| import OAI.Combinatorics.CompleteCrossing.Hill |
| import Mathlib.Data.Fintype.Card |
| import Mathlib.Data.Nat.ModEq |
| import Mathlib.Data.Nat.Prime.Basic |
|
|
| namespace OAI |
|
|
| /-! Residue fibers, same-page residues, and parity counts. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| theorem sameColorResidues_eq_intervals {n k : ℕ} (hn : 3 ≤ n) |
| (hk : k ≤ n / 2) : |
| sameColorResidues n k = |
| Finset.Ico 0 (n / 2 - k) ∪ Finset.Ico (n / 2) (n - k) := by |
| ext r |
| simp only [sameColorResidues, Finset.mem_filter, Finset.mem_range, |
| residueColor, decide_eq_decide, Finset.mem_union, Finset.mem_Ico] |
| by_cases hr : r < n |
| · rw [Nat.mod_eq_of_lt hr] |
| by_cases hrs : r + k < n |
| · rw [Nat.mod_eq_of_lt hrs] |
| omega |
| · rw [Nat.mod_eq_sub_mod (by omega : n ≤ r + k), |
| Nat.mod_eq_of_lt (by omega : r + k - n < n)] |
| omega |
| · omega |
|
|
| theorem sameColorResidues_eq_intervals_large {n k : ℕ} (hn : 3 ≤ n) |
| (hk : n / 2 < k) (hkn : k < n) : |
| sameColorResidues n k = |
| Finset.Ico (n - k) (n / 2) ∪ Finset.Ico (n + n / 2 - k) n := by |
| ext r |
| simp only [sameColorResidues, Finset.mem_filter, Finset.mem_range, |
| residueColor, decide_eq_decide, Finset.mem_union, Finset.mem_Ico] |
| by_cases hr : r < n |
| · rw [Nat.mod_eq_of_lt hr] |
| by_cases hrs : r + k < n |
| · rw [Nat.mod_eq_of_lt hrs] |
| omega |
| · rw [Nat.mod_eq_sub_mod (by omega : n ≤ r + k), |
| Nat.mod_eq_of_lt (by omega : r + k - n < n)] |
| omega |
| · omega |
|
|
| theorem sameColorResidues_card {n k : ℕ} (hn : 3 ≤ n) (hk : 2 ≤ k) |
| (hkn : k ≤ n - 2) : |
| (sameColorResidues n k).card = n - 2 * min k (n - k) := by |
| by_cases hh : k ≤ n / 2 |
| · rw [sameColorResidues_eq_intervals hn hh] |
| have hd : Disjoint (Finset.Ico 0 (n / 2 - k)) (Finset.Ico (n / 2) (n - k)) := by |
| apply Finset.disjoint_left.mpr |
| simp only [Finset.mem_Ico] |
| omega |
| rw [Finset.card_union_of_disjoint hd, Nat.card_Ico, Nat.card_Ico, |
| min_eq_left (by omega : k ≤ n - k)] |
| omega |
| · rw [sameColorResidues_eq_intervals_large hn (by omega) (by omega)] |
| have hd : Disjoint (Finset.Ico (n - k) (n / 2)) (Finset.Ico (n + n / 2 - k) n) := by |
| apply Finset.disjoint_left.mpr |
| simp only [Finset.mem_Ico] |
| omega |
| rw [Finset.card_union_of_disjoint hd, Nat.card_Ico, Nat.card_Ico, |
| min_eq_right (by omega : n - k ≤ k)] |
| omega |
|
|
| @[simp] theorem residueColor_mod (n r : ℕ) : residueColor n (r % n) = residueColor n r := by |
| simp [residueColor] |
|
|
| theorem residueColor_add_half {m : ℕ} (hm : 0 < m) (r : ℕ) : |
| residueColor (2 * m) (r + m) = !residueColor (2 * m) r := by |
| have hr := Nat.mod_lt r (by omega : 0 < 2 * m) |
| simp only [residueColor, Nat.mul_div_right m (by decide : 0 < 2)] |
| rw [Nat.add_mod, Nat.mod_eq_of_lt (by omega : m < 2 * m)] |
| by_cases hh : r % (2 * m) < m |
| · rw [Nat.mod_eq_of_lt (by omega : r % (2 * m) + m < 2 * m)] |
| simp [hh, show ¬r % (2 * m) + m < m by omega] |
| · rw [Nat.mod_eq_sub_mod (by omega : 2 * m ≤ r % (2 * m) + m), |
| Nat.mod_eq_of_lt (by omega : r % (2 * m) + m - 2 * m < 2 * m)] |
| simp [hh, show r % (2 * m) + m - 2 * m < m by omega] |
|
|
| theorem sameColorResidues_half_shift {m k r : ℕ} (hm : 0 < m) |
| (hr : r < 2 * m) : |
| r ∈ sameColorResidues (2 * m) k ↔ |
| (r + m) % (2 * m) ∈ sameColorResidues (2 * m) k := by |
| simp only [sameColorResidues, Finset.mem_filter, Finset.mem_range, hr, |
| Nat.mod_lt _ (by omega : 0 < 2 * m), true_and, residueColor_mod] |
| rw [show residueColor (2 * m) ((r + m) % (2 * m) + k) = |
| residueColor (2 * m) (r + k + m) by simp [residueColor, Nat.add_mod, Nat.add_right_comm]] |
| rw [residueColor_add_half hm, residueColor_add_half hm] |
| simp |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| theorem residueFiber_card_odd {n b r : ℕ} (hn : Odd n) (hr : r < n) : |
| (residueFiber n b r).card = 1 := by |
| let f : Fin n → Fin n := fun a => ⟨(2 * a.val + b) % n, Nat.mod_lt _ (by omega)⟩ |
| have hi : Function.Injective f := by |
| intro a c h |
| have hmod : Nat.ModEq n (2 * a.val + b) (2 * c.val + b) := congrArg Fin.val h |
| have he := (hmod.add_right_cancel' b).cancel_left_of_coprime hn.coprime_two_right |
| apply Fin.ext |
| simpa only [Nat.ModEq, Nat.mod_eq_of_lt a.isLt, Nat.mod_eq_of_lt c.isLt] using he |
| obtain ⟨a, ha⟩ := (Finite.surjective_of_injective hi) ⟨r, hr⟩ |
| have hset : residueFiber n b r = {a.val} := by |
| ext c |
| simp only [residueFiber, Finset.mem_filter, Finset.mem_range, Finset.mem_singleton] |
| constructor |
| · rintro ⟨hc, hcr⟩ |
| have he : f ⟨c, hc⟩ = f a := by |
| rw [ha] |
| exact Fin.ext hcr |
| exact congrArg Fin.val (hi he) |
| · rintro rfl |
| exact ⟨a.isLt, congrArg Fin.val ha⟩ |
| rw [hset, Finset.card_singleton] |
|
|
| theorem add_mod_inj {n a c b : ℕ} (ha : a < n) (hc : c < n) |
| (h : (a + b) % n = (c + b) % n) : a = c := by |
| have he := Nat.ModEq.add_right_cancel' b h |
| simpa only [Nat.ModEq, Nat.mod_eq_of_lt ha, Nat.mod_eq_of_lt hc] using he |
|
|
| theorem add_mod_surj {n r : ℕ} (hr : r < n) (b : ℕ) : |
| ∃ a < n, (a + b) % n = r := by |
| let f : Fin n → Fin n := fun a => ⟨(a.val + b) % n, Nat.mod_lt _ (by omega)⟩ |
| have hi : Function.Injective f := by |
| intro a c h |
| exact Fin.ext (add_mod_inj a.isLt c.isLt (congrArg Fin.val h)) |
| obtain ⟨a, ha⟩ := (Finite.surjective_of_injective hi) ⟨r, hr⟩ |
| exact ⟨a.val, a.isLt, congrArg Fin.val ha⟩ |
|
|
| theorem doubled_residue {m a b r : ℕ} : |
| (2 * a + b) % (2 * m) = r ↔ |
| (a + b / 2) % m = r / 2 ∧ b % 2 = r % 2 := by |
| have hh : (2 * a + b) % (2 * m) = b % 2 + 2 * ((a + b / 2) % m) := by |
| rw [Nat.mod_mul] |
| congr 1 |
| · omega |
| · rw [show (2 * a + b) / 2 = a + b / 2 by omega] |
| rw [hh] |
| omega |
|
|
| theorem residueFiber_card_even {m b r : ℕ} (hm : 0 < m) (hr : r < 2 * m) : |
| (residueFiber (2 * m) b r).card = if r % 2 = b % 2 then 2 else 0 := by |
| by_cases hp : r % 2 = b % 2 |
| · rw [ite_eq_left hp] |
| obtain ⟨a, ha, har⟩ := add_mod_surj (show r / 2 < m by omega) (b / 2) |
| have hset : residueFiber (2 * m) b r = {a, a + m} := by |
| ext c |
| simp only [residueFiber, Finset.mem_filter, Finset.mem_range, |
| Finset.mem_insert, Finset.mem_singleton, doubled_residue] |
| constructor |
| · rintro ⟨hc, hcr, _⟩ |
| have he : c % m = a := add_mod_inj (Nat.mod_lt _ hm) ha |
| (by simpa only [Nat.add_mod, Nat.mod_mod] using hcr.trans har.symm) |
| by_cases hcm : c < m |
| · left |
| simpa only [Nat.mod_eq_of_lt hcm] using he |
| · right |
| rw [Nat.mod_eq_sub_mod (by omega : m ≤ c), |
| Nat.mod_eq_of_lt (by omega : c - m < m)] at he |
| omega |
| · rintro (rfl | rfl) |
| · exact ⟨by omega, har, hp.symm⟩ |
| · refine ⟨by omega, ?_, hp.symm⟩ |
| simpa only [Nat.add_right_comm a m (b / 2), Nat.add_mod_right] using har |
| rw [hset, Finset.card_insert_of_notMem (by simp; omega), Finset.card_singleton] |
| · rw [ite_eq_right hp] |
| apply Finset.card_eq_zero.mpr |
| apply Finset.eq_empty_iff_forall_notMem.mpr |
| intro a |
| simp only [residueFiber, Finset.mem_filter, Finset.mem_range, doubled_residue] |
| rintro ⟨_, _, he⟩ |
| exact hp he.symm |
| |
| end Paper170 |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| namespace Paper170 |
| |
| theorem parity_balance_of_involution (s : Finset ℕ) (f : ℕ → ℕ) (c : ℕ) |
| (hmap : ∀ r ∈ s, f r ∈ s) (hinv : ∀ r ∈ s, f (f r) = r) |
| (hpar : ∀ r ∈ s, (f r + c) % 2 ≠ (r + c) % 2) : |
| (s.filter (fun r => (r + c) % 2 = 0)).card = |
| (s.filter (fun r => (r + c) % 2 = 1)).card := by |
| apply Finset.card_nbij' f f |
| · intro r hr |
| obtain ⟨hs, he⟩ := Finset.mem_filter.mp hr |
| exact Finset.mem_filter.mpr ⟨hmap r hs, by have := hpar r hs; omega⟩ |
| · intro r hr |
| obtain ⟨hs, he⟩ := Finset.mem_filter.mp hr |
| exact Finset.mem_filter.mpr ⟨hmap r hs, by have := hpar r hs; omega⟩ |
| · intro r hr |
| exact hinv r (Finset.mem_filter.mp hr).1 |
| · intro r hr |
| exact hinv r (Finset.mem_filter.mp hr).1 |
| |
| theorem interval_parity_balance {a b : ℕ} (heven : Even (b - a)) (c : ℕ) : |
| ((Finset.Ico a b).filter (fun r => (r + c) % 2 = 0)).card = |
| ((Finset.Ico a b).filter (fun r => (r + c) % 2 = 1)).card := by |
| obtain ⟨t, ht⟩ := heven |
| let f : ℕ → ℕ := fun r => if (r - a) % 2 = 0 then r + 1 else r - 1 |
| apply parity_balance_of_involution (Finset.Ico a b) f c |
| · intro r hr |
| simp only [Finset.mem_Ico] at hr ⊢ |
| dsimp [f] |
| split_ifs <;> omega |
| · intro r hr |
| simp only [Finset.mem_Ico] at hr |
| dsimp [f] |
| split_ifs <;> omega |
| · intro r hr |
| simp only [Finset.mem_Ico] at hr |
| dsimp [f] |
| split_ifs <;> omega |
| |
| theorem firstGapParity_balance {k : ℕ} (hk : Odd k) (b : ℕ) : |
| (firstGapParity k b 0).card = (firstGapParity k b 1).card := by |
| have heven : Even (k - 1) := by |
| rw [Nat.even_iff] |
| obtain ⟨t, ht⟩ := hk |
| omega |
| have heq : Finset.Icc 1 (k - 1) = Finset.Ico 1 k := by |
| ext r |
| simp only [Finset.mem_Icc, Finset.mem_Ico] |
| obtain ⟨t, ht⟩ := hk |
| omega |
| simpa only [firstGapParity, heq] using interval_parity_balance heven b |
|
|
| theorem parity_balance_union (s t : Finset ℕ) (hd : Disjoint s t) |
| (hs : (s.filter (fun r => r % 2 = 0)).card = (s.filter (fun r => r % 2 = 1)).card) |
| (ht : (t.filter (fun r => r % 2 = 0)).card = (t.filter (fun r => r % 2 = 1)).card) : |
| ((s ∪ t).filter (fun r => r % 2 = 0)).card = |
| ((s ∪ t).filter (fun r => r % 2 = 1)).card := by |
| have hdis (p : ℕ) : Disjoint (s.filter (fun r => r % 2 = p)) |
| (t.filter (fun r => r % 2 = p)) := hd.mono (Finset.filter_subset _ _) (Finset.filter_subset _ _) |
| simp only [Finset.filter_union, Finset.card_union_of_disjoint (hdis 0), |
| Finset.card_union_of_disjoint (hdis 1), hs, ht] |
|
|
| theorem sameColorResidues_parity_balance {m k : ℕ} (hm : 0 < m) |
| (hk : 2 ≤ k) (hkn : k ≤ 2 * m - 2) (hparity : Odd m ∨ Even k) : |
| (sameColorResiduesParity (2 * m) k 0).card = |
| (sameColorResiduesParity (2 * m) k 1).card := by |
| by_cases hmo : Odd m |
| · obtain ⟨t, ht⟩ := hmo |
| let f : ℕ → ℕ := fun r => if r < m then r + m else r - m |
| have hform (r : ℕ) (hr : r < 2 * m) : f r = (r + m) % (2 * m) := by |
| dsimp [f] |
| split_ifs with h |
| · rw [Nat.mod_eq_of_lt (by omega)] |
| · rw [Nat.mod_eq_sub_mod (by omega), Nat.mod_eq_of_lt (by omega)] |
| omega |
| apply parity_balance_of_involution (sameColorResidues (2 * m) k) f 0 |
| · intro r hr |
| have hrl : r < 2 * m := (Finset.mem_filter.mp hr).1 |> Finset.mem_range.mp |
| rw [hform r hrl] |
| exact (sameColorResidues_half_shift hm hrl).mp hr |
| · intro r hr |
| have hrl : r < 2 * m := (Finset.mem_filter.mp hr).1 |> Finset.mem_range.mp |
| dsimp [f] |
| split_ifs <;> omega |
| · intro r hr |
| have hrl : r < 2 * m := (Finset.mem_filter.mp hr).1 |> Finset.mem_range.mp |
| dsimp [f] |
| split_ifs <;> omega |
| · have hme : Even m := Nat.not_odd_iff_even.mp hmo |
| have hke : Even k := hparity.resolve_left hmo |
| have hm₂ : 3 ≤ 2 * m := by omega |
| have even_diff (a b : ℕ) (ha : Even a) (hb : Even b) : Even (a - b) := by |
| rw [Nat.even_iff] at * |
| omega |
| unfold sameColorResiduesParity |
| by_cases hhalf : k ≤ m |
| · rw [sameColorResidues_eq_intervals hm₂ (by simpa using hhalf)] |
| simp only [Nat.mul_div_right m (by decide : 0 < 2)] |
| apply parity_balance_union |
| · apply Finset.disjoint_left.mpr |
| simp only [Finset.mem_Ico] |
| omega |
| · simpa using interval_parity_balance (a := 0) (b := m - k) |
| (by simpa using even_diff m k hme hke) 0 |
| · have he : Even (2 * m - k - m) := by |
| apply even_diff _ _ (even_diff _ _ (by simp) hke) hme |
| simpa using interval_parity_balance he 0 |
| · rw [sameColorResidues_eq_intervals_large hm₂ (by simpa using Nat.lt_of_not_ge hhalf) (by omega)] |
| simp only [Nat.mul_div_right m (by decide : 0 < 2)] |
| apply parity_balance_union |
| · apply Finset.disjoint_left.mpr |
| simp only [Finset.mem_Ico] |
| omega |
| · have he : Even (m - (2 * m - k)) := even_diff _ _ hme (even_diff _ _ (by simp) hke) |
| simpa using interval_parity_balance he 0 |
| · have he : Even (2 * m - (2 * m + m - k)) := |
| even_diff _ _ (by simp) (even_diff _ _ |
| (by simpa only [two_mul, add_assoc] using hme.add hme |>.add hme) hke) |
| simpa using interval_parity_balance he 0 |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
