| import OAI.NumberTheory.Ostmann.Conclusion.PartitionPermutations |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Ostmann.Conclusion |
|
|
| def overlapRelation {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) (a b : Fin r) : Prop := |
| ∃ u v : Fin m, (σ (a,u)).1 = (σ (b,v)).1 |
|
|
| def OverlapComponent {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) := |
| Quotient (Relation.EqvGen.setoid (overlapRelation σ)) |
| |
| instance {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : Finite (OverlapComponent σ) := by |
| unfold OverlapComponent |
| infer_instance |
| |
| instance {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : Fintype (OverlapComponent σ) := |
| Fintype.ofFinite _ |
| |
| instance {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : DecidableEq (OverlapComponent σ) := |
| Classical.decEq _ |
| |
| def leftComponent {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) |
| (a : Fin r) : OverlapComponent σ := Quotient.mk _ a |
| |
| def rightComponent {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) (hm : 0 < m) |
| (b : Fin r) : OverlapComponent σ := |
| leftComponent σ (σ.symm (b,⟨0,hm⟩)).1 |
| |
| theorem leftComponent_surjective {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : |
| Function.Surjective (leftComponent σ) := by |
| exact Quotient.mk_surjective |
| |
| theorem slot_component {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) (hm : 0 < m) |
| (x : Fin r × Fin m) : leftComponent σ x.1 = rightComponent σ hm (σ x).1 := by |
| apply Quotient.sound |
| apply Relation.EqvGen.rel |
| refine ⟨x.2,(σ.symm ((σ x).1,⟨0,hm⟩)).2,?_⟩ |
| simp only [Prod.mk.eta, Equiv.apply_symm_apply] |
| |
| theorem component_card_balance {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) |
| (hm : 0 < m) (i : OverlapComponent σ) : |
| Fintype.card {a // leftComponent σ a = i} = |
| Fintype.card {b // rightComponent σ hm b = i} := by |
| let e : PartitionMatching (fun x : Fin r × Fin m => leftComponent σ x.1) |
| (fun x : Fin r × Fin m => rightComponent σ hm x.1) := |
| ⟨σ,fun x => (slot_component σ hm x).symm⟩ |
| have h := Fintype.card_congr (partitionMatchingEquiv _ _ e i) |
| simp only [card_slotFiber] at h |
| exact Nat.eq_of_mul_eq_mul_right hm h |
| |
| theorem component_card_pos {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) |
| (i : OverlapComponent σ) : 0 < Fintype.card {a // leftComponent σ a = i} := by |
| obtain ⟨a,ha⟩ := leftComponent_surjective σ i |
| exact Fintype.card_pos_iff.mpr ⟨⟨a,ha⟩⟩ |
| |
| theorem component_cards_sum {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : |
| ∑ i : OverlapComponent σ, Fintype.card {a // leftComponent σ a = i} = r := by |
| have h := Fintype.card_congr (Equiv.sigmaFiberEquiv (leftComponent σ)) |
| simpa only [Fintype.card_sigma, Fintype.card_fin] using h |
| |
| theorem overlapComponent_card_le {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : |
| Fintype.card (OverlapComponent σ) ≤ r := by |
| simpa only [Fintype.card_fin] using |
| Fintype.card_le_of_surjective (leftComponent σ) (leftComponent_surjective σ) |
| |
| theorem component_matching_card {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) |
| (hm : 0 < m) : |
| Nat.card (PartitionMatching (fun x : Fin r × Fin m => leftComponent σ x.1) |
| (fun x : Fin r × Fin m => rightComponent σ hm x.1)) = |
| ∏ i : OverlapComponent σ, (Fintype.card {a // leftComponent σ a = i} * m).factorial := |
| card_slotPartitionMatching _ _ (component_card_balance σ hm) |
| |
| def BadArrangement {r m : ℕ} (σ : Equiv.Perm (Fin r × Fin m)) : Prop := |
| 3*r < 4*Fintype.card (OverlapComponent σ) |
| |
| end Ostmann.Conclusion |
| |
| end |
| |
| end OAI |
| |
