Nuprl Lemma : bag-member-filter
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[x:T]. ∀[bs:bag(T)]. uiff(x ↓∈ [x∈bs|P[x]];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,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bag-filter: [x∈b|p[x]],
bag-member: x ↓∈ bs,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
rev_implies: P ⇐ Q,
label: ...$L... t
Lemmas referenced :
bag_to_squash_list,
sq_stable__bag-member,
bag-member_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
assert_witness,
bag_wf,
bool_wf,
eqtt_to_assert,
sq_stable_from_decidable,
decidable__assert,
member-permutation,
member_filter_2,
l_member_wf,
equal_wf,
list-subtype-bag,
member_wf,
list_wf,
filter_wf5,
permutation_wf,
member_filter,
iff_imp_equal_bool,
true_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
hypothesis,
independent_functionElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
independent_isectElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
independent_pairEquality,
productEquality,
isect_memberEquality,
equalityTransitivity,
functionEquality,
dependent_functionElimination,
pertypeElimination,
dependent_pairFormation,
lambdaFormation,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. uiff(x \mdownarrow{}\mmember{} [x\mmember{}bs|P[x]];x \mdownarrow{}\mmember{} bs \mwedge{} (\muparrow{}P[x]))
Date html generated:
2017_10_01-AM-08_54_12
Last ObjectModification:
2017_07_26-PM-04_35_57
Theory : bags
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