Nuprl Lemma : filter-upto-length
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. (filter(P;L) ~ map(λi.L[i];filter(λi.(P L[i]);upto(||L||))))
Proof
Definitions occuring in Statement :
upto: upto(n),
select: L[n],
length: ||as||,
filter: filter(P;l),
map: map(f;as),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
btrue: tt,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
subtract: n - m,
compose: f o g,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
filter_nil_lemma,
stuck-spread,
base_wf,
length_of_nil_lemma,
map_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
filter_cons_lemma,
length_of_cons_lemma,
list_wf,
bool_wf,
upto_decomp,
length_wf,
add_nat_wf,
length_wf_nat,
false_wf,
add-is-int-iff,
non_neg_length,
decidable__lt,
lelt_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
eqtt_to_assert,
map_cons_lemma,
add-subtract-cancel,
upto_wf,
int_seg_wf,
filter_map,
add-member-int_seg2,
select_wf,
cons_wf,
int_seg_properties,
map-map,
select_cons_tl_sq,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
uiff_transitivity,
assert_wf,
bnot_wf,
not_wf,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
imageElimination,
functionEquality,
universeEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionExtensionality,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. (filter(P;L) \msim{} map(\mlambda{}i.L[i];filter(\mlambda{}i.(P L[i]);upto(||L||))))
Date html generated:
2017_04_17-AM-08_36_57
Last ObjectModification:
2017_02_27-PM-04_55_49
Theory : list_1
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