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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