Nuprl Lemma : length-filter-pos-iff
∀[A:Type]. ∀P:A ⟶ 𝔹. ∀L:A List. ((∃x∈L. ↑(P x)) ⇐⇒ 0 < ||filter(P;L)||)
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
length: ||as||,
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
l_exists: (∃x∈L. P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A
Lemmas referenced :
l_exists_wf,
assert_wf,
l_member_wf,
less_than_wf,
length_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
list_wf,
length-filter-pos,
equal_wf,
member_exists,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
cons_wf,
cons_neq_nil,
member_filter,
lelt_wf,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. ((\mexists{}x\mmember{}L. \muparrow{}(P x)) \mLeftarrow{}{}\mRightarrow{} 0 < ||filter(P;L)||)
Date html generated:
2017_04_17-AM-07_48_14
Last ObjectModification:
2017_02_27-PM-04_22_02
Theory : list_1
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