Nuprl Lemma : bag-member-iff-hd
∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T].  uiff(x ↓∈ bs;↓∃L:T List. (bs = [x / L] ∈ bag(T)))
Proof
Definitions occuring in Statement : 
bag-member: x ↓∈ bs
, 
bag: bag(T)
, 
cons: [a / b]
, 
list: T List
, 
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]
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s]
, 
exists: ∃x:A. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
true: True
, 
bag: bag(T)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
append: as @ bs
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
top: Top
, 
so_apply: x[s1;s2;s3]
, 
rev_implies: P 
⇐ Q
, 
cand: A c∧ B
, 
or: P ∨ Q
Lemmas referenced : 
bag-member_wf, 
squash_wf, 
exists_wf, 
list_wf, 
equal_wf, 
bag_wf, 
cons_wf, 
list-subtype-bag, 
l_member_decomp, 
append_wf, 
true_wf, 
quotient-member-eq, 
permutation_wf, 
permutation-equiv, 
nil_wf, 
list_ind_cons_lemma, 
list_ind_nil_lemma, 
permutation_functionality_wrt_permutation, 
cons_functionality_wrt_permutation, 
permutation-rotate, 
permutation_weakening, 
cons_member, 
l_member_wf
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, 
applyEquality, 
because_Cache, 
independent_isectElimination, 
productElimination, 
independent_pairEquality, 
isect_memberEquality, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
dependent_functionElimination, 
independent_functionElimination, 
dependent_pairFormation, 
hyp_replacement, 
natural_numberEquality, 
applyLambdaEquality, 
voidElimination, 
voidEquality, 
inlFormation, 
productEquality
Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].  \mforall{}[x:T].    uiff(x  \mdownarrow{}\mmember{}  bs;\mdownarrow{}\mexists{}L:T  List.  (bs  =  [x  /  L]))
Date html generated:
2017_10_01-AM-08_53_49
Last ObjectModification:
2017_07_26-PM-04_35_30
Theory : bags
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