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
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