| import OAI.NumberTheory.TwoPoint.Walks.BlockEndpointSum |
|
|
| namespace OAI |
|
|
| /-! The finite directed matrix form is the overlapping edge-block sum |
| used in the prefix comparison, with no residual multiplicity assumption. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| theorem directed_block_form_eq {E : Type*} [Fintype E] (r : E → ℕ) |
| (F : E → ℤ → ℤ → ℂ) |
| (hsupport : ∀ (e : E) (n m : ℤ), m ≠ n + r e → F e n m = 0) (M t : ℕ) : |
| (∑ e : E, ∑ i : Fin M, ∑ j : Fin M, |
| F e ((i.val : ℤ) + (t + 2 : ℕ)) ((j.val : ℤ) + (t + 2 : ℕ))) = |
| edgeBlockSum univ r (fun e n => F e n ((n : ℤ) + r e)) M t := by |
| let g (i : ℕ) := ∑ e : E, if i + r e < M then |
| F e ((i : ℤ) + (t + 2 : ℕ)) ((i : ℤ) + (t + 2 : ℕ) + r e) else 0 |
| calc |
| _ = ∑ i : Fin M, ∑ e : E, ∑ j : Fin M, |
| F e ((i.val : ℤ) + (t + 2 : ℕ)) ((j.val : ℤ) + (t + 2 : ℕ)) := sum_comm |
| _ = ∑ i : Fin M, g i.val := by |
| apply sum_congr rfl |
| intro i _ |
| apply sum_congr rfl |
| intro e _ |
| exact block_supported_edge_sum M (r e) (t + 2 : ℕ) (F e) (hsupport e) i |
| _ = ∑ i ∈ range M, g i := Fin.sum_univ_eq_sum_range g M |
| _ = _ := by |
| unfold edgeBlockSum g |
| apply sum_congr rfl |
| intro i _ |
| apply sum_congr rfl |
| intro e _ |
| have he : i + r e < M ↔ i + 1 + r e ≤ M := by omega |
| simp only [he, Nat.cast_add, Nat.cast_ofNat] |
| simp only [add_assoc, add_left_comm, add_comm] |
|
|
| theorem directed_block_form_prefix_error {E : Type*} [Fintype E] |
| (r : E → ℕ) (F : E → ℤ → ℤ → ℂ) |
| (hsupport : ∀ (e : E) (n m : ℤ), m ≠ n + r e → F e n m = 0) |
| (M R N : ℕ) (hM : 0 < M) (hN : 0 < N) (D : ℝ) (hD : 0 ≤ D) |
| (hr : ∀ e, r e ≤ R) |
| (hrow : ∀ n : ℕ, 0 < n → (∑ e : E, ‖F e n ((n : ℤ) + r e)‖) ≤ D) : |
| ‖((∑ t ∈ range N, ∑ e : E, ∑ i : Fin M, ∑ j : Fin M, |
| F e ((i.val : ℤ) + (t + 2 : ℕ)) ((j.val : ℤ) + (t + 2 : ℕ))) / (M : ℂ)) / (N : ℂ) - |
| positivePrefix (fun n => ∑ e : E, F e n ((n : ℤ) + r e)) N / (N : ℂ)‖ ≤ |
| (R : ℝ) / M * D + 2 * (M : ℝ) / N * D := by |
| have hb := edgeBlockSum_prefix_error univ r (fun e n => F e n ((n : ℤ) + r e)) |
| M R N hM hN D hD (fun e _ => hr e) hrow |
| simpa only [directed_block_form_eq r F hsupport] using hb |
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|
| end TwoPointCorrelations |
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|
| end OAI |
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|
