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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 | import Mathlib
import OAI.Probability.SKGap.Matrix.ClosedErrorTrace
import OAI.Probability.SKGap.Localization.StartedGraph
import OAI.Probability.SKGap.Stability.LocalImplicitNode
import OAI.Probability.SKGap.Stability.ImplicitTraceControl
import OAI.Probability.SKGap.Matrix.StartedNodeTrace
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SKGapCutoff.Recipe.OrdinaryData
open Primary Matrix SKGap SKGap.Noncrossing SKGap.Noncrossing.Primary SKGap.Noncrossing.Primary.Tensor.Series
variable {n : ℕ} {ι κ σ κ₀ σ₀ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] [Fintype σ]
variable [Fintype κ₀] [DecidableEq κ₀] [Fintype σ₀]
variable (D : OrdinaryData n ι κ σ) (I : OrdinaryData n Unit κ₀ σ₀)
variable (w y : VectorFields n) (T : ι→SourceTree (Fin n→ℝ)) (t : SourceTree (Fin n→ℝ))
lemma started_implicit_initial_errors_local (hj : D.j=I.j) (hJ : D.J=I.J)
(x : Spin n)
(hW : w x=I.sourceOf 1 (fun _=>y) x)
(hWf : ∀i,w (flip x i)=I.sourceOf 1 (fun _=>y) (flip x i))
(hY : y x=I.fieldOf 1 (fun _=>w) (fun _=>y) x)
(hYf : ∀i,y (flip x i)=I.fieldOf 1 (fun _=>w) (fun _=>y) (flip x i))
(ha : ∀i,0≤I.implicitCoefficient x i)
(hS : (1-SKGap.pathDiagonal (I.implicitCoefficient x) 1*
(I.J-(I.j*siteMean I.implicitCoefficient x) • 1)*SKGap.pathDiagonal (I.implicitCoefficient x) 1).PosDef) :
let t₀:=ClosedTree.implicit (I.implicitPartial y x) t
let a₀:=I.implicitCoefficient x
D.startedSourceError w y T t₀ a₀ x 0=
WordLetter.inverse.exactEval D.j a₀ D.J*
(I.implicitSourcePrimitive t y x+Matrix.diagonal a₀*I.implicitFieldPrimitive t w y x) ∧
D.startedFieldError w y T t₀ a₀ x 0=
(D.J-(I.j*siteMean I.implicitCoefficient x) • 1)*D.startedSourceError w y T t₀ a₀ x 0+
I.implicitFieldPrimitive t w y x := by
have H:=I.implicit_error_solved_local t w y x hW hWf hY hYf (I.implicit_inverse_left x ha hS)
dsimp only
simp only [startedSourceError_zero,startedFieldError_zero,hj,hJ,ClosedTree.sourceMatrix,
ClosedTree.fieldMatrix,ClosedTree.words]
exact H
theorem implicit_started_error_words_local (hj : D.j=I.j) (hJ : D.J=I.J)
(x : Spin n)
(hW : w x=I.sourceOf 1 (fun _=>y) x)
(hWf : ∀i,w (flip x i)=I.sourceOf 1 (fun _=>y) (flip x i))
(hY : y x=I.fieldOf 1 (fun _=>w) (fun _=>y) x)
(hYf : ∀i,y (flip x i)=I.fieldOf 1 (fun _=>w) (fun _=>y) (flip x i))
(ha0 : ∀i,0≤I.implicitCoefficient x i)
(hS : (1-SKGap.pathDiagonal (I.implicitCoefficient x) 1*
(I.J-(I.j*siteMean I.implicitCoefficient x) • 1)*SKGap.pathDiagonal (I.implicitCoefficient x) 1).PosDef)
{C F A R M : ℝ} {Cs Cf : ℕ→ℝ} {N L : ℕ}
(hP : TraceControl (I.implicitSourcePrimitive t y x) C)
(hQ : TraceControl (I.implicitFieldPrimitive t w y x) F)
(hR : 0≤R) (hM : 0≤M) (ha : ∀i,|I.implicitCoefficient x i|≤A)
(hp : ∀a≤N,∀b i,|D.auxCoefficient a b x i|≤A)
(hr : ∀a≤N,∀b,|D.j*siteMean (D.auxCoefficient a b) x|≤R)
(hs : ∀a≤N,0<a→TraceControl (D.startedNodeSourceError w y T x a) (Cs a))
(hf : ∀a≤N,0<a→TraceControl (D.startedNodeFieldError w y T x a) (Cf a))
(hw : ClosedWordTestBound D.j (I.implicitCoefficient x) D.J A M (L+2)) :
let t₀:=ClosedTree.implicit (I.implicitPartial y x) t
let a₀:=I.implicitCoefficient x
let q:=I.j*siteMean I.implicitCoefficient x
let Bs:=fun a=>if a=0 then C+F else Cs a
let Bf:=fun a=>if a=0 then |q| *(C+F)+F else Cf a
∀a≤N,
(∀v:OrdinaryWord n,v.length+2*a+1≤L→wordBounded A v→
|trace (matrixWord D.J v*D.startedSourceError w y T t₀ a₀ x a)|≤(errorBudget R Bs Bf a).1*M) ∧
(∀v:OrdinaryWord n,v.length+2*a+2≤L→wordBounded A v→
|trace (matrixWord D.J v*D.startedFieldError w y T t₀ a₀ x a)|≤(errorBudget R Bs Bf a).2*M) := by
obtain ⟨HE,HF⟩:=D.started_implicit_initial_errors_local I w y T t hj hJ x hW hWf hY hYf ha0 hS
exact D.started_derivative_error_words w y T _ _ x
(I.implicitSourcePrimitive t y x) (I.implicitFieldPrimitive t w y x)
(I.j*siteMean I.implicitCoefficient x) HE HF hP hQ hR hM ha hp hr hs hf hw
end SKGapCutoff.Recipe.OrdinaryData
end
end
end OAI
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