Nuprl Lemma : slln-lemma1
∀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. ((0 ≤ B) ∧ (∀n:ℕ. (E(f[n];(x.(x * x) * x * x) o rv-partial-sum(n;i.X[i])) ≤ (B * n * n)))))) 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),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
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,
qmul: r * s,
rationals: ℚ
Lemmas :
int-subtype-rationals,
Error :qle_wf,
qmul_wf,
rv-compose_wf,
rv-const_wf,
expectation-monotone,
iff_weakening_equal,
le_wf,
false_wf,
expectation-rv-const,
rationals_wf,
p-outcome_wf,
nat_wf,
int_seg_wf,
random-variable_wf,
Error :q-square-non-neg,
Error :decidable__qle,
Error :qle_weakening_eq_qorder,
Error :qle_complement_qorder,
Error :qle_weakening_lt_qorder,
Error :qle_reflexivity,
primrec-wf2,
less_than_wf,
set_wf,
Error :mon_ident_q,
Error :qmul_zero_qrng,
Error :qmul_over_plus_qrng,
true_wf,
squash_wf,
qadd_wf,
expectation_wf,
Error :qsum_wf,
equal_wf,
subtype_rel-int_seg,
less_than_irreflexivity,
less_than_transitivity1,
int_seg_subtype-nat,
Error :sum_unroll_base_q,
length_wf,
top_wf,
subtype_rel_dep_function,
expectation-constant,
subtract-is-less,
lelt_wf,
le-add-cancel2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
le_weakening2,
Error :sum_unroll_hi_q,
subtype_rel-random-variable,
rv-add_wf,
Error :qle_witness,
rv-partial-sum_wf,
finite-prob-space_wf,
qmul_ident,
Error :q_distrib,
Error :qadd_comm_q,
Error :qadd_ac_1_q,
Error :mon_assoc_q,
Error :qmul_ac_1_qrng,
Error :qmul_comm_qrng,
Error :qmul_assoc_qrng,
expectation-rv-add,
rv-scale_wf,
rv-mul_wf,
expectation-rv-disjoint,
rv-disjoint-compose,
rv-disjoint_wf,
rv-disjoint-rv-partial-sum,
length_wf_nat,
rv-disjoint-monotone-in-first,
equal-wf-T-base,
expectation-rv-scale,
expectation-monotone-in-first,
rv-disjoint-symmetry,
expectation-qsum,
Error :qsum-const,
Error :qadd_functionality_wrt_qle,
Error :qle_functionality_wrt_implies,
Error :qmul_functionality_wrt_qle,
Error :qle-int,
Error :non-neg-qmul,
expectation-non-neg,
Error :qsub-sub,
qmul_assoc,
Error :qinv_inv_q,
Error :qmul_one_qrng,
Error :qmul_over_minus_qrng,
Error :qinverse_q,
Error :qadd_inv_assoc_q,
Error :qadd_preserves_qle,
Error :qle_transitivity_qorder,
qsub_wf,
rv-disjoint-rv-scale,
member_wf,
subtype_rel_set,
all_wf,
Error :qmul_preserves_qle2
\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{}
((0 \mleq{} B)
\mwedge{} (\mforall{}n:\mBbbN{}
(E(f[n];(x.(x * x) * x * x) o rv-partial-sum(n;i.X[i])) \mleq{} (B * n * n)))))) 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_25
Last ObjectModification:
2015_07_17-AM-06_48_39
Home
Index