Nuprl Lemma : bag-filter-filter2
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[q:{x:T| ↑p[x]} ⟶ 𝔹]. ∀[bs:bag(T)].
([x∈[x∈bs|p[x]]|q[x]] = [x∈bs|p[x] ∧b q[x]] ∈ bag({x:T| ↑(p[x] ∧b q[x])} ))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag: bag(T),
band: p ∧b q,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
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],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
true: True,
bag-filter: [x∈b|p[x]],
top: Top
Lemmas referenced :
bag_to_squash_list,
equal_wf,
bag_wf,
assert_wf,
bool_wf,
eqtt_to_assert,
bag-filter_wf,
subtype_rel_bag,
subtype_rel_sets,
assert_of_band,
assert_elim,
subtype_base_sq,
bool_subtype_base,
set_wf,
filter-filter,
filter_wf4_2,
list-subtype-bag,
l_member_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
independent_isectElimination,
dependent_set_memberEquality,
equalityTransitivity,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
setElimination,
addLevel,
levelHypothesis,
instantiate,
natural_numberEquality,
independent_pairFormation,
functionEquality,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[q:\{x:T| \muparrow{}p[x]\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)].
([x\mmember{}[x\mmember{}bs|p[x]]|q[x]] = [x\mmember{}bs|p[x] \mwedge{}\msubb{} q[x]])
Date html generated:
2017_10_01-AM-08_45_26
Last ObjectModification:
2017_07_26-PM-04_30_42
Theory : bags
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