Nuprl Lemma : member_filter
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ∀x:T. ((x ∈ filter(P;L)) ⇐⇒ (x ∈ L) ∧ (↑(P x)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀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],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
true: True,
cand: A c∧ B
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
l_member_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
assert_wf,
list_wf,
filter_nil_lemma,
filter_cons_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
assert_witness,
eqtt_to_assert,
cons_member,
cons_wf,
or_wf,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
ifthenelse_wf,
and_wf,
assert_elim,
not_assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
setEquality,
independent_isectElimination,
setElimination,
rename,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
Error :universeIsType,
universeEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
cumulativity,
dependent_pairFormation,
promote_hyp,
instantiate,
Error :inhabitedIsType,
inlFormation,
inrFormation,
dependent_set_memberEquality,
applyLambdaEquality,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} filter(P;L)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} (\muparrow{}(P x)))
Date html generated:
2019_06_20-PM-00_43_56
Last ObjectModification:
2018_09_26-PM-02_25_23
Theory : list_0
Home
Index