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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