Nuprl Lemma : bag-mapfilter-combine
∀[A,B,C:Type]. ∀[b:bag(A)]. ∀[f:A ⟶ bag(B)]. ∀[P:B ⟶ 𝔹]. ∀[g:{x:B| ↑P[x]} ⟶ C].
(bag-mapfilter(g;P;⋃x∈b.f[x]) = ⋃x∈b.bag-mapfilter(g;P;f[x]) ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag-mapfilter: bag-mapfilter(f;P;bs),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
assert_wf,
bool_wf,
bag_wf,
set_wf,
bag-combine_wf,
bag-map_wf,
bag-filter_wf,
equal_wf,
squash_wf,
true_wf,
bag-filter-combine2,
iff_weakening_equal,
bag-map-combine
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
functionEquality,
setEquality,
cumulativity,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
lambdaEquality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[b:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[P:B {}\mrightarrow{} \mBbbB{}]. \mforall{}[g:\{x:B| \muparrow{}P[x]\} {}\mrightarrow{} C].
(bag-mapfilter(g;P;\mcup{}x\mmember{}b.f[x]) = \mcup{}x\mmember{}b.bag-mapfilter(g;P;f[x]))
Date html generated:
2017_10_01-AM-08_58_19
Last ObjectModification:
2017_07_26-PM-04_40_18
Theory : bags
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