Nuprl Lemma : bag-filter-trivial2
∀[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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
bag-filter: [x∈b|p[x]],
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
cons-bag: x.b,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
bag_to_squash_list,
all_wf,
bag-member_wf,
assert_wf,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
filter_wf5,
l_member_wf,
list_induction,
list-subtype-bag,
filter_nil_lemma,
permutation-nil,
nil_wf,
filter_cons_lemma,
bag-member-cons,
cons_wf,
equal_wf,
bag_wf,
bag-filter_wf,
subtype_rel_bag,
bool_wf,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
btrue_wf,
true_wf,
permutation-cons2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
cumulativity,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
rename,
independent_isectElimination,
dependent_functionElimination,
setElimination,
setEquality,
independent_functionElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
inlFormation,
imageMemberEquality,
baseClosed,
universeEquality,
isect_memberFormation,
axiomEquality,
equalityTransitivity,
instantiate,
independent_pairFormation,
natural_numberEquality,
inrFormation
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_30_14
Last ObjectModification:
2016_07_12-AM-06_46_17
Theory : bags
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