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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 | 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
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