Nuprl Lemma : rv-partial-sum-monotone
∀[p:FinProbSpace]. ∀[f:ℕ ─→ ℕ]. ∀[X:n:ℕ ─→ RandomVariable(p;f[n])].
(∀[m:ℕ]. ∀[n:ℕm + 1]. rv-partial-sum(n;i.X[i]) ≤ rv-partial-sum(m;i.X[i])) supposing
((∀n:ℕ. 0 ≤ X[n]) 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-le: X ≤ Y,
rv-const: a,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ─→ B[x],
add: n + m,
natural_number: $n
Lemmas :
decidable__equal_int,
le_wf,
and_wf,
equal_wf,
nat_wf,
int_seg_subtype-nat,
false_wf,
le_weakening2,
rv-partial-sum_wf,
subtype_rel-random-variable,
lelt_wf,
rv-le_witness,
int_seg_wf,
all_wf,
rv-le_wf,
rv-const_wf,
int-subtype-rationals,
less_than_wf,
random-variable_wf,
finite-prob-space_wf,
complete_nat_ind,
subtype_rel-int_seg,
rationals_wf,
length_wf,
p-outcome_wf,
subtype_rel_dep_function,
Error :qsum_wf,
Error :qle_reflexivity,
int_subtype_base,
subtype_base_sq,
le-add-cancel2,
add-commutes,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
le-add-cancel,
zero-add,
add-swap,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
decidable__lt,
rv-partial-sum-unroll,
subtract_wf,
minus-minus,
sq_stable__le,
not-le-2,
decidable__le,
le_weakening,
le-add-cancel-alt,
Error :qle_weakening_eq_qorder,
Error :qle_functionality_wrt_implies,
qadd_wf,
subtract-is-less,
Error :qle_wf,
subtype_rel_self,
iff_weakening_equal,
Error :qadd_inv_assoc_q,
Error :qinverse_q,
Error :qadd_ac_1_q,
Error :qadd_comm_q,
true_wf,
squash_wf,
qmul_wf,
Error :qadd_preserves_qle
\mforall{}[p:FinProbSpace]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])].
(\mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1]. rv-partial-sum(n;i.X[i]) \mleq{} rv-partial-sum(m;i.X[i])) supposing
((\mforall{}n:\mBbbN{}. 0 \mleq{} X[n]) and
(\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]))
Date html generated:
2015_07_17-AM-08_02_34
Last ObjectModification:
2015_07_16-AM-11_19_35
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