| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section FreshEvenBank |
|
|
| theorem even_move_embedding {α : Type*} [Fintype α] [DecidableEq α] {n : ℕ} |
| (hn : n + 2 ≤ Nat.card α) (s t : Fin n ↪ α) : |
| ∃ σ : alternatingGroup α, ∀ i, σ.val (s i) = t i := by |
| let := alternatingGroup.isMultiplyPretransitive α |
| let : MulAction.IsMultiplyPretransitive (alternatingGroup α) α n := |
| MulAction.isMultiplyPretransitive_of_le (by omega : n ≤ Nat.card α - 2) |
| (Nat.sub_le _ _) |
| obtain ⟨σ,hσ⟩ := MulAction.exists_smul_eq (alternatingGroup α) s t |
| exact ⟨σ,fun i => congrArg (fun e : Fin n ↪ α => e i) hσ⟩ |
|
|
| theorem fresh_five_bank {α : Type*} [Fintype α] [DecidableEq α] (J : Finset α) |
| (hJ : J.card + 5 ≤ Fintype.card α) : |
| ∃ t : Fin 5 ↪ α, Disjoint (Finset.univ.map t) J := by |
| classical |
| have hc : 5 ≤ Jᶜ.card := by rw [Finset.card_compl]; omega |
| have he : Nonempty (Fin 5 ↪ {x : α // x ∈ Jᶜ}) := by |
| rw [Function.Embedding.nonempty_iff_card_le,Fintype.card_fin,Fintype.card_coe] |
| exact hc |
| obtain ⟨e⟩ := he |
| refine ⟨e.trans (Function.Embedding.subtype _),?_⟩ |
| apply Finset.disjoint_left.mpr |
| intro x hx hj |
| obtain ⟨i,_,rfl⟩ := Finset.mem_map.mp hx |
| exact Finset.mem_compl.mp (e i).property hj |
|
|
| theorem even_move_disjoint_banks {α : Type*} [Fintype α] [DecidableEq α] |
| (s t : Fin 5 ↪ α) (hd : Disjoint (Finset.univ.map s) (Finset.univ.map t)) : |
| ∃ σ : alternatingGroup α, |
| (∀ i, σ.val (s i) = t i) ∧ |
| σ.val.support ⊆ (Finset.univ.map s) ∪ (Finset.univ.map t) := by |
| classical |
| let S := (Finset.univ.map s) ∪ (Finset.univ.map t) |
| let T := {x : α // x ∈ S} |
| let s' : Fin 5 ↪ T := ⟨fun i => ⟨s i,Finset.mem_union_left _ (Finset.mem_map.mpr |
| ⟨i,Finset.mem_univ _,rfl⟩)⟩,fun _ _ h => s.injective (congrArg Subtype.val h)⟩ |
| let t' : Fin 5 ↪ T := ⟨fun i => ⟨t i,Finset.mem_union_right _ (Finset.mem_map.mpr |
| ⟨i,Finset.mem_univ _,rfl⟩)⟩,fun _ _ h => t.injective (congrArg Subtype.val h)⟩ |
| have hT : Nat.card T = 10 := by |
| rw [Nat.card_eq_fintype_card,Fintype.card_coe] |
| dsimp only [S] |
| rw [Finset.card_union_of_disjoint hd,Finset.card_map,Finset.card_map] |
| simp |
| obtain ⟨ρ,hρ⟩ := even_move_embedding (by omega : 5+2 ≤ Nat.card T) s' t' |
| let e : T ↪ α := Function.Embedding.subtype _ |
| let σ := ρ.val.viaEmbedding e |
| have hσ : σ ∈ alternatingGroup α := by |
| change Equiv.Perm.sign (ρ.val.viaEmbedding e) = 1 |
| rw [Equiv.Perm.viaEmbedding,Equiv.Perm.sign_extendDomain] |
| exact ρ.property |
| refine ⟨⟨σ,hσ⟩,?_,?_⟩ |
| · intro i |
| change ρ.val.viaEmbedding e (e (s' i)) = t i |
| rw [Equiv.Perm.viaEmbedding_apply,hρ] |
| rfl |
| · intro x hx |
| by_contra hn |
| have he : x ∉ Set.range e := by |
| rintro ⟨y,rfl⟩ |
| exact hn y.property |
| exact Equiv.Perm.mem_support.mp hx (Equiv.Perm.viaEmbedding_apply_of_notMem _ _ _ he) |
|
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| end FreshEvenBank |
|
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| end SimpleAmenable |
| end |
| end |
|
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| end OAI |
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