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37bcf52 | 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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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