Nuprl Lemma : bl-exists-filter
∀[P,Q,L:Top]. ((∃x∈filter(P;L).Q[x])_b ~ (∃x∈L.(P x) ∧b Q[x])_b)
Proof
Definitions occuring in Statement :
bl-exists: (∃x∈L.P[x])_b,
filter: filter(P;l),
band: p ∧b q,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
apply: f a,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
filter: filter(P;l),
reduce: reduce(f;k;as),
bl-exists: (∃x∈L.P[x])_b,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strict1: strict1(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
list_ind: list_ind,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
guard: {T},
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
top: Top,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bor: p ∨bq,
bfalse: ff,
sq_type: SQType(T),
uiff: uiff(P;Q),
btrue: tt
Lemmas referenced :
sqequal-list_ind,
has-value_wf_base,
base_wf,
is-exception_wf,
sqle_wf_base,
bl-exists-nil,
top_wf,
has-value-ifthenelse-type,
bl-exists-cons,
equal_wf,
injection-eta,
isl_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
has-value-bor-type,
value-type-has-value,
union-value-type,
has-value-band-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueCallbyvalue,
hypothesis,
callbyvalueReduce,
callbyvalueExceptionCases,
inlFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inrFormation,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalAxiom,
divergentSqle,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleRule,
sqleReflexivity,
independent_functionElimination,
axiomSqleEquality,
instantiate,
cumulativity,
productElimination,
decideExceptionCases,
applyExceptionCases
Latex:
\mforall{}[P,Q,L:Top]. ((\mexists{}x\mmember{}filter(P;L).Q[x])\_b \msim{} (\mexists{}x\mmember{}L.(P x) \mwedge{}\msubb{} Q[x])\_b)
Date html generated:
2017_04_17-AM-08_04_22
Last ObjectModification:
2017_02_27-PM-04_34_21
Theory : list_1
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