Nuprl Lemma : bag-partitions-assoc
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[bs:bag(T)].
(⋃x∈bag-partitions(eq;bs).bag-map(λy.<fst(x), y>;bag-partitions(eq;snd(x)))
= bag-map(λ2p.<fst(snd(p)), snd(snd(p)), fst(p)>;⋃x∈bag-partitions(eq;bs).bag-map(λy.<snd(x), y>;bag-partitions(eq;f\000Cst(x))))
∈ bag(bag(T) × bag(T) × bag(T)))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
product-deq: product-deq(A;B;a;b),
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
true: True,
inject: Inj(A;B;f),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
product-deq_wf,
bag_wf,
bag-deq_wf,
deq_wf,
valueall-type_wf,
bag-extensionality-no-repeats,
decidable__equal_product,
decidable__equal_bag,
decidable-equal-deq,
bag-combine_wf,
bag-partitions_wf,
bag-map_wf,
bag-member_wf,
bag-combine-no-repeats,
pi1_wf_top,
pi2_wf,
bag-member-map,
bag-member-partitions,
and_wf,
equal_wf,
subtype_rel_product,
top_wf,
bag-append_wf,
squash_wf,
true_wf,
bag-map-no-repeats,
no-repeats-bag-partitions,
bag-member-combine,
bag-append-assoc,
iff_weakening_equal,
bag-member-subtype,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
productEquality,
comment,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
lambdaFormation,
lambdaEquality,
dependent_functionElimination,
independent_isectElimination,
independent_pairEquality,
productElimination,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
voidElimination,
voidEquality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
setElimination,
rename,
applyEquality,
natural_numberEquality,
dependent_pairFormation,
addLevel,
existsFunctionality,
andLevelFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)].
(\mcup{}x\mmember{}bag-partitions(eq;bs).bag-map(\mlambda{}y.<fst(x), y>bag-partitions(eq;snd(x)))
= bag-map(\mlambda{}\msubtwo{}p.<fst(snd(p)), snd(snd(p)), fst(p)>
\mcup{}x\mmember{}bag-partitions(eq;bs).bag-map(\mlambda{}y.<snd(x), y>bag-partitions(eq;fst(x)))))
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_49_50
Last ObjectModification:
2017_07_26-PM-06_31_17
Theory : bags_2
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