Nuprl Lemma : bag-union-union-as-combine
∀[X:Type]. ∀[x:bag(bag(bag(X)))]. (bag-union(bag-union(x)) = ⋃z∈x.bag-union(z) ∈ bag(X))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
bag: bag(T),
uall: ∀[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],
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
bag_wf,
bag-union_wf,
bag-map_wf,
bag-combine_wf,
subtype_rel_bag,
top_wf,
equal_wf,
bag-combine-map,
iff_weakening_equal,
bag-union-as-combine,
squash_wf,
true_wf,
subtype_rel_self,
bag-combine-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
lambdaEquality,
natural_numberEquality,
voidElimination,
voidEquality,
applyEquality,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
instantiate,
functionEquality
Latex:
\mforall{}[X:Type]. \mforall{}[x:bag(bag(bag(X)))]. (bag-union(bag-union(x)) = \mcup{}z\mmember{}x.bag-union(z))
Date html generated:
2018_05_21-PM-06_24_24
Last ObjectModification:
2018_05_19-PM-05_15_18
Theory : bags
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