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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.NumberTheory.TwoPoint.Bounds.IntegerShiftGeometry
namespace OAI
/-! Exact normalization of a closed numerical word, counting repeated visits. -/
namespace TwoPointCorrelations
open scoped Classical
/-- Multiply departure-indexed edge coefficients along the actual integer walk. -/
noncomputable def scalarWalkProduct (h : ℕ) (weight : SignedStep → ℤ → ℝ)
(n : ℤ) : List SignedStep → ℝ
| [] => 1
| a :: w => weight a n * scalarWalkProduct h weight (n + a.displacement h) w
lemma scalarWalkProduct_mul (h : ℕ) (a b : SignedStep → ℤ → ℝ)
(n : ℤ) (w : List SignedStep) :
scalarWalkProduct h (fun t x => a t x * b t x) n w =
scalarWalkProduct h a n w * scalarWalkProduct h b n w := by
induction w generalizing n with
| nil => simp [scalarWalkProduct]
| cons t w ih => simp only [scalarWalkProduct, ih]; ring
lemma scalarWalkProduct_congr (h : ℕ) {a b : SignedStep → ℤ → ℝ}
(hab : ∀ t x, a t x = b t x) (n : ℤ) (w : List SignedStep) :
scalarWalkProduct h a n w = scalarWalkProduct h b n w := by
have he : a = b := funext (fun t => funext (hab t))
rw [he]
/-- Each visit contributes its denominator once at departure and once
at arrival; the only uncancelled factors lie at the two endpoints. -/
theorem scalarWalkProduct_normalize (h : ℕ) (a : SignedStep → ℤ → ℝ)
(g : ℤ → ℝ) (hg : ∀ n, g n ≠ 0) (n : ℤ) (w : List SignedStep) :
scalarWalkProduct h (fun t x => a t x / (g x * g (x + t.displacement h))) n w =
scalarWalkProduct h (fun t x => a t x / (g x) ^ 2) n w *
g n / g (n + wordDisplacement h w) := by
induction w generalizing n with
| nil => simp [scalarWalkProduct, hg]
| cons t w ih =>
rw [scalarWalkProduct, ih, scalarWalkProduct, wordDisplacement_cons]
rw [← add_assoc]
field_simp [hg n, hg (n + t.displacement h),
hg (n + t.displacement h + wordDisplacement h w)]
/-- On a closed word, the exact original edge denominators become the
product of departure weights. Repeated sites remain repeated factors. -/
theorem scalarWalkProduct_closed_normalize (h : ℕ) (a : SignedStep → ℤ → ℝ)
(g : ℤ → ℝ) (hg : ∀ n, g n ≠ 0) (n : ℤ) (w : List SignedStep)
(hclosed : wordDisplacement h w = 0) :
scalarWalkProduct h (fun t x => a t x / (g x * g (x + t.displacement h))) n w =
scalarWalkProduct h (fun t x => a t x / (g x) ^ 2) n w := by
rw [scalarWalkProduct_normalize h a g hg, hclosed, add_zero, mul_div_cancel_right₀ _ (hg n)]
/-- Keep every arithmetic, vertex, and orientation condition while moving
the exact matrix denominator to the departure site. -/
theorem signedIntegerWeight_closed_normalize (Q : Finset ℕ) (u : ℕ → ℝ)
(eligible : SignedStep → ℕ → Prop) (g : ℤ → ℝ) (center : SignedStep → ℤ → ℝ) (L K : ℝ)
(extra : SignedStep → ℤ → Prop) (h : ℕ) (n : ℤ) (w : List SignedStep)
(hg : ∀ n, g n ≠ 0)
(hperiod : ∀ t, ∀ q ∈ Q, ∀ z, center t (z + (h * q * t.tuple : ℕ)) = center t z)
(hclosed : wordDisplacement h w = 0) :
scalarWalkProduct h (fun t x =>
signedIntegerWeight Q u (eligible t) g (center t) L K (extra t) h t x) n w =
scalarWalkProduct h (fun t x =>
if t.padding ∈ Q ∧ eligible t t.padding ∧ (t.padding : ℤ) ∣ x ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) x ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) (x + t.displacement h)
then L * u t.padding * center t x / (g x) ^ 2 else 0) n w := by
let a : SignedStep → ℤ → ℝ := fun t x =>
if t.padding ∈ Q ∧ eligible t t.padding ∧ (t.padding : ℤ) ∣ x ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) x ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) (x + t.displacement h)
then L * u t.padding * center t x else 0
calc
_ = scalarWalkProduct h (fun t x => a t x / (g x * g (x + t.displacement h))) n w := by
apply scalarWalkProduct_congr
intro t x
rw [signedIntegerWeight_departure Q u (eligible t) g (center t) L K (extra t) h t x
(hperiod t)]
dsimp only [a]
split_ifs <;> simp
_ = scalarWalkProduct h (fun t x => a t x / (g x) ^ 2) n w :=
scalarWalkProduct_closed_normalize h a g hg n w hclosed
_ = _ := by
apply scalarWalkProduct_congr
intro t x
dsimp only [a]
split_ifs <;> simp
end TwoPointCorrelations
end OAI
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