| import OAI.Geometry.PeriodicTiling.WordRule |
| import Mathlib.Data.Nat.Prime.Basic |
| import Mathlib.Data.Nat.GCD.BigOperators |
| import Mathlib.Algebra.Order.BigOperators.Ring.Finset |
|
|
| namespace OAI |
|
|
| namespace PeriodicTilingThree |
|
|
| open scoped BigOperators |
|
|
| abbrev PrimeTag (p : ℕ) := |
| Channel p ⊕ (Channel p ⊕ (Σ i : Channel p, Label p i)) |
|
|
| structure EncodingParameters (p : ℕ) where |
| p_prime : Nat.Prime p |
| p_large : 200 < p |
| primeAt : PrimeTag p → ℕ |
| prime_isPrime : ∀ t, Nat.Prime (primeAt t) |
| prime_injective : Function.Injective primeAt |
| prime_ne_p : ∀ t, primeAt t ≠ p |
| seed_a_eq : ∀ t : Fin 2, primeAt (.inl (.inr t)) = t.val + 2 |
| seed_b_large : ∀ t : Fin 2, p ^ 4 < primeAt (.inr (.inl (.inr t))) |
| blockA : ∀ i : Channel p, Label p i → ℕ |
| blockB : ∀ i : Channel p, Label p i → ℕ |
| blockA_pos : ∀ i j, 0 < blockA i j |
| blockB_pos : ∀ i j, 0 < blockB i j |
| digit_representation : ∀ i j, |
| primeAt (.inr (.inr ⟨i, j⟩)) = |
| blockA i j * primeAt (.inl i) + blockB i j * primeAt (.inr (.inl i)) |
|
|
| namespace EncodingParameters |
|
|
| variable {p : ℕ} (E : EncodingParameters p) |
|
|
| def a (i : Channel p) : ℕ := E.primeAt (.inl i) |
| def b (i : Channel p) : ℕ := E.primeAt (.inr (.inl i)) |
| def digitPrime (i : Channel p) (j : Label p i) : ℕ := |
| E.primeAt (.inr (.inr ⟨i, j⟩)) |
|
|
| theorem a_prime (i : Channel p) : Nat.Prime (E.a i) := E.prime_isPrime _ |
| theorem b_prime (i : Channel p) : Nat.Prime (E.b i) := E.prime_isPrime _ |
| theorem digitPrime_prime (i : Channel p) (j : Label p i) : |
| Nat.Prime (E.digitPrime i j) := E.prime_isPrime _ |
|
|
| theorem a_pos (i : Channel p) : 0 < E.a i := (E.a_prime i).pos |
| theorem b_pos (i : Channel p) : 0 < E.b i := (E.b_prime i).pos |
| theorem digitPrime_pos (i : Channel p) (j : Label p i) : |
| 0 < E.digitPrime i j := (E.digitPrime_prime i j).pos |
|
|
| theorem a_two_le (i : Channel p) : 2 ≤ E.a i := (E.a_prime i).two_le |
| theorem b_two_le (i : Channel p) : 2 ≤ E.b i := (E.b_prime i).two_le |
|
|
| theorem seed_a (t : Fin 2) : E.a (.inr t) = t.val + 2 := E.seed_a_eq t |
|
|
| theorem digit_eq (i : Channel p) (j : Label p i) : |
| E.digitPrime i j = E.blockA i j * E.a i + E.blockB i j * E.b i := |
| E.digit_representation i j |
|
|
| theorem prime_coprime {s t : PrimeTag p} (hst : s ≠ t) : |
| Nat.Coprime (E.primeAt s) (E.primeAt t) := |
| (Nat.coprime_primes (E.prime_isPrime s) (E.prime_isPrime t)).2 |
| (fun h => hst (E.prime_injective h)) |
|
|
| theorem prime_coprime_p (t : PrimeTag p) : Nat.Coprime (E.primeAt t) p := |
| (Nat.coprime_primes (E.prime_isPrime t) E.p_prime).2 (E.prime_ne_p t) |
|
|
| theorem a_coprime_b (i : Channel p) : Nat.Coprime (E.a i) (E.b i) := by |
| apply E.prime_coprime |
| simp |
|
|
| theorem digitPrime_pairwise (i : Channel p) : |
| Pairwise (fun j j' : Label p i => Nat.Coprime (E.digitPrime i j) (E.digitPrime i j')) := by |
| intro j j' hj |
| apply E.prime_coprime |
| intro he |
| exact hj (by simpa using he) |
|
|
| section Finite |
|
|
| variable [NeZero p] |
|
|
| def r (i : Channel p) : ℕ := ∏ j : Label p i, E.digitPrime i j |
|
|
| theorem r_pos (i : Channel p) : 0 < E.r i := by |
| exact Finset.prod_pos (fun j _ => E.digitPrime_pos i j) |
|
|
| instance a_neZero (i : Channel p) : NeZero (E.a i) := ⟨(E.a_pos i).ne'⟩ |
| instance b_neZero (i : Channel p) : NeZero (E.b i) := ⟨(E.b_pos i).ne'⟩ |
| instance digitPrime_neZero (i : Channel p) (j : Label p i) : |
| NeZero (E.digitPrime i j) := ⟨(E.digitPrime_pos i j).ne'⟩ |
| instance r_neZero (i : Channel p) : NeZero (E.r i) := ⟨(E.r_pos i).ne'⟩ |
|
|
| theorem r_coprime_a (i j : Channel p) : Nat.Coprime (E.r i) (E.a j) := by |
| apply Nat.coprime_fintype_prod_left_iff.mpr |
| intro k |
| apply E.prime_coprime |
| simp |
|
|
| theorem r_coprime_b (i j : Channel p) : Nat.Coprime (E.r i) (E.b j) := by |
| apply Nat.coprime_fintype_prod_left_iff.mpr |
| intro k |
| apply E.prime_coprime |
| simp |
|
|
| theorem r_coprime_p (i : Channel p) : Nat.Coprime (E.r i) p := by |
| apply Nat.coprime_fintype_prod_left_iff.mpr |
| intro k |
| exact E.prime_coprime_p _ |
|
|
| theorem r_pairwise_coprime : Pairwise (fun i j : Channel p => Nat.Coprime (E.r i) (E.r j)) := by |
| intro i j hij |
| apply Nat.coprime_fintype_prod_left_iff.mpr |
| intro k |
| apply Nat.coprime_fintype_prod_right_iff.mpr |
| intro l |
| apply E.prime_coprime |
| intro he |
| have hs : (⟨i, k⟩ : Σ i : Channel p, Label p i) = ⟨j, l⟩ := |
| Sum.inr.inj (Sum.inr.inj he) |
| exact hij (congrArg Sigma.fst hs) |
|
|
| end Finite |
| end EncodingParameters |
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
