| import Mathlib |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace CylinderCovering.BeamCost |
|
|
| def S (p : ℝ) : ℝ := p^2+(1-p)^2 |
| def T (p : ℝ) : ℝ := p^4+(1-p)^4 |
|
|
| def beamDensity (a p : ℝ) : ℝ := |
| 1 / Real.sqrt (1-a*(2-a)*S p/2+a^2*(2-a)^2*T p/4) |
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|
| def beamCost (a : ℝ) : ℝ := (2-a)^2 * ∫ p in (0:ℝ)..1, beamDensity a p |
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| def stripWidth (a t : ℝ) : ℝ := |
| max (a*(1-2*t^2)+a^2*t^2*(1+(1-|t|)^2+t^2)+25*a^3) 0 |
|
|
| def stripDensity (a t : ℝ) : ℝ := |
| stripWidth a t / Real.sqrt (1+(a*t-a^2*t^3)^2) |
|
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| def stripCost (a : ℝ) : ℝ := |
| 2*Real.sqrt 2 * ∫ t in (-1:ℝ)..1, stripDensity a t |
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|
| def stripPolynomial (a t : ℝ) : ℝ := |
| a*(1-2*t^2)+a^2*t^2*(1+(1-t)^2+t^2) |
|
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| def cutoff : ℝ := Real.sqrt 2/2 |
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|
| def CostStatement : Prop := |
| (∀ a : ℝ, 0 ≤ a → a ≤ 1/4 → |
| |beamCost a-(4-(8/3)*a-(11/10)*a^2)| ≤ 300*a^3) ∧ |
| (∀ a : ℝ, 0 ≤ a → a ≤ 1/100 → |
| |stripCost a-((8/3)*a+((26/15)-Real.sqrt 2/2)*a^2)| ≤ 300*a^3) ∧ |
| (∀ a : ℝ, 0 ≤ a → a ≤ 1/100 → |
| |beamCost a+stripCost a-(4+((19/30)-Real.sqrt 2/2)*a^2)| ≤ 600*a^3) ∧ |
| (∀ a : ℝ, 0 < a → a ≤ 1/20000 → beamCost a+stripCost a < 4) |
|
|
| end CylinderCovering.BeamCost |
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|
| end |
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| end OAI |
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|
