Nuprl Lemma : fps-moebius-inversion
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)].
g = (f*fps-moebius(eq;r)) ∈ PowerSeries(X;r) supposing f = (g*λb.1) ∈ PowerSeries(X;r)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-moebius: fps-moebius(eq;r),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
power-series: PowerSeries(X;r),
crng: CRng,
rng: Rng,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fps-coeff: f[b]
Lemmas referenced :
fps-moebius-eq,
equal_wf,
power-series_wf,
fps-mul_wf,
rng_one_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
fps-mul-assoc,
iff_weakening_equal,
fps-one_wf,
fps-div-property,
rng_car_wf,
rng_times_one,
mul_one_fps
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
cumulativity,
because_Cache,
sqequalRule,
lambdaEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
hyp_replacement,
applyLambdaEquality,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)].
g = (f*fps-moebius(eq;r)) supposing f = (g*\mlambda{}b.1)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_56_52
Last ObjectModification:
2017_07_26-PM-06_33_07
Theory : power!series
Home
Index