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,
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
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
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],
nat: ℕ,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
guard: {T},
rv-compose: (x.F[x]) o X,
rv-const: a,
rv-le: X ≤ Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
p-outcome: Outcome,
cand: A c∧ B,
uiff: uiff(P;Q),
rv-partial-sum: rv-partial-sum(n;i.X[i]),
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
grp_id: e,
pi1: fst(t),
pi2: snd(t),
add_grp_of_rng: r↓+gp,
rng_zero: 0,
qrng: <ℚ+*>,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
btrue: tt,
true: True,
qle: r ≤ s,
grp_leq: a ≤ b,
assert: ↑b,
infix_ap: x f y,
grp_le: ≤b,
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qeq: qeq(r;s),
eq_int: (i =z j),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rv-add: X + Y,
rv-mul: X * Y,
rv-scale: q*X,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
member-less_than,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-nat,
int_seg_wf,
rv-disjoint_wf,
int_seg_subtype_nat,
istype-false,
expectation_wf,
int-subtype-rationals,
rv-compose_wf,
qmul_wf,
istype-less_than,
rationals_wf,
random-variable_wf,
finite-prob-space_wf,
expectation-rv-const,
expectation-monotone,
rv-const_wf,
qle_wf,
q-square-non-neg,
subtype_rel_self,
length_wf,
p-outcome_wf,
decidable__qle,
qle_reflexivity,
qle_complement_qorder,
qle_weakening_lt_qorder,
primrec-wf2,
expectation-constant,
istype-top,
subtype_rel_dep_function,
top_wf,
equal_wf,
qsum_wf,
intformless_wf,
int_formula_prop_less_lemma,
qadd_wf,
and_wf,
squash_wf,
true_wf,
qmul_over_plus_qrng,
qmul_zero_qrng,
mon_ident_q,
iff_weakening_equal,
rv-partial-sum_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
istype-universe,
sum_unroll_hi_q,
int_seg_subtype,
le_weakening2,
qle_witness,
rv-add_wf,
qmul_assoc_qrng,
qmul_comm_qrng,
qmul_ac_1_qrng,
mon_assoc_q,
qadd_ac_1_q,
qadd_comm_q,
q_distrib,
qmul_ident,
subtype_rel-random-variable,
rv-mul_wf,
rv-scale_wf,
expectation-rv-add,
expectation-rv-disjoint,
rv-disjoint-compose,
length_wf_nat,
rv-disjoint-rv-partial-sum,
rv-disjoint-monotone-in-first,
expectation-rv-scale,
expectation-monotone-in-first,
rv-disjoint-symmetry,
expectation-qsum,
nat_wf,
qsum-const,
qle_functionality_wrt_implies,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
qle-int,
qmul_functionality_wrt_qle,
non-neg-qmul,
expectation-non-neg,
qsub-sub,
qmul_over_minus_qrng,
qmul_one_qrng,
qinv_inv_q,
qmul_assoc,
qadd_preserves_qle,
qadd_inv_assoc_q,
qinverse_q,
qle_transitivity_qorder,
qsub_wf,
rv-disjoint-rv-scale,
istype_wf,
le_weakening,
member_wf,
subtype_rel_set,
le_wf,
qmul_preserves_qle2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
natural_numberEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
functionIsTypeImplies,
inhabitedIsType,
independent_pairEquality,
axiomEquality,
functionIsType,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
functionEquality,
setIsType,
intEquality,
functionExtensionality_alt,
closedConclusion,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
promote_hyp,
functionExtensionality,
minusEquality,
productEquality,
cumulativity
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{}
((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:
2019_10_16-PM-00_40_12
Last ObjectModification:
2018_12_08-AM-11_55_39
Theory : randomness
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