Nuprl Lemma : member-filter3
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ∀x:{x:T| ↑(P x)} . ((x ∈ L) ⇒ (x ∈ filter(P;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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
prop: ℙ
Lemmas referenced :
l_member_set2,
assert_wf,
filter_type,
member_filter,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
l_member_wf,
set_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
dependent_functionElimination,
cumulativity,
setElimination,
rename,
independent_functionElimination,
because_Cache,
productElimination,
independent_pairFormation,
addLevel,
independent_isectElimination,
levelHypothesis,
instantiate,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. \mforall{}x:\{x:T| \muparrow{}(P x)\} . ((x \mmember{} L) {}\mRightarrow{} (x \mmember{} filter(P;L)))
Date html generated:
2016_05_14-AM-07_50_53
Last ObjectModification:
2015_12_26-PM-04_46_14
Theory : list_1
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