| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleRetainedStats |
| import OAI.Probability.SKBarriers.Replicas.TripleScheduleScale |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| @[simp] theorem rawVariance_scale (β : ℝ) (l : List (ℝ × ℝ)) : |
| rawVariance (scaleIncrementChain β l)=β^2*rawVariance l := by |
| induction l with |
| | nil => simp [rawVariance,scaleIncrementChain] |
| | cons p l ih => |
| change (β*p.2)^2+rawVariance (scaleIncrementChain β l)=β^2*(p.2^2+rawVariance l) |
| rw [ih]; ring |
|
|
| @[simp] theorem rawArea_scale (β : ℝ) (l : List (ℝ × ℝ)) : |
| rawArea (scaleIncrementChain β l)=β^2*rawArea l := by |
| induction l with |
| | nil => simp [rawArea,scaleIncrementChain] |
| | cons p l ih => |
| change p.1*(β*p.2)^2+rawArea (scaleIncrementChain β l)=β^2*(p.1*p.2^2+rawArea l) |
| rw [ih]; ring |
|
|
| @[simp] theorem weightedVariance_scale (β : ℝ) (w : List (ℝ × (ℝ × ℝ))) : |
| weightedVariance (scaleIncrementChain β w)=β^2*weightedVariance w := by |
| induction w with |
| | nil => simp [weightedVariance,scaleIncrementChain] |
| | cons p w ih => |
| change (β*p.2.2)^2+weightedVariance (scaleIncrementChain β w)=β^2*(p.2.2^2+weightedVariance w) |
| rw [ih]; ring |
|
|
| @[simp] theorem weightedCross_scale (β : ℝ) (w : List (ℝ × (ℝ × ℝ))) : |
| weightedCross (scaleIncrementChain β w)=β^2*weightedCross w := by |
| induction w with |
| | nil => simp [weightedCross,scaleIncrementChain] |
| | cons p w ih => |
| change (β*p.2.1)*(β*p.2.2)+weightedCross (scaleIncrementChain β w)=β^2*(p.2.1*p.2.2+weightedCross w) |
| rw [ih]; ring |
|
|
| @[simp] theorem weightedAbsCross_scale (β : ℝ) (w : List (ℝ × (ℝ × ℝ))) : |
| weightedAbsCross (scaleIncrementChain β w)=β^2*weightedAbsCross w := by |
| induction w with |
| | nil => simp [weightedAbsCross,scaleIncrementChain] |
| | cons p w ih => |
| change |β*p.2.2| * |β*p.2.1|+weightedAbsCross (scaleIncrementChain β w)=β^2*(|p.2.2| * |p.2.1|+weightedAbsCross w) |
| rw [ih,abs_mul,abs_mul,← sq_abs β]; ring |
|
|
| theorem weightedCrossPenalty_scale (β g : ℝ) (w : List (ℝ × (ℝ × ℝ))) : |
| weightedCrossPenalty (scaleIncrementChain β w) (β^2*g)=β^4*weightedCrossPenalty w g := by |
| induction w generalizing g with |
| | nil => simp [weightedCrossPenalty,scaleIncrementChain,chainPotentialPenalty] |
| | cons p w ih => |
| change p.1*((β^2*g+(β*p.2.1)*(β*p.2.2))^2-(β^2*g)^2)+ |
| weightedCrossPenalty (scaleIncrementChain β w) (β^2*g+(β*p.2.1)*(β*p.2.2))= |
| β^4*(p.1*((g+p.2.1*p.2.2)^2-g^2)+weightedCrossPenalty w (g+p.2.1*p.2.2)) |
| rw [show β^2*g+(β*p.2.1)*(β*p.2.2)=β^2*(g+p.2.1*p.2.2) by ring,ih] |
| ring |
|
|
| @[simp] theorem weightedCrossPenalty_scale_zero (β : ℝ) (w : List (ℝ × (ℝ × ℝ))) : |
| weightedCrossPenalty (scaleIncrementChain β w) 0=β^4*weightedCrossPenalty w 0 := by |
| simpa only [mul_zero] using weightedCrossPenalty_scale β 0 w |
|
|
| theorem scaleIncrementChain_mass {E : Type} [SMul ℝ E] (β : ℝ) (l : List (ℝ × E)) {P : ℝ → Prop} |
| (hm : ∀ p∈l,P p.1) : ∀ p∈scaleIncrementChain β l,P p.1 := by |
| intro p hp |
| obtain ⟨q,hq,rfl⟩ := List.mem_map.mp hp |
| exact hm q hq |
|
|
| theorem scaleIncrementChain_sorted {E : Type} [SMul ℝ E] (β : ℝ) (l : List (ℝ × E)) |
| (hs : l.Pairwise (fun p q => p.1≤q.1)) : (scaleIncrementChain β l).Pairwise (fun p q => p.1≤q.1) := by |
| simpa only [scaleIncrementChain,List.pairwise_map] using hs |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
