File size: 2,718 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 | 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
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