Nuprl Lemma : rv-unbounded_wf
∀[p:FinProbSpace]. ∀[f:ℕ ⟶ ℕ]. ∀[X:n:ℕ ⟶ RandomVariable(p;f[n])]. ((X[n]⟶∞ as n⟶∞) ∈ (ℕ ⟶ Outcome) ⟶ ℙ)
Proof
Definitions occuring in Statement :
rv-unbounded: (X[n]⟶∞ as n⟶∞),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rv-unbounded: (X[n]⟶∞ as n⟶∞),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
finite-prob-space: FinProbSpace,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
p-outcome: Outcome
Lemmas referenced :
all_wf,
rationals_wf,
exists_wf,
nat_wf,
le_wf,
qle_wf,
subtype_rel_dep_function,
p-outcome_wf,
int_seg_wf,
length_wf,
int_seg_subtype_nat,
false_wf,
random-variable_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
because_Cache,
functionEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])].
((X[n]{}\mrightarrow{}\minfty{} as n{}\mrightarrow{}\minfty{}) \mmember{} (\mBbbN{} {}\mrightarrow{} Outcome) {}\mrightarrow{} \mBbbP{})
Date html generated:
2016_05_15-PM-11_50_39
Last ObjectModification:
2015_12_28-PM-07_14_38
Theory : randomness
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