Nuprl Lemma : filter_wf2
∀[T:Type]. ∀[l:T List]. ∀[P:{x:T| (x ∈ l)} ⟶ 𝔹]. (filter(P;l) ∈ {x:T| (x ∈ l)} List)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
l_member_wf,
bool_wf,
list_wf,
filter_wf5,
list-subtype,
subtype_rel_dep_function,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
hypothesisEquality,
lemma_by_obid,
isectElimination,
thin,
isect_memberEquality,
because_Cache,
universeEquality,
applyEquality,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. \mforall{}[P:\{x:T| (x \mmember{} l)\} {}\mrightarrow{} \mBbbB{}]. (filter(P;l) \mmember{} \{x:T| (x \mmember{} l)\} List)
Date html generated:
2016_05_14-AM-06_39_48
Last ObjectModification:
2015_12_26-PM-00_31_52
Theory : list_0
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