File size: 3,733 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Combinatorics.Crossing.FinitePathChain
namespace OAI
noncomputable section
namespace Zarankiewicz
def alternatingPath (a b : ℕ → Plane) (p : ∀ i, Path (a i) (b i))
(q : ∀ i, Path (b i) (a (i+1))) : (n : ℕ) → Path (a 0) (b n)
| 0 => p 0
| n+1 => ((alternatingPath a b p q n).trans (q n)).trans (p (n+1))
theorem alternatingPath_range_iff (a b : ℕ → Plane) (p : ∀ i, Path (a i) (b i))
(q : ∀ i, Path (b i) (a (i+1))) : (n : ℕ) → ∀ x,
x ∈ Set.range (alternatingPath a b p q n) ↔
(∃ i, i ≤ n ∧ x ∈ Set.range (p i)) ∨ (∃ i, i < n ∧ x ∈ Set.range (q i))
| 0, x => by
constructor
· intro h
exact Or.inl ⟨0,le_rfl,h⟩
· rintro (⟨i,hi,h⟩ | ⟨i,hi,h⟩)
· have he : i=0 := by omega
subst i
exact h
· omega
| n+1, x => by
rw [alternatingPath,Path.trans_range,Path.trans_range,Set.mem_union,Set.mem_union,
alternatingPath_range_iff a b p q n]
constructor
· intro hx
rcases hx with hx | hx
· rcases hx with hx | hx
· rcases hx with ⟨i,hi,h⟩ | ⟨i,hi,h⟩
· exact Or.inl ⟨i,by omega,h⟩
· exact Or.inr ⟨i,by omega,h⟩
· exact Or.inr ⟨n,by omega,hx⟩
· exact Or.inl ⟨n+1,le_rfl,hx⟩
· rintro (⟨i,hi,h⟩ | ⟨i,hi,h⟩)
· by_cases he : i=n+1
· subst i
exact Or.inr h
· exact Or.inl (Or.inl (Or.inl ⟨i,by omega,h⟩))
· by_cases he : i=n
· subst i
exact Or.inl (Or.inr h)
· exact Or.inl (Or.inl (Or.inr ⟨i,by omega,h⟩))
theorem alternatingPath_injective (a b : ℕ → Plane) (p : ∀ i, Path (a i) (b i))
(q : ∀ i, Path (b i) (a (i+1))) : (n : ℕ) →
(∀ i, i ≤ n → Function.Injective (p i)) → (∀ i, i < n → Function.Injective (q i)) →
(∀ i j, i ≤ n → j ≤ n → i ≠ j → Disjoint (Set.range (p i)) (Set.range (p j))) →
(∀ i j, i < n → j < n → i ≠ j → Disjoint (Set.range (q i)) (Set.range (q j))) →
(∀ i j, i ≤ n → j < n → ∀ x, x ∈ Set.range (p i) → x ∈ Set.range (q j) →
(i=j ∧ x=b j) ∨ (i=j+1 ∧ x=a (j+1))) →
Function.Injective (alternatingPath a b p q n)
| 0, hp, _, _, _, _ => hp 0 le_rfl
| n+1, hp, hq, hpp, hqq, hpq => by
have hprev : Function.Injective (alternatingPath a b p q n) :=
alternatingPath_injective a b p q n (fun i hi => hp i (by omega))
(fun i hi => hq i (by omega)) (fun i j hi hj => hpp i j (by omega) (by omega))
(fun i j hi hj => hqq i j (by omega) (by omega))
(fun i j hi hj => hpq i j (by omega) (by omega))
have hfirst : Function.Injective ((alternatingPath a b p q n).trans (q n)) := by
apply path_trans_injective _ _ hprev (hq n (by omega))
intro x hx hy
rcases (alternatingPath_range_iff a b p q n x).mp hx with ⟨i,hi,hh⟩ | ⟨i,hi,hh⟩
· rcases hpq i n (by omega) (by omega) x hh hy with ⟨he,hx⟩ | ⟨he,hx⟩
· exact hx
· omega
· exact False.elim (Set.disjoint_left.mp (hqq i n (by omega) (by omega) (by omega)) hh hy)
apply path_trans_injective _ _ hfirst (hp (n+1) le_rfl)
intro x hx hy
rw [Path.trans_range] at hx
rcases hx with hx | hx
· rcases (alternatingPath_range_iff a b p q n x).mp hx with ⟨i,hi,hh⟩ | ⟨i,hi,hh⟩
· exact False.elim (Set.disjoint_left.mp (hpp i (n+1) (by omega) le_rfl (by omega)) hh hy)
· rcases hpq (n+1) i le_rfl (by omega) x hy hh with ⟨he,hx⟩ | ⟨he,hx⟩ <;> omega
· rcases hpq (n+1) n le_rfl (by omega) x hy hx with ⟨he,hx⟩ | ⟨he,hx⟩
· omega
· exact hx
end Zarankiewicz
end
end OAI
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