Nuprl Lemma : bag-filter-empty-iff
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[b:bag(T)]. uiff(∀x:T. (x ↓∈ b ⇒ (¬↑P[x]));↑bag-null([x∈b|P[x]]))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-null: bag-null(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],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
empty-bag: {},
bag-null: bag-null(bs),
null: null(as),
bag-filter: [x∈b|p[x]],
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
btrue: tt,
assert: ↑b,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
true: True,
false: False,
not: ¬A,
cons-bag: x.b,
top: Top,
bool: 𝔹,
unit: Unit,
cons: [a / b],
bfalse: ff,
sq_or: a ↓∨ b,
or: P ∨ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P)
Lemmas referenced :
all_wf,
bag-member_wf,
not_wf,
assert_wf,
bag-null_wf,
bag-filter_wf,
squash_wf,
false_wf,
true_wf,
bag-member-empty-iff,
empty-bag_wf,
uiff_wf,
bag_filter_cons_lemma,
bool_wf,
eqtt_to_assert,
list-subtype-bag,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sq_or_wf,
iff_weakening_uiff,
rev_implies_wf,
assert_witness,
bag-member-cons,
cons-bag_wf,
bag_wf,
bag_to_squash_list,
sq_stable__uiff,
sq_stable__all,
sq_stable__not,
sq_stable_from_decidable,
decidable__assert,
list_induction,
list_wf,
assert_elim,
and_wf,
not_assert_elim,
btrue_neq_bfalse
Rules used in proof :
because_Cache,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
functionExtensionality,
dependent_functionElimination,
setEquality,
independent_pairFormation,
isect_memberFormation,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
voidElimination,
addLevel,
productElimination,
independent_isectElimination,
allFunctionality,
independent_functionElimination,
isect_memberEquality,
voidEquality,
unionElimination,
equalityElimination,
inlFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
promote_hyp,
instantiate,
inrFormation,
dependent_set_memberEquality,
universeEquality,
independent_pairEquality,
imageElimination,
rename,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:bag(T)]. uiff(\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\mneg{}\muparrow{}P[x]));\muparrow{}bag-null([x\mmember{}b|P[x]]))
Date html generated:
2017_10_01-AM-08_56_15
Last ObjectModification:
2017_07_26-PM-04_38_16
Theory : bags
Home
Index