| import Mathlib |
| import OAI.Probability.SKGap.Stability.OrdinaryRecipe |
| import OAI.Probability.SKGap.Localization.RecipeError |
| import OAI.Probability.SKGap.Matrix.ClosedErrorTrace |
| import OAI.Probability.SKGap.Matrix.ClosedMatrix |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe.OrdinaryData |
| open Primary Matrix SKGap SKGap.Noncrossing SKGap.Noncrossing.Primary |
| open SKGap.Noncrossing.Primary.Tensor.Series |
| variable {n : ℕ} {κ σ : Type*} [Fintype κ] [DecidableEq κ] [Fintype σ] |
| variable (D : OrdinaryData n Unit κ σ) |
|
|
| def implicitPartial (y : VectorFields n) : VectorFields n := D.partialOf 1 () (fun _=>y) |
| def implicitCoefficient : VectorFields n := D.auxCoefficient 1 0 |
|
|
| def implicitSourcePrimitive (t : SourceTree (Fin n→ℝ)) (y : VectorFields n) (x : Spin n) : Interaction n := |
| D.sourceAtoms 1 (fun _=>y) x+Matrix.diagonal (D.implicitPartial y x)* |
| (derivativeMatrix (D.H ()) x-t.fieldMatrix D.j D.J) |
| def implicitFieldPrimitive (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) (x : Spin n) : Interaction n := |
| -(D.j*siteMean (D.implicitPartial y) x) • |
| (derivativeMatrix (D.predecessor ()) x-t.sourceMatrix D.j D.J)- |
| D.j • (averageError (siteMean (D.implicitPartial y)) (D.predecessor ()) x+ |
| averageError (siteMean D.implicitCoefficient) w x) |
|
|
| def implicitRetainedSource (t : SourceTree (Fin n→ℝ)) (y : VectorFields n) (x : Spin n) : Interaction n := |
| GradedWords.matrixValue D.j (D.implicitCoefficient x) D.J |
| (GradedWords.implicitSource D.j (D.implicitCoefficient x) (D.implicitPartial y x) t) |
| def implicitRetainedField (t : SourceTree (Fin n→ℝ)) (y : VectorFields n) (x : Spin n) : Interaction n := |
| GradedWords.matrixValue D.j (D.implicitCoefficient x) D.J |
| (GradedWords.implicitField D.j (D.implicitCoefficient x) (D.implicitPartial y x) t) |
|
|
| lemma mean_implicit (F : VectorFields n) (x : Spin n) : Diagram.mean (F x)=siteMean F x := by |
| simp [Diagram.mean,siteMean] |
|
|
| lemma implicit_source_derivative (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) |
| (hw : w=D.sourceOf 1 (fun _=>y)) (x : Spin n) : |
| derivativeMatrix w x=Matrix.diagonal (D.implicitCoefficient x)*derivativeMatrix y x+ |
| Matrix.diagonal (D.implicitPartial y x)*t.fieldMatrix D.j D.J+D.implicitSourcePrimitive t y x := by |
| rw [hw,D.sourceOf_derivative] |
| simp only [Fintype.sum_unique,implicitCoefficient,implicitPartial,implicitSourcePrimitive,mul_sub] |
| have hd : (default : Fin 1)=0 := Subsingleton.elim _ _ |
| rw [hd] |
| abel |
| lemma implicit_field_derivative (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) : |
| derivativeMatrix y x=(D.J-(D.j*siteMean D.implicitCoefficient x) • 1)*derivativeMatrix w x- |
| (D.j*siteMean (D.implicitPartial y) x) • t.sourceMatrix D.j D.J+D.implicitFieldPrimitive t w y x := by |
| have H:=congrArg (fun Q:VectorFields n=>derivativeMatrix Q x) hy |
| rw [D.fieldOf_derivative,←hw] at H |
| simp only [Fintype.sum_unique] at H |
| rw [H] |
| simp only [implicitCoefficient,implicitPartial,implicitFieldPrimitive,neg_smul,smul_sub, |
| sub_mul,smul_mul_assoc,one_mul] |
| abel |
|
|
| theorem implicit_error_solved (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) |
| (hK : 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) : |
| derivativeMatrix w x-D.implicitRetainedSource t y x= |
| WordLetter.inverse.exactEval D.j (D.implicitCoefficient x) D.J* |
| (D.implicitSourcePrimitive t y x+Matrix.diagonal (D.implicitCoefficient x)*D.implicitFieldPrimitive t w y x) ∧ |
| derivativeMatrix y x-D.implicitRetainedField t y x= |
| (D.J-(D.j*siteMean D.implicitCoefficient x) • 1)* |
| (derivativeMatrix w x-D.implicitRetainedSource t y x)+D.implicitFieldPrimitive t w y x := by |
| have hsource:=D.implicit_source_derivative t w y hw x |
| have hfield:=D.implicit_field_derivative t w y hw hy x |
| have hs : derivativeMatrix w x=Matrix.diagonal (D.implicitCoefficient x)*derivativeMatrix y x+ |
| (Matrix.diagonal (D.implicitPartial y x)*t.fieldMatrix D.j D.J+D.implicitSourcePrimitive t y x) := by rw [hsource];abel |
| have hf : derivativeMatrix y x=(D.J-(D.j*siteMean D.implicitCoefficient x) • 1)*derivativeMatrix w x+ |
| (-(D.j*siteMean (D.implicitPartial y) x) • t.sourceMatrix D.j D.J+D.implicitFieldPrimitive t w y x) := by |
| rw [hfield];simp only [neg_smul];abel |
| have H:=implicit_error_elimination _ _ _ _ _ _ _ _ hK hs hf |
| constructor |
| · rw [H,implicitRetainedSource,GradedWords.matrixValue_implicitSource,mean_implicit] |
| simp only [mul_add,mul_sub,neg_smul,mul_neg,mul_smul_comm] |
| abel |
| · rw [hfield,implicitRetainedField,GradedWords.matrixValue_implicitField,mean_implicit,mean_implicit] |
| change _-((D.J-(D.j*siteMean D.implicitCoefficient x) • 1)*D.implicitRetainedSource t y x- |
| (D.j*siteMean (D.implicitPartial y) x) • t.sourceMatrix D.j D.J)=_ |
| rw [mul_sub] |
| abel |
|
|
| end SKGapCutoff.Recipe.OrdinaryData |
|
|
| end |
| end |
|
|
| end OAI |
|
|
