| import Mathlib |
| import OAI.Probability.SKGap.Matrix.ClosedMatrix |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGap.Noncrossing.Primary.Tensor.Series.ClosedTree |
| open Matrix GradedWords |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def ofOrdinary : SourceTree (ι→ℝ)→ClosedTree (ι→ℝ) |
| | .leaf c => .leaf c |
| | .branch p t u => .branch p (ofOrdinary t) (ofOrdinary u) |
| omit [DecidableEq ι] in |
| lemma words_ofOrdinary (j : ℝ) (a : ι→ℝ) (t : SourceTree (ι→ℝ)) : |
| (ofOrdinary t).words j a=ordinaryWords j t := by |
| induction t with |
| | leaf c => rfl |
| | branch p t u ht hu => simp only [ofOrdinary,words,ordinaryWords,ht,hu] |
|
|
| def sourceMatrix (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (t : ClosedTree (ι→ℝ)) := |
| GradedWords.matrixValue j a J (t.words j a).1 |
| def fieldMatrix (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (t : ClosedTree (ι→ℝ)) := |
| GradedWords.matrixValue j a J (t.words j a).2 |
| lemma sourceMatrix_ofOrdinary (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) : |
| (ofOrdinary t).sourceMatrix j a J=t.sourceMatrix j J := by |
| rw [sourceMatrix,words_ofOrdinary] |
| exact congrArg Prod.fst (matrixValue_ordinaryWords j a J t) |
| lemma fieldMatrix_ofOrdinary (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) : |
| (ofOrdinary t).fieldMatrix j a J=t.fieldMatrix j J := by |
| rw [fieldMatrix,words_ofOrdinary] |
| exact congrArg Prod.snd (matrixValue_ordinaryWords j a J t) |
| lemma sourceMatrix_leaf (j c : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) : |
| (ClosedTree.leaf c).sourceMatrix j a J=c • 1 := by |
| simp [sourceMatrix,words,ordinaryWords,matrixValue_scalar] |
| lemma fieldMatrix_leaf (j c : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) : |
| (ClosedTree.leaf c).fieldMatrix j a J=c • J := by |
| simp [fieldMatrix,words,ordinaryWords,matrixValue_bump,matrixValue_prepend, |
| matrixValue_scalar,WordLetter.exactEval] |
| lemma sourceMatrix_branch (j : ℝ) (a p : ι→ℝ) (J : Matrix ι ι ℝ) (t u : ClosedTree (ι→ℝ)) : |
| (ClosedTree.branch p t u).sourceMatrix j a J= |
| Matrix.diagonal p*t.fieldMatrix j a J+u.sourceMatrix j a J := by |
| simp [sourceMatrix,fieldMatrix,words,GradedWords.branch,matrixValue_append,matrixValue_prepend, |
| WordLetter.exactEval] |
| lemma fieldMatrix_branch (j : ℝ) (a p : ι→ℝ) (J : Matrix ι ι ℝ) (t u : ClosedTree (ι→ℝ)) : |
| (ClosedTree.branch p t u).fieldMatrix j a J= |
| J*(Matrix.diagonal p*t.fieldMatrix j a J)- |
| (j*Diagram.mean p) • t.sourceMatrix j a J+u.fieldMatrix j a J := by |
| simp [sourceMatrix,fieldMatrix,words,GradedWords.branch,matrixValue_append,matrixValue_prepend, |
| matrixValue_bump,matrixValue_scale,WordLetter.exactEval,sub_eq_add_neg] |
| abel |
|
|
| def branches (xs : List ((ι→ℝ)×ClosedTree (ι→ℝ))) : ClosedTree (ι→ℝ) := |
| xs.foldr (fun t v=>.branch t.1 t.2 v) (.leaf 0) |
| omit [Fintype ι] [DecidableEq ι] in |
| lemma branches_leafMass (xs : List ((ι→ℝ)×ClosedTree (ι→ℝ))) : (branches xs).leafMass=0 := by |
| induction xs with |
| | nil => rfl |
| | cons t xs ih => exact ih |
| lemma branches_source (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) |
| (xs : List ((ι→ℝ)×ClosedTree (ι→ℝ))) : |
| (branches xs).sourceMatrix j a J= |
| (xs.map (fun t=>Matrix.diagonal t.1*t.2.fieldMatrix j a J)).sum := by |
| induction xs with |
| | nil => simp [branches,sourceMatrix_leaf] |
| | cons t xs ih => |
| simpa only [branches,List.foldr_cons,sourceMatrix_branch,List.map_cons,List.sum_cons] using |
| (congrArg (fun Z=>Matrix.diagonal t.1*t.2.fieldMatrix j a J+Z) ih) |
| lemma branches_field (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) |
| (xs : List ((ι→ℝ)×ClosedTree (ι→ℝ))) : |
| (branches xs).fieldMatrix j a J=J*(branches xs).sourceMatrix j a J- |
| (xs.map (fun t=>(j*Diagram.mean t.1) • t.2.sourceMatrix j a J)).sum := by |
| induction xs with |
| | nil => simp [branches,sourceMatrix_leaf,fieldMatrix_leaf] |
| | cons t xs ih => |
| change (ClosedTree.branch t.1 t.2 (branches xs)).fieldMatrix j a J=_ |
| rw [fieldMatrix_branch,ih] |
| change _=J*(ClosedTree.branch t.1 t.2 (branches xs)).sourceMatrix j a J-_ |
| rw [sourceMatrix_branch,mul_add] |
| simp only [List.map_cons,List.sum_cons] |
| abel |
|
|
| end SKGap.Noncrossing.Primary.Tensor.Series.ClosedTree |
|
|
| end |
| end |
|
|
| end OAI |
|
|
