Nuprl Lemma : l_exists_filter
∀[T:Type]. ∀[Q:T ⟶ ℙ]. ∀P:T ⟶ 𝔹. ∀L:T List. ((∃x∈filter(P;L). Q[x]) ⇐⇒ ∃x:T. ((x ∈ L) ∧ (↑(P x)) ∧ Q[x]))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
istype: istype(T)
Lemmas referenced :
l_member_wf,
istype-assert,
subtype_rel_self,
l_exists_iff,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
member_filter,
l_exists_wf,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
instantiate,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
lambdaEquality_alt,
setEquality,
setIsType,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
promote_hyp,
functionIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}].
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. ((\mexists{}x\mmember{}filter(P;L). Q[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. ((x \mmember{} L) \mwedge{} (\muparrow{}(P x)) \mwedge{} Q[x]))
Date html generated:
2020_05_19-PM-09_45_26
Last ObjectModification:
2019_10_23-PM-03_58_14
Theory : list_1
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