Nuprl Lemma : fps-mul-single-general
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[c:bag(X)]. ∀[f:PowerSeries(X;r)].
((<c>*f) = (λb.case bag-diff(eq;b;c) of inl(d) => f[d] | inr(z) => 0) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-mul: (f*g),
fps-single: <c>,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
bag-diff: bag-diff(eq;bs;as),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
comm: Comm(T;op),
rng: Rng,
power-series: PowerSeries(X;r),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
fps-coeff: f[b],
fps-mul: (f*g),
so_lambda: λ2x.t[x],
so_apply: x[s],
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
squash: ↓T,
top: Top,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fps-single: <c>,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
false: False,
not: ¬A,
infix_ap: x f y
Lemmas referenced :
rng_plus_comm,
crng_properties,
rng_properties,
fps-ext,
fps-mul_wf,
fps-single_wf,
subtype_rel_self,
bag_wf,
rng_car_wf,
bag-diff_wf,
unit_wf2,
fps-coeff_wf,
rng_zero_wf,
equal_wf,
bag-diff-property,
bag-append_wf,
all_wf,
not_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag-partitions-with-one-given,
bag-summation_wf,
rng_plus_wf,
rng_times_wf,
crng_all_properties,
squash_wf,
true_wf,
bag-summation-single,
pi1_wf_top,
pi2_wf,
iff_weakening_equal,
bag-eq_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
member_wf,
rng_times_one,
bag-summation-filter,
bag-partitions_wf,
bag-summation-equal,
ifthenelse_wf,
bag-member_wf,
rng_one_wf,
rng_times_zero,
bag-summation-is-zero,
bag-member-partitions,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
applyEquality,
sqequalRule,
functionEquality,
lambdaEquality,
unionEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
productElimination,
because_Cache,
isect_memberEquality,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
productEquality,
cumulativity,
independent_pairFormation,
imageElimination,
independent_pairEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[c:bag(X)]. \mforall{}[f:PowerSeries(X;r)].
((<c>*f) = (\mlambda{}b.case bag-diff(eq;b;c) of inl(d) => f[d] | inr(z) => 0))
supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_34_30
Last ObjectModification:
2018_08_21-PM-01_59_49
Theory : power!series
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