| 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 | |