Nuprl Lemma : int-moebius-inversion-general
∀[r:CRng]. ∀[f,g:ℕ+ ⟶ |r|].
∀n:ℕ+. (g[n] = Σ i|n. f[i] * int-to-ring(r;int-moebius(n ÷ i)) ∈ |r|) supposing ∀n:ℕ+. (f[n] = Σ i|n. g[i] ∈ |r|)
Proof
Definitions occuring in Statement :
int-moebius: int-moebius(n),
gen-divisors-sum: Σ i|n. f[i],
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
divide: n ÷ m,
equal: s = t ∈ T,
int-to-ring: int-to-ring(r;n),
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
Prime: Prime,
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
nat_plus: ℕ+,
prop: ℙ,
crng: CRng,
rng: Rng,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
cand: A c∧ B,
guard: {T},
sub-bags: sub-bags(eq;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
let: let,
gen-divisors-sum: Σ i|n. f[i],
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
int-bag-product: Π(b),
bag-product: Πx ∈ b. f[x],
label: ...$L... t,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
infix_ap: x f y,
int-moebius: int-moebius(n),
pi2: snd(t),
nequal: a ≠ b ∈ T
Lemmas referenced :
int-deq_wf,
strong-subtype-deq-subtype,
Prime_wf,
strong-subtype-set3,
int_upper_wf,
prime_wf,
le_wf,
strong-subtype-self,
bag-moebius-inversion,
set-valueall-type,
int-valueall-type,
int-bag-product_wf,
subtype_rel_bag,
bag-product-primes,
less_than_wf,
bag_wf,
factors_wf,
nat_plus_wf,
all_wf,
equal_wf,
rng_car_wf,
gen-divisors-sum_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
int_seg_wf,
crng_wf,
squash_wf,
true_wf,
bag-summation_wf,
rng_plus_wf,
rng_zero_wf,
sub-bags_wf,
rng_all_properties,
rng_plus_comm2,
iff_weakening_equal,
bag-summation-map,
bag-partitions_wf,
bag-summation-partitions-primes-general,
product-factors,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
rng_times_wf,
int-to-ring_wf,
bag-moebius_wf,
infix_ap_wf,
pi1_wf_top,
rng_wf,
deq_wf,
factors-prime-product,
pi2_wf,
int-moebius_wf,
div-positive-1,
int_seg_properties,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
hypothesis,
applyEquality,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
axiomEquality,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
voidEquality,
addEquality,
functionEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
dependent_pairFormation,
int_eqEquality,
computeAll,
productEquality,
independent_pairEquality,
divideEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[f,g:\mBbbN{}\msupplus{} {}\mrightarrow{} |r|].
\mforall{}n:\mBbbN{}\msupplus{}. (g[n] = \mSigma{} i|n. f[i] * int-to-ring(r;int-moebius(n \mdiv{} i)))
supposing \mforall{}n:\mBbbN{}\msupplus{}. (f[n] = \mSigma{} i|n. g[i])
Date html generated:
2018_05_21-PM-09_57_03
Last ObjectModification:
2017_07_26-PM-06_33_10
Theory : power!series
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