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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
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