Nuprl Lemma : bag-moebius-inversion
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:bag(X) ⟶ |r|].
∀b:bag(X). ((g b) = Σ(p∈bag-partitions(eq;b)). (f (fst(p))) * int-to-ring(r;bag-moebius(eq;snd(p))) ∈ |r|)
supposing ∀b:bag(X). ((f b) = Σ(s∈sub-bags(eq;b)). g s ∈ |r|)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
bag-moebius: bag-moebius(eq;b),
sub-bags: sub-bags(eq;bs),
bag-partitions: bag-partitions(eq;bs),
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
int-to-ring: int-to-ring(r;n),
crng: CRng,
rng_times: *,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
power-series: PowerSeries(X;r),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
fps-mul: (f*g),
fps-coeff: f[b],
squash: ↓T,
pi1: fst(t),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sub-bags: sub-bags(eq;bs),
top: Top,
infix_ap: x f y,
fps-moebius: fps-moebius(eq;r),
has-value: (a)↓,
pi2: snd(t)
Lemmas referenced :
fps-moebius-inversion,
bag_wf,
all_wf,
equal_wf,
rng_car_wf,
bag-summation_wf,
rng_plus_wf,
rng_zero_wf,
sub-bags_wf,
rng_all_properties,
rng_plus_comm2,
crng_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
rng_times_wf,
rng_one_wf,
bag-partitions_wf,
iff_weakening_equal,
bag-summation-map,
bag-subtype-list,
assoc_wf,
comm_wf,
rng_times_one,
fps-coeff_wf,
power-series_wf,
value-type-has-value,
int-value-type,
bag-moebius_wf,
pi2_wf,
int-to-ring_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
cumulativity,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
because_Cache,
setElimination,
rename,
applyEquality,
functionExtensionality,
productElimination,
independent_pairFormation,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageElimination,
productEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
voidElimination,
voidEquality,
hyp_replacement,
callbyvalueReduce,
intEquality,
independent_pairEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:bag(X) {}\mrightarrow{} |r|].
\mforall{}b:bag(X)
((g b) = \mSigma{}(p\mmember{}bag-partitions(eq;b)). (f (fst(p))) * int-to-ring(r;bag-moebius(eq;snd(p))))
supposing \mforall{}b:bag(X). ((f b) = \mSigma{}(s\mmember{}sub-bags(eq;b)). g s)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_56_58
Last ObjectModification:
2017_07_26-PM-06_33_09
Theory : power!series
Home
Index