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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