Nuprl Lemma : bag-combine-map
∀[A,B,C:Type]. ∀[g:B ⟶ bag(C)]. ∀[f:A ⟶ B]. ∀[bs:bag(A)]. (⋃x∈bag-map(f;bs).g[x] = ⋃x∈bs.g[f 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],
apply: f a,
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,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
reduce: reduce(f;k;as),
list_ind: list_ind,
bag-map: bag-map(f;bs),
map: map(f;as),
nil: [],
it: ⋅,
prop: ℙ,
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,
list_induction,
equal_wf,
bag-combine_wf,
bag-map_wf,
list-subtype-bag,
nil_wf,
equal-wf-base,
map_cons_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
single-bag_wf,
squash_wf,
true_wf,
bag-combine-append-left,
iff_weakening_equal,
bag-append_wf,
bag-combine-unit-left
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,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
voidEquality,
voidElimination,
hyp_replacement,
applyLambdaEquality,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:B {}\mrightarrow{} bag(C)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[bs:bag(A)]. (\mcup{}x\mmember{}bag-map(f;bs).g[x] = \mcup{}x\mmember{}bs.g[f x])
Date html generated:
2017_10_01-AM-08_47_37
Last ObjectModification:
2017_07_26-PM-04_32_03
Theory : bags
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