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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 | import OAI.NumberTheory.TwoPoint.Walks.ColumnClosedEncoding
namespace OAI
/-! The actual closed-pair catalog has length `2*k`, as in the trace bounds. -/
namespace TwoPointCorrelations
open Finset
lemma columnTupleWord_reindex {J R S : ℕ} {P : Fin J → Finset ℕ}
(he : R = S) (w : ColumnPrimeAssignment J R P)
(forward : Fin R → Bool) (padding : Fin R → ℕ) :
columnTupleWord (fun j i => w j (Fin.cast he.symm i))
(fun i => forward (Fin.cast he.symm i)) (fun i => padding (Fin.cast he.symm i)) =
columnTupleWord w forward padding := by
subst S
rfl
def closedTraceEncoding {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ)
(p : (Fin k → ((j : Fin J) → P j) × (Q × Bool)) ×
(Fin k → ((j : Fin J) → P j) × (Q × Bool))) :
(Fin (2 * k) → Bool) × (ColumnPrimeAssignment J (2 * k) P × (Fin (2 * k) → Q)) :=
let castIndex : Fin (2 * k) → Fin (k + k) := Fin.cast (by omega)
(fun i => closedColumnForward Q p.1 p.2 (castIndex i),
(fun j i => closedColumnAssignment Q p.1 p.2 j (castIndex i),
fun i => closedColumnPadding Q p.1 p.2 (castIndex i)))
theorem closedTraceEncoding_word {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ)
(p : (Fin k → ((j : Fin J) → P j) × (Q × Bool)) ×
(Fin k → ((j : Fin J) → P j) × (Q × Bool))) :
columnTupleWord (closedTraceEncoding Q p).2.1 (closedTraceEncoding Q p).1
(fun i => ((closedTraceEncoding Q p).2.2 i).val) =
integerClosedWordCode Q (fun d => ∏ j, (d j).val) p := by
dsimp only [closedTraceEncoding]
exact (columnTupleWord_reindex (show k + k = 2 * k by omega)
(closedColumnAssignment Q p.1 p.2) (closedColumnForward Q p.1 p.2)
(fun i => (closedColumnPadding Q p.1 p.2 i).val)).trans
(closedColumn_encoding_word Q p.1 p.2)
theorem closedTraceEncoding_injective {J k : ℕ} {P : Fin J → Finset ℕ} (Q : Finset ℕ)
(hprime : ∀ j, ∀ p ∈ P j, p.Prime)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l)) :
Function.Injective (closedTraceEncoding (k := k) (P := P) Q) := by
intro a b hab
apply integerClosedWordCode_injective Q _ (primeTuple_injective hprime hdisjoint) k
rw [← closedTraceEncoding_word Q a, ← closedTraceEncoding_word Q b, hab]
end TwoPointCorrelations
end OAI
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