Nuprl Lemma : bag-moebius-property
∀[T:Type]
  ∀[eq:EqDecider(T)]. ∀[b:bag(T)].
    (bag-moebius(eq;b)
    = if bag-null(b) then 1 else -Σ(p∈[p∈bag-partitions(eq;b)|¬bbag-null(fst(p))]). bag-moebius(eq;snd(p)) fi 
    ∈ ℤ) 
  supposing valueall-type(T)
Proof
Definitions occuring in Statement : 
bag-moebius: bag-moebius(eq;b)
, 
bag-partitions: bag-partitions(eq;bs)
, 
bag-summation: Σ(x∈b). f[x]
, 
bag-null: bag-null(bs)
, 
bag-filter: [x∈b|p[x]]
, 
bag: bag(T)
, 
deq: EqDecider(T)
, 
valueall-type: valueall-type(T)
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
lambda: λx.A[x]
, 
add: n + m
, 
minus: -n
, 
natural_number: $n
, 
int: ℤ
, 
universe: Type
, 
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]
, 
implies: P 
⇒ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
prop: ℙ
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
false: False
, 
not: ¬A
, 
bag-moebius: bag-moebius(eq;b)
, 
bag-has-no-repeats: bag-has-no-repeats(eq;b)
, 
eq_int: (i =z j)
, 
bag-size: #(bs)
, 
length: ||as||
, 
list_ind: list_ind, 
bag-remove-repeats: bag-remove-repeats(eq;bs)
, 
list-to-set: list-to-set(eq;L)
, 
l-union: as ⋃ bs
, 
reduce: reduce(f;k;as)
, 
empty-bag: {}
, 
nil: []
, 
infix_ap: x f y
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
assoc: Assoc(T;op)
, 
comm: Comm(T;op)
, 
monoid_p: IsMonoid(T;op;id)
, 
ident: Ident(T;op;id)
, 
cand: A c∧ B
, 
so_lambda: λ2x.t[x]
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
so_apply: x[s]
, 
squash: ↓T
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
bag-member: x ↓∈ bs
, 
sub-bag: sub-bag(T;as;bs)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
inject: Inj(A;B;f)
, 
sq_stable: SqStable(P)
Lemmas referenced : 
bag-null_wf, 
bool_wf, 
eqtt_to_assert, 
assert-bag-null, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
equal-wf-T-base, 
bag_wf, 
deq_wf, 
valueall-type_wf, 
bag-moebius_wf, 
bag-moebius-property1, 
decidable__equal_int, 
satisfiable-full-omega-tt, 
intformnot_wf, 
intformeq_wf, 
itermAdd_wf, 
itermVar_wf, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
add-is-int-iff, 
intformand_wf, 
itermMinus_wf, 
itermConstant_wf, 
int_formula_prop_and_lemma, 
int_term_value_minus_lemma, 
int_term_value_constant_lemma, 
false_wf, 
int_subtype_base, 
bag-summation-split, 
sub-bags_wf, 
eq_int_wf, 
bag-size_wf, 
nat_wf, 
squash_wf, 
true_wf, 
bag-summation_wf, 
assert_wf, 
bnot_wf, 
pi1_wf_top, 
bag-filter_wf, 
bag-partitions_wf, 
iff_weakening_equal, 
bag-summation-single, 
bag-extensionality-no-repeats, 
decidable__equal_bag, 
decidable-equal-deq, 
subtype_rel_bag, 
single-bag_wf, 
bag-single-no-repeats, 
bag-member-single, 
bag-member_wf, 
bag-no-repeats-filter, 
sub-bags-no-repeats, 
bag-member-filter, 
assert_of_eq_int, 
member-sub-bags, 
bag-size-append, 
bag-size-is-zero, 
bag-append-empty, 
bag-subtype-list, 
sub-bag_weakening, 
bag-summation-map, 
assoc_wf, 
comm_wf, 
bag-map_wf, 
bag-map-no-repeats, 
bag-append_wf, 
pi2_wf, 
bag-settype, 
bag-member-partitions, 
top_wf, 
subtype_rel_product, 
subtype_rel_self, 
set_wf, 
bag-append-comm, 
bag-append-cancel, 
no-repeats-bag-partitions, 
bag-no-repeats-subtype, 
strong-subtype-set2, 
bag-member-map, 
sq_stable__bag-member, 
bag-member-filter-set, 
iff_transitivity, 
not_wf, 
iff_weakening_uiff, 
assert_of_bnot, 
null-bag-size
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
cumulativity, 
hypothesisEquality, 
hypothesis, 
lambdaFormation, 
unionElimination, 
equalityElimination, 
isectElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
independent_isectElimination, 
sqequalRule, 
dependent_pairFormation, 
promote_hyp, 
instantiate, 
because_Cache, 
independent_functionElimination, 
voidElimination, 
baseClosed, 
isect_memberEquality, 
axiomEquality, 
universeEquality, 
natural_numberEquality, 
hyp_replacement, 
applyLambdaEquality, 
intEquality, 
lambdaEquality, 
int_eqEquality, 
voidEquality, 
computeAll, 
independent_pairFormation, 
pointwiseFunctionality, 
rename, 
baseApply, 
closedConclusion, 
independent_pairEquality, 
addEquality, 
applyEquality, 
setElimination, 
imageElimination, 
setEquality, 
productEquality, 
imageMemberEquality, 
functionExtensionality, 
functionEquality, 
equalityUniverse, 
levelHypothesis, 
dependent_set_memberEquality, 
impliesFunctionality
Latex:
\mforall{}[T:Type]
    \mforall{}[eq:EqDecider(T)].  \mforall{}[b:bag(T)].
        (bag-moebius(eq;b)
        =  if  bag-null(b)
            then  1
            else  -\mSigma{}(p\mmember{}[p\mmember{}bag-partitions(eq;b)|\mneg{}\msubb{}bag-null(fst(p))]).  bag-moebius(eq;snd(p))
            fi  ) 
    supposing  valueall-type(T)
Date html generated:
2018_05_21-PM-09_54_40
Last ObjectModification:
2017_07_26-PM-06_32_29
Theory : bags_2
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