openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ClockedLipschitz.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.ClockedBounds | |
| namespace OAI | |
| /-! A computable modulus on each finite time slab for the detector's | |
| clocked expressions. Spatial coordinates remain unrestricted. -/ | |
| namespace ForcedComputation.ClockedExpr | |
| open ShearFlows Set | |
| open scoped ContDiff BigOperators | |
| def derivativeBound (e : ClockedExpr) (M : ℚ) : ℚ := | |
| ∑ j : Fin 4, (e.diff j).bound M | |
| theorem derivativeBound_nonneg (e : ClockedExpr) (M : ℚ) : 0 ≤ e.derivativeBound M := | |
| Finset.sum_nonneg (fun j _ => (e.diff j).bound_nonneg M) | |
| theorem fderiv_bound {e : ClockedExpr} (he : e.Valid) (M : ℚ) (y : SpaceTime) | |
| (hy : |y.1| ≤ |(M : ℝ)|) : ‖fderiv ℝ e.val y‖ ≤ (e.derivativeBound M : ℝ) := by | |
| apply ContinuousLinearMap.opNorm_le_bound _ (Rat.cast_nonneg.mpr (e.derivativeBound_nonneg M)) | |
| intro v | |
| conv_lhs => rw [← spaceTimeDirections_sum v] | |
| rw [map_sum] | |
| calc | |
| _ ≤ ∑ j : Fin 4, ‖fderiv ℝ e.val y (timeSpaceCoord j v • spaceTimeDirection j)‖ := | |
| norm_sum_le _ _ | |
| _ ≤ ∑ j : Fin 4, ‖v‖ * ((e.diff j).bound M : ℝ) := by | |
| apply Finset.sum_le_sum | |
| intro j _ | |
| rw [map_smul, norm_smul] | |
| have hj : ‖fderiv ℝ e.val y (spaceTimeDirection j)‖ ≤ ((e.diff j).bound M : ℝ) := by | |
| rw [← e.val_diff he j y, Real.norm_eq_abs] | |
| exact val_bound (valid_diff he j) M y hy | |
| exact mul_le_mul (timeSpaceCoord_norm j v) hj (norm_nonneg _) (norm_nonneg _) | |
| _ = _ := by | |
| rw [derivativeBound, Rat.cast_sum, Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ring | |
| theorem lipschitz_bound {e : ClockedExpr} (he : e.Valid) (M : ℚ) (x y : SpaceTime) | |
| (hx : |x.1| ≤ |(M : ℝ)|) (hy : |y.1| ≤ |(M : ℝ)|) : | |
| |e.val x - e.val y| ≤ (e.derivativeBound M : ℝ) * ‖x - y‖ := by | |
| let S : Set SpaceTime := Icc (-|(M : ℝ)|) |(M : ℝ)| ×ˢ (univ : Set Space) | |
| have hc : Convex ℝ S := (convex_Icc (-|(M : ℝ)|) |(M : ℝ)|).prod convex_univ | |
| have hxS : x ∈ S := ⟨abs_le.mp hx, mem_univ _⟩ | |
| have hyS : y ∈ S := ⟨abs_le.mp hy, mem_univ _⟩ | |
| have h := Convex.norm_image_sub_le_of_norm_fderiv_le | |
| (fun z (_ : z ∈ S) => (e.smooth he).differentiable (by simp) z) | |
| (fun z hz => fderiv_bound he M z (abs_le.mpr hz.1)) hc hyS hxS | |
| simpa only [Real.norm_eq_abs] using h | |
| end ForcedComputation.ClockedExpr | |
| end OAI | |