Nuprl Lemma : bag-append-union
∀[T:Type]. ∀as,bs:bag(bag(T)). (bag-union(as + bs) = (bag-union(as) + bag-union(bs)) ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag-append: as + bs,
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
bag-union: bag-union(bbs),
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
concat: concat(ll)
Lemmas referenced :
bag_wf,
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-union_wf,
bag-append_wf,
equal-wf-base,
list_induction,
list-subtype-bag,
subtype_rel_self,
list_ind_nil_lemma,
reduce_nil_lemma,
list_ind_cons_lemma,
reduce_cons_lemma,
bag-append-assoc2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
productEquality,
axiomEquality,
universeEquality,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}as,bs:bag(bag(T)). (bag-union(as + bs) = (bag-union(as) + bag-union(bs)))
Date html generated:
2017_10_01-AM-08_46_45
Last ObjectModification:
2017_07_26-PM-04_31_27
Theory : bags
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