Nuprl Lemma : bag-filter-same
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. [x∈bs|p[x]] = bs ∈ bag(T) supposing ∀x:T. (x ↓∈ bs ⇒ (↑p[x]))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: 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,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
bag-filter-trivial,
bag-member_wf,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
sq_stable__bag-member,
bag-settype,
all_wf,
assert_wf,
bag_wf,
equal_wf,
subtype_rel_bag
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. [x\mmember{}bs|p[x]] = bs supposing \mforall{}x:T. (x \mdownarrow{}\mmember{} bs {}\mRightarrow{} (\muparrow{}p[x]))
Date html generated:
2016_10_25-AM-10_32_45
Last ObjectModification:
2016_07_12-AM-06_49_15
Theory : bags
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