| import OAI.Combinatorics.EditDistance.PrimeCharacters |
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| namespace OAI |
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| /-! |
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| -/ |
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| namespace EditPhase |
| open scoped BigOperators |
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| lemma few_divisors {J : Type*} [Fintype J] [DecidableEq J] |
| (p : J → ℕ) (hp : Pairwise (fun i j => (p i).Coprime (p j))) |
| {P r n : ℕ} (hP : 1 ≤ P) (hpP : ∀ j, P ≤ p j) |
| (hn : 0 < n) (hnr : n < P ^ r) : |
| (Finset.univ.filter (fun j => p j ∣ n)).card < r := by |
| let S := Finset.univ.filter (fun j => p j ∣ n) |
| have hdiv : (∏ j ∈ S, p j) ∣ n := by |
| apply Finset.prod_dvd_of_isRelPrime |
| · intro i hi j hj hij |
| exact Nat.coprime_iff_isRelPrime.mp (hp hij) |
| · intro j hj |
| exact (Finset.mem_filter.mp hj).2 |
| have hlow : P ^ S.card ≤ ∏ j ∈ S, p j := |
| Finset.pow_card_le_prod S p P (fun j _ => hpP j) |
| have hle : P ^ S.card ≤ n := hlow.trans (Nat.le_of_dvd hn hdiv) |
| by_contra h |
| have hcard : r ≤ S.card := Nat.le_of_not_gt h |
| have hpow := pow_le_pow_right' hP hcard |
| exact (not_lt_of_ge (hpow.trans hle)) hnr |
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| lemma few_prime_divisors {J : Type*} [Fintype J] [DecidableEq J] |
| (p : J → ℕ) (hp : ∀ j, (p j).Prime) (hinj : Function.Injective p) |
| {P r n : ℕ} (hP : 1 ≤ P) (hpP : ∀ j, P ≤ p j) |
| (hn : 0 < n) (hnr : n < P ^ r) : |
| (Finset.univ.filter (fun j => p j ∣ n)).card < r := by |
| apply few_divisors p (fun i j hij => ?_) hP hpP hn hnr |
| exact (Nat.coprime_primes (hp i) (hp j)).mpr (fun h => hij (hinj h)) |
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| lemma many_nondivisors_except {J : Type*} [Fintype J] [DecidableEq J] |
| (p : J → ℕ) (hp : ∀ j, (p j).Prime) (hinj : Function.Injective p) |
| {P r n : ℕ} (hP : 1 ≤ P) (hpP : ∀ j, P ≤ p j) |
| (hn : 0 < n) (hnr : n < P ^ r) (i : J) : |
| Fintype.card J - r ≤ |
| (Finset.univ.filter (fun j => j ≠ i ∧ ¬p j ∣ n)).card := by |
| have hbad := few_prime_divisors p hp hinj hP hpP hn hnr |
| have hsum : (Finset.univ.filter (fun j : J => p j ∣ n)).card + |
| (Finset.univ.filter (fun j : J => ¬p j ∣ n)).card = Fintype.card J := by |
| exact Finset.card_filter_add_card_filter_not _ |
| have hins : (Finset.univ.filter (fun j : J => ¬p j ∣ n)) ⊆ |
| insert i (Finset.univ.filter (fun j => j ≠ i ∧ ¬p j ∣ n)) := by |
| intro j hj |
| by_cases hji : j = i |
| · simp [hji] |
| · simp only [Finset.mem_insert, Finset.mem_filter, Finset.mem_univ, true_and] |
| exact Or.inr ⟨hji, (Finset.mem_filter.mp hj).2⟩ |
| have hcard := Finset.card_le_card hins |
| have hle := Finset.card_insert_le i (Finset.univ.filter (fun j => j ≠ i ∧ ¬p j ∣ n)) |
| omega |
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| lemma many_nonzero_phases {J : Type*} [Fintype J] [DecidableEq J] |
| (p : J → ℕ) (hp : ∀ j, (p j).Prime) (hinj : Function.Injective p) |
| {P r : ℕ} (hP : 1 ≤ P) (hpP : ∀ j, P ≤ p j) |
| {q : ℤ} (hq : q ≠ 0) (hqr : q.natAbs < P ^ r) (i : J) : |
| Fintype.card J - r ≤ |
| (Finset.univ.filter (fun j => j ≠ i ∧ ¬(p j : ℤ) ∣ q)).card := by |
| simpa only [Int.natCast_dvd] using |
| many_nondivisors_except p hp hinj hP hpP (Int.natAbs_pos.mpr hq) hqr i |
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| end EditPhase |
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| end OAI |
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