Nuprl Lemma : ws-linear
∀[a,b:ℚ]. ∀[p:FinProbSpace]. ∀[F,G:Outcome ⟶ ℚ].
(weighted-sum(p;λx.((a * (F x)) + (b * (G x)))) = ((a * weighted-sum(p;F)) + (b * weighted-sum(p;G))) ∈ ℚ)
Proof
Definitions occuring in Statement :
weighted-sum: weighted-sum(p;F),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
qmul: r * s,
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
finite-prob-space: FinProbSpace,
p-outcome: Outcome
Lemmas referenced :
weighted-sum-linear,
p-outcome_wf,
rationals_wf,
finite-prob-space_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
functionEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mforall{}[p:FinProbSpace]. \mforall{}[F,G:Outcome {}\mrightarrow{} \mBbbQ{}].
(weighted-sum(p;\mlambda{}x.((a * (F x)) + (b * (G x))))
= ((a * weighted-sum(p;F)) + (b * weighted-sum(p;G))))
Date html generated:
2016_05_15-PM-11_45_43
Last ObjectModification:
2015_12_28-PM-07_16_30
Theory : randomness
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