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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 | 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
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