| 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 |
|
|
