Nuprl Lemma : bag-map-union2
∀[A,B:Type]. ∀[g:A ⟶ B]. ∀[x:bag(bag(A))]. (bag-map(g;bag-union(x)) = bag-union(bag-map(λz.bag-map(g;z);x)) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag-map: bag-map(f;bs),
bag: bag(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
squash: ↓T,
true: True,
bag-combine: ⋃x∈bs.f[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
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-map-union,
single-bag_wf,
bag-union_wf,
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-combine-single-right-as-map,
bag-map_wf,
eta_conv,
subtype_rel_self,
iff_weakening_equal,
bag-map-combine,
bag-map-trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
applyEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
voidElimination,
voidEquality,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
lambdaFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[g:A {}\mrightarrow{} B]. \mforall{}[x:bag(bag(A))].
(bag-map(g;bag-union(x)) = bag-union(bag-map(\mlambda{}z.bag-map(g;z);x)))
Date html generated:
2018_05_21-PM-06_24_09
Last ObjectModification:
2018_05_19-PM-05_15_08
Theory : bags
Home
Index