| import OAI.NumberTheory.TwoPoint.Bounds.IntegerClosedCatalog |
| import OAI.NumberTheory.TwoPoint.Bounds.TraceCatalogMaps |
|
|
| namespace OAI |
|
|
| /-! Recover the column data of the actual joined matrix paths. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma ofFn_rev_eq_reverse {α : Type*} {n : ℕ} (f : Fin n → α) : |
| List.ofFn (fun i => f i.rev) = (List.ofFn f).reverse := by |
| apply List.ext_getElem |
| · simp only [List.length_ofFn, List.length_reverse] |
| · intro i hi hj |
| simp only [List.getElem_ofFn, List.getElem_reverse, List.length_ofFn] |
| congr 1 |
| apply Fin.ext |
| simp only [Fin.val_rev] |
| omega |
|
|
| def closedColumnRows {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : |
| Fin (k + k) → ((j : Fin J) → P j) × (Q × Bool) := |
| Fin.append a (fun i => ((b i.rev).1, (b i.rev).2.1, !(b i.rev).2.2)) |
|
|
| lemma closedColumnRows_word {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : |
| integerStepWord Q (fun d => ∏ j, (d j).val) (closedColumnRows Q a b) = |
| integerClosedWordCode Q (fun d => ∏ j, (d j).val) (a, b) := by |
| unfold integerStepWord closedColumnRows integerClosedWordCode |
| rw [List.ofFn_fin_append, List.map_append] |
| congr 1 |
| rw [reverseWord] |
| simp only [integerStepWord, List.map_reverse, List.map_ofFn, Function.comp_def] |
| rw [← ofFn_rev_eq_reverse] |
| simp only [SignedStep.flip] |
|
|
| def closedColumnAssignment {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : |
| ColumnPrimeAssignment J (k + k) P := |
| fun j i => (closedColumnRows Q a b i).1 j |
|
|
| def closedColumnPadding {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : Fin (k + k) → Q := |
| fun i => (closedColumnRows Q a b i).2.1 |
|
|
| def closedColumnForward {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : Fin (k + k) → Bool := |
| fun i => (closedColumnRows Q a b i).2.2 |
|
|
| theorem closedColumn_encoding_word {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ) |
| (a b : Fin k → ((j : Fin J) → P j) × (Q × Bool)) : |
| columnTupleWord (closedColumnAssignment Q a b) (closedColumnForward Q a b) |
| (fun i => (closedColumnPadding Q a b i).val) = |
| integerClosedWordCode Q (fun d => ∏ j, (d j).val) (a, b) := by |
| rw [← closedColumnRows_word] |
| simp only [columnTupleWord, columnTuple, closedColumnAssignment, closedColumnForward, |
| closedColumnPadding, integerStepWord, List.map_ofFn, Function.comp_def] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
