Nuprl Lemma : non_null_filter_iff
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. uiff((∃x∈L. ↑P[x]);¬↑null(filter(P;L)))
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,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
top: Top,
bfalse: ff,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
non_null_filter,
assert_wf,
null_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
l_exists_iff,
not_wf,
list_wf,
filter_type,
member_filter,
hd_wf,
subtype_rel_list,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__le,
false_wf,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
equal_wf,
hd_member
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_pairFormation,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
voidElimination,
cumulativity,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
functionExtensionality,
productElimination,
independent_functionElimination,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
minusEquality,
dependent_pairFormation,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. uiff((\mexists{}x\mmember{}L. \muparrow{}P[x]);\mneg{}\muparrow{}null(filter(P;L)))
Date html generated:
2017_04_14-AM-08_52_57
Last ObjectModification:
2017_02_27-PM-03_38_15
Theory : list_0
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