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62291ee | 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 | 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
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