Nuprl Lemma : fps-summation-coeff
∀[X:Type]. ∀[r:CRng]. ∀[T:Type]. ∀[f:T ⟶ PowerSeries(X;r)]. ∀[b:bag(T)].
∀m:bag(X). (fps-summation(r;b;x.f[x])[m] = Σ(x∈b). f[x][m] ∈ |r|)
Proof
Definitions occuring in Statement :
fps-summation: fps-summation(r;b;x.f[x]),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
fps-coeff: f[b],
bag-summation: Σ(x∈b). f[x],
fps-summation: fps-summation(r;b;x.f[x]),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
fps-add: (f+g),
fps-zero: 0,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
power-series: PowerSeries(X;r)
Lemmas referenced :
bag_to_squash_list,
equal_wf,
rng_car_wf,
fps-coeff_wf,
fps-summation_wf,
bag-summation_wf,
rng_plus_wf,
rng_zero_wf,
rng_all_properties,
rng_plus_comm2,
bag_wf,
power-series_wf,
crng_wf,
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,
le_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
set_wf,
lelt_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
list-cases,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
list_wf,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
length_wf_nat,
list_accum_nil_lemma,
list_accum_cons_lemma,
last_wf,
list_accum_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
setElimination,
because_Cache,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
unionElimination,
equalityTransitivity,
hypothesis_subsumption,
dependent_set_memberEquality,
baseClosed,
impliesFunctionality,
productEquality,
addEquality
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} PowerSeries(X;r)]. \mforall{}[b:bag(T)].
\mforall{}m:bag(X). (fps-summation(r;b;x.f[x])[m] = \mSigma{}(x\mmember{}b). f[x][m])
Date html generated:
2016_10_25-AM-11_34_15
Last ObjectModification:
2016_07_12-AM-07_37_52
Theory : power!series
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