Nuprl Lemma : bag-member-not-bag-null
∀[T:Type]. ∀[bs:bag(T)]. uiff(↓∃x:T. x ↓∈ bs;¬↑bag-null(bs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-null: bag-null(bs),
bag: bag(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
squash: ↓T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-null: bag-null(bs),
bag-member: x ↓∈ bs,
cand: A c∧ B
Lemmas referenced :
assert-bag-null,
bag-member_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag-member-empty-iff,
assert_wf,
bag-null_wf,
exists_wf,
bag_to_squash_list,
not_wf,
assert_of_null,
equal-wf-T-base,
list_wf,
member_exists,
list-subtype-bag,
equal_wf,
bag_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
imageElimination,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
applyEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
cumulativity,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
rename,
dependent_pairFormation,
productEquality,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(\mdownarrow{}\mexists{}x:T. x \mdownarrow{}\mmember{} bs;\mneg{}\muparrow{}bag-null(bs))
Date html generated:
2017_10_01-AM-08_54_32
Last ObjectModification:
2017_07_26-PM-04_36_21
Theory : bags
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