Nuprl Lemma : l_sum-split
∀[A:Type]. ∀[L:A List]. ∀[f:{a:A| (a ∈ L)} ⟶ ℤ]. ∀[P:{a:A| (a ∈ L)} ⟶ 𝔹].
(l_sum(map(f;L)) = (l_sum(map(f;filter(P;L))) + l_sum(map(f;filter(λx.(¬b(P x));L)))) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
l_member: (x ∈ l),
filter: filter(P;l),
map: map(f;as),
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
int: ℤ,
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
istype: istype(T),
l_member: (x ∈ l),
select: L[n],
cand: A c∧ B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
assert: ↑b
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,
istype-less_than,
list-cases,
map_nil_lemma,
filter_nil_lemma,
l_sum_nil_lemma,
l_member_wf,
nil_wf,
bool_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
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,
map_cons_lemma,
filter_cons_lemma,
l_sum_cons_lemma,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets_simple,
cons_member,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
decidable__lt,
nat_plus_properties,
add-is-int-iff,
false_wf,
length_wf,
select_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
istype-nat,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
functionIsType,
setIsType,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
setEquality,
inrFormation_alt,
cumulativity,
pointwiseFunctionality,
productIsType,
equalityElimination,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[L:A List]. \mforall{}[f:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[P:\{a:A| (a \mmember{} L)\} {}\mrightarrow{} \mBbbB{}].
(l\_sum(map(f;L)) = (l\_sum(map(f;filter(P;L))) + l\_sum(map(f;filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L)))))
Date html generated:
2020_05_19-PM-09_46_16
Last ObjectModification:
2019_11_12-PM-03_16_45
Theory : list_1
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