Nuprl Lemma : bag-moebius-no-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[b:bag(T)].
bag-moebius(eq;b) ~ if (#(b) rem 2 =z 0) then 1 else -1 fi supposing ↑bag-has-no-repeats(eq;b)
Proof
Definitions occuring in Statement :
bag-moebius: bag-moebius(eq;b),
bag-has-no-repeats: bag-has-no-repeats(eq;b),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
minus: -n,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-moebius: bag-moebius(eq;b),
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 ,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
bag-has-no-repeats_wf,
bool_wf,
eqtt_to_assert,
eq_int_wf,
equal-wf-base,
true_wf,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
assert_wf,
bag_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
sqequalRule,
remainderEquality,
because_Cache,
natural_numberEquality,
addLevel,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
baseClosed,
dependent_pairFormation,
promote_hyp,
minusEquality,
sqequalAxiom,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[b:bag(T)].
bag-moebius(eq;b) \msim{} if (\#(b) rem 2 =\msubz{} 0) then 1 else -1 fi supposing \muparrow{}bag-has-no-repeats(eq;b)
Date html generated:
2018_05_21-PM-09_53_42
Last ObjectModification:
2017_07_26-PM-06_32_24
Theory : bags_2
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