Nuprl Lemma : bag-combine-append-left
∀[A,B:Type]. ∀[ba,bb:bag(A)]. ∀[f:A ⟶ bag(B)]. (⋃x∈ba + bb.f[x] = (⋃x∈ba.f[x] + ⋃x∈bb.f[x]) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-append: as + bs,
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag-combine: ⋃x∈bs.f[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
squash: ↓T,
exists: ∃x:A. B[x],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
bag-append: as + bs,
so_apply: x[s],
prop: ℙ
Lemmas referenced :
bag-map-append,
subtype_rel_bag,
top_wf,
bag_to_squash_list,
concat_append,
bag-append_wf,
bag-union_wf,
bag-map_wf,
bag_wf,
list-subtype-bag,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
imageElimination,
productElimination,
promote_hyp,
rename,
cumulativity,
functionExtensionality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
functionEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[ba,bb:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. (\mcup{}x\mmember{}ba + bb.f[x] = (\mcup{}x\mmember{}ba.f[x] + \mcup{}x\mmember{}bb.f[x]))
Date html generated:
2016_10_25-AM-10_23_29
Last ObjectModification:
2016_07_12-AM-06_40_13
Theory : bags
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