| import Mathlib |
| import OAI.Probability.SKGap.Stability.RootDiagonalTransfer |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace SKGap.ObservationBridge |
| open MeasureTheory ProbabilityTheory Matrix Real Set |
| open scoped BigOperators |
| variable {n : ℕ} |
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| lemma tapField_field_difference (j : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) |
| (h h' y : Field n) : tapField j J h' y=tapField j J h y+(h-h') := by |
| ext i |
| simp only [tapField,Pi.add_apply,Pi.sub_apply] |
| ring |
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| lemma rootBad_field_transfer {j A ε c ρ κ : ℝ} |
| (J : Matrix (Fin n) (Fin n) ℝ) (h h' : Field n) |
| (hd : vectorNorm (h-h') ≤ κ*sqrt (n:ℝ)) : |
| rootBad j A ε c ρ J h → rootBad j A ε c (ρ+κ) J h' := by |
| rintro ⟨y,hy,a,ha,hdist,x,hx,hq⟩ |
| refine ⟨y,?_,a,ha,hdist,x,hx,hq⟩ |
| rw [tapField_field_difference] |
| exact (vectorNorm_add_le _ _).trans ((add_le_add hy hd).trans_eq (by ring)) |
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| theorem rootBad_mesh_transfer {ι τ : Type*} (mesh : ι→τ) (h : τ→Field n) |
| (j A ε c ρ κ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) |
| (hcover : ∀ t : τ, ∃ k : ι,vectorNorm (h t-h (mesh k)) ≤ κ*sqrt (n:ℝ)) : |
| (∃ t,rootBad j A ε c ρ J (h t)) → ∃ k,rootBad j A ε c (ρ+κ) J (h (mesh k)) := by |
| rintro ⟨t,ht⟩ |
| obtain ⟨k,hk⟩ := hcover t |
| exact ⟨k,rootBad_field_transfer J (h t) (h (mesh k)) hk ht⟩ |
| end SKGap.ObservationBridge |
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| end |
| end |
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| end OAI |
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