Nuprl Lemma : deq-member-length-filter2
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[L:A List]. ∀[x:A].  (x ∈b L ~ 0 <z ||filter(λy.(eq x y);L)||)
Proof
Definitions occuring in Statement : 
length: ||as||
, 
deq-member: x ∈b L
, 
filter: filter(P;l)
, 
list: T List
, 
deq: EqDecider(T)
, 
lt_int: i <z j
, 
uall: ∀[x:A]. B[x]
, 
apply: f a
, 
lambda: λx.A[x]
, 
natural_number: $n
, 
universe: Type
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
squash: ↓T
, 
deq: EqDecider(T)
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
rev_implies: P 
⇐ Q
, 
eqof: eqof(d)
, 
prop: ℙ
, 
true: True
, 
sq_type: SQType(T)
, 
all: ∀x:A. B[x]
, 
guard: {T}
Lemmas referenced : 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
deq-member-length-filter, 
lt_int_wf, 
length_wf, 
filter_wf5, 
iff_imp_equal_bool, 
iff_weakening_uiff, 
assert_wf, 
eqof_wf, 
equal_wf, 
safe-assert-deq, 
istype-assert, 
l_member_wf, 
list_wf, 
deq_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
thin, 
instantiate, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
cumulativity, 
hypothesis, 
independent_isectElimination, 
sqequalRule, 
hypothesisEquality, 
applyEquality, 
lambdaEquality_alt, 
imageElimination, 
because_Cache, 
natural_numberEquality, 
functionExtensionality_alt, 
setElimination, 
rename, 
independent_pairFormation, 
lambdaFormation_alt, 
equalitySymmetry, 
equalityIstype, 
inhabitedIsType, 
productElimination, 
independent_functionElimination, 
promote_hyp, 
setIsType, 
universeIsType, 
imageMemberEquality, 
baseClosed, 
dependent_functionElimination, 
equalityTransitivity, 
axiomSqEquality, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
universeEquality
Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[L:A  List].  \mforall{}[x:A].    (x  \mmember{}\msubb{}  L  \msim{}  0  <z  ||filter(\mlambda{}y.(eq  x  y);L)||)
Date html generated:
2020_05_19-PM-09_52_15
Last ObjectModification:
2020_01_04-PM-08_00_02
Theory : decidable!equality
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