| import Mathlib |
| import OAI.Probability.SKBarriers.Locking.NarrowTimeBlocks |
| import OAI.Probability.SKBarriers.Locking.NarrowBlockPenalty |
| import OAI.Probability.SKBarriers.Scalar.NegativePenalty |
| import OAI.Probability.SKBarriers.Locking.FourBlockTent |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open Set |
| namespace SK.Analytic |
|
|
| theorem rawVariance_pos_length {l : List (ℝ × ℝ)} (h : 0<rawVariance l) : 0<l.length := by |
| cases l with |
| | nil => simp [rawVariance] at h |
| | cons p l => simp |
|
|
| namespace NarrowTimeBlocks |
| variable {α : ℝ → ℝ} {r h q : ℝ} (C : NarrowTimeBlocks α r h q) |
|
|
| theorem v_absCross (hh : 0<h) {θ : ℝ} (hθ : 0≤θ) : weightedAbsCross (C.v θ)=θ := by |
| simp only [v,weightedAbsCross_append,weightedAbsCross_zeroWeight,constantWeightChain_absCross,C.j_variance,add_zero] |
| rw [abs_of_nonneg (div_nonneg hθ hh.le)] |
| field_simp |
|
|
| theorem v_mean_bounds (hα : Monotone α) (hh : 0<h) {θ : ℝ} (hθ : 0≤θ) : |
| weightedMean (C.v θ)∈Icc (θ*α r) (θ*α (r+h)) := by |
| have HA := rawArea_bounds C.j (C.right.raw_mass_interval hα) |
| rw [C.j_variance] at HA |
| rw [C.v_mean] |
| have hl := mul_le_mul_of_nonneg_left HA.1 (div_nonneg hθ hh.le) |
| have hu := mul_le_mul_of_nonneg_left HA.2 (div_nonneg hθ hh.le) |
| constructor |
| · convert hl using 1; field_simp |
| · convert hu using 1; field_simp |
|
|
| theorem scaled_cw (β a : ℝ) : |
| scaleIncrementChain β (C.c a)++scaleIncrementChain β (C.w a)= |
| fourBlockTent (a/h) (scaleIncrementChain β C.b) (scaleIncrementChain β C.l) |
| (scaleIncrementChain β C.j) (scaleIncrementChain β C.k) := by |
| simp only [c,w,fourBlockTent,scaleIncrementChain_append,zeroWeightChain_scale,constantWeightChain_scale,neg_div] |
|
|
| end NarrowTimeBlocks |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
