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
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