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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 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.NumberTheory.TwoPoint.Bounds.ActualPaddingProduct
namespace OAI
/-! Separate the actual closed matrix word into centered factors and its retained padding weight. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
noncomputable def retainedEdgeDeparture (Q : Finset ℕ) (u : ℕ → ℝ)
(eligible : SignedStep → ℕ → Prop) (g : ℤ → ℝ) (L K : ℝ)
(extra : SignedStep → ℤ → Prop) (h : ℕ) (t : SignedStep) (n : ℤ) : ℝ :=
if t.padding ∈ Q ∧ eligible t t.padding ∧ (t.padding : ℤ) ∣ n ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) n ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) (n + t.displacement h)
then L * u t.padding / (g n) ^ 2 else 0
lemma retainedEdgeDeparture_nonneg (Q : Finset ℕ) (u : ℕ → ℝ)
(eligible : SignedStep → ℕ → Prop) (g : ℤ → ℝ) (L K : ℝ)
(extra : SignedStep → ℤ → Prop) (h : ℕ) (hL : 0 ≤ L) (hu : ∀ q, 0 ≤ u q)
(t : SignedStep) (n : ℤ) :
0 ≤ retainedEdgeDeparture Q u eligible g L K extra h t n := by
unfold retainedEdgeDeparture
split_ifs
· exact div_nonneg (mul_nonneg hL (hu _)) (sq_nonneg _)
· exact le_rfl
lemma retainedEdgeDeparture_le_atom (Q : Finset ℕ) (u : ℕ → ℝ)
(eligible : SignedStep → ℕ → Prop) (g : ℤ → ℝ) (L K : ℝ)
(extra : SignedStep → ℤ → Prop) (h : ℕ) (hL : 0 ≤ L) (hu : ∀ q, 0 ≤ u q)
(t : SignedStep) (n : ℤ) :
retainedEdgeDeparture Q u eligible g L K extra h t n ≤
retainedPaddingAtom Q u (eligible t) g L K (extra t) n t.padding := by
unfold retainedEdgeDeparture
split_ifs with ht
· have hs : eligible t t.padding ∧ (t.padding : ℤ) ∣ n ∧
integerEdgeKeep Q u (eligible t) g L K (extra t) n :=
⟨ht.2.1, ht.2.2.1, ht.2.2.2.1⟩
simp only [retainedPaddingAtom, ite_eq_left hs, le_refl]
· exact retainedPaddingAtom_nonneg Q u (eligible t) g L K (extra t) n t.padding hL (hu _)
theorem closed_signed_word_factorization (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) (word : 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 word = 0) :
scalarWalkProduct h (fun t x =>
signedIntegerWeight Q u (eligible t) g (center t) L K (extra t) h t x) n word =
scalarWalkProduct h (retainedEdgeDeparture Q u eligible g L K extra h) n word *
scalarWalkProduct h center n word := by
rw [signedIntegerWeight_closed_normalize Q u eligible g center L K extra h n word
hg hperiod hclosed]
rw [← scalarWalkProduct_mul]
apply scalarWalkProduct_congr
intro t x
unfold retainedEdgeDeparture
split_ifs <;> ring
lemma scalarWalkProduct_nonneg (h : ℕ) (weight : SignedStep → ℤ → ℝ)
(hw : ∀ t n, 0 ≤ weight t n) (n : ℤ) (word : List SignedStep) :
0 ≤ scalarWalkProduct h weight n word := by
induction word generalizing n with
| nil => exact zero_le_one
| cons t word ih => exact mul_nonneg (hw t n) (ih _)
lemma scalarWalkProduct_mono (h : ℕ) (a b : SignedStep → ℤ → ℝ)
(ha : ∀ t n, 0 ≤ a t n) (hab : ∀ t n, a t n ≤ b t n)
(n : ℤ) (word : List SignedStep) :
scalarWalkProduct h a n word ≤ scalarWalkProduct h b n word := by
induction word generalizing n with
| nil => exact le_rfl
| cons t word ih =>
exact mul_le_mul (hab t n) (ih _) (scalarWalkProduct_nonneg h a ha _ _)
((ha t n).trans (hab t n))
theorem closed_word_padding_majorant (Q : Finset ℕ) (u : ℕ → ℝ)
(eligible : SignedStep → ℕ → Prop) (g : ℤ → ℝ) (L K : ℝ)
(extra : SignedStep → ℤ → Prop) (h : ℕ) (hL : 0 ≤ L) (hu : ∀ q, 0 ≤ u q)
(n : ℤ) (word : List SignedStep) :
scalarWalkProduct h (retainedEdgeDeparture Q u eligible g L K extra h) n word ≤
scalarWalkProduct h
(fun t x => retainedPaddingAtom Q u (eligible t) g L K (extra t) x t.padding) n word :=
scalarWalkProduct_mono h _ _ (retainedEdgeDeparture_nonneg Q u eligible g L K extra h hL hu)
(retainedEdgeDeparture_le_atom Q u eligible g L K extra h hL hu) n word
theorem scalarWalkProduct_ofFn {m : ℕ} (h : ℕ) (weight : SignedStep → ℤ → ℝ)
(step : Fin m → SignedStep) (n : ℤ) :
scalarWalkProduct h weight n (List.ofFn step) =
∏ i : Fin m, weight (step i) (n + wordDisplacement h ((List.ofFn step).take i.val)) := by
induction m generalizing n with
| zero => simp [scalarWalkProduct]
| succ m ih =>
rw [List.ofFn_succ, scalarWalkProduct, ih, Fin.prod_univ_succ]
simp only [Fin.val_zero, List.take_zero, wordDisplacement_nil, add_zero]
congr 1
apply prod_congr rfl
intro i _
simp only [Fin.val_succ, List.take_succ_cons, wordDisplacement_cons,
add_assoc]
end TwoPointCorrelations
end OAI
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