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)

end FreshEvenBank

end SimpleAmenable
end
end

end OAI