Nuprl Lemma : rv-partial-sum_wf
∀[p:FinProbSpace]. ∀[f:ℕ ⟶ ℕ]. ∀[X:n:ℕ ⟶ RandomVariable(p;f[n])]. ∀[n:ℕ].
rv-partial-sum(n;i.X[i]) ∈ RandomVariable(p;f[n]) supposing ∀n:ℕ. ∀i:ℕn. f[i] < f[n]
Proof
Definitions occuring in Statement :
rv-partial-sum: rv-partial-sum(n;i.X[i]),
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],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rv-partial-sum: rv-partial-sum(n;i.X[i]),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
qsum_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
int_seg_wf,
p-outcome_wf,
int_seg_subtype,
le_weakening2,
subtype_rel_self,
all_wf,
nat_wf,
less_than_wf,
random-variable_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lambdaEquality,
lemma_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
because_Cache,
dependent_functionElimination,
functionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])]. \mforall{}[n:\mBbbN{}].
rv-partial-sum(n;i.X[i]) \mmember{} RandomVariable(p;f[n]) supposing \mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]
Date html generated:
2016_05_15-PM-11_51_57
Last ObjectModification:
2015_12_28-PM-07_14_25
Theory : randomness
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