Nuprl Lemma : filter-less
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  ||filter(P;L)|| < ||L|| supposing ∃x:T. ((x ∈ L) ∧ (¬↑(P x)))
Proof
Definitions occuring in Statement : 
l_member: (x ∈ l)
, 
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]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
and: 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]
, 
uimplies: b supposing a
, 
prop: ℙ
, 
and: P ∧ Q
, 
so_apply: x[s]
, 
exists: ∃x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
top: Top
, 
iff: P 
⇐⇒ Q
, 
false: False
, 
or: P ∨ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
not: ¬A
, 
bfalse: ff
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
decidable: Dec(P)
, 
le: A ≤ B
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
less_than: a < b
, 
squash: ↓T
Lemmas referenced : 
list_induction, 
isect_wf, 
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, 
nil_wf, 
filter_cons_lemma, 
length_of_cons_lemma, 
cons_wf, 
member-less_than, 
nil_member, 
cons_member, 
eqtt_to_assert, 
assert_elim, 
and_wf, 
equal_wf, 
not_assert_elim, 
btrue_neq_bfalse, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
filter-le, 
decidable__lt, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformless_wf, 
itermVar_wf, 
itermAdd_wf, 
itermConstant_wf, 
intformle_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
cumulativity, 
productEquality, 
because_Cache, 
hypothesis, 
applyEquality, 
functionExtensionality, 
setEquality, 
independent_isectElimination, 
setElimination, 
rename, 
lambdaFormation, 
independent_functionElimination, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
productElimination, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
universeEquality, 
unionElimination, 
equalityElimination, 
dependent_set_memberEquality, 
independent_pairFormation, 
applyLambdaEquality, 
dependent_pairFormation, 
promote_hyp, 
instantiate, 
addEquality, 
natural_numberEquality, 
int_eqEquality, 
intEquality, 
computeAll, 
imageElimination
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].    ||filter(P;L)||  <  ||L||  supposing  \mexists{}x:T.  ((x  \mmember{}  L)  \mwedge{}  (\mneg{}\muparrow{}(P  x)))
Date html generated:
2017_04_17-AM-08_36_41
Last ObjectModification:
2017_02_27-PM-04_55_31
Theory : list_1
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