Nuprl Lemma : bag-remove-repeats-append
∀[T:Type]. ∀[as,bs:bag(T)]. ∀[eq:EqDecider(T)].
  (bag-remove-repeats(eq;as + bs)
  = (bag-remove-repeats(eq;as) + [x∈bag-remove-repeats(eq;bs)|¬bbag-deq-member(eq;x;as)])
  ∈ bag(T))
Proof
Definitions occuring in Statement : 
bag-remove-repeats: bag-remove-repeats(eq;bs)
, 
bag-deq-member: bag-deq-member(eq;x;b)
, 
bag-filter: [x∈b|p[x]]
, 
bag-append: as + bs
, 
bag: bag(T)
, 
deq: EqDecider(T)
, 
bnot: ¬bb
, 
uall: ∀[x:A]. B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
uimplies: b supposing a
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
uiff: uiff(P;Q)
, 
bag-member: x ↓∈ bs
, 
squash: ↓T
, 
not: ¬A
, 
false: False
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
sq_or: a ↓∨ b
, 
sq_stable: SqStable(P)
, 
guard: {T}
Lemmas referenced : 
sq_stable__sq_or, 
sq_stable__bag-member, 
bag-member-append, 
decidable__bag-member, 
assert-bag-deq-member, 
assert_of_bnot, 
iff_weakening_uiff, 
iff_transitivity, 
and_wf, 
not_wf, 
bag-member-filter, 
bag-member-remove-repeats, 
bag-no-repeats-filter, 
bag-no-repeats-append, 
bag_wf, 
deq_wf, 
bag-member_wf, 
bag-remove-repeats-no-repeats, 
assert_wf, 
subtype_rel_bag, 
bag-deq-member_wf, 
bnot_wf, 
bag-filter_wf, 
bag-append_wf, 
bag-remove-repeats_wf, 
decidable-equal-deq, 
bag-extensionality-no-repeats
Rules used in proof : 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isectElimination, 
thin, 
hypothesisEquality, 
independent_functionElimination, 
lambdaFormation, 
because_Cache, 
hypothesis, 
dependent_functionElimination, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
setEquality, 
independent_isectElimination, 
setElimination, 
rename, 
independent_pairFormation, 
productElimination, 
isect_memberFormation, 
introduction, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
universeEquality, 
isect_memberEquality, 
axiomEquality, 
voidElimination, 
addLevel, 
impliesFunctionality, 
unionElimination, 
inlFormation, 
inrFormation
Latex:
\mforall{}[T:Type].  \mforall{}[as,bs:bag(T)].  \mforall{}[eq:EqDecider(T)].
    (bag-remove-repeats(eq;as  +  bs)
    =  (bag-remove-repeats(eq;as)  +  [x\mmember{}bag-remove-repeats(eq;bs)|\mneg{}\msubb{}bag-deq-member(eq;x;as)]))
Date html generated:
2016_05_15-PM-08_02_54
Last ObjectModification:
2016_01_16-PM-01_30_00
Theory : bags_2
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