Nuprl Lemma : length-filter-pos
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List]. 0 < ||filter(P;L)|| supposing (∃x∈L. ↑(P x))
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
top: Top,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
list_induction,
isect_wf,
l_exists_wf,
l_member_wf,
assert_wf,
less_than_wf,
length_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
false_wf,
l_exists_nil,
l_exists_wf_nil,
filter_cons_lemma,
eqtt_to_assert,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
or_wf,
true_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_exists_cons,
cons_wf,
ifthenelse_wf,
member-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
lambdaFormation,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
productElimination,
isectEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
instantiate,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List]. 0 < ||filter(P;L)|| supposing (\mexists{}x\mmember{}L. \muparrow{}(P x))
Date html generated:
2017_04_17-AM-07_48_04
Last ObjectModification:
2017_02_27-PM-04_22_20
Theory : list_1
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