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
Lemmas :
member-less_than,
int_seg_subtype-nat,
false_wf,
int_seg_wf,
less_than_transitivity1,
less_than_irreflexivity,
all_wf,
subtract_wf,
rv-disjoint_wf,
subtype_rel-random-variable,
sq_stable__le,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-associates,
zero-add,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
less_than_wf,
length_wf_nat,
rationals_wf,
le_weakening2,
isect_wf,
set_wf,
primrec-wf2,
random-variable_wf,
nat_wf,
finite-prob-space_wf,
decidable__lt,
add-swap,
add-commutes,
lelt_wf,
decidable__equal_int,
int-subtype-rationals,
rv-disjoint-const,
true_wf,
squash_wf,
iff_weakening_equal,
subtype_rel-int_seg,
subtype_rel_dep_function,
Error :sum_unroll_base_q,
int_subtype_base,
subtype_base_sq,
le-add-cancel-alt,
le_wf,
minus-minus,
minus-zero,
le-add-cancel,
not-equal-2,
not-le-2,
decidable__le,
rv-disjoint-rv-add2,
rv-partial-sum_wf,
subtype_rel_self,
Error :qsum_wf,
Error :sum_unroll_hi_q,
qadd_wf,
less_than_transitivity2
\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:
2015_07_17-AM-08_02_29
Last ObjectModification:
2015_07_16-AM-09_52_06
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