Nuprl Lemma : bag-remove-repeats-filter
∀[T:Type]. ∀[b:bag(T)]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹].
(bag-remove-repeats(eq;[x∈b|P[x]]) = [x∈bag-remove-repeats(eq;b)|P[x]] ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(T),
bool: 𝔹,
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,
squash: ↓T,
exists: ∃x:A. B[x],
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-filter: [x∈b|p[x]],
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x]
Lemmas referenced :
bag_to_squash_list,
list-to-set-filter,
filter_wf5,
list-to-set_wf,
l_member_wf,
list-subtype-bag,
equal_wf,
bag_wf,
bag-remove-repeats_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
bool_wf,
deq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
sqequalRule,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
lambdaFormation,
setElimination,
setEquality,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
functionEquality,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
(bag-remove-repeats(eq;[x\mmember{}b|P[x]]) = [x\mmember{}bag-remove-repeats(eq;b)|P[x]])
Date html generated:
2016_10_25-AM-11_26_55
Last ObjectModification:
2016_07_12-AM-07_33_26
Theory : bags_2
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