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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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import Mathlib
import OAI.Probability.SKGap.Localization.GraphPrediction
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SKGap.Noncrossing.Primary.Tensor
open Diagram
variable {ι : Type*} [Fintype ι]
abbrev Space := ι→List ι→ℝ
def tensorValue : List (ι→ℝ)→List ι→ℝ
| [],[] => 1
| a::A,i::I => a i*tensorValue A I
| _,_ => 0
def realize : Primary.Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) :=
Finsupp.linearCombination ℝ (fun s i I =>s.1 i*tensorValue s.2 I)
omit [Fintype ι] in
lemma realize_ket (s : Stack (ι:=ι)) (i : ι) (I : List ι) :
realize (ket s) i I=s.1 i*tensorValue s.2 I := by simp [realize,ket]
def diagonal (a : ι→ℝ) : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where
toFun v i I := a i*v i I
map_add' v w := by ext i I; simp [mul_add]
map_smul' c v := by ext i I; simp [mul_left_comm]
def creation : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where
toFun v i I := match I with | [] => 0 | q::Q => v q Q
map_add' v w := by ext i I; cases I <;> simp
map_smul' c v := by ext i I; cases I <;> simp
def annihilation : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where
toFun v i I := Diagram.mean (fun q =>v q (i::I))
map_add' v w := by ext i I; simp [Diagram.mean,Finset.sum_add_distrib,add_div]
map_smul' c v := by ext i I; simp [Diagram.mean,←Finset.mul_sum,mul_div_assoc]
def noise (j : ℝ) : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) := creation+j • annihilation
def project : Space (ι:=ι)→ₗ[ℝ] (ι→ℝ) where
toFun v i := v i []
map_add' _ _ := rfl
map_smul' _ _ := rfl
def vacuum : Space (ι:=ι) := realize Primary.vacuum
def letter (j : ℝ) : Letter (ι→ℝ)→Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι)
| .diag a => diagonal a
| .noise => noise j
def word (j : ℝ) : List (Letter (ι→ℝ))→Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι)
| [] => LinearMap.id
| a::F => (letter j a).comp (word j F)
omit [Fintype ι] in
lemma realize_diagonal (a : ι→ℝ) (v : Primary.Space (ι:=ι)) :
realize (Primary.diagonal a v)=diagonal a (realize v) := by
have h : realize.comp (Primary.diagonal a)=(diagonal a).comp realize := by
apply Finsupp.lhom_ext'
intro s
ext i I
change realize (Primary.diagonal a (ket s)) i I = diagonal a (realize (ket s)) i I
rw [Primary.diagonal_ket,realize_ket]
change (a i*s.1 i)*tensorValue s.2 I=a i*realize (ket s) i I
rw [realize_ket]; ring
exact LinearMap.congr_fun h v
omit [Fintype ι] in
lemma realize_creation (v : Primary.Space (ι:=ι)) :
realize (Primary.creation v)=creation (realize v) := by
have h : (realize (ι:=ι)).comp Primary.creation=creation.comp realize := by
apply Finsupp.lhom_ext'
intro s
ext i I
change realize (Primary.creation (ket s)) i I = creation (realize (ket s)) i I
rw [Primary.creation_ket,realize_ket]
cases I <;> simp [creation,tensorValue,realize_ket]
exact LinearMap.congr_fun h v
lemma realize_annihilation (v : Primary.Space (ι:=ι)) :
realize (Primary.annihilation v)=annihilation (realize v) := by
have h : (realize (ι:=ι)).comp Primary.annihilation=annihilation.comp realize := by
apply Finsupp.lhom_ext'
rintro ⟨a,L⟩
ext i I
change realize (Primary.annihilation (ket (a,L))) i I = annihilation (realize (ket (a,L))) i I
cases L <;> simp [annihilation,realize_ket,tensorValue,Diagram.mean,←Finset.sum_mul,div_mul_eq_mul_div]
exact LinearMap.congr_fun h v
lemma realize_noise (j : ℝ) (v : Primary.Space (ι:=ι)) :
realize (Primary.noise j v)=noise j (realize v) := by
simp [Primary.noise,noise,realize_annihilation,realize_creation]
omit [Fintype ι] in
lemma project_realize (v : Primary.Space (ι:=ι)) : project (realize v)=Primary.project v := by
have h : (project (ι:=ι)).comp realize=Primary.project := by
apply Finsupp.lhom_ext'
rintro ⟨a,L⟩
ext i
change project (realize (ket (a,L))) i = Primary.project (ket (a,L)) i
cases L <;> simp [project,realize_ket,tensorValue]
exact LinearMap.congr_fun h v
lemma realize_word (j : ℝ) (F : List (Letter (ι→ℝ))) (v : Primary.Space (ι:=ι)) :
realize (Primary.word j F v)=word j F (realize v) := by
induction F generalizing v with
| nil => rfl
| cons a F ih =>
cases a <;> simp [Primary.word,Primary.letter,word,letter,realize_noise,realize_diagonal,ih]
omit [Fintype ι] in
lemma diagram_map_id (d : Diagram (ι→ℝ)) : d.map id=d := by
induction d <;> simp_all [Diagram.map]
lemma project_word (j : ℝ) (F : List (Letter (ι→ℝ))) (i : ι) :
project (word j F vacuum) i=WordSeries.ordinary j F i := by
rw [vacuum,←realize_word,project_realize,Primary.project_word]
simp [WordSeries.ordinary,Diagram.expect,Diagram.evaluate,Diagram.prediction,diagram_map_id]
lemma word_append (j : ℝ) (F G : List (Letter (ι→ℝ))) :
word j (F++G)=(word j F).comp (word j G) := by
induction F with
| nil => simp [word]
| cons a F ih => simp [word,ih,LinearMap.comp_assoc]
omit [Fintype ι] in
lemma diagonal_scale (c : ℝ) (a : ι→ℝ) : diagonal (fun i=>c*a i)=c • diagonal a := by
ext v i I
change (c*a i)*v i I=c*(a i*v i I)
ring
lemma annihilation_diagonal_creation (a : ι→ℝ) (v : Space (ι:=ι)) :
annihilation (diagonal a (creation v))=Diagram.mean a • v := by
ext i I
simp [annihilation,diagonal,creation,Diagram.mean,←Finset.sum_mul,mul_div_right_comm]
omit [Fintype ι] in
lemma project_diagonal_creation (a : ι→ℝ) (v : Space (ι:=ι)) :
project (diagonal a (creation v))=0 := by ext i; simp [project,diagonal,creation]
omit [Fintype ι] in
lemma project_creation (v : Space (ι:=ι)) : project (creation v)=0 := rfl
end SKGap.Noncrossing.Primary.Tensor
end
end
end OAI
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