Nuprl Lemma : bag-combine-com
∀[A,B,C:Type]. ∀[f:A ⟶ B ⟶ bag(C)]. ∀[ba:bag(A)]. ∀[bb:bag(B)]. (⋃a∈ba.⋃b∈bb.f[a;b] = ⋃b∈bb.⋃a∈ba.f[a;b] ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
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[s1;s2],
so_apply: x[s],
empty-bag: {},
top: Top,
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
bag_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
bag-combine_wf,
list-subtype-bag,
list_induction,
bag-combine-empty-left,
bag-combine-empty-right,
empty-bag_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_wf,
nil_wf,
bag-combine-unit-left,
bag-combine-append-right,
bag-append_wf,
iff_weakening_equal,
squash_wf,
true_wf,
bag-combine-append-left,
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,
dependent_functionElimination,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
hyp_replacement,
applyLambdaEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
functionExtensionality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} bag(C)]. \mforall{}[ba:bag(A)]. \mforall{}[bb:bag(B)].
(\mcup{}a\mmember{}ba.\mcup{}b\mmember{}bb.f[a;b] = \mcup{}b\mmember{}bb.\mcup{}a\mmember{}ba.f[a;b])
Date html generated:
2017_10_01-AM-08_47_24
Last ObjectModification:
2017_07_26-PM-04_31_57
Theory : bags
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