| import Mathlib |
| import OAI.Probability.SKBarriers.Locking.FourBlockTent |
| import OAI.Probability.SKBarriers.Locking.NarrowTentBound |
| import OAI.Probability.SKBarriers.Hierarchy.WeightedPrefixBound |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open Set |
| namespace SK.Analytic |
|
|
| abbrev fourBlockCommon (κ : ℝ) (b l : List (ℝ × ℝ)) := zeroWeightChain b++constantWeightChain κ l |
| abbrev fourBlockFuture (κ : ℝ) (j k : List (ℝ × ℝ)) := constantWeightChain κ j++zeroWeightChain k |
|
|
| theorem fourBlockQuadratic_tent_le (κ η : ℝ) (b l j k t : List (ℝ × ℝ)) |
| (hm : ∀ p∈(b++l)++(j++k)++t,p.1∈Icc (0:ℝ) 1) |
| (hs : ((b++l)++(j++k)++t).Pairwise (fun p q => p.1≤q.1)) |
| (hmean : 0≤weightedMean (fourBlockFuture η j k)) |
| {d B : ℝ} (hd : 0<d) (hB : 0≤B) (hκ : κ*d=1) |
| (hl : rawVariance l=B*d) (hj : rawVariance j=B*d) : |
| narrowBaseQuadratic (fourBlockCommon κ b l) (fourBlockFuture η j k) (fourBlockFuture (-κ) j k) t≤ |
| narrowVariance (fourBlockCommon κ b l) (fourBlockFuture η j k) (fourBlockFuture (-κ) j k)* |
| narrowSusceptibility (fourBlockCommon κ b l) (fourBlockFuture (-κ) j k) t+ |
| 2*weightedMean (fourBlockFuture η j k)*B*narrowJointMoment (fourBlockCommon κ b l) (fourBlockFuture (-κ) j k) t+ |
| 6*(weightedAbsCross (fourBlockFuture η j k))^2+ |
| 2*weightedMean (fourBlockFuture η j k)*B*scalarTentAtomMass (fourBlockTent κ b l j k).length |
| (fun i => ((fourBlockTent κ b l j k).get i).1) (fourBlockLeft κ b l j k) (fourBlockRight κ b l j k)+ |
| (B*scalarTentAtomMass (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).1) |
| (fourBlockLeft κ b l j k) (fourBlockRight κ b l j k))^2+ |
| B^2*scalarTentAtomMass (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).1) |
| (fourBlockLeft κ b l j k) (fourBlockRight κ b l j k) := by |
| let c := fourBlockCommon κ b l |
| let v := fourBlockFuture η j k |
| let w := fourBlockFuture (-κ) j k |
| have hc : weightedUnderlying c=b++l := by simp [c,fourBlockCommon] |
| have hv : weightedUnderlying v=j++k := by simp [v,fourBlockFuture] |
| have hw : weightedUnderlying w=j++k := by simp [w,fourBlockFuture] |
| have hcw : weightedUnderlying (c++w)=(b++l)++(j++k) := by rw [weightedUnderlying_append,hc,hw] |
| have hmcw : ∀ p∈c++w,p.1∈Icc (0:ℝ) 1 := by |
| apply (weighted_mass_iff _ (P:=fun x => x∈Icc (0:ℝ) 1)).mpr |
| rw [hcw] |
| intro p hp |
| have hscw : (c++w).Pairwise (fun p q => p.1≤q.1) := by |
| apply (weighted_sorted_iff _).mpr |
| rw [hcw] |
| exact (List.pairwise_append.mp hs).1 |
| have hmv : ∀ p∈v,p.1∈Icc (0:ℝ) 1 := by |
| apply (weighted_mass_iff _ (P:=fun x => x∈Icc (0:ℝ) 1)).mpr |
| rw [hv] |
| intro p hp |
| have hsv : v.Pairwise (fun p q => p.1≤q.1) := by |
| apply (weighted_sorted_iff _).mpr |
| rw [hv] |
| exact (List.pairwise_append.mp (List.pairwise_append.mp hs).1).2.1 |
| have hR : 0≤weightedAbsCross v := by |
| unfold weightedAbsCross |
| apply List.sum_nonneg |
| intro x hx |
| have he : (⟨c.length,by simp only [List.length_append] |
| fourBlockMiddle κ b l j k := by |
| apply Fin.ext |
| simp [c,fourBlockCommon,zeroWeightChain,constantWeightChain] |
| apply narrowBaseQuadratic_tent_le c v w t (hv.trans hw.symm) hmcw hscw |
| (fun p hp => hm p (List.mem_append_right _ hp)) (List.pairwise_append.mp hs).2.1 hmv hsv hmean |
| (fourBlockLeft κ b l j k) (fourBlockRight κ b l j k) |
| · rw [he] |
| · rw [he] |
| · exact hd |
| · exact hB |
| · exact hR |
| · rw [he,fourBlockTent_middle_variance,fourBlockTent_left_variance,hl] |
| · rw [he,fourBlockTent_middle_variance,fourBlockTent_right_variance,hj] |
| · rw [he] |
| · exact weightedList_field_bound v |
|
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| end SK.Analytic |
|
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| end |
| end |
|
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| end OAI |
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