Nuprl Lemma : bag-filter-split
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. (([x∈bs|p[x]] + [x∈bs|¬bp[x]]) = bs ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-append: as + bs,
bag: bag(T),
bnot: ¬bb,
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-filter: [x∈b|p[x]],
bag-append: as + bs,
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B
Lemmas referenced :
bag_to_squash_list,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
permutation-split,
equal_wf,
bag_wf,
bag-append_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
sqequalRule,
cumulativity,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
functionExtensionality,
setElimination,
setEquality,
because_Cache,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. (([x\mmember{}bs|p[x]] + [x\mmember{}bs|\mneg{}\msubb{}p[x]]) = bs)
Date html generated:
2016_10_25-AM-10_22_08
Last ObjectModification:
2016_07_12-AM-06_38_50
Theory : bags
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