Nuprl Lemma : slln-lemma2
∀p:FinProbSpace. ∀f:ℕ ─→ ℕ. ∀X:n:ℕ ─→ RandomVariable(p;f[n]). ∀s,k:ℚ.
((∀n:ℕ. ∀i:ℕn. rv-disjoint(p;f[n];X[i];X[n]))
⇒ (∃B:ℚ
∀n:ℕ
(E(f[n];rv-partial-sum(n;k.if (k =z 0)
then 0
else (x.(x * x) * x * x) o (1/k)*rv-partial-sum(k;i.X[i])
fi )) ≤ B))) supposing
((∀n:ℕ
((E(f[n];X[n]) = 0 ∈ ℚ)
∧ (E(f[n];(x.x * x) o X[n]) = s ∈ ℚ)
∧ (E(f[n];(x.(x * x) * x * x) o X[n]) = k ∈ ℚ))) and
(∀n:ℕ. ∀i:ℕn. f[i] < f[n]))
Proof
Definitions occuring in Statement :
rv-partial-sum: rv-partial-sum(n;i.X[i]),
rv-compose: (x.F[x]) o X,
rv-disjoint: rv-disjoint(p;n;X;Y),
expectation: E(n;F),
rv-const: a,
rv-scale: q*X,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ
Lemmas :
member-less_than,
int_seg_subtype-nat,
false_wf,
nat_wf,
int_seg_wf,
slln-lemma1,
qmul_wf,
int-subtype-rationals,
all_wf,
Error :qle_wf,
expectation_wf,
rv-partial-sum_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rv-const_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
iff_weakening_uiff,
assert_of_bnot,
int_upper_subtype_nat,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
rv-compose_wf,
rv-scale_wf,
qdiv_wf,
subtype_rel_set,
Error :int_nzero-rational,
subtype_rel_sets,
nequal_wf,
not-equal-2,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-associates,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
or_wf,
equal-wf-base,
int_subtype_base,
rv-disjoint_wf,
subtype_rel-random-variable,
le_weakening2,
rationals_wf,
less_than_wf,
random-variable_wf,
finite-prob-space_wf,
expectation-qsum,
assert-bnot,
neg_assert_of_eq_int,
lelt_wf,
int-equal-in-rationals,
int_entire_a,
qmul-mul,
Error :qsum-qle,
uiff_transitivity,
Error :qle_reflexivity,
Error :non-neg-qmul,
iff_weakening_equal,
expectation-rv-const,
true_wf,
squash_wf,
expectation-monotone-in-first,
p-outcome_wf,
mul_nzero,
Error :qmul-qdiv,
Error :qmul_comm_qrng,
Error :qmul_ac_1_qrng,
Error :qmul_assoc_qrng,
le_antisymmetry_iff,
decidable__equal_int,
zero_ann_a,
mul-associates,
minus-zero,
le-add-cancel,
add-zero,
expectation-rv-scale,
Error :qle_weakening_eq_qorder,
Error :qmul_functionality_wrt_qle,
Error :qle_functionality_wrt_implies,
Error :qless_wf,
decidable__lt,
Error :qless-int,
Error :qmul-positive,
Error :qinv-nonneg,
expectation-non-neg,
Error :q-square-non-neg,
Error :qmul_preserves_qle,
qmul_assoc,
qmul_com,
Error :qmul-qdiv-cancel,
Error :qsum_wf,
Error :prod_sum_l_q,
sq_stable__le,
Error :sum_unroll_base_q,
less_than_irreflexivity,
less_than_transitivity1,
subtype_rel-equal,
Error :qle_witness,
Error :qmul_preserves_qle2,
Error :qmul_zero_qrng,
Error :sum_unroll_lo_q,
Error :qsum-reciprocal-squares-bound,
Error :decidable__equal_rationals,
Error :mon_ident_q,
Error :qmul_over_plus_qrng,
qadd_wf,
Error :qmul_one_qrng,
Error :qmul-qdiv-cancel2,
Error :qle_weakening_lt_qorder,
Error :qinv-positive,
Error :q-square-positive,
Error :qmul_preserves_qless,
Error :qle-iff
\mforall{}p:FinProbSpace. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]). \mforall{}s,k:\mBbbQ{}.
((\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. rv-disjoint(p;f[n];X[i];X[n]))
{}\mRightarrow{} (\mexists{}B:\mBbbQ{}
\mforall{}n:\mBbbN{}
(E(f[n];rv-partial-sum(n;k.if (k =\msubz{} 0)
then 0
else (x.(x * x) * x * x) o (1/k)*rv-partial-sum(k;i.X[i])
fi )) \mleq{} B))) supposing
((\mforall{}n:\mBbbN{}
((E(f[n];X[n]) = 0)
\mwedge{} (E(f[n];(x.x * x) o X[n]) = s)
\mwedge{} (E(f[n];(x.(x * x) * x * x) o X[n]) = k))) and
(\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]))
Date html generated:
2015_07_17-AM-08_03_40
Last ObjectModification:
2015_07_17-AM-07_31_03
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