Nuprl Lemma : bag-partitions-with-one-given
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[as,bs,cs:bag(T)].
([p∈bag-partitions(eq;cs)|bag-eq(eq;fst(p);as)] = {<as, bs>} ∈ bag(bag(T) × bag(T)))
∧ ([p∈bag-partitions(eq;cs)|bag-eq(eq;snd(p);bs)] = {<as, bs>} ∈ bag(bag(T) × bag(T)))
supposing (as + bs) = cs ∈ bag(T)
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
bag-eq: bag-eq(eq;as;bs),
bag-filter: [x∈b|p[x]],
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
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,
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
pi1: fst(t),
rev_uimplies: rev_uimplies(P;Q),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi2: snd(t)
Lemmas referenced :
bag-extensionality-no-repeats,
bag_wf,
decidable__equal_product,
decidable__equal_bag,
decidable-equal-deq,
bag-filter_wf,
bag-eq_wf,
pi1_wf_top,
bag-partitions_wf,
subtype_rel_bag,
assert_wf,
single-bag_wf,
bag-filter-no-repeats,
no-repeats-bag-partitions,
bag-single-no-repeats,
bag-member-single,
bag-member_wf,
bag-member-filter,
squash_wf,
true_wf,
iff_weakening_equal,
bag-member-partitions,
assert-bag-eq,
and_wf,
equal_wf,
subtype_rel_product,
top_wf,
pi2_wf,
bag-append_wf,
deq_wf,
valueall-type_wf,
bag-append-cancel,
bag-append-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_functionElimination,
lambdaFormation,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
setElimination,
rename,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
universeEquality,
dependent_set_memberEquality,
applyLambdaEquality,
axiomEquality
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs,cs:bag(T)].
([p\mmember{}bag-partitions(eq;cs)|bag-eq(eq;fst(p);as)] = \{<as, bs>\})
\mwedge{} ([p\mmember{}bag-partitions(eq;cs)|bag-eq(eq;snd(p);bs)] = \{<as, bs>\})
supposing (as + bs) = cs
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-09_49_37
Last ObjectModification:
2017_07_26-PM-06_31_14
Theory : bags_2
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