Nuprl Lemma : l_sum-mapfilter
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. ∀[f:{x:T| (x ∈ L) ∧ (↑(P x))} ⟶ ℤ].
(l_sum(mapfilter(f;P;L)) = Σ(if P L[i] then f L[i] else 0 fi | i < ||L||) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
mapfilter: mapfilter(f;P;L),
sum: Σ(f[x] | x < k),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
mapfilter: mapfilter(f;P;L),
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
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,
l_sum: l_sum(L),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
filter_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
map_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
istype-universe,
l_member_wf,
assert_wf,
bool_wf,
subtract-1-ge-0,
subtype_base_sq,
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,
filter_cons_lemma,
length_of_cons_lemma,
nat_wf,
list_wf,
nil_wf,
reduce_nil_lemma,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
sum_split,
length_wf,
add_nat_wf,
length_wf_nat,
add-is-int-iff,
false_wf,
select_wf,
int_seg_properties,
non_neg_length,
decidable__lt,
eqtt_to_assert,
select_member,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
int_seg_wf,
sum1,
int_seg_subtype_special,
int_seg_cases,
map_cons_lemma,
reduce_cons_lemma,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
not_wf,
assert_of_bnot,
sum_wf,
squash_wf,
true_wf,
select-cons-tl,
add-subtract-cancel,
satisfiable-full-omega-tt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
Error :functionIsType,
Error :setIsType,
Error :productIsType,
applyEquality,
instantiate,
imageElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
intEquality,
universeEquality,
functionExtensionality,
productEquality,
cumulativity,
setEquality,
functionEquality,
isect_memberFormation,
voidEquality,
isect_memberEquality,
Error :inrFormation_alt,
addEquality,
pointwiseFunctionality,
equalityElimination,
Error :inlFormation_alt,
imageMemberEquality,
computeAll,
lambdaEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:T| (x \mmember{} L) \mwedge{} (\muparrow{}(P x))\} {}\mrightarrow{} \mBbbZ{}].
(l\_sum(mapfilter(f;P;L)) = \mSigma{}(if P L[i] then f L[i] else 0 fi | i < ||L||))
Date html generated:
2019_06_20-PM-01_44_12
Last ObjectModification:
2018_10_06-PM-11_55_51
Theory : list_1
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