openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ArrayBounds.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArrayGate | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.UniformDerivativesNormSumLeSparse | |
| 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) | |
| theorem gate_jet_zero_outside (hν : 0 < ν) (n d : ℕ) | |
| (c : Active hb table (capacity b m n)) (q : ℝ × Coord 2) (i : Fin 2) | |
| (hq : ∃ j, 3 * radius ν (K₀ N b m) (D b) n < | |
| |q.2 j - path hb table terminal ν m n c q.1 j|) : | |
| iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n c q i) q = 0 := by | |
| obtain ⟨j, hj⟩ := hq | |
| have hcont : Continuous (fun p : ℝ × Coord 2 => | |
| |p.2 j - path hb table terminal ν m n c p.1 j|) := | |
| ((continuous_apply j).comp continuous_snd |>.sub | |
| ((continuous_apply j).comp ((contDiff_path hb table terminal ν m n c).continuous.comp | |
| continuous_fst))).abs | |
| have heq : (fun p => gate hb table terminal ν m n c p i) =ᶠ[𝓝 q] 0 := by | |
| filter_upwards [(isOpen_lt continuous_const hcont).mem_nhds hj] with p hp | |
| have h := TranslationGates.field_zero_outside | |
| (lt_of_lt_of_le zero_lt_one (radius_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) n)) | |
| (path hb table terminal ν m n c p.1) (deriv (path hb table terminal ν m n c) p.1) p.2 ⟨j, hp.le⟩ | |
| exact congrFun h i | |
| simpa using (heq.iteratedFDeriv ℝ d).eq_of_nhds | |
| theorem gate_jets_disjoint (hν : 0 < ν) (hm : 1 ≤ m) | |
| (hdir : IncomingDirection table) (hin : IncomingRule table) (n d : ℕ) | |
| (c e : Active hb table (capacity b m n)) (hce : c ≠ e) (q : ℝ × Coord 2) (i : Fin 2) : | |
| iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n c q i) q = 0 ∨ | |
| iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n e q i) q = 0 := by | |
| obtain ⟨j, hj⟩ := paths_separated hb table terminal ν m hν hm hdir hin n c e hce q.1 | |
| have hR : 0 < radius ν (K₀ N b m) (D b) n := | |
| lt_of_lt_of_le zero_lt_one (radius_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) n) | |
| by_cases hc : 3 * radius ν (K₀ N b m) (D b) n < | |
| |q.2 j - path hb table terminal ν m n c q.1 j| | |
| · exact Or.inl (gate_jet_zero_outside hb table terminal ν m hν n d c q i ⟨j, hc⟩) | |
| · right | |
| apply gate_jet_zero_outside hb table terminal ν m hν n d e q i | |
| refine ⟨j, ?_⟩ | |
| have htri : |path hb table terminal ν m n c q.1 j - path hb table terminal ν m n e q.1 j| ≤ | |
| |q.2 j - path hb table terminal ν m n c q.1 j| + | |
| |q.2 j - path hb table terminal ν m n e q.1 j| := by | |
| simpa only [abs_sub_comm (path hb table terminal ν m n c q.1 j) (q.2 j)] | |
| using abs_sub_le (path hb table terminal ν m n c q.1 j) (q.2 j) | |
| (path hb table terminal ν m n e q.1 j) | |
| linarith | |
| theorem stages_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m) | |
| (hdir : IncomingDirection table) (hin : IncomingRule table) (i : Fin 2) : | |
| Bounded (fun n (q : ℝ × Coord 2) => stage hb table terminal ν m n q.1 q.2 i) := by | |
| intro d | |
| obtain ⟨C, hC, hc⟩ := gates_uniformly_bounded hb table terminal ν m hν hm i d | |
| refine ⟨C, hC, fun n q => ?_⟩ | |
| have heq : (fun q : ℝ × Coord 2 => stage hb table terminal ν m n q.1 q.2 i) = | |
| ∑ c : Active hb table (capacity b m n), fun q => gate hb table terminal ν m n c q i := by | |
| ext q | |
| simp only [stage, gate, Finset.sum_apply] | |
| change ‖iteratedFDeriv ℝ d (fun q : ℝ × Coord 2 => stage hb table terminal ν m n q.1 q.2 i) q‖ ≤ C | |
| rw [heq, iteratedFDeriv_sum_apply] | |
| · apply norm_sum_le_sparse _ _ hC | |
| · intro c _ | |
| exact hc ⟨n, c⟩ q | |
| · intro c _ e _ hce | |
| exact gate_jets_disjoint hb table terminal ν m hν hm hdir hin n d c e hce q i | |
| · intro c _ | |
| exact (contDiff_infty.mp ((contDiff_apply ℝ ℝ i).comp | |
| (contDiff_gate hb table terminal ν m n c)) d).contDiffAt | |
| theorem stage_jet_zero_before (hν : 0 < ν) (n d : ℕ) (q : ℝ × Coord 2) (i : Fin 2) | |
| (hq : q.1 ≤ startTime ν (K₀ N b m) (D b) n) : | |
| iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 := by | |
| exact jet_zero_before ((contDiff_apply ℝ ℝ i).comp (contDiff_stage hb table terminal ν m n)) | |
| (fun t ht X => congrFun (stage_zero_before hb table terminal ν m hν n ht X) i) d hq q.2 | |
| theorem stage_jet_zero_after (hν : 0 < ν) (n d : ℕ) (q : ℝ × Coord 2) (i : Fin 2) | |
| (hq : startTime ν (K₀ N b m) (D b) (n + 1) ≤ q.1) : | |
| iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 := by | |
| exact jet_zero_after ((contDiff_apply ℝ ℝ i).comp (contDiff_stage hb table terminal ν m n)) | |
| (fun t ht X => congrFun (stage_zero_after hb table terminal ν m hν n ht X) i) d hq q.2 | |
| theorem stage_jets_disjoint (hν : 0 < ν) (n k d : ℕ) (hnk : n ≠ k) (q : ℝ × Coord 2) (i : Fin 2) : | |
| iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 ∨ | |
| iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m k q.1 q.2 i) q = 0 := by | |
| wlog hnk' : n < k generalizing n k | |
| · exact (this k n hnk.symm (by omega)).symm | |
| by_cases hq : q.1 ≤ startTime ν (K₀ N b m) (D b) k | |
| · exact Or.inr (stage_jet_zero_before hb table terminal ν m hν k d q i hq) | |
| · left | |
| apply stage_jet_zero_after hb table terminal ν m hν n d q i | |
| have hm := (startTime_strictMono hν (K₀_nonneg N b m) (D_ge_one hb)).monotone (show n + 1 ≤ k by omega) | |
| linarith | |
| theorem field_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m) | |
| (hdir : IncomingDirection table) (hin : IncomingRule table) (i : Fin 2) : | |
| Bounded (fun _ : Unit => fun q : ℝ × Coord 2 => field hb table terminal ν m q.1 q.2 i) := by | |
| intro d | |
| obtain ⟨C, hC, hc⟩ := stages_uniformly_bounded hb table terminal ν m hν hm hdir hin i d | |
| refine ⟨C, hC, fun _ q => ?_⟩ | |
| obtain ⟨Q, hQ⟩ := exists_nat_gt q.1 | |
| have heq : (fun q : ℝ × Coord 2 => field hb table terminal ν m q.1 q.2 i) =ᶠ[𝓝 q] | |
| ∑ n ∈ Finset.range Q, fun q => stage hb table terminal ν m n q.1 q.2 i := by | |
| filter_upwards [(isOpen_lt continuous_fst continuous_const).mem_nhds hQ] with p hp | |
| simpa only [Finset.sum_apply] using congrFun (field_eq_finite hb table terminal ν m hν hp p.2) i | |
| rw [(heq.iteratedFDeriv ℝ d).eq_of_nhds, iteratedFDeriv_sum_apply] | |
| · apply norm_sum_le_sparse _ _ hC | |
| · intro n _ | |
| exact hc n q | |
| · intro n _ k _ hnk | |
| exact stage_jets_disjoint hb table terminal ν m hν n k d hnk q i | |
| · intro n _ | |
| exact (contDiff_infty.mp ((contDiff_apply ℝ ℝ i).comp | |
| (contDiff_stage hb table terminal ν m n)) d).contDiffAt | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |