Nuprl Lemma : bag-moebius-property1
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[b:bag(T)]. (Σ(c∈sub-bags(eq;b)). bag-moebius(eq;c) = if bag-null(b) then 1 else 0 fi ∈ ℤ)
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-moebius: bag-moebius(eq;b),
sub-bags: sub-bags(eq;bs),
bag-summation: Σ(x∈b). f[x],
bag-null: bag-null(bs),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
add: n + m,
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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
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 ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
assoc: Assoc(T;op),
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
comm: Comm(T;op),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
top: Top,
pi1: fst(t),
length: ||as||,
bag-size: #(bs),
eq_int: (i =z j),
bag-has-no-repeats: bag-has-no-repeats(eq;b),
bag-moebius: bag-moebius(eq;b),
deq-member: x ∈b L,
eval_list: eval_list(t),
insert: insert(a;L),
reduce: reduce(f;k;as),
l-union: as ⋃ bs,
list-to-set: list-to-set(eq;L),
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-to-set: bag-to-set(eq;bs),
cons: [a / b],
single-bag: {x},
nil: [],
empty-bag: {},
bag-splits: bag-splits(b),
evalall: evalall(t),
callbyvalueall: callbyvalueall,
bag-partitions: bag-partitions(eq;bs),
list_ind: list_ind,
map: map(f;as),
bag-map: bag-map(f;bs),
sub-bags: sub-bags(eq;bs),
list_accum: list_accum,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-summation: Σ(x∈b). f[x],
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id),
true: True,
ge: i ≥ j ,
le: A ≤ B,
nat: ℕ,
integ_dom: IntegDom{i},
crng: CRng,
rng_car: |r|,
int_ring: ℤ-rng,
rng: Rng,
rng_zero: 0,
pi2: snd(t),
rng_plus: +r,
rng_minus: -r,
sq_stable: SqStable(P),
bag-member: x ↓∈ bs,
inject: Inj(A;B;f),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
sub-bag: sub-bag(T;as;bs),
deq: EqDecider(T),
eqof: eqof(d),
sq_or: a ↓∨ b,
less_than': less_than'(a;b),
nat_plus: ℕ+,
subtract: n - m,
int_seg: {i..j-},
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k
Lemmas referenced :
bag_to_squash_list,
bag-null_wf,
eqtt_to_assert,
assert-bag-null,
equal_wf,
bag-summation_wf,
istype-int,
sub-bags_wf,
decidable__equal_int,
full-omega-unsat,
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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
bag_wf,
empty-bag_wf,
deq_wf,
valueall-type_wf,
istype-universe,
list-subtype-bag,
bag_qinc,
satisfiable-full-omega-tt,
bag-summation-split,
bag-has-no-repeats_wf,
itermConstant_wf,
int_term_value_constant_lemma,
squash_wf,
true_wf,
bag-moebius_wf,
subtype_rel_self,
iff_weakening_equal,
int_subtype_base,
bag-summation-is-zero,
bag-filter_wf,
bag-member_wf,
bag-member-filter,
bnot_wf,
assert_of_bnot,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
non_neg_length,
decidable__le,
length_wf,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
subtype_rel_bag,
istype-assert,
lt_int_wf,
bag-count_wf,
hd_wf,
add_functionality_wrt_eq,
bag-summation-minus,
int_ring_wf,
rng_car_wf,
equal-wf-base,
bag-no-repeats_wf,
assert-bag-has-no-repeats,
bag-no-repeats-filter,
sub-bags-no-repeats,
bag-no-repeats-subtype,
strong-subtype-set2,
bag-append_wf,
single-bag_wf,
bag-no-repeats-append,
bag-single-no-repeats,
sq_stable__bag-no-repeats,
bag-member-single,
not_wf,
less_than_wf,
istype-less_than,
istype-void,
iff_transitivity,
assert_of_lt_int,
bag-member-count,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
bag-extensionality-no-repeats,
decidable__equal_set,
decidable__equal_bag,
decidable-equal-deq,
bag-map_wf,
bag-map-no-repeats,
bag-append-cancel,
sq_stable__bag-member,
bag-member-map,
bag-drop-property2,
bag-drop_wf,
bag-member-filter-set,
bag-subtype2,
member-sub-bags,
bag-append-comm,
sub-bag_transitivity,
sub-bag_wf,
sub-bag-iff,
le_wf,
bag-count-single,
safe-assert-deq,
length_wf_nat,
nat_properties,
bag-member-list,
hd_member,
not_functionality_wrt_implies,
list_wf,
equal_functionality_wrt_subtype_rel2,
nil_wf,
null_wf3,
subtype_rel_list,
top_wf,
assert_of_null,
bag-member-append,
bag-append-assoc2,
bag-count-append,
cons_wf,
ifthenelse_wf,
member_wf,
add_nat_plus,
istype-false,
istype-le,
nat_plus_properties,
add-is-int-iff,
false_wf,
bag-summation-map,
assoc_wf,
comm_wf,
bag-size-append,
bag_size_single_lemma,
rem_add1,
bag-size_wf,
rem_bounds_1,
add-commutes,
int_seg_properties,
bfalse_wf,
int_seg_subtype_special,
int_seg_cases,
btrue_wf,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
independent_pairFormation,
closedConclusion,
equalityTransitivity,
equalityIstype,
instantiate,
cumulativity,
baseClosed,
universeEquality,
applyEquality,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
dependent_pairFormation,
isect_memberFormation,
lambdaEquality,
addEquality,
independent_pairEquality,
imageMemberEquality,
hypothesis_subsumption,
setEquality,
setElimination,
setIsType,
dependent_set_memberEquality_alt,
functionIsType,
productIsType,
pointwiseFunctionality,
baseApply,
productEquality,
remainderEquality,
sqequalBase,
minusEquality
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[b:bag(T)].
(\mSigma{}(c\mmember{}sub-bags(eq;b)). bag-moebius(eq;c) = if bag-null(b) then 1 else 0 fi )
supposing valueall-type(T)
Date html generated:
2020_05_20-AM-09_05_19
Last ObjectModification:
2020_01_04-PM-11_09_51
Theory : bags_2
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