;x.f[x])` fps-mul: `(f*g)` power-series: `PowerSeries(X;r)` upto: `upto(n)` deq: `EqDecider(T)` nat: `ℕ` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` cand: `A c∧ B` crng: `CRng` power-series: `PowerSeries(X;r)` nat: `ℕ` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` fps-coeff: `f[b]` squash: `↓T` rng: `Rng` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` fps-slice: `[f]_n` fps-mul: `(f*g)` infix_ap: `x f y` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` top: `Top` pi1: `fst(t)` pi2: `snd(t)` band: `p ∧b q` int_seg: `{i..j-}` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` sq_exists: `∃x:A [B[x]]` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` bag-member: `x ↓∈ bs` l_member: `(x ∈ l)` bag-no-repeats: `bag-no-repeats(T;bs)` nequal: `a ≠ b ∈ T `
Lemmas referenced :  rng_all_properties rng_plus_comm2 upto_wf list-subtype-bag int_seg_wf nat_wf int_seg_subtype_nat false_wf equal_wf squash_wf true_wf rng_car_wf fps-coeff_wf fps-slice_wf fps-mul_wf fps-summation-coeff subtract_wf iff_weakening_equal bag_wf power-series_wf crng_wf deq_wf valueall-type_wf crng_properties rng_properties eq_int_wf bag-size_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int bag-summation_wf assoc_wf comm_wf rng_plus_wf rng_zero_wf bag-summation-filter bag-partitions_wf band_wf pi1_wf_top pi2_wf rng_times_wf infix_ap_wf bag-summation-equal iff_transitivity assert_wf iff_weakening_uiff assert_of_band rng_times_zero bag-member_wf bag-summation-partition decidable__int_equal bag-subtype set_wf member_wf subtype_rel_bag bag-member-partitions bag-size-append nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_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 decidable__lt intformless_wf itermAdd_wf intformeq_wf int_formula_prop_less_lemma int_term_value_add_lemma int_formula_prop_eq_lemma lelt_wf subtype_rel_list equal-wf-base-T list_subtype_base int_subtype_base l_member_wf member_upto int_seg_properties le_wf decidable__equal_int less_than_wf length_wf select_wf add-is-int-iff itermSubtract_wf int_term_value_subtract_lemma no_repeats_upto no_repeats_wf equal-wf-T-base bag-summation-is-zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis productElimination independent_pairFormation because_Cache functionExtensionality addEquality natural_numberEquality applyEquality independent_isectElimination sqequalRule lambdaFormation lambdaEquality imageElimination equalityTransitivity equalitySymmetry cumulativity dependent_functionElimination imageMemberEquality baseClosed universeEquality independent_functionElimination isect_memberEquality axiomEquality unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate voidElimination productEquality functionEquality independent_pairEquality voidEquality intEquality hyp_replacement setEquality dependent_set_memberEquality dependent_set_memberFormation applyLambdaEquality int_eqEquality computeAll comment pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[n:\mBbbN{}].  \mforall{}[f,g:PowerSeries(X;r)].
([(f*g)]\_n  =  fps-summation(r;upto(n  +  1);k.([f]\_k*[g]\_n  -  k)))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_56_21
Last ObjectModification: 2017_07_26-PM-06_32_56

Theory : power!series

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