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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