Nuprl Lemma : null-filter2
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. uiff(↑null(filter(P;L));(∀x∈L.¬↑(P x)))
Proof
Definitions occuring in Statement :
l_all: (∀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],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
not: ¬A,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T
Lemmas referenced :
list_induction,
assert_wf,
null_wf,
filter_wf5,
l_all_wf,
not_wf,
l_member_wf,
list_wf,
filter_nil_lemma,
null_nil_lemma,
l_all_nil,
true_wf,
filter_cons_lemma,
l_all_cons,
assert_elim,
bool_wf,
eqtt_to_assert,
cons_wf,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
null_cons_lemma,
bfalse_wf,
and_wf,
ifthenelse_wf,
btrue_neq_bfalse,
assert-bnot,
null_filter,
assert_witness,
not_assert_elim,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
because_Cache,
hypothesis,
functionExtensionality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
productElimination,
addLevel,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
levelHypothesis,
dependent_set_memberEquality,
applyLambdaEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
independent_pairEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. uiff(\muparrow{}null(filter(P;L));(\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)))
Date html generated:
2017_04_14-AM-08_52_20
Last ObjectModification:
2017_02_27-PM-03_36_49
Theory : list_0
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