Nuprl Lemma : bag-map-combine
∀[A,B,C:Type]. ∀[g:A ⟶ bag(B)]. ∀[f:B ⟶ C]. ∀[bs:bag(A)]. (bag-map(f;⋃x∈bs.g[x]) = ⋃x∈bs.bag-map(f;g[x]) ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;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,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
empty-bag: {},
top: Top,
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag_wf,
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-map_wf,
bag-combine_wf,
list-subtype-bag,
equal-wf-base,
list_induction,
bag_combine_empty_lemma,
bag_map_empty_lemma,
empty-bag_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
top_wf,
single-bag_wf,
subtype_rel_bag,
bag-append_wf,
squash_wf,
true_wf,
bag-combine-single-left,
bag-map-append,
bag-combine-append-left,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
applyEquality,
productEquality,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} bag(B)]. \mforall{}[f:B {}\mrightarrow{} C]. \mforall{}[bs:bag(A)].
(bag-map(f;\mcup{}x\mmember{}bs.g[x]) = \mcup{}x\mmember{}bs.bag-map(f;g[x]))
Date html generated:
2017_10_01-AM-08_47_33
Last ObjectModification:
2017_07_26-PM-04_32_01
Theory : bags
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