| import OAI.Combinatorics.Crossing.FinitePathChain |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz |
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| 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)) |
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| 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⟩)) |
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| 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 |
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| end Zarankiewicz |
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| end |
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| end OAI |
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