openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ArrayScalars.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArrayForce | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ArrayMass | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArrayLocalSupportField | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.TailSpaceToC0 | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.JointCalculusTimeDEqDeriv | |
| 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 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) | |
| theorem scalar_properties_of_C1Tails (hν : 0 < ν) (p : Coord 2) {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))) | |
| (heq : ∀ t ≥ 0, ∀ X, timeD ρ t X + advection (field hb table terminal ν m) ρ t X = | |
| ν * laplacian ρ t X + impulse p t X) | |
| (hzero : ∀ X, ρ 0 X = 0) : | |
| (∀ t ≥ 0, ∀ X, 0 ≤ ρ t X) ∧ (∀ t, Integrable (ρ t)) ∧ | |
| (∀ t ≥ 0, (∫ X, ρ t X) = step t) := by | |
| have hd (t : ℝ) (ht : 0 ≤ t) (X : Coord 2) : timeD ρ t X = deriv (fun s => ρ s X) t := | |
| JointCalculus.timeD_eq_deriv ht X (hρ.differentiable (by simp) (t, X)) | |
| have heq' (t : ℝ) (ht : 0 ≤ t) (X : Coord 2) : | |
| 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 := by | |
| rw [← hd t ht X] | |
| exact heq t ht X | |
| obtain ⟨F, hFc, hFr⟩ := hr.continuous.C0_rep | |
| have hF2 : ContDiff ℝ 2 (fun q : ℝ × Coord 2 => F q.1 q.2) := by | |
| simpa only [hFr, Function.uncurry_def] using hρ.of_le (show (2 : ℕ∞ω) ≤ ∞ by exact WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top)) | |
| have hp := ParabolicComparison.nonnegative_of_continuous_C0 F hν.le hFc hF2 | |
| (by simpa only [hFr] using heq') (fun X => by simpa only [hFr, hzero] using (le_refl (0 : ℝ))) | |
| (fun t _ X => impulse_nonneg p X t) | |
| obtain ⟨G, dG, -, -, hGa, hdGa, hGd⟩ := hr.L1_rep | |
| have hi (t : ℝ) (_ht : 0 ≤ t) (i : Fin 2) : Integrable (SpatialCalculus.partialD i (ρ t)) := | |
| (hD i).integrable t | |
| have hii (t : ℝ) (_ht : 0 ≤ t) (i : Fin 2) : | |
| Integrable (SpatialCalculus.partialD i (SpatialCalculus.partialD i (ρ t))) := | |
| (hDD i i).integrable t | |
| refine ⟨?_, fun t => hr.continuous.integrable t, ?_⟩ | |
| · simpa only [hFr] using hp | |
| · apply scalar_mass hb table terminal ν m hν p (hρ.of_le (by exact WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))) hi hii heq' hGa ?_ hGd hzero | |
| intro t ht | |
| have hh := hdGa t ht | |
| exact hh.trans (Filter.Eventually.of_forall (hd t ht)) | |
| theorem detection_of_C1Tails (hν : 0 < ν) (c₀ : Configuration (Fin N)) {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))) | |
| (heq : ∀ t ≥ 0, ∀ X, timeD ρ t X + advection (field hb table terminal ν m) ρ t X = | |
| ν * laplacian ρ t X + impulse (point hb table terminal ν m c₀ 0) t X) | |
| (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) ρ) ↔ | |
| Reaches hb table terminal c₀ := by | |
| obtain ⟨hp, hi, hm'⟩ := scalar_properties_of_C1Tails hb table terminal ν m hν | |
| (point hb table terminal ν m c₀ 0) hρ hr hD hDD heq hzero | |
| apply scalar_detection hb table terminal ν m c₀ hν (hρ.of_le (by exact WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))) ?_ hzero | |
| (fun t _ => hi t) hp hm' hm hc₀ hdir hin hterm hcont hinit | |
| intro t ht X | |
| rw [← JointCalculus.timeD_eq_deriv ht X (hρ.differentiable (by simp) (t, X))] | |
| exact heq t ht X | |
| theorem scalar_lift_unique (hν : 0 < ν) (p : Coord 2) {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))) | |
| (heq : ∀ t ≥ 0, ∀ X, timeD ρ t X + advection (field hb table terminal ν m) ρ t X = | |
| ν * laplacian ρ t X + impulse p t X) | |
| (hzero : ∀ X, ρ 0 X = 0) | |
| {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 = liftVelocity (field hb table terminal ν m) ρ t x) ∧ | |
| (∀ x, q t x = 0) := by | |
| exact Cylinder.comparison_unique_coordinates hν.le | |
| (comparisonClass_of_scalar_tails hb table terminal ν m hν hρ hr hD hDD) hv | |
| (force_NavierStokes hb table terminal ν m hν p ρ heq hzero) hNS | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |