| import Mathlib |
| import OAI.Probability.SKGap.Localization.ClosedPrediction |
| import OAI.Probability.SKGap.Matrix.ClosedTrace |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGap.Noncrossing.Primary.Tensor.Series.GradedWords |
| open Matrix Diagram InverseDiagram |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| lemma matrixValue_append (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (P Q : GradedWords ι) : |
| matrixValue j a J (P++Q)=matrixValue j a J P+matrixValue j a J Q := by |
| simp [matrixValue] |
| lemma matrixValue_scalar (j c : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) : |
| matrixValue j a J (scalar c)=c • 1 := by |
| simp [matrixValue,scalar,exactWord,matrixFactorProduct] |
| lemma matrixValue_bump (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (P : GradedWords ι) : |
| matrixValue j a J (bump P)=matrixValue j a J P := by |
| simp [matrixValue,bump,List.map_map,Function.comp_def] |
| lemma matrixValue_scale (j c : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (P : GradedWords ι) : |
| matrixValue j a J (scale c P)=c • matrixValue j a J P := by |
| induction P with |
| | nil => simp [matrixValue,scale] |
| | cons t P ih => |
| simp only [scale,List.map_cons,matrixValue,List.sum_cons] |
| change (c*t.1) • _+matrixValue j a J (scale c P)=c • (t.1 • _+matrixValue j a J P) |
| rw [ih,smul_add,smul_smul] |
| lemma matrixValue_prepend (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (P : GradedWords ι) (l : WordLetter ι) : |
| matrixValue j a J (prepend l P)=l.exactEval j a J*matrixValue j a J P := by |
| induction P with |
| | nil => simp [matrixValue,prepend] |
| | cons t P ih => |
| simp only [prepend,List.map_cons,matrixValue,List.sum_cons] |
| change t.1 • (l.exactEval j a J*exactWord j a t.2.2 J)+matrixValue j a J (prepend l P)=_ |
| rw [ih,mul_add,mul_smul_comm] |
| rfl |
| lemma matrixValue_ordinaryWords (j : ℝ) (a : ι→ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) : |
| (matrixValue j a J (ordinaryWords j t).1,matrixValue j a J (ordinaryWords j t).2)= |
| (t.sourceMatrix j J,t.fieldMatrix j J) := by |
| induction t with |
| | leaf c => simp [ordinaryWords,matrixValue_bump,matrixValue_prepend,matrixValue_scalar, |
| SourceTree.sourceMatrix_leaf,SourceTree.fieldMatrix_leaf,WordLetter.exactEval] |
| | branch p t u ht hu => |
| have hs:=congrArg Prod.fst ht |
| have hf:=congrArg Prod.snd ht |
| have us:=congrArg Prod.fst hu |
| have uf:=congrArg Prod.snd hu |
| dsimp only at hs hf us uf |
| simp only [ordinaryWords,branch,matrixValue_append,matrixValue_bump,matrixValue_prepend, |
| matrixValue_scale,WordLetter.exactEval,hs,hf,us,uf, |
| SourceTree.sourceMatrix_branch,SourceTree.fieldMatrix_branch,neg_smul,sub_eq_add_neg] |
| lemma matrixValue_implicitSource (j : ℝ) (a p : ι→ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) : |
| matrixValue j a J (implicitSource j a p t)=WordLetter.inverse.exactEval j a J* |
| (Matrix.diagonal p*t.fieldMatrix j J-(j*Diagram.mean p) • (Matrix.diagonal a*t.sourceMatrix j J)) := by |
| have hs:=congrArg Prod.fst (matrixValue_ordinaryWords j a J t) |
| have hf:=congrArg Prod.snd (matrixValue_ordinaryWords j a J t) |
| dsimp only at hs hf |
| simp only [implicitSource,matrixValue_prepend,matrixValue_append,matrixValue_scale, |
| matrixValue_bump,WordLetter.exactEval,hs,hf,neg_smul,sub_eq_add_neg] |
| lemma matrixValue_implicitField (j : ℝ) (a p : ι→ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) : |
| matrixValue j a J (implicitField j a p t)= |
| (J-(j*Diagram.mean a) • 1)*matrixValue j a J (implicitSource j a p t)- |
| (j*Diagram.mean p) • t.sourceMatrix j J := by |
| have hs:=congrArg Prod.fst (matrixValue_ordinaryWords j a J t) |
| dsimp only at hs |
| simp only [implicitField,matrixValue_append,matrixValue_bump,matrixValue_prepend, |
| matrixValue_scale,WordLetter.exactEval,hs,neg_smul,smul_mul_assoc,one_mul, |
| sub_eq_add_neg,add_mul,neg_mul] |
|
|
| end SKGap.Noncrossing.Primary.Tensor.Series.GradedWords |
|
|
| end |
| end |
|
|
| end OAI |
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|
