Nuprl Lemma : non_null_filter
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ¬↑null(filter(P;L)) supposing (∃x∈L. ↑P[x])
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
null: null(as),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
l_all: (∀x∈L.P[x])
Lemmas referenced :
null-filter,
assert_wf,
null_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
l_exists_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
independent_isectElimination,
hypothesis,
because_Cache,
independent_functionElimination,
voidElimination,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
setElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mneg{}\muparrow{}null(filter(P;L)) supposing (\mexists{}x\mmember{}L. \muparrow{}P[x])
Date html generated:
2016_05_14-AM-06_51_51
Last ObjectModification:
2015_12_26-PM-00_21_32
Theory : list_0
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