Nuprl Lemma : member-filter
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ∀x:{x:T| ↑P[x]} . ((x ∈ filter(λx.P[x];L)) ⇐⇒ (x ∈ L))
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],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
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],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
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 ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
true: True
Lemmas referenced :
list_induction,
assert_wf,
iff_wf,
l_member_wf,
filter_type,
list_wf,
filter_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
istype-assert,
btrue_neq_bfalse,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
istype-universe,
equal_functionality_wrt_subtype_rel2,
subtype_rel_set_simple,
cons_member,
cons_wf,
assert_elim,
equal_wf,
and_wf,
not_assert_elim,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
setEquality,
applyEquality,
hypothesis,
universeIsType,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
dependent_set_memberEquality_alt,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
because_Cache,
setIsType,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
functionIsType,
productIsType,
universeEquality,
inlFormation_alt,
inrFormation_alt,
unionIsType,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inlFormation,
natural_numberEquality,
dependent_set_memberEquality,
inrFormation,
lambdaFormation,
lambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. \mforall{}x:\{x:T| \muparrow{}P[x]\} . ((x \mmember{} filter(\mlambda{}x.P[x];L)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2020_05_19-PM-09_37_46
Last ObjectModification:
2020_01_04-PM-07_57_53
Theory : list_0
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