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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