Nuprl Lemma : bag-filter-empty
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[b:bag(T)]. ∀x:T. (x ↓∈ b ⇒ (¬↑P[x])) supposing ↑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,
uimplies: b supposing a,
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 :
not: ¬A,
implies: P ⇒ Q,
bag-member: x ↓∈ bs,
squash: ↓T,
false: False,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
bag-filter: [x∈b|p[x]],
bag-null: bag-null(bs),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
true_wf,
squash_wf,
assert_functionality_wrt_uiff,
bool_wf,
bag_wf,
bag-member_wf,
nil_member,
sqequal-nil,
member-filter,
l_member_wf,
filter_wf5,
assert_of_null,
list-subtype-bag,
bag-filter_wf,
assert_wf,
bag-null_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
imageElimination,
cut,
productElimination,
thin,
hypothesis,
because_Cache,
lemma_by_obid,
dependent_functionElimination,
setEquality,
hypothesisEquality,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
functionEquality,
universeEquality,
isect_memberFormation,
introduction,
isect_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:bag(T)]. \mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\mneg{}\muparrow{}P[x])) supposing \muparrow{}bag-null([x\mmember{}b|P[x]])
Date html generated:
2016_05_15-PM-02_44_28
Last ObjectModification:
2016_01_16-AM-08_45_08
Theory : bags
Home
Index