File size: 1,499 Bytes
dd14c26 | 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 | 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
|