Nuprl Lemma : expectation-non-neg
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[Y:RandomVariable(p;n)]. 0 ≤ E(n;Y) supposing 0 ≤ Y
Proof
Definitions occuring in Statement :
rv-le: X ≤ Y,
expectation: E(n;F),
rv-const: a,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qle: r ≤ s,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q
Lemmas referenced :
finite-prob-space_wf,
nat_wf,
random-variable_wf,
rv-le_wf,
qle_witness,
iff_weakening_equal,
expectation_wf,
expectation-rv-const,
rationals_wf,
true_wf,
squash_wf,
qle_wf,
int-subtype-rationals,
rv-const_wf,
expectation-monotone
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
sqequalRule,
introduction,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[Y:RandomVariable(p;n)]. 0 \mleq{} E(n;Y) supposing 0 \mleq{} Y
Date html generated:
2016_05_15-PM-11_48_39
Last ObjectModification:
2016_01_17-AM-10_05_53
Theory : randomness
Home
Index