File size: 3,059 Bytes
180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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