Nuprl Lemma : bag-member-remove-repeats
∀[T:Type]. ∀[b:bag(T)]. ∀[eq:EqDecider(T)]. ∀[x:T]. uiff(x ↓∈ b;x ↓∈ bag-remove-repeats(eq;b))
Proof
Definitions occuring in Statement :
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
bag-remove-repeats: bag-remove-repeats(eq;bs),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs
Lemmas referenced :
bag_to_squash_list,
sq_stable__bag-member,
bag-remove-repeats_wf,
bag-member_wf,
bag-member-list,
decidable-equal-deq,
list-to-set_wf,
member-list-to-set,
deq_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
cumulativity,
hypothesis,
independent_functionElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
rename,
lambdaFormation,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. uiff(x \mdownarrow{}\mmember{} b;x \mdownarrow{}\mmember{} bag-remove-repeats(eq;b))
Date html generated:
2016_10_25-AM-11_26_54
Last ObjectModification:
2016_07_12-AM-07_33_27
Theory : bags_2
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