Nuprl Lemma : slln-lemma3
∀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]))
⇒ nullset(p;λs.∃q:ℚ. (0 < q ∧ (∀n:ℕ. ∃m:ℕ. (n < m ∧ (q ≤ |Σ0 ≤ i < m. (1/m) * (X[i] s)|)))))) 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 :
nullset: nullset(p;S),
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,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ
Lemmas :
slln-lemma2,
member-less_than,
int_seg_subtype-nat,
false_wf,
nat_wf,
int_seg_wf,
all_wf,
rv-disjoint_wf,
subtype_rel-random-variable,
le_weakening2,
equal-wf-T-base,
rationals_wf,
expectation_wf,
rv-compose_wf,
qmul_wf,
less_than_wf,
random-variable_wf,
finite-prob-space_wf,
rv-partial-sum_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rv-const_wf,
int-subtype-rationals,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
int_upper_subtype_nat,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
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,
Error :qle_wf,
squash_wf,
true_wf,
p-outcome_wf,
top_wf,
assert-bnot,
neg_assert_of_eq_int,
less_than_transitivity1,
less_than_irreflexivity,
Error :sum_unroll_base_q,
iff_weakening_equal,
expectation-rv-const,
Error :qadd_preserves_qless,
qadd_wf,
Error :qless_wf,
Error :qadd_comm_q,
Error :qadd_ac_1_q,
Error :qinverse_q,
Error :mon_ident_q,
rv-partial-sum-monotone,
decidable__lt,
less-iff-le,
le-add-cancel,
lelt_wf,
Error :q-square-non-neg,
Error :qsum_wf,
subtype_rel_dep_function,
length_wf,
subtype_rel-int_seg,
uiff_transitivity,
bounded-expectation,
Error :qless_transitivity_1_qorder,
decidable__equal_int,
Error :qle_reflexivity,
add-zero,
minus-zero,
rv-le_wf,
le_weakening,
nat_plus_wf,
nullset-monotone,
rv-unbounded_wf,
exists_wf,
Error :qabs_wf,
less_than_transitivity2,
Error :q-not-limit-zero-diverges-2,
Error :qmul-positive,
Error :qmul_preserves_qle2,
Error :qle_weakening_lt_qorder,
Error :qle_witness,
Error :qless_transitivity_2_qorder,
Error :qmul_comm_qrng,
Error :qle_transitivity_qorder,
Error :qmul_functionality_wrt_qle,
Error :qsum-linearity1,
Error :qsum-non-neg2,
Error :qle_functionality_wrt_implies,
Error :qle_weakening_eq_qorder,
set_wf,
sq_stable_from_decidable,
Error :decidable__qle,
Error :qabs-squared,
Error :sum_split_q,
sq_stable__le,
Error :qadd_preserves_qle,
Error :qadd_inv_assoc_q,
Error :sum_unroll_lo_q,
Error :qadd_ident,
le_antisymmetry_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{} nullset(p;\mlambda{}s.\mexists{}q:\mBbbQ{}
(0 < q
\mwedge{} (\mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}. (n < m \mwedge{} (q \mleq{} |\mSigma{}0 \mleq{} i < m. (1/m) * (X[i] s)|)))))) 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_04_16
Last ObjectModification:
2015_02_03-PM-09_50_07
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