| import Mathlib |
| import OAI.Probability.SKGap.Stability.ImplicitNode |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Primary |
| variable {n : ℕ} |
| lemma derivativeMatrix_congr_on_flips (F G : VectorFields n) (x : Spin n) |
| (hx : F x=G x) (hf : ∀i,F (flip x i)=G (flip x i)) : |
| derivativeMatrix F x=derivativeMatrix G x := by |
| ext i k |
| simp only [derivativeMatrix,halfDiff_as_flip,hx,hf] |
| end SKGapCutoff.Primary |
|
|
| 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) |
|
|
| lemma implicit_source_derivative_local (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) (x : Spin n) |
| (hw : w x=D.sourceOf 1 (fun _=>y) x) |
| (hwf : ∀i,w (flip x i)=D.sourceOf 1 (fun _=>y) (flip x i)) : |
| 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 [derivativeMatrix_congr_on_flips w _ x hw hwf,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_local (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) (x : Spin n) |
| (hw : w x=D.sourceOf 1 (fun _=>y) x) |
| (hwf : ∀i,w (flip x i)=D.sourceOf 1 (fun _=>y) (flip x i)) |
| (hy : y x=D.fieldOf 1 (fun _=>w) (fun _=>y) x) |
| (hyf : ∀i,y (flip x i)=D.fieldOf 1 (fun _=>w) (fun _=>y) (flip x i)) : |
| 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:=derivativeMatrix_congr_on_flips y _ x hy hyf |
| have HW:=derivativeMatrix_congr_on_flips w _ x hw hwf |
| 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_local (t : SourceTree (Fin n→ℝ)) (w y : VectorFields n) |
| (x : Spin n) |
| (hw : w x=D.sourceOf 1 (fun _=>y) x) |
| (hwf : ∀i,w (flip x i)=D.sourceOf 1 (fun _=>y) (flip x i)) |
| (hy : y x=D.fieldOf 1 (fun _=>w) (fun _=>y) x) |
| (hyf : ∀i,y (flip x i)=D.fieldOf 1 (fun _=>w) (fun _=>y) (flip x i)) |
| (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_local t w y x hw hwf |
| have hfield:=D.implicit_field_derivative_local t w y x hw hwf hy hyf |
| 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 |
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|
