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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | 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; exact hm p (List.mem_append_left _ 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; exact hm p (List.mem_append_left _ (List.mem_append_right _ 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; obtain ⟨p,hp,rfl⟩ := List.mem_map.mp hx; positivity
have he : (⟨c.length,by simp only [List.length_append]; omega⟩ : Fin ((c++w).length+1))=
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]; change b.length≤b.length+l.length; omega
· rw [he]; change b.length+l.length≤b.length+l.length+j.length; omega
· exact hd
· exact hB
· exact hR
· rw [he,fourBlockTent_middle_variance,fourBlockTent_left_variance,hl]; ring
· rw [he,fourBlockTent_middle_variance,fourBlockTent_right_variance,hj]; ring
· rw [he]; exact fourBlockTent_vector (ne_of_gt hd) hκ b l j k
· exact weightedList_field_bound v
end SK.Analytic
end
end
end OAI
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