Nuprl Lemma : mapfilter-wf2
∀[A,B:Type]. ∀[L:A List]. ∀[P:{x:A| (x ∈ L)} ⟶ 𝔹]. ∀[f:{x:A| (x ∈ L) ∧ (↑(P x))} ⟶ B]. (mapfilter(f;P;L) ∈ B List)
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
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,
and: P ∧ Q,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
mapfilter_wf,
l_member_wf,
list-subtype,
subtype_rel_dep_function,
assert_wf,
set_wf,
bool_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
because_Cache,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyEquality,
productEquality,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[L:A List]. \mforall{}[P:\{x:A| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:A| (x \mmember{} L) \mwedge{} (\muparrow{}(P x))\} {}\mrightarrow{} B].
(mapfilter(f;P;L) \mmember{} B List)
Date html generated:
2016_05_14-AM-07_50_05
Last ObjectModification:
2015_12_26-PM-04_45_52
Theory : list_1
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