Nuprl Lemma : bag-filter-is-empty
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[b:bag(T)]. uiff(∀x:T. (x ↓∈ b ⇒ (¬↑P[x]));[x∈b|P[x]] = {} ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
empty-bag: {},
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
bag-null: bag-null(bs),
null: null(as),
empty-bag: {},
nil: [],
it: ⋅,
btrue: tt,
true: True
Lemmas referenced :
bag-filter-empty-iff,
all_wf,
bag-member_wf,
not_wf,
assert_wf,
equal-wf-T-base,
bag_wf,
bag-filter_wf,
subtype_rel_bag,
bool_wf,
assert-bag-null,
equal_functionality_wrt_subtype_rel2,
bag-null_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
independent_isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
dependent_functionElimination,
voidElimination,
setEquality,
setElimination,
rename,
because_Cache,
baseClosed,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
hyp_replacement,
Error :applyLambdaEquality
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]));[x\mmember{}b|P[x]] = \{\})
Date html generated:
2016_10_25-AM-10_30_09
Last ObjectModification:
2016_07_12-AM-06_46_11
Theory : bags
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