| import Mathlib |
| import OAI.Probability.SKGap.Stability.StableFields |
| import OAI.Probability.SKGap.Localization.RecipeError |
| import OAI.Probability.SKGap.Stability.ImplicitNode |
|
|
| namespace OAI |
|
|
| section |
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|
| 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 |
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|
| end |
| end |
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| end OAI |
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