Nuprl Lemma : count-pos-iff
∀[A:Type]. ∀P:A ⟶ 𝔹. ∀L:A List. ((∃x∈L. ↑(P x)) ⇐⇒ 0 < count(P;L))
Proof
Definitions occuring in Statement :
count: count(P;L),
l_exists: (∃x∈L. P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
length-filter-pos-iff,
count-length-filter,
list_wf,
bool_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
sqequalRule,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. ((\mexists{}x\mmember{}L. \muparrow{}(P x)) \mLeftarrow{}{}\mRightarrow{} 0 < count(P;L))
Date html generated:
2016_10_21-AM-10_13_08
Last ObjectModification:
2016_08_05-PM-06_13_46
Theory : list_1
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