Nuprl Lemma : bag-member-iff
∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T]. uiff(x ↓∈ bs;↓∃as:bag(T). (bs = ({x} + as) ∈ bag(T)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
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,
squash: ↓T,
prop: ℙ,
bag-member: x ↓∈ bs,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
sq_stable: SqStable(P),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
or: P ∨ Q
Lemmas referenced :
bag-member_wf,
squash_wf,
exists_wf,
bag_wf,
equal_wf,
bag-append_wf,
single-bag_wf,
bag-member-iff-hd,
list-subtype-bag,
list_ind_cons_lemma,
list_ind_nil_lemma,
sq_stable__bag-member,
bag-member-append,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
dependent_pairFormation,
applyEquality,
voidElimination,
voidEquality,
independent_functionElimination,
hyp_replacement,
Error :applyLambdaEquality,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[x:T]. uiff(x \mdownarrow{}\mmember{} bs;\mdownarrow{}\mexists{}as:bag(T). (bs = (\{x\} + as)))
Date html generated:
2016_10_25-AM-10_27_34
Last ObjectModification:
2016_07_12-AM-06_43_34
Theory : bags
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