Nuprl Lemma : mapfilter-nil
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. ∀[f:Top]. mapfilter(f;P;L) ~ [] supposing (∀x∈L.¬↑(P x))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
mapfilter: mapfilter(f;P;L),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top
Lemmas referenced :
l_all_wf,
not_wf,
assert_wf,
l_member_wf,
top_wf,
bool_wf,
list_wf,
filter_is_nil3,
map_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
setEquality,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
introduction,
sqequalAxiom,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:Top].
mapfilter(f;P;L) \msim{} [] supposing (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x))
Date html generated:
2016_05_14-PM-01_28_24
Last ObjectModification:
2015_12_26-PM-05_21_15
Theory : list_1
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