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,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
equal: s = t ∈ T,
qmul: r * s,
qadd: r + s,
rationals: ℚ
Lemmas :
weighted-sum-linear,
p-outcome_wf,
finite-prob-space_wf,
rationals_wf
\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:
2015_07_17-AM-07_58_27
Last ObjectModification:
2015_01_27-AM-11_24_00
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