File size: 8,007 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | import Mathlib
import OAI.Probability.SKGap.Matrix.WordFockSeries
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SKGap.Noncrossing.Primary.Tensor.Series
open Diagram InverseDiagram WordSeries Loop
variable {ι : Type*} [Fintype ι]
namespace GradedWords
def branch (j : ℝ) (p : ι→ℝ) (U V : GradedWords ι×GradedWords ι) : GradedWords ι×GradedWords ι :=
(prepend (.diag p) U.2++V.1,
(bump (prepend .noise (prepend (.diag p) U.2))++
scale (-(j*Diagram.mean p)) (bump (bump U.1)))++V.2)
def ordinaryWords (j : ℝ) : SourceTree (ι→ℝ)→GradedWords ι×GradedWords ι
| .leaf c => (scalar c,bump (prepend .noise (scalar c)))
| .branch p t u => branch j p (ordinaryWords j t) (ordinaryWords j u)
def implicitSource (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) : GradedWords ι :=
prepend .inverse (prepend (.diag p) (ordinaryWords j t).2++
scale (-(j*Diagram.mean p)) (bump (bump (prepend (.diag a) (ordinaryWords j t).1))))
def implicitField (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) : GradedWords ι :=
(bump (prepend .noise (implicitSource j a p t))++
scale (-(j*Diagram.mean a)) (bump (bump (implicitSource j a p t))))++
scale (-(j*Diagram.mean p)) (bump (bump (ordinaryWords j t).1))
def pairSeq (j : ℝ) (a : ι→ℝ) (P : GradedWords ι×GradedWords ι) :=
(seq j a P.1,seq j a P.2)
lemma pairSeq_branch (j : ℝ) (a p : ι→ℝ) (U V : GradedWords ι×GradedWords ι) :
pairSeq j a (branch j p U V)=
(branchSource p (pairSeq j a U) (pairSeq j a V),
branchField j p (pairSeq j a U) (pairSeq j a V)) := by
simp [pairSeq,branch,branchSource,branchField,seq_append,seq_bump,seq_prepend,
seq_scale,letterOperator,sub_eq_add_neg,add_assoc]
lemma pairSeq_ordinary (j : ℝ) (a : ι→ℝ) (t : SourceTree (ι→ℝ)) :
pairSeq j a (ordinaryWords j t)=ordinary j t := by
induction t with
| leaf c =>
simp [ordinaryWords,pairSeq,seq_scalar,seq_bump,seq_prepend,letterOperator,
ordinary,act_ofVector]
| branch p t u ih1 ih2 => simp only [ordinaryWords,pairSeq_branch,ih1,ih2,ordinary]
lemma seq_implicitSource (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) :
seq j a (implicitSource j a p t)=loopSource j a p t := by
have h1:=congrArg Prod.fst (pairSeq_ordinary j a t)
have h2:=congrArg Prod.snd (pairSeq_ordinary j a t)
simp only [pairSeq] at h1 h2
simp [implicitSource,seq_prepend,seq_append,seq_scale,seq_bump,letterOperator,
h1,h2,loopSource,sub_eq_add_neg]
lemma seq_implicitField (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) :
seq j a (implicitField j a p t)=loopField j a p t := by
have h1:=congrArg Prod.fst (pairSeq_ordinary j a t)
simp only [pairSeq] at h1
simp [implicitField,seq_prepend,seq_append,seq_scale,seq_bump,letterOperator,
h1,seq_implicitSource,loopField,sub_eq_add_neg,add_assoc]
section
omit [Fintype ι]
def bounded (k : ℕ) (P : GradedWords ι) : Prop := ∀ t∈P,inverseCount t.2.2≤k
lemma bounded_mono {k l : ℕ} (hkl : k≤l) {P : GradedWords ι} (h : bounded k P) : bounded l P :=
fun t ht=>(h t ht).trans hkl
lemma bounded_scalar (k : ℕ) (c : ℝ) : bounded k (scalar (ι:=ι) c) := by
rintro t ht
simp only [scalar,List.mem_singleton] at ht
subst t
simp [inverseCount]
lemma bounded_append {k : ℕ} {P Q : GradedWords ι} (hP : bounded k P) (hQ : bounded k Q) :
bounded k (P++Q) := by
intro t ht
rcases List.mem_append.mp ht with h|h
· exact hP t h
· exact hQ t h
lemma bounded_scale {k : ℕ} {P : GradedWords ι} (c : ℝ) (hP : bounded k P) : bounded k (scale c P) := by
intro t ht
obtain ⟨s,hs,rfl⟩:=List.mem_map.mp ht
exact hP s hs
lemma bounded_bump {k : ℕ} {P : GradedWords ι} (hP : bounded k P) : bounded k (bump P) := by
intro t ht
obtain ⟨s,hs,rfl⟩:=List.mem_map.mp ht
exact hP s hs
lemma bounded_diag {k : ℕ} {P : GradedWords ι} (d : ι→ℝ) (hP : bounded k P) :
bounded k (prepend (.diag d) P) := by
intro t ht
obtain ⟨s,hs,rfl⟩:=List.mem_map.mp ht
exact hP s hs
lemma bounded_noise {k : ℕ} {P : GradedWords ι} (hP : bounded k P) :
bounded k (prepend .noise P) := by
intro t ht
obtain ⟨s,hs,rfl⟩:=List.mem_map.mp ht
exact hP s hs
lemma bounded_inverse {k : ℕ} {P : GradedWords ι} (hP : bounded k P) :
bounded (k+1) (prepend .inverse P) := by
intro t ht
obtain ⟨s,hs,rfl⟩:=List.mem_map.mp ht
simpa only [inverseCount_inverse] using Nat.add_le_add_right (hP s hs) 1
end
lemma bounded_branch {k : ℕ} (j : ℝ) (p : ι→ℝ) {U V : GradedWords ι×GradedWords ι}
(hU : bounded k U.1 ∧ bounded k U.2) (hV : bounded k V.1 ∧ bounded k V.2) :
bounded k (branch j p U V).1 ∧ bounded k (branch j p U V).2 := by
exact ⟨bounded_append (bounded_diag _ hU.2) hV.1,
bounded_append (bounded_append (bounded_bump (bounded_noise (bounded_diag _ hU.2)))
(bounded_scale _ (bounded_bump (bounded_bump hU.1)))) hV.2⟩
lemma bounded_ordinary (j : ℝ) (t : SourceTree (ι→ℝ)) :
bounded 0 (ordinaryWords j t).1 ∧ bounded 0 (ordinaryWords j t).2 := by
induction t with
| leaf c => exact ⟨bounded_scalar 0 c,bounded_bump (bounded_noise (bounded_scalar 0 c))⟩
| branch p t u ih1 ih2 => exact bounded_branch j p ih1 ih2
lemma bounded_implicitSource (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) :
bounded 1 (implicitSource j a p t) := by
have h:=bounded_ordinary j t
exact bounded_inverse (bounded_append (bounded_diag _ h.2)
(bounded_scale _ (bounded_bump (bounded_bump (bounded_diag _ h.1)))))
lemma bounded_implicitField (j : ℝ) (a p : ι→ℝ) (t : SourceTree (ι→ℝ)) :
bounded 1 (implicitField j a p t) := by
have h:=bounded_implicitSource j a p t
exact bounded_append (bounded_append (bounded_bump (bounded_noise h))
(bounded_scale _ (bounded_bump (bounded_bump h))))
(bounded_scale _ (bounded_bump (bounded_bump (bounded_mono (by omega) (bounded_ordinary j t).1))))
end GradedWords
namespace ClosedTree
open GradedWords
def words (j : ℝ) (a : ι→ℝ) : ClosedTree (ι→ℝ)→GradedWords ι×GradedWords ι
| .leaf c => ordinaryWords j (.leaf c)
| .implicit p t => (implicitSource j a p t,implicitField j a p t)
| .branch p t u => GradedWords.branch j p (t.words j a) (u.words j a)
lemma words_series (j : ℝ) (a : ι→ℝ) (t : ClosedTree (ι→ℝ)) :
pairSeq j a (t.words j a)=t.series j a := by
induction t with
| leaf c => exact pairSeq_ordinary j a (.leaf c)
| implicit p t => exact Prod.ext (seq_implicitSource j a p t) (seq_implicitField j a p t)
| branch p t u ih1 ih2 => simp only [words,pairSeq_branch,ih1,ih2,series]
lemma words_bounded (j : ℝ) (a : ι→ℝ) (t : ClosedTree (ι→ℝ)) :
bounded 1 (t.words j a).1 ∧ bounded 1 (t.words j a).2 := by
induction t with
| leaf c =>
exact ⟨bounded_mono (by omega) (bounded_ordinary j (.leaf c)).1,
bounded_mono (by omega) (bounded_ordinary j (.leaf c)).2⟩
| implicit p t => exact ⟨bounded_implicitSource j a p t,bounded_implicitField j a p t⟩
| branch p t u ih1 ih2 => exact bounded_branch j p ih1 ih2
theorem field_prediction_zero (j : ℝ) (a : ι→ℝ) (t : ClosedTree (ι→ℝ)) (z : ℝ) (i : ι) :
prediction j a z (t.words j a).2 i=0 := by
apply prediction_zero_of_project j a _ (t.words_bounded j a).2
intro n
have h:=congrArg Prod.snd (t.words_series j a)
change seq j a (t.words j a).2=(t.series j a).2 at h
rw [h]
exact t.project_field j a n ▸ rfl
theorem source_prediction_zero (j : ℝ) (a : ι→ℝ) (t : ClosedTree (ι→ℝ))
(ht : t.leafMass=0) (z : ℝ) (i : ι) : prediction j a z (t.words j a).1 i=0 := by
apply prediction_zero_of_project j a _ (t.words_bounded j a).1
intro n
have h:=congrArg Prod.fst (t.words_series j a)
change seq j a (t.words j a).1=(t.series j a).1 at h
rw [h,t.project_source,ht]
split_ifs <;> rfl
end ClosedTree
end SKGap.Noncrossing.Primary.Tensor.Series
end
end
end OAI
|