Nuprl Lemma : bag-count-single
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ((#x in [y]) = if eq x y then 1 else 0 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
cons: [a / b],
nil: [],
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-filter: [x∈b|p[x]],
all: ∀x:A. B[x],
top: Top,
deq: EqDecider(T),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bag-size: #(bs),
length: ||as||,
list_ind: list_ind,
cons: [a / b],
nil: [],
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
Lemmas referenced :
bag-count-sqequal,
cons_wf,
nil_wf,
list-subtype-bag,
filter_cons_lemma,
filter_nil_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
cumulativity,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
natural_numberEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. ((\#x in [y]) = if eq x y then 1 else 0 fi )
Date html generated:
2018_05_21-PM-09_45_50
Last ObjectModification:
2017_07_26-PM-06_29_53
Theory : bags_2
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