openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ArrayMass.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArrayLoss | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ParabolicComparison | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ScalarMass | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.SmoothProfilesIntegrableImpulse | |
| 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 Stacks SmoothProfiles | |
| open scoped ZeroAtInfty | |
| 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 compactSupport_field_component (hν : 0 < ν) {t : ℝ} (ht : 0 ≤ t) (i : Fin 2) : | |
| HasCompactSupport (fun X => field hb table terminal ν m t X i) := by | |
| obtain ⟨K, hK, hzero⟩ := finiteCylinderSupport hb table terminal ν m hν t | |
| apply HasCompactSupport.of_support_subset_isCompact hK | |
| intro X hX | |
| by_contra hnot | |
| exact hX (congrFun (hzero t ht le_rfl X hnot) i) | |
| theorem scalar_mass (hν : 0 < ν) (p : Coord 2) {ρ : ScalarField 2} | |
| {r dr : ℝ → Lp ℝ 1 (volume : Measure (Coord 2))} | |
| (hρ : ContDiff ℝ 2 (uncurry ρ)) | |
| (hi : ∀ t, 0 ≤ t → ∀ i, Integrable (SpatialCalculus.partialD i (ρ t))) | |
| (hii : ∀ t, 0 ≤ t → ∀ i, | |
| Integrable (SpatialCalculus.partialD i (SpatialCalculus.partialD i (ρ t)))) | |
| (heq : ∀ t, 0 ≤ t → ∀ X, | |
| deriv (fun s => ρ s X) t + advection (field hb table terminal ν m) ρ t X = | |
| ν * laplacian ρ t X + impulse p t X) | |
| (hrep : ∀ t, (fun X => r t X) =ᵐ[volume] ρ t) | |
| (hdrep : ∀ t, 0 ≤ t → (fun X => dr t X) =ᵐ[volume] (fun X => deriv (fun s => ρ s X) t)) | |
| (hr : ∀ t, 0 ≤ t → HasDerivWithinAt r (dr t) (Ici 0) t) | |
| (hzero : ∀ X, ρ 0 X = 0) : ∀ t, 0 ≤ t → (∫ X, ρ t X) = step t := by | |
| refine ScalarMass.mass_eq_primitive hρ hi hii (fun t _ => integrable_impulse p t) | |
| (fun t _ i => field_slice_component hb table terminal ν m hν t i) | |
| (fun t ht i => compactSupport_field_component hb table terminal ν m hν ht i) | |
| (fun t _ X => divergence_field hb table terminal ν m hν t X) | |
| heq hrep hdrep hr ?_ ?_ | |
| · intro t _ | |
| rw [integral_impulse] | |
| exact (differentiable_step t).hasDerivAt.hasDerivWithinAt | |
| · simp only [hzero, integral_zero] | |
| exact Real.smoothTransition.zero.symm | |
| theorem scalar_detection_of_tail_regular (c₀ : Configuration (Fin N)) | |
| (hν : 0 < ν) (ρ : ℝ → C₀(Coord 2, ℝ)) | |
| {r dr : ℝ → Lp ℝ 1 (volume : Measure (Coord 2))} | |
| (hρc : ContinuousOn ρ (Ici 0)) | |
| (hρ : ContDiff ℝ 2 (fun q : ℝ × Coord 2 => ρ q.1 q.2)) | |
| (hi : ∀ t, 0 ≤ t → ∀ i, Integrable (SpatialCalculus.partialD i (fun X => ρ t X))) | |
| (hii : ∀ t, 0 ≤ t → ∀ i, Integrable | |
| (SpatialCalculus.partialD i (SpatialCalculus.partialD i (fun X => ρ t X)))) | |
| (heq : ∀ t, 0 ≤ t → ∀ X, | |
| deriv (fun s => ρ s X) t + advection (field hb table terminal ν m) (fun s Y => ρ s Y) t X = | |
| ν * laplacian (fun s Y => ρ s Y) t X + impulse (point hb table terminal ν m c₀ 0) t X) | |
| (hrep : ∀ t, (fun X => r t X) =ᵐ[volume] (fun X => ρ t X)) | |
| (hdrep : ∀ t, 0 ≤ t → (fun X => dr t X) =ᵐ[volume] (fun X => deriv (fun s => ρ s X) t)) | |
| (hr : ∀ t, 0 ≤ t → HasDerivWithinAt r (dr t) (Ici 0) t) | |
| (hzero : ∀ X, ρ 0 X = 0) (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 (liftVelocity (field hb table terminal ν m) (fun s Y => ρ s Y)) ↔ | |
| Reaches hb table terminal c₀ := by | |
| have hp := ParabolicComparison.nonnegative_of_continuous_C0 ρ hν.le hρc hρ heq | |
| (fun X => (hzero X).ge) | |
| (fun t _ X => impulse_nonneg (point hb table terminal ν m c₀ 0) X t) | |
| have hmass := scalar_mass (ρ := fun t X => ρ t X) hb table terminal ν m hν (point hb table terminal ν m c₀ 0) | |
| hρ hi hii heq hrep hdrep hr hzero | |
| exact scalar_detection (ρ := fun t X => ρ t X) hb table terminal ν m c₀ hν hρ heq hzero | |
| (fun t _ => ScalarMass.integrable_of_L1_rep (hrep t)) hp hmass hm hc₀ hdir hin hterm hcont hinit | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |