| import OAI.RingTheory.DirectFiniteness.PartitionMerging |
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|
| namespace OAI |
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|
| universe u_1 |
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| noncomputable section |
| open scoped Classical |
| open scoped BigOperators |
| open Equiv Equiv.Perm |
| namespace KaplanskyCounterexample.Planar |
| open Equiv Equiv.Perm |
| variable {α : Type u_1} [Fintype α] |
| |
| lemma quotient_card_antitone {s t : Setoid α} (h : s ≤ t) : |
| Nat.card (Quotient t) ≤ Nat.card (Quotient s) := by |
| exact Nat.card_le_card_of_surjective (Quotient.map id (fun _ _ hxy => h hxy)) |
| (Quotient.map_surjective _ Function.surjective_id) |
|
|
| /-- Connected components of a bipartite rotation system, including edge |
| components consisting of a single edge. Isolated vertices are not encoded. -/ |
| def Ribbon.components (R : Ribbon α) : Setoid α := |
| SameCycle.setoid R.black ⊔ SameCycle.setoid R.white |
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|
| omit [Fintype α] in |
| lemma cycles_inv_card (p : Perm α) : Nat.card (Cycles p⁻¹) = Nat.card (Cycles p) := by |
| have h : SameCycle.setoid p⁻¹ = SameCycle.setoid p := by |
| ext a b |
| exact sameCycle_inv |
| unfold Cycles |
| rw [h] |
|
|
| /-- The elementary genus inequality for an arbitrary finite ribbon system. |
| This supplies the topological constraint needed when components are discarded |
| in the planar minimality argument; it is not assumed as an extra hypothesis. -/ |
| theorem Ribbon.euler_le (R : Ribbon α) : |
| Nat.card (Cycles R.black) + Nat.card (Cycles R.white) + Nat.card (Cycles R.face) ≤ |
| Fintype.card α + 2 * Nat.card (Quotient R.components) := by |
| obtain ⟨lb, hb, hblen, _⟩ := exists_cycle_factorization R.black |
| obtain ⟨lw, hw, hwlen, _⟩ := exists_cycle_factorization R.white |
| obtain ⟨lf, hf, hflen, _⟩ := exists_cycle_factorization R.face⁻¹ |
| rw [cycles_inv_card] at hflen |
| let l := lw ++ lb ++ lf |
| have hl : (l.map fun ab => swap ab.1 ab.2).prod = 1 := by |
| simp only [l, List.map_append, List.prod_append, hb, hw, hf] |
| change R.face * R.face⁻¹ = 1 |
| exact mul_inv_cancel _ |
| have hbl : edgePartition lb ≤ edgePartition l := by |
| apply edgePartition_le |
| intro ab hab |
| apply edgePartition_pair |
| simp [l, hab] |
| have hwl : edgePartition lw ≤ edgePartition l := by |
| apply edgePartition_le |
| intro ab hab |
| apply edgePartition_pair |
| simp [l, hab] |
| have hcomp : R.components ≤ edgePartition l := by |
| apply sup_le |
| · intro x y hxy |
| apply setoid_sameCycle_of_step (edgePartition l) R.black _ hxy |
| intro x |
| rw [← hb] |
| exact hbl (edgePartition_prod lb x) |
| · intro x y hxy |
| apply setoid_sameCycle_of_step (edgePartition l) R.white _ hxy |
| intro x |
| rw [← hw] |
| exact hwl (edgePartition_prod lw x) |
| have hc := quotient_card_antitone hcomp |
| have hbound := transposition_euler_bound l |
| rw [hl, cycles_one_card] at hbound |
| have hlen : l.length = lw.length + lb.length + lf.length := by simp [l, Nat.add_assoc] |
| omega |
|
|
| end KaplanskyCounterexample.Planar |
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| end |
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| end OAI |
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|
