Nuprl Lemma : bag-count-ap-map
∀[T1,T2:Type]. ∀[f:T1 ⟶ T2]. ∀[eq1:EqDecider(T1)]. ∀[eq2:EqDecider(T2)]. ∀[x:T1]. ∀[bs:bag(T1)].
(#f x in bag-map(f;bs)) ~ (#x in bs) supposing Inj(T1;T2;f)
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-map: bag-map(f;bs),
bag: bag(T),
deq: EqDecider(T),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
top: Top,
squash: ↓T,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
eqof: eqof(d),
true: True,
inject: Inj(A;B;f)
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
bag-count-sqequal,
bag-map_wf,
inject_wf,
bag_wf,
deq_wf,
bag-size_wf,
set_wf,
assert_wf,
bag-filter_wf,
iff_imp_equal_bool,
equal_wf,
and_wf,
safe-assert-deq,
eqof_wf,
iff_wf,
bag-filter-map,
bag-size-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
isect_memberEquality,
functionEquality,
universeEquality,
voidElimination,
voidEquality,
imageElimination,
setElimination,
rename,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
applyLambdaEquality,
productElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} T2]. \mforall{}[eq1:EqDecider(T1)]. \mforall{}[eq2:EqDecider(T2)]. \mforall{}[x:T1]. \mforall{}[bs:bag(T1)].
(\#f x in bag-map(f;bs)) \msim{} (\#x in bs) supposing Inj(T1;T2;f)
Date html generated:
2018_05_21-PM-09_46_02
Last ObjectModification:
2017_07_26-PM-06_29_55
Theory : bags_2
Home
Index