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import OAI.Probability.SKGap.Localization.GraphPrediction
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SKGap.Noncrossing.Primary
open Diagram
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
inductive SourceTree (D : Type*) where
| leaf : ℝ → SourceTree D
| branch : D → SourceTree D → SourceTree D → SourceTree D
namespace SourceTree
variable {D : Type*}
def leafMass : SourceTree D→ℝ
| .leaf c => c
| .branch _ _ b => leafMass b
def words (j : ℝ) : SourceTree (ι→ℝ)→WordPolynomial (ι:=ι)×WordPolynomial (ι:=ι)
| .leaf c => ([(c,[])],[(c,[.noise])])
| .branch p a b =>
(prependWords [.diag p] (words j a).2 ++ (words j b).1,
(prependWords [.noise,.diag p] (words j a).2 ++
scale (-(j*Diagram.mean p)) (words j a).1) ++ (words j b).2)
def sourceOperator (j : ℝ) (t : SourceTree (ι→ℝ)) := polynomialOperator j (t.words j).1
def fieldOperator (j : ℝ) (t : SourceTree (ι→ℝ)) := polynomialOperator j (t.words j).2
section
omit [DecidableEq ι]
lemma source_leaf (j c : ℝ) : sourceOperator (ι:=ι) j (.leaf c)=c • LinearMap.id := by
simp [sourceOperator,words,polynomialOperator,word]
lemma field_leaf (j c : ℝ) : fieldOperator (ι:=ι) j (.leaf c)=c • noise j := by
simp [fieldOperator,words,polynomialOperator,word,letter]
lemma source_branch (j : ℝ) (p : ι→ℝ) (a b : SourceTree (ι→ℝ)) :
sourceOperator j (.branch p a b)=(diagonal p).comp (fieldOperator j a)+sourceOperator j b := by
simp [sourceOperator,fieldOperator,words,polynomialOperator_append,
polynomialOperator_prefix,word,letter]
lemma field_branch (j : ℝ) (p : ι→ℝ) (a b : SourceTree (ι→ℝ)) :
fieldOperator j (.branch p a b)=
(noise j).comp ((diagonal p).comp (fieldOperator j a))-
(j*Diagram.mean p) • sourceOperator j a+fieldOperator j b := by
simp only [sourceOperator,fieldOperator,words,polynomialOperator_append,
polynomialOperator_prefix,polynomialOperator_scale,word,letter,LinearMap.comp_id,
LinearMap.comp_assoc]
ext v s
simp [LinearMap.sub_apply,LinearMap.smul_apply,sub_eq_add_neg]
theorem field_vacuum (j : ℝ) (t : SourceTree (ι→ℝ)) :
fieldOperator j t vacuum=creation (sourceOperator j t vacuum) := by
induction t with
| leaf c => simp [field_leaf,source_leaf,noise,vacuum]
| branch p a b ia ib =>
rw [field_branch,source_branch]
simp only [LinearMap.add_apply,LinearMap.sub_apply,LinearMap.comp_apply,
LinearMap.smul_apply,ia,ib,map_add,noise]
simp only [annihilation_diagonal_creation,smul_smul]
abel
theorem field_prediction_zero (j : ℝ) (t : SourceTree (ι→ℝ)) (i : ι) :
polynomialPrediction j (t.words j).2 i=0 := by
rw [polynomialPrediction_eq_project]
change project (fieldOperator j t vacuum) i=0
rw [field_vacuum,project_creation]; rfl
theorem source_prediction (j : ℝ) (t : SourceTree (ι→ℝ)) (i : ι) :
polynomialPrediction j (t.words j).1 i=t.leafMass := by
rw [polynomialPrediction_eq_project]
change project (sourceOperator j t vacuum) i=t.leafMass
induction t with
| leaf c => simp [source_leaf,leafMass,vacuum,project,ket]
| branch p a b ia ib =>
rw [source_branch]
simp only [LinearMap.add_apply,LinearMap.comp_apply,map_add,field_vacuum,
project_diagonal_creation,zero_add,leafMass]
exact ib
end
def sourceMatrix (j : ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) :=
matrixPolynomial J (t.words j).1
def fieldMatrix (j : ℝ) (J : Matrix ι ι ℝ) (t : SourceTree (ι→ℝ)) :=
matrixPolynomial J (t.words j).2
lemma sourceMatrix_leaf (j c : ℝ) (J : Matrix ι ι ℝ) :
sourceMatrix j J (.leaf c)=c • 1 := by simp [sourceMatrix,words,matrixPolynomial,matrixWord]
lemma fieldMatrix_leaf (j c : ℝ) (J : Matrix ι ι ℝ) :
fieldMatrix j J (.leaf c)=c • J := by
simp [fieldMatrix,words,matrixPolynomial,matrixWord,matrixLetter]
lemma sourceMatrix_branch (j : ℝ) (J : Matrix ι ι ℝ) (p : ι→ℝ) (a b : SourceTree (ι→ℝ)) :
sourceMatrix j J (.branch p a b)=Matrix.diagonal p*fieldMatrix j J a+sourceMatrix j J b := by
simp [sourceMatrix,fieldMatrix,words,matrixPolynomial_append,matrixPolynomial_prefix,
matrixWord,matrixLetter]
lemma fieldMatrix_branch (j : ℝ) (J : Matrix ι ι ℝ) (p : ι→ℝ) (a b : SourceTree (ι→ℝ)) :
fieldMatrix j J (.branch p a b)=J*(Matrix.diagonal p*fieldMatrix j J a)-
(j*Diagram.mean p) • sourceMatrix j J a+fieldMatrix j J b := by
simp [sourceMatrix,fieldMatrix,words,matrixPolynomial_append,matrixPolynomial_prefix,
matrixPolynomial_scale,matrixWord,matrixLetter,mul_assoc,sub_eq_add_neg,add_assoc]
end SourceTree
end SKGap.Noncrossing.Primary
end
end
end OAI
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