Nuprl Lemma : mapfilter-not-nil
∀[T,B:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹]. ∀[f:{x:{x:T| (x ∈ L)} | ↑(P x)}  ⟶ B].
  ¬(mapfilter(f;P;L) = [] ∈ (B List)) supposing (∃x∈L. ↑(P x))
Proof
Definitions occuring in Statement : 
mapfilter: mapfilter(f;P;L)
, 
l_exists: (∃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]
, 
not: ¬A
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
l_exists_iff, 
assert_wf, 
l_member_wf, 
member_not_nil, 
mapfilter_wf, 
list-subtype, 
member-mapfilter, 
equal_wf, 
equal-wf-T-base, 
list_wf, 
l_exists_wf, 
bool_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
thin, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
dependent_functionElimination, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
setEquality, 
cumulativity, 
hypothesis, 
setElimination, 
rename, 
dependent_set_memberEquality, 
productElimination, 
independent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
dependent_pairFormation, 
productEquality, 
because_Cache, 
independent_pairFormation, 
voidElimination, 
baseClosed, 
isect_memberEquality, 
functionEquality, 
universeEquality
Latex:
\mforall{}[T,B:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].  \mforall{}[f:\{x:\{x:T|  (x  \mmember{}  L)\}  |  \muparrow{}(P  x)\}    {}\mrightarrow{}  B].
    \mneg{}(mapfilter(f;P;L)  =  [])  supposing  (\mexists{}x\mmember{}L.  \muparrow{}(P  x))
Date html generated:
2017_04_17-AM-07_29_59
Last ObjectModification:
2017_02_27-PM-04_07_32
Theory : list_1
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