Nuprl Lemma : bag-filter-combine
∀[T,U:Type]. ∀[P:T ⟶ 𝔹]. ∀[f:U ⟶ bag(T)]. ∀[b:bag(U)]. ([x∈⋃z∈b.f[z]|P[x]] = ⋃z∈b.[x∈f[z]|P[x]] ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-filter: [x∈b|p[x]],
bag: bag(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bag-combine: ⋃x∈bs.f[x],
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
top: Top,
compose: f o g
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-filter-union2,
bag-map_wf,
bag-union_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
iff_weakening_equal,
bag-map-map,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
sqequalRule,
functionExtensionality,
cumulativity,
setEquality,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
isect_memberFormation,
axiomEquality
Latex:
\mforall{}[T,U:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:U {}\mrightarrow{} bag(T)]. \mforall{}[b:bag(U)]. ([x\mmember{}\mcup{}z\mmember{}b.f[z]|P[x]] = \mcup{}z\mmember{}b.[x\mmember{}f[z]|P[x]])
Date html generated:
2017_10_01-AM-08_58_01
Last ObjectModification:
2017_07_26-PM-04_40_09
Theory : bags
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