openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ExpandingArrayLocalSupportField.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ArrayTrajectories | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.CylinderDLift | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.CompactTails | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.ExpandingArray | |
| open scoped BigOperators Topology ContDiff | |
| open Set Function Filter | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology BigOperators ContDiff | |
| open scoped Topology ContDiff BigOperators | |
| open scoped Topology ContDiff ZeroAtInfty | |
| open Stacks TailSpace | |
| variable {N b : ℕ} (hb : 0 < b) (table : Fin N → Fin b → Option (Rule (Fin N) b)) | |
| (terminal : Fin N) (ν : ℝ) (m : ℕ) | |
| theorem localSupport_field (hν : 0 < ν) (i : Fin 2) : | |
| LocalHorizontalSupport (fun t X => field hb table terminal ν m t X i) := by | |
| intro T hT | |
| obtain ⟨K, hK, hzero⟩ := finiteCylinderSupport hb table terminal ν m hν T | |
| refine ⟨K, hK, ?_⟩ | |
| intro t ht X hX | |
| by_cases ht0 : 0 ≤ t | |
| · exact congrFun (hzero t ht0 ((le_abs_self t).trans ht) X hX) i | |
| · have h := field_at_rest hb table terminal ν m hν (show t ≤ 1 by linarith) X | |
| exact congrFun h i | |
| theorem field_C1Tails (hν : 0 < ν) (i : Fin 2) : | |
| C1Tails (fun t X => field hb table terminal ν m t X i) := | |
| c1Tails_of_smooth_support ((contDiff_apply ℝ ℝ i).comp (contDiff_field hb table terminal ν m hν)) | |
| (localSupport_field hb table terminal ν m hν i) | |
| theorem field_D_Tails (hν : 0 < ν) (i k : Fin 2) : | |
| ContinuousTails (spatialD k (fun t X => field hb table terminal ν m t X i)) := | |
| (spatial_c1Tails_of_smooth_support (f := fun t X => field hb table terminal ν m t X i) | |
| ((contDiff_apply ℝ ℝ i).comp (contDiff_field hb table terminal ν m hν)) | |
| (localSupport_field hb table terminal ν m hν i) k).continuous | |
| theorem field_DD_Tails (hν : 0 < ν) (i k l : Fin 2) : | |
| ContinuousTails (spatialD l (spatialD k (fun t X => field hb table terminal ν m t X i))) := | |
| second_continuousTails_of_smooth_support (f := fun t X => field hb table terminal ν m t X i) | |
| ((contDiff_apply ℝ ℝ i).comp (contDiff_field hb table terminal ν m hν)) | |
| (localSupport_field hb table terminal ν m hν i) k l | |
| theorem comparisonClass_of_scalar_tails (hν : 0 < ν) {ρ : ScalarField 2} | |
| (hρ : ContDiff ℝ ∞ (uncurry ρ)) (hr : C1Tails ρ) | |
| (hD : ∀ i, ContinuousTails (spatialD i ρ)) | |
| (hDD : ∀ i k, ContinuousTails (spatialD k (spatialD i ρ))) : | |
| Cylinder.ComparisonClass | |
| (fun t x => liftVelocity (field hb table terminal ν m) ρ t (Cylinder.join x)) | |
| (fun _ _ => 0) := by | |
| let v : ℝ → Coord 2 → Cylinder.Vect := fun t X => | |
| ![field hb table terminal ν m t X 0, field hb table terminal ν m t X 1, ρ t X] | |
| have heq : (fun t x => liftVelocity (field hb table terminal ν m) ρ t (Cylinder.join x)) = | |
| (fun (t : ℝ) (x : Cylinder.Space) => v t x.1) := by | |
| funext t x | |
| have hh : horizontal (Cylinder.join x) = x.1 := | |
| congrArg Prod.fst (Cylinder.split_join x) | |
| simp only [liftVelocity, hh, v] | |
| rw [heq] | |
| apply Cylinder.comparisonClass_lift_of_tails v | |
| · intro i | |
| fin_cases i | |
| · exact (contDiff_apply ℝ ℝ (0 : Fin 2)).comp (contDiff_field hb table terminal ν m hν) | |
| · exact (contDiff_apply ℝ ℝ (1 : Fin 2)).comp (contDiff_field hb table terminal ν m hν) | |
| · exact hρ | |
| · intro i | |
| fin_cases i | |
| · exact field_C1Tails hb table terminal ν m hν 0 | |
| · exact field_C1Tails hb table terminal ν m hν 1 | |
| · exact hr | |
| · intro i k | |
| fin_cases i | |
| · exact field_D_Tails hb table terminal ν m hν 0 k | |
| · exact field_D_Tails hb table terminal ν m hν 1 k | |
| · exact hD k | |
| · intro i k l | |
| fin_cases i | |
| · exact field_DD_Tails hb table terminal ν m hν 0 k l | |
| · exact field_DD_Tails hb table terminal ν m hν 1 k l | |
| · exact hDD k l | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |