openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ExpandingArrayGate.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArray | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.GateBounds | |
| 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 Expanding Stacks FiniteAddresses UniformDerivatives | |
| 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 gate (n : ℕ) (c : Active hb table (capacity b m n)) (q : ℝ × Coord 2) : Coord 2 := | |
| TranslationGates.field (radius ν (K₀ N b m) (D b) n) | |
| (path hb table terminal ν m n c q.1) (deriv (path hb table terminal ν m n c) q.1) q.2 | |
| @[fun_prop] theorem contDiff_gate (n : ℕ) (c : Active hb table (capacity b m n)) : | |
| ContDiff ℝ ∞ (gate hb table terminal ν m n c) := | |
| TranslationGates.contDiff_movingField _ (contDiff_path hb table terminal ν m n c) | |
| ((contDiff_infty_iff_deriv.mp (contDiff_path hb table terminal ν m n c)).2) | |
| theorem gates_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m) (i : Fin 2) : | |
| Bounded (fun (a : (n : ℕ) × Active hb table (capacity b m n)) q => | |
| gate hb table terminal ν m a.1 a.2 q i) := by | |
| unfold gate | |
| refine TranslationGates.bounded_movingField | |
| (I := (n : ℕ) × Active hb table (capacity b m n)) | |
| (fun a => radius ν (K₀ N b m) (D b) a.1) | |
| (fun a => radius_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) a.1) | |
| (fun a => path hb table terminal ν m a.1 a.2) | |
| (fun a => contDiff_path hb table terminal ν m a.1 a.2) ?_ i | |
| intro j n hn | |
| obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne' | |
| obtain ⟨C, hC, hc⟩ := path_derivative_bounds hb table terminal ν m hν hm n | |
| refine ⟨C, hC, fun a t => ?_⟩ | |
| rw [norm_iteratedFDeriv_eq_norm_iteratedDeriv, Real.norm_eq_abs] | |
| exact (hc a.1 a.2 t j).trans | |
| (div_le_self hC (one_le_pow₀ (duration_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) a.1))) | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |