Nuprl Lemma : bag-member-remove
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x,z:T]. uiff(z ↓∈ bs - x;z ↓∈ bs ∧ (¬(z = x ∈ T)))
Proof
Definitions occuring in Statement :
bag-remove: bs - x,
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bag-remove: bs - x,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
bag-member: x ↓∈ bs,
squash: ↓T,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqof: eqof(d),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B
Lemmas referenced :
equal_wf,
bag-member_wf,
assert_wf,
bnot_wf,
iff_transitivity,
eqof_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
assert_witness,
bag-filter_wf,
subtype_rel_bag,
bag-member-filter,
uiff_wf,
bag-remove_wf,
bag_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
independent_pairFormation,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lambdaFormation,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaEquality,
dependent_functionElimination,
productEquality,
applyEquality,
setElimination,
rename,
equalitySymmetry,
addLevel,
because_Cache,
impliesFunctionality,
independent_isectElimination,
setEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x,z:T]. uiff(z \mdownarrow{}\mmember{} bs - x;z \mdownarrow{}\mmember{} bs \mwedge{} (\mneg{}(z = x)))
Date html generated:
2018_05_21-PM-09_47_41
Last ObjectModification:
2017_07_26-PM-06_30_23
Theory : bags_2
Home
Index