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,
qle: r ≤ s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
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
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rv-partial-sum: rv-partial-sum(n;i.X[i]),
int_nzero: ℤ-o,
squash: ↓T,
true: True,
rv-compose: (x.F[x]) o X,
rv-scale: q*X,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
cand: A c∧ B,
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b,
rv-const: a,
rv-le: X ≤ Y,
qle: r ≤ s,
grp_leq: a ≤ b,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
lt_int: i <z j,
eq_int: (i =z j),
qdiv: (r/s),
qinv: 1/r,
qeq: qeq(r;s)
Lemmas referenced :
member-less_than,
le_wf,
nat_wf,
int_seg_wf,
slln-lemma1,
qmul_wf,
int-subtype-rationals,
rv-partial-sum_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel-random-variable,
le_weakening2,
all_wf,
qle_wf,
expectation_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,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
rv-compose_wf,
rv-scale_wf,
qdiv_wf,
subtype_rel_set,
rationals_wf,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int-equal-in-rationals,
rv-disjoint_wf,
less_than_wf,
random-variable_wf,
finite-prob-space_wf,
expectation-qsum,
assert-bnot,
neg_assert_of_eq_int,
lelt_wf,
int_nzero-rational,
nequal_wf,
qsum-qle,
qmul-mul,
int_entire_a,
equal-wf-base,
int_subtype_base,
uiff_transitivity,
squash_wf,
true_wf,
expectation-rv-const,
subtype_rel_self,
iff_weakening_equal,
non-neg-qmul,
qle_reflexivity,
expectation-monotone-in-first,
p-outcome_wf,
mul_nzero,
itermMultiply_wf,
intformnot_wf,
int_term_value_mul_lemma,
int_formula_prop_not_lemma,
zero_ann_a,
qmul_assoc_qrng,
qmul_ac_1_qrng,
qmul_comm_qrng,
qmul-qdiv,
expectation-rv-scale,
qinv-nonneg,
qmul-positive,
qless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
qless_wf,
qle_functionality_wrt_implies,
qmul_functionality_wrt_qle,
qle_weakening_eq_qorder,
expectation-non-neg,
q-square-non-neg,
qmul_preserves_qle,
qmul_com,
qmul_assoc,
qmul-qdiv-cancel,
qsum_wf,
prod_sum_l_q,
decidable__le,
decidable__equal_int,
int_seg_properties,
sum_unroll_base_q,
qmul_zero_qrng,
qmul_preserves_qle2,
qle_witness,
sum_unroll_lo_q,
qsum-reciprocal-squares-bound,
decidable__equal_rationals,
qadd_wf,
qmul_over_plus_qrng,
mon_ident_q,
qinv-positive,
qle_weakening_lt_qorder,
qmul-qdiv-cancel2,
qmul_one_qrng,
qmul_preserves_qless,
q-square-positive,
qle-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
hypothesis,
natural_numberEquality,
because_Cache,
independent_isectElimination,
independent_pairEquality,
axiomEquality,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
intEquality,
baseClosed,
impliesFunctionality,
hypothesis_subsumption,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
addLevel,
productEquality,
functionEquality,
hyp_replacement,
applyLambdaEquality,
multiplyEquality,
baseApply,
closedConclusion,
imageElimination,
imageMemberEquality,
universeEquality,
inlFormation,
minusEquality
Latex:
\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:
2018_05_22-AM-00_42_30
Last ObjectModification:
2018_05_19-PM-04_00_04
Theory : randomness
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