Step
*
1
2
1
1
2
1
2
of Lemma
bounded-expectation
1. p : FinProbSpace@i
2. f : ℕ ─→ ℕ@i
3. X : n:ℕ ─→ RandomVariable(p;f[n])@i
4. B : ℚ@i
5. ∀n:ℕ. ∀i:ℕn. f[i] < f[n]
6. ∀n:ℕ. ∀i:ℕn. X[i] ≤ X[n]
7. 0 < B
8. ∀n:ℕ. (0 ≤ X[n] ∧ E(f[n];X[n]) < B)
9. ∀q:ℚ. (0 < q
⇒ (∃b:ℚ. ∀n:ℕ. E(f[n];b ≤ X[n]) < q))
10. q : ℚ@i
11. 0 < q@i
12. b : ℚ
13. ∀n:ℕ. E(f[n];b ≤ X[n]) < q
14. g : ∀n:ℕ. ∀s:ℕn ─→ Outcome. Dec(∃i:ℕn. (f[i] < n c∧ (b ≤ (X[i] s))))
⊢ ∀n:ℕ. ∀s:ℕn ─→ Outcome. (((λp.case g (fst(p)) (snd(p)) of inl(x) => 1 | inr(x) => 0) <n, s>) = 1 ∈ ℤ
⇐⇒ ∃i:ℕn. (f[i]\000C < n c∧ (b ≤ (X[i] s))))
BY
{ (Reduce 0
THEN (UnivCD THENA Auto)
THEN ((GenConclAtAddr [1; 2; 1])
THENA (Auto THEN Try ((Fold `random-variable` 0 THEN Auto)) THEN Try ((Fold `p-outcome` 0 THEN Auto)))
)
THEN D -2
THEN Reduce 0
THEN Auto
THEN Try ((Fold `random-variable` 0 THEN Auto))
THEN Try ((Fold `p-outcome` 0 THEN Auto))) }
Latex:
1. p : FinProbSpace@i
2. f : \mBbbN{} {}\mrightarrow{} \mBbbN{}@i
3. X : n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])@i
4. B : \mBbbQ{}@i
5. \mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]
6. \mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. X[i] \mleq{} X[n]
7. 0 < B
8. \mforall{}n:\mBbbN{}. (0 \mleq{} X[n] \mwedge{} E(f[n];X[n]) < B)
9. \mforall{}q:\mBbbQ{}. (0 < q {}\mRightarrow{} (\mexists{}b:\mBbbQ{}. \mforall{}n:\mBbbN{}. E(f[n];b \mleq{} X[n]) < q))
10. q : \mBbbQ{}@i
11. 0 < q@i
12. b : \mBbbQ{}
13. \mforall{}n:\mBbbN{}. E(f[n];b \mleq{} X[n]) < q
14. g : \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} Outcome. Dec(\mexists{}i:\mBbbN{}n. (f[i] < n c\mwedge{} (b \mleq{} (X[i] s))))
\mvdash{} \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} Outcome. (((\mlambda{}p.case g (fst(p)) (snd(p)) of inl(x) => 1 | inr(x) => 0) <n, s>) = 1 \000C\mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (f[i] < n c\mwedge{} (b \mleq{} (X[i] s))))
By
(Reduce 0
THEN (UnivCD THENA Auto)
THEN ((GenConclAtAddr [1; 2; 1])
THENA (Auto
THEN Try ((Fold `random-variable` 0 THEN Auto))
THEN Try ((Fold `p-outcome` 0 THEN Auto)))
)
THEN D -2
THEN Reduce 0
THEN Auto
THEN Try ((Fold `random-variable` 0 THEN Auto))
THEN Try ((Fold `p-outcome` 0 THEN Auto)))
Home
Index