| import Mathlib |
| import OAI.Analysis.CoulombIonization.Ionization.IntegratedStability |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| namespace CoulombAnalysis |
| open CoulombAtom |
|
|
| private theorem grid_increment_error {G d P D : ℝ → ℝ} |
| (hG : DeficitSecants G d) (hP : DeficitSecants P D) |
| {a h η : ℝ} (ha : 0 < a) (hh : 0 < h) {N : ℕ} |
| (hclose : ∀ i ≤ N, |d (a + i * h) - D (a + i * h)| ≤ η) : |
| |(G (a + N * h) - G a) - (P (a + N * h) - P a)| ≤ |
| h * (D (a + N * h) - D a) + N * h * η := by |
| have hind : ∀ i ≤ N, |
| |(G (a + i * h) - G a) - (P (a + i * h) - P a)| ≤ |
| h * (D (a + i * h) - D a) + i * h * η := by |
| intro i hi |
| induction i with |
| | zero => simp |
| | succ i ih => |
| have hiN : i ≤ N := Nat.le_trans (Nat.le_succ i) hi |
| have hix : 0 < a + i * h := by positivity |
| have hxy : a + i * h < a + (i + 1 : ℕ) * h := by |
| push_cast |
| nlinarith |
| have hsG := hG (a + i * h) (a + (i + 1 : ℕ) * h) hix hxy |
| have hsP := hP (a + i * h) (a + (i + 1 : ℕ) * h) hix hxy |
| have hcl := (abs_le.mp (hclose i hiN)) |
| have hcr := (abs_le.mp (hclose (i + 1) hi)) |
| have hcl' := mul_le_mul_of_nonneg_left hcl.1 hh.le |
| have hcr' := mul_le_mul_of_nonneg_left hcr.2 hh.le |
| have heq : a + (i + 1 : ℕ) * h - (a + i * h) = h := by |
| push_cast |
| ring |
| rw [heq] at hsG hsP |
| have hstep : |(G (a + (i + 1 : ℕ) * h) - G (a + i * h)) - |
| (P (a + (i + 1 : ℕ) * h) - P (a + i * h))| ≤ |
| h * (D (a + (i + 1 : ℕ) * h) - D (a + i * h)) + h * η := by |
| apply abs_le.mpr |
| constructor <;> nlinarith [hsG.1, hsG.2, hsP.1, hsP.2] |
| have htel : (G (a + (i + 1 : ℕ) * h) - G a) - |
| (P (a + (i + 1 : ℕ) * h) - P a) = |
| ((G (a + i * h) - G a) - (P (a + i * h) - P a)) + |
| ((G (a + (i + 1 : ℕ) * h) - G (a + i * h)) - |
| (P (a + (i + 1 : ℕ) * h) - P (a + i * h))) := by ring |
| rw [htel] |
| calc |
| _ ≤ |(G (a + i * h) - G a) - (P (a + i * h) - P a)| + |
| |(G (a + (i + 1 : ℕ) * h) - G (a + i * h)) - |
| (P (a + (i + 1 : ℕ) * h) - P (a + i * h))| := abs_add_le _ _ |
| _ ≤ (h * (D (a + i * h) - D a) + i * h * η) + |
| (h * (D (a + (i + 1 : ℕ) * h) - D (a + i * h)) + h * η) := |
| add_le_add (ih hiN) hstep |
| _ = _ := by push_cast; ring |
| exact hind N le_rfl |
|
|
| theorem deficit_secant_increments_tendsto_filter {ι : Type*} {l : Filter ι} {G d : ι → ℝ → ℝ} {P D : ℝ → ℝ} |
| (hG : ∀ n, DeficitSecants (G n) (d n)) (hP : DeficitSecants P D) |
| (hd : ∀ t : ℝ, 0 < t → Tendsto (fun n => d n t) l (𝓝 (D t))) |
| {a b : ℝ} (ha : 0 < a) (hab : a < b) : |
| Tendsto (fun n => G n b - G n a) l (𝓝 (P b - P a)) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| obtain ⟨N, hN⟩ := exists_nat_gt (max 1 (2 * ((b - a) * (D b - D a)) / ε)) |
| have hN1 : (1 : ℝ) < N := (le_max_left _ _).trans_lt hN |
| have hNpos : (0 : ℝ) < N := lt_trans zero_lt_one hN1 |
| have hwidth : 0 < b - a := sub_pos.mpr hab |
| let h : ℝ := (b - a) / N |
| let η : ℝ := ε / (2 * (b - a)) |
| have hh : 0 < h := div_pos hwidth hNpos |
| have hη : 0 < η := div_pos hε (by positivity) |
| have hxN : a + N * h = b := by dsimp [h]; field_simp; ring |
| have hmesh : h * (D b - D a) < ε / 2 := by |
| have hnr : 2 * ((b - a) * (D b - D a)) / ε < N := |
| (le_max_right _ _).trans_lt hN |
| have hnr' := (div_lt_iff₀ hε).mp hnr |
| dsimp [h] |
| rw [div_mul_eq_mul_div] |
| apply (div_lt_div_iff₀ hNpos (show (0 : ℝ) < 2 by norm_num)).mpr |
| nlinarith |
| have hηeq : (N : ℝ) * h * η = ε / 2 := by |
| dsimp [h, η] |
| field_simp |
| have hev : ∀ i : Fin (N + 1), ∀ᶠ n in l, |
| |d n (a + (i : ℕ) * h) - D (a + (i : ℕ) * h)| ≤ η := by |
| intro i |
| have hxi : 0 < a + (i : ℕ) * h := by positivity |
| exact ((Metric.tendsto_nhds.mp (hd _ hxi)) η hη).mono fun n hn => |
| le_of_lt (by simpa only [Real.dist_eq] using hn) |
| filter_upwards [Filter.eventually_all.mpr hev] with n hn |
| rw [Real.dist_eq] |
| have he := grid_increment_error (hG n) hP ha hh |
| (N := N) (η := η) (fun i hi => hn ⟨i, Nat.lt_succ_iff.mpr hi⟩) |
| rw [hxN, hηeq] at he |
| linarith |
| |
| theorem deficit_secant_energy_tendsto_filter {ι : Type*} {l : Filter ι} {G d : ι → ℝ → ℝ} {P D : ℝ → ℝ} |
| {err : ι → ℝ} (hG : ∀ n, DeficitSecants (G n) (d n)) |
| (hP : DeficitSecants P D) |
| (hd : ∀ t : ℝ, 0 < t → Tendsto (fun n => d n t) l (𝓝 (D t))) |
| (he : Tendsto err l (𝓝 0)) |
| (hzero : ∀ v, 0 < v → ∀ᶠ n in l, -err n - v * d n v ≤ G n v ∧ G n v ≤ 0) |
| (hPc : ContinuousAt P 0) (hP0 : P 0 = 0) (hDc : ContinuousAt D 0) |
| {t : ℝ} (ht : 0 < t) : Tendsto (fun n => G n t) l (𝓝 (P t)) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| have hc : ContinuousAt (fun v : ℝ => |P v| + |v * D v|) 0 := |
| hPc.abs.add ((continuousAt_id.mul hDc).abs) |
| have hevent : ∀ᶠ v in 𝓝 (0 : ℝ), |P v| + |v * D v| < ε / 4 := by |
| have hconv : Tendsto (fun v : ℝ => |P v| + |v * D v|) (𝓝 0) (𝓝 0) := by |
| simpa [hP0] using hc.tendsto |
| exact hconv.eventually (gt_mem_nhds (show (0 : ℝ) < ε / 4 by positivity)) |
| obtain ⟨δ, hδ, hδevent⟩ := Metric.eventually_nhds_iff.mp hevent |
| let v : ℝ := min (t / 2) (δ / 2) |
| have hv : 0 < v := lt_min (by positivity) (by positivity) |
| have hvt : v < t := lt_of_le_of_lt (min_le_left _ _) (by linarith) |
| have hvδ : dist v 0 < δ := by |
| rw [Real.dist_eq, sub_zero, abs_of_pos hv] |
| exact lt_of_le_of_lt (min_le_right _ _) (by linarith) |
| have hsmall := hδevent hvδ |
| have hinc := deficit_secant_increments_tendsto_filter hG hP hd hv hvt |
| have hbound : Tendsto (fun n => err n + v * d n v) l (𝓝 (v * D v)) := by |
| simpa using he.add ((hd v hv).const_mul v) |
| filter_upwards [(Metric.tendsto_nhds.mp hinc) (ε / 4) (by positivity), |
| (Metric.tendsto_nhds.mp hbound) (ε / 4) (by positivity), hzero v hv] with n hn hb hz |
| rw [Real.dist_eq] at hn hb ⊢ |
| have hGv : |G n v| ≤ err n + v * d n v := by |
| rw [abs_of_nonpos hz.2] |
| linarith [hz.1] |
| have hbound' : err n + v * d n v < |v * D v| + ε / 4 := by |
| have hb' := (abs_lt.mp hb).2 |
| linarith [le_abs_self (v * D v)] |
| have halg : G n t - P t = |
| ((G n t - G n v) - (P t - P v)) + (G n v - P v) := by ring |
| rw [halg] |
| have htri := abs_add_le ((G n t - G n v) - (P t - P v)) (G n v - P v) |
| have hsub := abs_sub (G n v) (P v) |
| linarith |
| |
| theorem priced_energy_limits_filter {ι : Type*} {l : Filter ι} {G d : ι → ℝ → ℝ} {err : ι → ℝ} {q : ℝ} |
| (hq : 0 ≤ q) (hG : ∀ n, DeficitSecants (G n) (d n)) |
| (hd : ∀ t : ℝ, 0 < t → Tendsto (fun n => d n t) l (𝓝 (limitDeficit q t))) |
| (he : Tendsto err l (𝓝 0)) |
| (hzero : ∀ n v, 0 < v → -err n - v * d n v ≤ G n v ∧ G n v ≤ 0) |
| {t : ℝ} (ht : 0 < t) : |
| Tendsto (fun n => G n t) l (𝓝 (limitPriceEnergy q t)) ∧ |
| Tendsto (fun n => G n t + t * d n t) l (𝓝 (limitSectorEnergy q t)) := by |
| have hprice := deficit_secant_energy_tendsto_filter hG (deficitSecants_limitPriceEnergy hq) |
| hd he (fun v hv => Eventually.of_forall fun n => hzero n v hv) (continuous_limitPriceEnergy q).continuousAt |
| (by simp [limitPriceEnergy]) (continuous_limitDeficit q).continuousAt ht |
| exact ⟨hprice, by simpa only [limitPriceEnergy_add ht] using hprice.add ((hd t ht).const_mul t)⟩ |
| |
| end CoulombAnalysis |
| |
| open MeasureTheory Filter Set Metric Topology |
| |
| end |
| |
| end OAI |
| |
