Nuprl Lemma : fps-mul-coeff-bag-rep-simple
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[n:ℕ]. ∀[k:ℕn + 1]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X].
(f*g)[bag-rep(n;x)] = (* f[bag-rep(k;x)] g[bag-rep(n - k;x)]) ∈ |r|
supposing ∀i:ℕn + 1. ((¬(i = k ∈ ℤ)) ⇒ (f[bag-rep(i;x)] = 0 ∈ |r|))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-mul: (f*g),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
bag-rep: bag-rep(n;x),
deq: EqDecider(T),
int_seg: {i..j-},
nat: ℕ,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-coeff: f[b],
fps-mul: (f*g),
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
infix_ap: x f y,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag-summation-single-non-zero-no-repeats,
bag_wf,
rng_car_wf,
product-deq_wf,
bag-deq_wf,
rng_plus_wf,
rng_zero_wf,
bag-partitions_wf,
bag-rep_wf,
list-subtype-bag,
rng_times_wf,
fps-coeff_wf,
pi1_wf_top,
pi2_wf,
rng_all_properties,
rng_plus_comm2,
int_seg_subtype_nat,
false_wf,
subtract_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
itermAdd_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_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_wf,
bag-member_wf,
all_wf,
int_seg_wf,
not_wf,
equal_wf,
power-series_wf,
crng_wf,
nat_wf,
deq_wf,
valueall-type_wf,
bag-member-partitions,
bag-append-equal-bag-rep,
decidable__equal_int,
bag-size_wf,
squash_wf,
true_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
add-is-int-iff,
decidable__lt,
lelt_wf,
and_wf,
rng_times_zero,
no-repeats-bag-partitions,
bag-size-rep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
cumulativity,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
applyEquality,
lambdaEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_functionElimination,
natural_numberEquality,
addEquality,
lambdaFormation,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
functionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inlFormation,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
instantiate,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
inrFormation,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}n + 1]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:X].
(f*g)[bag-rep(n;x)] = (* f[bag-rep(k;x)] g[bag-rep(n - k;x)])
supposing \mforall{}i:\mBbbN{}n + 1. ((\mneg{}(i = k)) {}\mRightarrow{} (f[bag-rep(i;x)] = 0))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_54_53
Last ObjectModification:
2017_07_26-PM-06_32_34
Theory : power!series
Home
Index