Nuprl Lemma : rv-disjoint-rv-partial-sum
∀p:FinProbSpace. ∀f:ℕ ⟶ ℕ. ∀X:n:ℕ ⟶ RandomVariable(p;f[n]). ∀N:ℕ. ∀Z:RandomVariable(p;N). ∀n:ℕ.
(∀i:ℕn - 1. rv-disjoint(p;N;X[i];Z)) ⇒ (∀k:ℕn. rv-disjoint(p;N;rv-partial-sum(k;i.X[i]);Z)) supposing ∀i:ℕn. f[i] < N
Proof
Definitions occuring in Statement :
rv-partial-sum: rv-partial-sum(n;i.X[i]),
rv-disjoint: rv-disjoint(p;n;X;Y),
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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
and: 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,
guard: {T},
subtract: n - m,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
finite-prob-space: FinProbSpace,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
random-variable: RandomVariable(p;n),
rv-partial-sum: rv-partial-sum(n;i.X[i]),
rv-const: a,
sq_type: SQType(T),
istype: istype(T),
rv-add: X + Y
Lemmas referenced :
member-less_than,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
length_wf_nat,
rationals_wf,
rv-disjoint_wf,
int_seg_subtype_nat,
istype-false,
istype-less_than,
primrec-wf2,
isect_wf,
all_wf,
random-variable_wf,
istype-nat,
finite-prob-space_wf,
decidable__equal_int,
int-subtype-rationals,
rv-disjoint-const,
nat_wf,
true_wf,
squash_wf,
length_wf,
iff_weakening_equal,
subtype_rel_self,
less_than_irreflexivity,
less_than_transitivity1,
int_seg_subtype,
subtype_rel_function,
sum_unroll_base_q,
istype-universe,
equal_wf,
int_subtype_base,
subtype_base_sq,
rv-disjoint-rv-add2,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel-random-variable,
le_weakening2,
decidable__lt,
rv-partial-sum_wf,
less_than_wf,
le_wf,
le_weakening,
subtype_rel_dep_function,
qsum_wf,
qadd_wf,
sum_unroll_hi_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
isect_memberFormation_alt,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
applyEquality,
dependent_set_memberEquality_alt,
unionElimination,
because_Cache,
isectIsType,
setIsType,
instantiate,
functionEquality,
baseClosed,
imageMemberEquality,
equalityTransitivity,
imageElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
universeEquality,
functionExtensionality_alt,
intEquality,
cumulativity,
productIsType
Latex:
\mforall{}p:FinProbSpace. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n]). \mforall{}N:\mBbbN{}. \mforall{}Z:RandomVariable(p;N). \mforall{}n:\mBbbN{}.
(\mforall{}i:\mBbbN{}n - 1. rv-disjoint(p;N;X[i];Z)) {}\mRightarrow{} (\mforall{}k:\mBbbN{}n. rv-disjoint(p;N;rv-partial-sum(k;i.X[i]);Z))
supposing \mforall{}i:\mBbbN{}n. f[i] < N
Date html generated:
2019_10_16-PM-00_39_25
Last ObjectModification:
2018_12_08-AM-11_56_06
Theory : randomness
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