Nuprl Lemma : length_filter
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List]. (||filter(P;L)|| = count(P;L) ∈ ℕ)
Proof
Definitions occuring in Statement :
count: count(P;L),
length: ||as||,
filter: filter(P;l),
list: T List,
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
count: count(P;L),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
ge: i ≥ j ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
equal_wf,
nat_wf,
length_wf_nat,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
set_wf,
count_wf,
list_wf,
filter_nil_lemma,
reduce_nil_lemma,
length_of_nil_lemma,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
filter_cons_lemma,
reduce_cons_lemma,
eqtt_to_assert,
length_of_cons_lemma,
nat_properties,
length_wf,
subtype_rel_self,
intformand_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
add_nat_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
zero-add,
reduce_wf,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
cumulativity,
because_Cache,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
intEquality,
computeAll,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
equalityElimination,
productElimination,
applyLambdaEquality,
addEquality,
int_eqEquality,
promote_hyp,
instantiate,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List]. (||filter(P;L)|| = count(P;L))
Date html generated:
2017_04_14-AM-09_31_21
Last ObjectModification:
2017_02_27-PM-04_03_17
Theory : list_1
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