| import OAI.NumberTheory.TwoPoint.Walks.RetainedLiouvilleEdges |
| import OAI.NumberTheory.TwoPoint.Bounds.OverlapPrefix |
|
|
| namespace OAI |
|
|
| /-! Summing a directed edge over the finite block keeps precisely those |
| starting positions whose terminal endpoint remains in the block. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma block_endpoint_sum (M r : ℕ) (i : Fin M) (c : ℤ) (F : ℤ → ℂ) : |
| (∑ j : Fin M, if (j.val : ℤ) + c = (i.val : ℤ) + c + r then F ((j.val : ℤ) + c) else 0) = |
| if i.val + r < M then F ((i.val : ℤ) + c + r) else 0 := by |
| by_cases hi : i.val + r < M |
| · let j₀ : Fin M := ⟨i.val + r, hi⟩ |
| have he (j : Fin M) : (j.val : ℤ) + c = (i.val : ℤ) + c + r ↔ j = j₀ := by |
| constructor |
| · intro hj |
| apply Fin.ext |
| dsimp only [j₀] |
| omega |
| · rintro rfl |
| dsimp only [j₀] |
| push_cast |
| ring |
| simp only [he, sum_ite_eq', mem_univ, ite_true, hi] |
| congr 1 |
| dsimp [j₀] |
| ring |
| · have he (j : Fin M) : (j.val : ℤ) + c ≠ (i.val : ℤ) + c + r := by |
| intro hj |
| have hjM := j.isLt |
| omega |
| simp [he, hi] |
|
|
| lemma block_supported_edge_sum (M r : ℕ) (c : ℤ) (F : ℤ → ℤ → ℂ) |
| (hsupport : ∀ n m : ℤ, m ≠ n + r → F n m = 0) (i : Fin M) : |
| (∑ j : Fin M, F ((i.val : ℤ) + c) ((j.val : ℤ) + c)) = |
| if i.val + r < M then F ((i.val : ℤ) + c) ((i.val : ℤ) + c + r) else 0 := by |
| have he (j : Fin M) : F ((i.val : ℤ) + c) ((j.val : ℤ) + c) = |
| if (j.val : ℤ) + c = (i.val : ℤ) + c + r then |
| F ((i.val : ℤ) + c) ((j.val : ℤ) + c) else 0 := by |
| split_ifs with h |
| · rfl |
| · exact hsupport _ _ h |
| calc |
| _ = ∑ j : Fin M, if (j.val : ℤ) + c = (i.val : ℤ) + c + r then |
| F ((i.val : ℤ) + c) ((j.val : ℤ) + c) else 0 := sum_congr rfl (fun j _ => he j) |
| _ = _ := block_endpoint_sum M r i c (F ((i.val : ℤ) + c)) |
|
|
| lemma retainedLiouvilleEdge_support (Q : Finset ℕ) (u : ℕ → ℝ) (eligible : ℕ → Prop) |
| (g center : ℤ → ℝ) (L K : ℝ) (extra keep : ℤ → Prop) |
| (h d q : ℕ) (n m : ℤ) (hm : m ≠ n + (h * q * d : ℕ)) : |
| retainedLiouvilleEdge Q u eligible g center L K extra keep h d q n m = 0 := by |
| unfold retainedLiouvilleEdge |
| split_ifs with he |
| · exact (hm he.2.2.2.1).elim |
| · rfl |
|
|
| lemma retainedLiouvilleEdge_block_sum (Q : Finset ℕ) (u : ℕ → ℝ) (eligible : ℕ → Prop) |
| (g center : ℤ → ℝ) (L K : ℝ) (extra keep : ℤ → Prop) |
| (h d q M : ℕ) (c : ℤ) (i : Fin M) : |
| (∑ j : Fin M, retainedLiouvilleEdge Q u eligible g center L K extra keep h d q |
| ((i.val : ℤ) + c) ((j.val : ℤ) + c)) = |
| if i.val + h * q * d < M then |
| retainedLiouvilleEdge Q u eligible g center L K extra keep h d q |
| ((i.val : ℤ) + c) ((i.val : ℤ) + c + (h * q * d : ℕ)) else 0 := |
| block_supported_edge_sum M (h * q * d) c _ |
| (retainedLiouvilleEdge_support Q u eligible g center L K extra keep h d q) i |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
