Nuprl Lemma : bag-combine-append-right
∀[A,B:Type]. ∀[F,G:A ⟶ bag(B)]. ∀[ba:bag(A)]. (⋃x∈ba.F[x] + G[x] = (⋃x∈ba.F[x] + ⋃x∈ba.G[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: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-combine: ⋃x∈bs.f[x],
bag-append: as + bs,
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
top: Top,
concat: concat(ll),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
empty-bag: {},
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
bag_wf,
list_wf,
equal_wf,
bag-append_wf,
bag-combine_wf,
list-subtype-bag,
permutation_wf,
equal-wf-base,
list_induction,
map_nil_lemma,
reduce_nil_lemma,
list_ind_nil_lemma,
empty-bag_wf,
map_cons_lemma,
reduce_cons_lemma,
bag-append-assoc2,
squash_wf,
true_wf,
bag-append-ac,
bag-append-comm,
iff_weakening_equal,
quotient-member-eq,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
hyp_replacement,
applyLambdaEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
functionExtensionality,
dependent_functionElimination,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality,
voidElimination,
voidEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[F,G:A {}\mrightarrow{} bag(B)]. \mforall{}[ba:bag(A)]. (\mcup{}x\mmember{}ba.F[x] + G[x] = (\mcup{}x\mmember{}ba.F[x] + \mcup{}x\mmember{}ba.G[x]))
Date html generated:
2017_10_01-AM-08_47_20
Last ObjectModification:
2017_07_26-PM-04_31_55
Theory : bags
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