Nuprl Lemma : filter-index_wf
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. (filter-index(P;L) ∈ i:{i:ℕ||L||| ↑(P L[i])} ⟶ {j:ℕ||filter(P;L)||| filter(P;L)[\000Cj] = L[i] ∈ T} )
Proof
Definitions occuring in Statement :
filter-index: filter-index(P;L),
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
filter-index: filter-index(P;L),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
istype: istype(T),
nat_plus: ℕ+,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
void_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
filter_nil_lemma,
int_seg_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
length_of_cons_lemma,
filter_cons_lemma,
length_wf,
istype-nat,
istype-assert,
select_wf,
int_seg_properties,
decidable__lt,
list_wf,
bool_wf,
istype-universe,
list_ind_cons_lemma,
istype-false,
add_nat_plus,
length_wf_nat,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
nat_plus_properties,
add-is-int-iff,
false_wf,
cons_wf,
non_neg_length,
assert_functionality_wrt_uiff,
select_cons_tl,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-member-int_seg2,
ifthenelse_wf,
select_cons_tl_sq2,
int_seg_subtype_nat,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
select-cons-tl,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
add-zero,
uiff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
voidElimination,
thin,
instantiate,
extract_by_obid,
hypothesis,
functionExtensionality_alt,
setElimination,
rename,
lambdaFormation_alt,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
applyEquality,
intEquality,
sqequalBase,
addEquality,
setIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
universeEquality,
cumulativity,
setEquality,
pointwiseFunctionality,
productIsType,
equalityElimination,
minusEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
(filter-index(P;L) \mmember{} i:\{i:\mBbbN{}||L||| \muparrow{}(P L[i])\} {}\mrightarrow{} \{j:\mBbbN{}||filter(P;L)||| filter(P;L)[j] = L[i]\} )
Date html generated:
2020_05_19-PM-09_42_29
Last ObjectModification:
2019_12_31-PM-00_12_36
Theory : list_1
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