| 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 |
|
|
