openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ExpandingArrayDriftData.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ArrayScalars | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.MildScalarContDiffScalar | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.SmoothProfilesLocalSupportImpulse | |
| 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 scoped Topology ContDiff ZeroAtInfty BigOperators | |
| open scoped Topology | |
| open Stacks TailSpace SmoothProfiles | |
| variable {N b : ℕ} (hb : 0 < b) (table : Fin N → Fin b → Option (Rule (Fin N) b)) | |
| (terminal : Fin N) (ν : ℝ) (m : ℕ) | |
| def driftData (hν : 0 < ν) (i : Fin 2) : MildScalar.SupportedData := | |
| ⟨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⟩ | |
| @[simp] theorem driftData_scalar (hν : 0 < ν) (i : Fin 2) (t : ℝ) (X : Coord 2) : | |
| (driftData hb table terminal ν m hν i).scalar t X = field hb table terminal ν m t X i := rfl | |
| def scalarSolution (hν : 0 < ν) (p : Coord 2) : ScalarField 2 := | |
| MildScalar.scalar hν (driftData hb table terminal ν m hν) (impulseData p) | |
| theorem contDiff_scalarSolution (hν : 0 < ν) (p : Coord 2) : | |
| ContDiff ℝ ∞ (uncurry (scalarSolution hb table terminal ν m hν p)) := | |
| MildScalar.contDiff_scalar hν _ _ (impulseData_nonpos p) | |
| theorem scalarSolution_C1Tails (hν : 0 < ν) (p : Coord 2) : | |
| C1Tails (scalarSolution hb table terminal ν m hν p) := | |
| MildScalar.scalar_C1Tails hν _ _ | |
| theorem scalarSolution_D_Tails (hν : 0 < ν) (p : Coord 2) (i : Fin 2) : | |
| ContinuousTails (spatialD i (scalarSolution hb table terminal ν m hν p)) := | |
| MildScalar.scalar_D_Tails hν _ _ i | |
| theorem scalarSolution_DD_Tails (hν : 0 < ν) (p : Coord 2) (i j : Fin 2) : | |
| ContinuousTails (spatialD j (spatialD i (scalarSolution hb table terminal ν m hν p))) := | |
| MildScalar.scalar_DD_Tails hν _ _ i j | |
| theorem scalarSolution_equation (hν : 0 < ν) (p : Coord 2) {t : ℝ} (ht : 0 ≤ t) (X : Coord 2) : | |
| timeD (scalarSolution hb table terminal ν m hν p) t X + | |
| advection (field hb table terminal ν m) (scalarSolution hb table terminal ν m hν p) t X = | |
| ν * laplacian (scalarSolution hb table terminal ν m hν p) t X + impulse p t X := by | |
| exact MildScalar.scalar_equation hν (driftData hb table terminal ν m hν) (impulseData p) | |
| (fun t _ X => divergence_field hb table terminal ν m hν t X) ht X | |
| @[simp] theorem scalarSolution_zero (hν : 0 < ν) (p X : Coord 2) : | |
| scalarSolution hb table terminal ν m hν p 0 X = 0 := | |
| MildScalar.scalar_zero hν _ _ X | |
| theorem scalarSolution_properties (hν : 0 < ν) (p : Coord 2) : | |
| (∀ t ≥ 0, ∀ X, 0 ≤ scalarSolution hb table terminal ν m hν p t X) ∧ | |
| (∀ t, Integrable (scalarSolution hb table terminal ν m hν p t)) ∧ | |
| (∀ t ≥ 0, (∫ X, scalarSolution hb table terminal ν m hν p t X) = step t) := | |
| scalar_properties_of_C1Tails hb table terminal ν m hν p | |
| (contDiff_scalarSolution hb table terminal ν m hν p) | |
| (scalarSolution_C1Tails hb table terminal ν m hν p) | |
| (scalarSolution_D_Tails hb table terminal ν m hν p) | |
| (scalarSolution_DD_Tails hb table terminal ν m hν p) | |
| (fun _ ht => scalarSolution_equation hb table terminal ν m hν p ht) | |
| (scalarSolution_zero hb table terminal ν m hν p) | |
| def velocity (hν : 0 < ν) (p : Coord 2) : VectorField 3 := | |
| liftVelocity (field hb table terminal ν m) (scalarSolution hb table terminal ν m hν p) | |
| theorem contDiff_velocity (hν : 0 < ν) (p : Coord 2) : | |
| ContDiff ℝ ∞ (uncurry (velocity hb table terminal ν m hν p)) := by | |
| convert JointCalculus.contDiff_lift (contDiff_field hb table terminal ν m hν) | |
| (contDiff_scalarSolution hb table terminal ν m hν p) using 1 | |
| funext q | |
| simp only [uncurry, velocity, liftVelocity, JointCalculus.lift, JointCalculus.projection_apply] | |
| theorem velocity_comparisonClass (hν : 0 < ν) (p : Coord 2) : | |
| Cylinder.ComparisonClass | |
| (fun t x => velocity hb table terminal ν m hν p t (Cylinder.join x)) (fun _ _ => 0) := | |
| comparisonClass_of_scalar_tails hb table terminal ν m hν | |
| (contDiff_scalarSolution hb table terminal ν m hν p) | |
| (scalarSolution_C1Tails hb table terminal ν m hν p) | |
| (scalarSolution_D_Tails hb table terminal ν m hν p) | |
| (scalarSolution_DD_Tails hb table terminal ν m hν p) | |
| theorem velocity_NavierStokes (hν : 0 < ν) (p : Coord 2) : | |
| NavierStokes ν (velocity hb table terminal ν m hν p) (fun _ _ => 0) | |
| (force hb table terminal ν m p) := | |
| force_NavierStokes hb table terminal ν m hν p _ | |
| (fun _ ht => scalarSolution_equation hb table terminal ν m hν p ht) | |
| (scalarSolution_zero hb table terminal ν m hν p) | |
| theorem velocity_unique (hν : 0 < ν) (p : Coord 2) {v : VectorField 3} {q : ScalarField 3} | |
| (hv : Cylinder.ComparisonClass (fun t x => v t (Cylinder.join x)) | |
| (fun t x => q t (Cylinder.join x))) | |
| (hNS : NavierStokes ν v q (force hb table terminal ν m p)) : | |
| ∀ t ≥ 0, (∀ x, v t x = velocity hb table terminal ν m hν p t x) ∧ (∀ x, q t x = 0) := | |
| Cylinder.comparison_unique_coordinates hν.le | |
| (velocity_comparisonClass hb table terminal ν m hν p) hv | |
| (velocity_NavierStokes hb table terminal ν m hν p) hNS | |
| theorem velocity_detection (hν : 0 < ν) (c₀ : Configuration (Fin N)) | |
| (hm : 1 ≤ m) (hc₀ : c₀.leftStack < capacity b m 0 ∧ c₀.rightStack < capacity b m 0) | |
| (hdir : IncomingDirection table) (hin : IncomingRule table) | |
| (hterm : NoOutgoing table terminal) (hcont : Continues hb table terminal c₀) | |
| (hinit : c₀.state ≠ terminal) : | |
| Observation.planeEvent | |
| (velocity hb table terminal ν m hν (point hb table terminal ν m c₀ 0)) ↔ | |
| Reaches hb table terminal c₀ := | |
| detection_of_C1Tails hb table terminal ν m hν c₀ | |
| (contDiff_scalarSolution hb table terminal ν m hν _) | |
| (scalarSolution_C1Tails hb table terminal ν m hν _) | |
| (scalarSolution_D_Tails hb table terminal ν m hν _) | |
| (scalarSolution_DD_Tails hb table terminal ν m hν _) | |
| (fun _ ht => scalarSolution_equation hb table terminal ν m hν _ ht) | |
| (scalarSolution_zero hb table terminal ν m hν _) hm hc₀ hdir hin hterm hcont hinit | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |