Nuprl Lemma : mapfilter-wf
∀[T,U:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. ∀[f:{x:T| (x ∈ L) ∧ (↑P[x])} ⟶ U]. (mapfilter(f;P;L) ∈ U 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],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mapfilter: mapfilter(f;P;L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
map_wf,
and_wf,
l_member_wf,
assert_wf,
filter_type,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
list-subtype,
subtype_rel_list,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
applyEquality,
cumulativity,
because_Cache,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
productEquality,
dependent_set_memberEquality,
independent_pairFormation,
functionEquality,
universeEquality,
isect_memberFormation,
introduction,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[T,U:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:T| (x \mmember{} L) \mwedge{} (\muparrow{}P[x])\} {}\mrightarrow{} U].
(mapfilter(f;P;L) \mmember{} U List)
Date html generated:
2016_05_14-PM-01_29_12
Last ObjectModification:
2015_12_26-PM-05_22_16
Theory : list_1
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