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7d9f027 | 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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import Mathlib
import OAI.Probability.SKGap.Stability.StableFields
import OAI.Probability.SKGap.Localization.RecipeError
import OAI.Probability.SKGap.Stability.ImplicitNode
namespace OAI
section
noncomputable section
open scoped BigOperators Matrix.Norms.Frobenius
namespace SKGapCutoff.Recipe.OrdinaryData
open Primary Matrix SKGap SKGap.Noncrossing SKGap.Noncrossing.Primary
variable {n : ℕ} {κ σ : Type*} [Fintype κ] [DecidableEq κ] [Fintype σ]
variable (D : OrdinaryData n Unit κ σ)
lemma implicitSource_control (t : SourceTree (Fin n→ℝ)) (y : VectorFields n) (x : Spin n)
{C Q R L V Z B Cp E : ℝ}
(hC : 0≤C) (hQ : 0≤Q) (hR : 0≤R) (hL : 0≤L) (hV : 0≤V)
(hY : SmallBound y x B) (hp : SmallBound (D.implicitPartial y) x Cp)
(hE : ‖derivativeMatrix (D.H ()) x-t.fieldMatrix D.j D.J‖≤E)
(hs : ∀s,SegmentRegular D.H D.θ (D.seedFunction 1 s) (D.seedDerivative 1 s) x C)
(ha : SegmentRegular D.H D.θ (D.auxFunction 1 0) (D.auxDerivative 1 0) x C)
(ht : vectorNorm (parameterIncrement D.θ x)≤V)
(hQ4 : (∑i,∑k,(primaryIncrement D.H x i k)^4)≤Q^4)
(hQr : ∀i,(∑k,(primaryIncrement D.H x i k)^2)≤R^2)
(hQd : (∑i,(primaryIncrement D.H x i i)^2)≤L^2)
(hZ : (∑α,‖derivativeVector (D.θ α) x‖)+4*(Q^2+R*V+V^2)+2*(Q+L+V)≤Z) :
TraceControl (D.implicitSourcePrimitive t y x)
(((∑s,C*vectorNorm (D.seed s)*Z)+C*B*Z)+Cp*E) := by
have hc (s : VectorFields n) {S : ℝ} (hS : SmallBound s x S)
(F : Fin n→Args (ι:=Unit) (κ:=κ)→ℝ)
(F' : Fin n→Args (ι:=Unit) (κ:=κ)→Args (ι:=Unit) (κ:=κ)→L[ℝ]ℝ)
(hF : SegmentRegular D.H D.θ F F' x C) : TraceControl (sourceError D.H D.θ F F' s x) (C*S*Z) :=
(sourceError_control D.H D.θ F F' s x hF hS.nonneg hV hQ hR hL hS.size hS.derivative ht hQ4 hQr hQd).mono
(mul_le_mul_of_nonneg_left hZ (mul_nonneg hC hS.nonneg))
have hs' := TraceControl.sum (fun s=>hc (fun _=>D.seed s) (SmallBound.seed _ x) _ _ (hs s))
have ha' := hc y hY _ _ ha
have hp' := (small_diagonal_control
(derivativeMatrix (D.H ()) x-t.fieldMatrix D.j D.J) (D.implicitPartial y x)).mono
(mul_le_mul hp.size hE (norm_nonneg _) hp.nonneg)
have H:=(hs'.add ha').add hp'
simpa only [implicitSourcePrimitive,sourceAtoms,Fin.sum_univ_one] using H
lemma implicitField_control (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) (x : Spin n) (hn : 0<n)
{P R B Cp E : ℝ} (hP : 0≤P) (hR : 0≤R)
(hW : SmallBound w x B) (hp : SmallBound (D.implicitPartial y) x Cp)
(hE : ‖derivativeMatrix (D.predecessor ()) x-t.sourceMatrix D.j D.J‖≤E)
(hm : ∀i,|D.predecessor () x i|≤1)
(hdm : SKGap.opNorm (derivativeMatrix (D.predecessor ()) x)≤P)
(hda : ∀i,∑k,(derivativeMatrix D.implicitCoefficient x i k)^2≤R^2) :
TraceControl (D.implicitFieldPrimitive t w y x)
(|D.j| *(Cp*E)+|D.j| *(Cp*(1+2*P)+3*B*R)) := by
have hmE : TraceControl
((D.j*siteMean (D.implicitPartial y) x) •
(derivativeMatrix (D.predecessor ()) x-t.sourceMatrix D.j D.J)) (|D.j| *(Cp*E)) := by
simpa only [smul_smul] using ((small_mean_control
(derivativeMatrix (D.predecessor ()) x-t.sourceMatrix D.j D.J)
(D.implicitPartial y) x hn).mono
(mul_le_mul hp.size hE (norm_nonneg _) hp.nonneg)).smul D.j
have hpE := primary_averageError_control (D.implicitPartial y) (D.predecessor ()) x hn
hp.nonneg hP hp.derivative hm hdm
have haE : TraceControl (averageError (siteMean D.implicitCoefficient) w x) (3*B*R) := by
apply (averageError_control (siteMean D.implicitCoefficient) w x).mono
have H:=coefficient_mean_difference D.implicitCoefficient x hR hda
have h1 := add_le_add hW.size (mul_le_mul_of_nonneg_left hW.derivative (by norm_num : (0:ℝ)≤2))
exact (mul_le_mul h1 H (norm_nonneg _) (by linarith [hW.nonneg])).trans_eq (by ring)
simpa only [implicitFieldPrimitive,neg_smul] using hmE.neg.sub ((hpE.add haE).smul D.j)
omit [Fintype σ] in
lemma implicit_inverse_left (x : Spin n) (ha : ∀i,0≤D.implicitCoefficient x i)
(hS : (1-SKGap.pathDiagonal (D.implicitCoefficient x) 1*
(D.J-(D.j*siteMean D.implicitCoefficient x) • 1)*SKGap.pathDiagonal (D.implicitCoefficient x) 1).PosDef) :
WordLetter.inverse.exactEval D.j (D.implicitCoefficient x) D.J*
(1-Matrix.diagonal (D.implicitCoefficient x)*(D.J-(D.j*siteMean D.implicitCoefficient x) • 1))=1 := by
have hu:=SKGap.ResolventIdentity.inverse_one_sub_square_mul_unit
(SKGap.pathDiagonal (D.implicitCoefficient x) 1) (D.J-(D.j*siteMean D.implicitCoefficient x) • 1) hS.isUnit
rw [SKGap.pathDiagonal_square ha] at hu
have hi:=Matrix.nonsing_inv_mul _ (((1-Matrix.diagonal (D.implicitCoefficient x)*(D.J-(D.j*siteMean D.implicitCoefficient x) • 1)).isUnit_iff_isUnit_det).mp hu)
have hq : (D.j/(n:ℝ))*(∑i,D.implicitCoefficient x i)=D.j*siteMean D.implicitCoefficient x := by
simp only [siteMean];ring
simpa only [WordLetter.exactEval,Fintype.card_fin,hq] using hi
theorem implicit_error_trace (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n)
(hW : w=D.sourceOf 1 (fun _=>y)) (hY : y=D.fieldOf 1 (fun _=>w) (fun _=>y)) (x : Spin n)
(ha0 : ∀i,0≤D.implicitCoefficient x i)
(hS : (1-SKGap.pathDiagonal (D.implicitCoefficient x) 1*
(D.J-(D.j*siteMean D.implicitCoefficient x) • 1)*SKGap.pathDiagonal (D.implicitCoefficient x) 1).PosDef)
{C F A M : ℝ} {L : ℕ}
(hs : TraceControl (D.implicitSourcePrimitive t y x) C)
(hf : TraceControl (D.implicitFieldPrimitive t w y x) F)
(ha : ∀i,|D.implicitCoefficient x i|≤A)
(hwords : ClosedWordTestBound D.j (D.implicitCoefficient x) D.J A M L)
(v : OrdinaryWord n) (hlen : v.length+3≤L) (hv : wordBounded A v) :
|trace (matrixWord D.J v*(derivativeMatrix w x-D.implicitRetainedSource t y x))|≤(C+F)*M ∧
|trace (matrixWord D.J v*(derivativeMatrix y x-D.implicitRetainedField t y x))|≤
((1+|D.j*siteMean D.implicitCoefficient x|)*(C+F)+F)*M := by
obtain ⟨he,he'⟩:=D.implicit_error_solved t w y hW hY x (D.implicit_inverse_left x ha0 hS)
exact implicit_source_field_trace _ _ _ _ _ _ _ _ he he' hs hf ha hwords v hlen hv
end SKGapCutoff.Recipe.OrdinaryData
end
end
end OAI
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