| import Mathlib | |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionDerivative | |
| namespace OAI | |
| namespace ProjectionComparison | |
| open Set MeasureTheory | |
| variable {V E : Type*} [Fintype V] | |
| lemma project_path_lipschitz (S : System V E) (b v₀ v₁ : V → ℝ) (hne : (feasible S b).Nonempty) (x : V) : | |
| LipschitzWith ‖v₁-v₀‖₊ (fun t : ℝ => project S b hne (fun y => v₀ y+t*(v₁ y-v₀ y)) x) := by | |
| apply LipschitzWith.of_dist_le_mul | |
| intro s t | |
| rw [Real.dist_eq,Real.dist_eq] | |
| apply projection_parameter_stability S b b _ _ _ _ (‖v₁-v₀‖*|s-t|) (mul_nonneg (norm_nonneg _) (abs_nonneg _)) | |
| (project_spec S b hne _) (project_spec S b hne _) | |
| · intro y hy;simpa using mul_nonneg (norm_nonneg (v₁-v₀)) (abs_nonneg (s-t)) | |
| · intro y hy | |
| have hn : |v₁ y-v₀ y|≤‖v₁-v₀‖ := by simpa only [Real.norm_eq_abs,Pi.sub_apply] using norm_le_pi_norm (v₁-v₀) y | |
| calc | |
| |(v₀ y+s*(v₁ y-v₀ y))-(v₀ y+t*(v₁ y-v₀ y))|=|s-t| * |v₁ y-v₀ y| := by rw [← abs_mul];congr 1;ring | |
| _ ≤ ‖v₁-v₀‖*|s-t| := by nlinarith [abs_nonneg (s-t)] | |
| lemma project_path_ae_derivative (S : System V E) (b v₀ v₁ : V → ℝ) (hne : (feasible S b).Nonempty) : | |
| ∀ᵐ t : ℝ,∀ x,DifferentiableAt ℝ (fun s => project S b hne (fun y => v₀ y+s*(v₁ y-v₀ y)) x) t := by | |
| apply MeasureTheory.ae_all_iff.mpr | |
| intro x | |
| exact (project_path_lipschitz S b v₀ v₁ hne x).ae_differentiableAt_real | |
| end ProjectionComparison | |
| end OAI | |