| import Mathlib.Algebra.BigOperators.Group.Finset.Defs |
| import Mathlib.Algebra.BigOperators.Intervals |
| import Mathlib.Algebra.Order.Archimedean.Real.Basic |
| import Mathlib.Algebra.Ring.Parity |
| import Mathlib.Data.Finset.Card |
| import Mathlib.Data.Finset.Prod |
| import Mathlib.Data.Fintype.Pi |
| import Mathlib.Data.Nat.Cast.Field |
| import Mathlib.Data.Nat.Choose.Basic |
| import Mathlib.Data.Nat.Choose.Cast |
| import Mathlib.Order.Interval.Finset.Nat |
| import Mathlib.Tactic.IntervalCases |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Ring |
| import Mathlib.Tactic.Zify |
|
|
| namespace OAI |
|
|
| /-! Harary–Hill arithmetic, residue-page definitions, and the even-order deletion inequality. -/ |
|
|
| section |
|
|
| /-! The Harary–Hill formula and two-page crossing counts. |
|
|
| -/ |
|
|
| open scoped BigOperators |
| noncomputable section |
| namespace Paper170 |
|
|
| / |
| def hill (n : ℕ) : ℕ := |
| (n / 2 * ((n - 1) / 2) * ((n - 2) / 2) * ((n - 3) / 2)) / 4 |
|
|
| / |
| def residueColor (n r : ℕ) : Bool := decide (r % n < n / 2) |
|
|
| / |
| def crossingTuples (n : ℕ) : Finset (Fin 4 → Fin n) := by |
| classical |
| exact Finset.univ.filter (fun v => StrictMono v ∧ |
| residueColor n ((v 0).val + (v 2).val) = |
| residueColor n ((v 1).val + (v 3).val)) |
|
|
| / |
| def sameColorResidues (n k : ℕ) : Finset ℕ := |
| (Finset.range n).filter (fun r => residueColor n r = residueColor n (r + k)) |
|
|
| / |
| def sameColorGapTuples (n k : ℕ) : Finset (ℕ × ℕ × ℕ) := |
| ((Finset.range n) ×ˢ ((Finset.Icc 1 (k - 1)) ×ˢ (Finset.Icc 1 (n - k - 1)))).filter |
| (fun v => residueColor n (2 * v.1 + v.2.1 + v.2.2) = |
| residueColor n (2 * v.1 + v.2.1 + v.2.2 + k)) |
|
|
| / |
| def residueFiber (n b r : ℕ) : Finset ℕ := |
| (Finset.range n).filter (fun a => (2 * a + b) % n = r) |
|
|
| / |
| def sameColorResiduesParity (n k p : ℕ) : Finset ℕ := |
| (sameColorResidues n k).filter (fun r => r % 2 = p) |
|
|
| / |
| def firstGapParity (k b p : ℕ) : Finset ℕ := |
| (Finset.Icc 1 (k - 1)).filter (fun g => (g + b) % 2 = p) |
|
|
| / |
| def crossingAntiderivative (n k : ℤ) : ℤ := |
| k * (k - 1) * (n - k - 1) * (n - k - 2) |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Paper170 |
|
|
| theorem crossingAntiderivative_sub (n k : ℤ) : |
| crossingAntiderivative n k - crossingAntiderivative n (k - 1) = |
| 2 * (k - 1) * (n - k - 1) * (n - 2 * k) := by |
| unfold crossingAntiderivative |
| ring |
|
|
| theorem hill_numerator_dvd (n : ℕ) : |
| 4 ∣ n / 2 * ((n - 1) / 2) * ((n - 2) / 2) * ((n - 3) / 2) := by |
| have h₂ : (n - 2) / 2 = n / 2 - 1 := by omega |
| have h₃ : (n - 3) / 2 = (n - 1) / 2 - 1 := by omega |
| rw [h₂, h₃] |
| convert Nat.mul_dvd_mul (Nat.two_dvd_mul_sub_one (n / 2)) |
| (Nat.two_dvd_mul_sub_one ((n - 1) / 2)) using 1 |
| ring |
|
|
| theorem hill_even (s : ℕ) : hill (2 * s) = s * (s - 1) ^ 2 * (s - 2) / 4 := by |
| have h₁ : (2 * s - 1) / 2 = s - 1 := by omega |
| have h₂ : (2 * s - 2) / 2 = s - 1 := by omega |
| have h₃ : (2 * s - 3) / 2 = s - 2 := by omega |
| simp only [hill, Nat.mul_div_right s (by decide : 0 < 2), h₁, h₂, h₃] |
| congr 1 |
| ring |
|
|
| theorem hill_odd (s : ℕ) : hill (2 * s + 1) = (s.choose 2) ^ 2 := by |
| have h₀ : (2 * s + 1) / 2 = s := by omega |
| have h₁ : (2 * s + 1 - 1) / 2 = s := by omega |
| have h₂ : (2 * s + 1 - 2) / 2 = s - 1 := by omega |
| have h₃ : (2 * s + 1 - 3) / 2 = s - 1 := by omega |
| have hdiv := Nat.two_dvd_mul_sub_one s |
| obtain ⟨a, ha⟩ := hdiv |
| rw [hill, h₀, h₁, h₂, h₃, Nat.choose_two_right] |
| have hp : s * s * (s - 1) * (s - 1) = 4 * a ^ 2 := by nlinarith [ha] |
| rw [hp, ha] |
| simp |
|
|
| theorem hill_eq_floor_product {n : ℕ} (hn : 1 ≤ n) : |
| (hill n : ℝ) = (1 / 4 : ℝ) * (⌊(n : ℝ) / 2⌋ : ℝ) * |
| (⌊((n : ℝ) - 1) / 2⌋ : ℝ) * (⌊((n : ℝ) - 2) / 2⌋ : ℝ) * |
| (⌊((n : ℝ) - 3) / 2⌋ : ℝ) := by |
| by_cases hsmall : n < 3 |
| · interval_cases n <;> norm_num [hill] |
| have hf (j : ℕ) (hj : j ≤ n) : |
| (⌊((n : ℝ) - j) / 2⌋ : ℝ) = ((n - j) / 2 : ℕ) := by |
| rw [← Nat.cast_sub hj, show (2 : ℝ) = ((2 : ℕ) : ℝ) by rfl, |
| Int.floor_div_natCast, Int.floor_natCast] |
| norm_cast |
| rw [hill, Nat.cast_div (hill_numerator_dvd n) (by norm_num : (4 : ℝ) ≠ 0)] |
| have hf₀ := hf 0 (by omega) |
| have hf₁ := hf 1 (by omega) |
| have hf₂ := hf 2 (by omega) |
| have hf₃ := hf 3 (by omega) |
| simp only [Nat.cast_zero, sub_zero, Nat.sub_zero] at hf₀ |
| norm_num only [Nat.cast_one, Nat.cast_ofNat] at hf₁ hf₂ hf₃ |
| rw [hf₀, hf₁, hf₂, hf₃] |
| push_cast |
| ring |
|
|
| theorem crossing_sum_partial {n m : ℕ} (hn : 3 ≤ n) (hm : 2 * m ≤ n) : |
| 2 * ∑ k ∈ Finset.Icc 1 m, (k - 1) * (n - k - 1) * (n - 2 * k) = |
| m * (m - 1) * (n - m - 1) * (n - m - 2) := by |
| induction m with |
| | zero => simp |
| | succ m ih => |
| rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ m + 1), mul_add, |
| ih (by omega)] |
| by_cases hzero : m = 0 |
| · subst m; simp |
| have hm₁ : 1 ≤ m := by omega |
| zify |
| simp (disch := omega) only [Nat.cast_sub, Nat.cast_add, Nat.cast_mul, |
| Nat.cast_one, Nat.cast_ofNat] |
| ring |
|
|
| theorem crossing_sum_telescope {n : ℕ} (hn : 3 ≤ n) : |
| (2 * ∑ k ∈ Finset.Icc 1 (n / 2), (k - 1) * (n - k - 1) * (n - 2 * k)) = |
| (n / 2) * (n / 2 - 1) * (n - n / 2 - 1) * (n - n / 2 - 2) := |
| crossing_sum_partial hn (by omega) |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Paper170 |
|
|
| theorem cyclic_count_pairing {n : ℕ} (hn : 3 ≤ n) : |
| (∑ k ∈ Finset.Icc 2 (n - 2), |
| (k - 1) * (n - k - 1) * (n - 2 * min k (n - k))) = |
| 2 * ∑ k ∈ Finset.Icc 1 (n / 2), (k - 1) * (n - k - 1) * (n - 2 * k) := by |
| let F : ℕ → ℕ := fun k => (k - 1) * (n - k - 1) * (n - 2 * min k (n - k)) |
| have hsym (k : ℕ) (hk : k ≤ n) : F (n - k) = F k := by |
| dsimp [F] |
| rw [Nat.sub_sub_self hk, min_comm] |
| ring |
| have hfull : (∑ k ∈ Finset.Icc 2 (n - 2), F k) = ∑ k ∈ Finset.Ico 1 n, F k := by |
| apply Finset.sum_subset |
| · intro k hk |
| simp only [Finset.mem_Icc, Finset.mem_Ico] at hk ⊢ |
| omega |
| · intro k hk hnot |
| simp only [Finset.mem_Ico, Finset.mem_Icc, not_and] at hk hnot |
| have he : k = 1 ∨ k = n - 1 := by omega |
| rcases he with rfl | rfl |
| · simp [F] |
| · have he : n - (n - 1) - 1 = 0 := by omega |
| simp [F, he] |
| have hsplit := Finset.sum_Ico_consecutive F (show 1 ≤ n / 2 + 1 by omega) |
| (show n / 2 + 1 ≤ n by omega) |
| have htail : (∑ k ∈ Finset.Ico (n / 2 + 1) n, F k) = |
| ∑ k ∈ Finset.Ico 1 (n - n / 2), F k := by |
| calc |
| _ = ∑ k ∈ Finset.Ico (n / 2 + 1) n, F (n - k) := by |
| apply Finset.sum_congr rfl |
| intro k hk |
| exact (hsym k (by have := (Finset.mem_Ico.mp hk).2; omega)).symm |
| _ = _ := by |
| simpa only [Nat.add_sub_cancel_left, Nat.add_sub_add_right] using |
| (Finset.sum_Ico_reflect F (n / 2 + 1) (m := n) (n := n) (by omega)) |
| have hlow : (∑ k ∈ Finset.Ico 1 (n - n / 2), F k) = |
| ∑ k ∈ Finset.Ico 1 (n / 2 + 1), F k := by |
| apply Finset.sum_subset |
| · intro k hk |
| simp only [Finset.mem_Ico] at hk ⊢ |
| omega |
| · intro k hk hnot |
| simp only [Finset.mem_Ico, not_and] at hk hnot |
| have he : n = 2 * k := by omega |
| subst n |
| simp [F, show 2 * k - k = k by omega] |
| have hterm : (∑ k ∈ Finset.Ico 1 (n / 2 + 1), F k) = |
| ∑ k ∈ Finset.Icc 1 (n / 2), (k - 1) * (n - k - 1) * (n - 2 * k) := by |
| rw [Finset.Ico_add_one_right_eq_Icc] |
| apply Finset.sum_congr rfl |
| intro k hk |
| dsimp [F] |
| rw [min_eq_left (by have := (Finset.mem_Icc.mp hk).2; omega)] |
| change (∑ k ∈ Finset.Icc 2 (n - 2), F k) = _ |
| rw [hfull, ← hsplit, htail, hlow, hterm] |
| omega |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| lemma four_mul_hill (n : ℕ) : |
| 4 * hill n = n / 2 * ((n - 1) / 2) * ((n - 2) / 2) * ((n - 3) / 2) := by |
| exact Nat.mul_div_cancel' (hill_numerator_dvd n) |
| |
| lemma four_mul_hill_even (s : ℕ) : |
| 4 * hill (2 * s) = s * (s - 1) ^ 2 * (s - 2) := by |
| rw [four_mul_hill] |
| have h0 : 2 * s / 2 = s := by omega |
| have h1 : (2 * s - 1) / 2 = s - 1 := by omega |
| have h2 : (2 * s - 2) / 2 = s - 1 := by omega |
| have h3 : (2 * s - 3) / 2 = s - 2 := by omega |
| rw [h0, h1, h2, h3] |
| ring |
| |
| lemma four_mul_hill_preceding (s : ℕ) : |
| 4 * hill (2 * s - 1) = (s - 1) ^ 2 * (s - 2) ^ 2 := by |
| rw [four_mul_hill] |
| have h0 : (2 * s - 1) / 2 = s - 1 := by omega |
| have h1 : (2 * s - 1 - 1) / 2 = s - 1 := by omega |
| have h2 : (2 * s - 1 - 2) / 2 = s - 2 := by omega |
| have h3 : (2 * s - 1 - 3) / 2 = s - 2 := by omega |
| rw [h0, h1, h2, h3] |
| ring |
| |
| lemma hill_deletion_relation (s : ℕ) : |
| (2 * s - 4) * hill (2 * s) = 2 * s * hill (2 * s - 1) := by |
| have hs : 2 * s - 4 = 2 * (s - 2) := by omega |
| apply Nat.eq_of_mul_eq_mul_left (by decide : 0 < 4) |
| calc |
| 4 * ((2 * s - 4) * hill (2 * s)) = |
| (2 * s - 4) * (4 * hill (2 * s)) := by ring |
| _ = (2 * s - 4) * (s * (s - 1) ^ 2 * (s - 2)) := by rw [four_mul_hill_even] |
| _ = 2 * s * ((s - 1) ^ 2 * (s - 2) ^ 2) := by rw [hs]; ring |
| _ = 4 * (2 * s * hill (2 * s - 1)) := by rw [← four_mul_hill_preceding]; ring |
| |
| lemma even_lower_bound_of_deletion (s c : ℕ) (hs : 3 ≤ s) |
| (h : 2 * s * hill (2 * s - 1) ≤ (2 * s - 4) * c) : hill (2 * s) ≤ c := by |
| rw [← hill_deletion_relation] at h |
| have hp : 0 < 2 * s - 4 := by omega |
| nlinarith |
| |
| end Paper170 |
| |
| end |
| end |
| |
| end OAI |
| |
