### Nuprl Lemma : int-moebius-inversion

`∀[f,g:ℕ+ ⟶ ℤ].  ∀n:ℕ+. (g[n] = Σ i|n. f[i] * int-moebius(n ÷ i)  ∈ ℤ) supposing ∀n:ℕ+. (f[n] = Σ i|n. g[i]  ∈ ℤ)`

Proof

Definitions occuring in Statement :  int-moebius: `int-moebius(n)` divisors-sum: `Σ i|n. f[i] ` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` divide: `n ÷ m` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
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` integ_dom: `IntegDom{i}` int_ring: `ℤ-rng` rng_car: `|r|` pi1: `fst(t)` squash: `↓T` prop: `ℙ` crng: `CRng` rng: `Rng` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` 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` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` guard: `{T}` nequal: `a ≠ b ∈ T ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` rng_times: `*` pi2: `snd(t)` infix_ap: `x f y` sq_type: `SQType(T)`
Lemmas referenced :  int-moebius-inversion-general int_ring_wf integ_dom_wf equal_wf squash_wf true_wf rng_car_wf gen-divisors-sum-int-ring decidable__lt false_wf not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel less_than_wf int_seg_wf iff_weakening_equal divisors-sum_wf infix_ap_wf rng_times_wf int-to-ring_wf int-moebius_wf int_seg_properties nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf equal-wf-base int_subtype_base div-positive-1 nat_plus_wf all_wf subtype_base_sq int-to-ring-int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis applyEquality lambdaEquality setElimination rename hypothesisEquality sqequalRule functionExtensionality because_Cache independent_isectElimination imageElimination equalityTransitivity equalitySymmetry universeEquality dependent_set_memberEquality dependent_functionElimination natural_numberEquality unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination addEquality imageMemberEquality baseClosed intEquality functionEquality divideEquality dependent_pairFormation int_eqEquality isect_memberEquality voidEquality computeAll axiomEquality instantiate cumulativity multiplyEquality

Latex:
\mforall{}[f,g:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].
\mforall{}n:\mBbbN{}\msupplus{}.  (g[n]  =  \mSigma{}  i|n.  f[i]  *  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_06
Last ObjectModification: 2017_07_26-PM-06_33_11

Theory : power!series

Home Index