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namespace OAI
noncomputable section
open Filter Finset
open scoped Topology
namespace Problem367
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
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
section Hilbert
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
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
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
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
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
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)
end Hilbert
end Problem367
end
end OAI
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