Nuprl Lemma : length-filter-decreases
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ((∃x∈L. ¬↑(P x)) ⇒ ||filter(P;L)|| < ||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],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
less_than: a < b,
squash: ↓T,
select: L[n],
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B
Lemmas referenced :
list_induction,
l_exists_wf,
l_member_wf,
not_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,
l_exists_wf_nil,
filter_cons_lemma,
length_of_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
member-less_than,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
decidable__equal_int,
int_subtype_base,
select-cons-tl,
intformeq_wf,
int_formula_prop_eq_lemma,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
lelt_wf,
select_wf,
length-filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
lambdaFormation,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
universeEquality,
natural_numberEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
addEquality,
imageElimination,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. ((\mexists{}x\mmember{}L. \mneg{}\muparrow{}(P x)) {}\mRightarrow{} ||filter(P;L)|| < ||L||)
Date html generated:
2017_04_17-AM-07_51_29
Last ObjectModification:
2017_02_27-PM-04_25_51
Theory : list_1
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