Nuprl Lemma : bag-combine-is-map
∀[A,B:Type]. ∀[b:bag(A)]. ∀[f:A ⟶ B]. (⋃x∈b.{f[x]} = bag-map(f;b) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
single-bag: {x},
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,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
bag-combine-single-right-as-map,
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-map_wf,
eta_conv,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
functionEquality,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
independent_isectElimination,
productElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[b:bag(A)]. \mforall{}[f:A {}\mrightarrow{} B]. (\mcup{}x\mmember{}b.\{f[x]\} = bag-map(f;b))
Date html generated:
2017_10_01-AM-08_47_47
Last ObjectModification:
2017_07_26-PM-04_32_10
Theory : bags
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