| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Filter Finset |
| open scoped Topology |
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| namespace Problem367 |
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| theorem hasSum_of_uniform_energy_tails |
| {ι : Type*} {a : ℕ → ι → ℝ} {b : ι → ℝ} {energy : ℕ → ℝ} {total : ℝ} |
| (ha : ∀ n i, 0 ≤ a n i) |
| (hpartial : ∀ n (s : Finset ι), ∑ i ∈ s, a n i ≤ energy n) |
| (hcoordinate : ∀ i, Tendsto (fun n => a n i) atTop (𝓝 (b i))) |
| (henergy : Tendsto energy atTop (𝓝 total)) |
| (htail : ∀ ε > 0, ∃ s : Finset ι, |
| ∀ n, energy n - ∑ i ∈ s, a n i ≤ ε) : |
| HasSum b total := by |
| classical |
| have hb : ∀ i, 0 ≤ b i := fun i => |
| ge_of_tendsto (hcoordinate i) (Eventually.of_forall fun n => ha n i) |
| have hsumlim (s : Finset ι) : |
| Tendsto (fun n => ∑ i ∈ s, a n i) atTop (𝓝 (∑ i ∈ s, b i)) := |
| tendsto_finsetSum s (fun i _ => hcoordinate i) |
| have hbound (s : Finset ι) : ∑ i ∈ s, b i ≤ total := |
| le_of_tendsto_of_tendsto (hsumlim s) henergy |
| (Eventually.of_forall fun n => hpartial n s) |
| have hsummable : Summable b := summable_of_sum_le hb hbound |
| have hle : ∑' i, b i ≤ total := hsummable.tsum_le_of_sum_le hbound |
| have hge : total ≤ ∑' i, b i := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| obtain ⟨s, hs⟩ := htail ε hε |
| have hrem : total - ∑ i ∈ s, b i ≤ ε := |
| le_of_tendsto (henergy.sub (hsumlim s)) (Eventually.of_forall hs) |
| have hfinite : ∑ i ∈ s, b i ≤ ∑' i, b i := |
| hsummable.sum_le_tsum s (fun i _ => hb i) |
| linarith |
| simpa only [le_antisymm hle hge] using hsummable.hasSum |
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| theorem uniform_energy_tails_of_two_stage_bound |
| {ι : Type*} {a : ℕ → ι → ℝ} {energy : ℕ → ℝ} |
| {sets : ℕ → Finset ι} {early : ℕ → ℝ} {late : ℕ → ℕ → ℝ} |
| (hearly : Tendsto early atTop (𝓝 0)) |
| (hlate : ∀ k, Tendsto (late k) atTop (𝓝 0)) |
| (hbound : ∀ n k L, |
| energy n - ∑ i ∈ sets L, a n i ≤ early k + late k L) : |
| ∀ ε > 0, ∃ s : Finset ι, ∀ n, energy n - ∑ i ∈ s, a n i ≤ ε := by |
| intro ε hε |
| have hhalf : 0 < ε / 2 := half_pos hε |
| obtain ⟨k, hk⟩ := ((tendsto_order.1 hearly).2 (ε / 2) hhalf).exists |
| obtain ⟨L, hL⟩ := ((tendsto_order.1 (hlate k)).2 (ε / 2) hhalf).exists |
| refine ⟨sets L, fun n => ?_⟩ |
| have := hbound n k L |
| linarith |
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| section Hilbert |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] |
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| theorem norm_sq_residual_of_testing (x y : H) |
| (htest : inner ℝ x y = ‖y‖ ^ 2) : |
| ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - ‖y‖ ^ 2 := by |
| rw [norm_sub_sq_real, htest] |
| ring |
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| theorem tendsto_of_testing_and_energy |
| {α : Type*} {l : Filter α} {s : α → H} {x : H} |
| (htest : ∀ n, inner ℝ x (s n) = ‖s n‖ ^ 2) |
| (henergy : Tendsto (fun n => ‖s n‖ ^ 2) l (𝓝 (‖x‖ ^ 2))) : |
| Tendsto s l (𝓝 x) := by |
| have hres (n : α) : ‖s n - x‖ ^ 2 = ‖x‖ ^ 2 - ‖s n‖ ^ 2 := by |
| rw [norm_sub_rev] |
| exact norm_sq_residual_of_testing x (s n) (htest n) |
| have hsq : Tendsto (fun n => ‖s n - x‖ ^ 2) l (𝓝 0) := by |
| simpa only [hres, sub_self] using |
| (tendsto_const_nhds : Tendsto (fun _ : α => ‖x‖ ^ 2) l (𝓝 (‖x‖ ^ 2))).sub henergy |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| simpa only [Real.sqrt_sq_eq_abs, abs_norm, Real.sqrt_zero] using hsq.sqrt |
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| theorem norm_sq_sum_of_pairwise_orthogonal |
| {ι : Type*} (u : ι → H) |
| (hortho : ∀ i j, i ≠ j → inner ℝ (u i) (u j) = 0) |
| (s : Finset ι) : ‖∑ i ∈ s, u i‖ ^ 2 = ∑ i ∈ s, ‖u i‖ ^ 2 := by |
| classical |
| rw [← real_inner_self_eq_norm_sq, sum_inner] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [inner_sum, Finset.sum_eq_single i] |
| · exact real_inner_self_eq_norm_sq (u i) |
| · intro j hj hji |
| exact hortho i j hji.symm |
| · intro hnot |
| exact (hnot hi).elim |
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| theorem hasSum_of_orthogonal_testing_and_energy |
| {ι : Type*} {u : ι → H} {x : H} |
| (hortho : ∀ i j, i ≠ j → inner ℝ (u i) (u j) = 0) |
| (htest : ∀ i, inner ℝ x (u i) = ‖u i‖ ^ 2) |
| (henergy : HasSum (fun i => ‖u i‖ ^ 2) (‖x‖ ^ 2)) : |
| HasSum u x := by |
| classical |
| apply tendsto_of_testing_and_energy |
| · intro s |
| rw [inner_sum, norm_sq_sum_of_pairwise_orthogonal u hortho s] |
| exact Finset.sum_congr rfl fun i _ => htest i |
| · simpa only [HasSum, norm_sq_sum_of_pairwise_orthogonal u hortho] using henergy |
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| theorem hasSum_of_uniform_energy_tails_and_testing |
| {ι : Type*} {a : ℕ → ι → ℝ} {energy : ℕ → ℝ} {u : ι → H} {x : H} |
| (ha : ∀ n i, 0 ≤ a n i) |
| (hpartial : ∀ n (s : Finset ι), ∑ i ∈ s, a n i ≤ energy n) |
| (hcoordinate : ∀ i, Tendsto (fun n => a n i) atTop (𝓝 (‖u i‖ ^ 2))) |
| (henergy : Tendsto energy atTop (𝓝 (‖x‖ ^ 2))) |
| (htail : ∀ ε > 0, ∃ s : Finset ι, |
| ∀ n, energy n - ∑ i ∈ s, a n i ≤ ε) |
| (hortho : ∀ i j, i ≠ j → inner ℝ (u i) (u j) = 0) |
| (htest : ∀ i, inner ℝ x (u i) = ‖u i‖ ^ 2) : |
| HasSum u x := |
| hasSum_of_orthogonal_testing_and_energy hortho htest |
| (hasSum_of_uniform_energy_tails ha hpartial hcoordinate henergy htail) |
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| end Hilbert |
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| end Problem367 |
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| end |
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| end OAI |
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