### 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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