rubin-lean4/Rubin/AlgebraicDisjointness.lean

import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.Subgroup.Actions
import Mathlib.GroupTheory.Commutator
import Mathlib.Topology.Basic
import Mathlib.Data.Fintype.Perm
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.IntervalCases

import Rubin.RigidStabilizer
import Rubin.SmulImage
import Rubin.Topological
import Rubin.FaithfulAction

namespace Rubin

class LocallyMoving (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :=
  locally_moving: ∀ U : Set α, IsOpen U → Set.Nonempty U → RigidStabilizer G U ≠ ⊥
#align is_locally_moving Rubin.LocallyMoving

namespace LocallyMoving

theorem get_nontrivial_rist_elem {G α : Type _}
  [Group G]
  [TopologicalSpace α]
  [MulAction G α]
  [h_lm : LocallyMoving G α]
  {U: Set α}
  (U_open : IsOpen U)
  (U_nonempty : U.Nonempty) :
  ∃ x : G, x ∈ RigidStabilizer G U ∧ x ≠ 1 :=
by
  have rist_ne_bot := h_lm.locally_moving U U_open U_nonempty
  exact (or_iff_right rist_ne_bot).mp (Subgroup.bot_or_exists_ne_one _)

end LocallyMoving

structure AlgebraicallyDisjointElem {G : Type _} [Group G] (f g h : G) :=
  non_commute: ¬Commute f h
  fst : G
  snd : G
  fst_commute : Commute fst g
  snd_commute : Commute snd g
  comm_elem_commute : Commute ⁅fst, ⁅snd, h⁆⁆ g
  comm_elem_nontrivial : ⁅fst, ⁅snd, h⁆⁆ ≠ 1

namespace AlgebraicallyDisjointElem

def comm_elem {G : Type _} [Group G] {f g h : G} (disj_elem : AlgebraicallyDisjointElem f g h) : G :=
  ⁅disj_elem.fst, ⁅disj_elem.snd, h⁆⁆

@[simp]
theorem comm_elem_eq {G : Type _} [Group G] {f g h : G} (disj_elem : AlgebraicallyDisjointElem f g h) :
  disj_elem.comm_elem = ⁅disj_elem.fst, ⁅disj_elem.snd, h⁆⁆ :=
by
  unfold comm_elem
  simp

end AlgebraicallyDisjointElem

/--
A pair (f, g) is said to be "algebraically disjoint" if it can produce an instance of
[`AlgebraicallyDisjointElem`] for any element `h ∈ G` such that `f` and `h` don't commute.

In other words, `g` is algebraically disjoint from `f` if `∀ h ∈ G`, with `⁅f, h⁆ ≠ 1`,
there exists a pair `f₁, f₂ ∈ Centralizer(g, G)`,
so that `⁅f₁, ⁅f₂, h⁆⁆` is a nontrivial element of `Centralizer(g, G)`.

Here the definition of `k ∈ Centralizer(g, G)` is directly unrolled as `Commute k g`.

This is a slightly stronger proposition that plain disjointness, a
-/
def AlgebraicallyDisjoint {G : Type _} [Group G] (f g : G) :=
  ∀ (h : G), ¬Commute f h → AlgebraicallyDisjointElem f g h

theorem AlgebraicallyDisjoint_mk {G : Type _} [Group G] {f g : G}
  (mk_thm : ∀ h : G, ¬Commute f h →
    ∃ f₁ f₂ : G, Commute f₁ g ∧ Commute f₂ g ∧ Commute ⁅f₁, ⁅f₂, h⁆⁆ g ∧ ⁅f₁, ⁅f₂, h⁆⁆ ≠ 1
  ) : AlgebraicallyDisjoint f g :=
fun (h : G) (nc : ¬Commute f h) => {
  non_commute := nc,
  fst := (mk_thm h nc).choose
  snd := (mk_thm h nc).choose_spec.choose
  fst_commute := (mk_thm h nc).choose_spec.choose_spec.left
  snd_commute := (mk_thm h nc).choose_spec.choose_spec.right.left
  comm_elem_commute := (mk_thm h nc).choose_spec.choose_spec.right.right.left
  comm_elem_nontrivial := by
    exact (mk_thm h nc).choose_spec.choose_spec.right.right.right
}

-- This is an idea of having a Prop version of AlgebraicallyDisjoint, but it sounds painful to work with
-- def IsAlgebraicallyDisjoint {G : Type _} [Group G] (f g : G): Prop :=
--   ∀ (h : G), ¬Commute f h → ∃ (f₁ f₂ : G), ∃ (elem : AlgebraicallyDisjointElem f g h), elem.fst = f₁ ∧ elem.snd = f₂

@[simp]
theorem orbit_bot (G : Type _) [Group G] [MulAction G α] (p : α) :
    MulAction.orbit (⊥ : Subgroup G) p = {p} :=
  by
  ext1
  rw [MulAction.mem_orbit_iff]
  constructor
  · rintro ⟨⟨_, g_bot⟩, g_to_x⟩
    rw [← g_to_x, Set.mem_singleton_iff, Subgroup.mk_smul]
    exact (Subgroup.mem_bot.mp g_bot).symm ▸ one_smul _ _
  exact fun h => ⟨1, Eq.trans (one_smul _ p) (Set.mem_singleton_iff.mp h).symm⟩
#align orbit_bot Rubin.orbit_bot

variable {G α : Type _}
variable [Group G]
variable [TopologicalSpace α]
variable [ContinuousMulAction G α]
variable [FaithfulSMul G α]

instance dense_locally_moving [T2Space α]
  (H_nip : has_no_isolated_points α)
  (H_ld : LocallyDense G α) :
  LocallyMoving G α
where
  locally_moving := by
    intros U _ H_nonempty
    by_contra h_rs
    have ⟨elem, ⟨_, some_in_orbit⟩⟩ := H_ld.nonEmpty H_nonempty
    have H_nebot := has_no_isolated_points_neBot H_nip elem
    rw [h_rs] at some_in_orbit
    simp at some_in_orbit

lemma disjoint_nbhd [T2Space α] {g : G} {x : α} (x_moved: g • x ≠ x) :
  ∃ U: Set α, IsOpen U ∧ x ∈ U ∧ Disjoint U (g •'' U) :=
by
  have ⟨V, W, V_open, W_open, gx_in_V, x_in_W, disjoint_V_W⟩ := T2Space.t2 (g • x) x x_moved
  let U := (g⁻¹ •'' V) ∩ W
  use U
  constructor
  {
    -- NOTE: if this is common, then we should make a tactic for solving IsOpen goals
    exact IsOpen.inter (img_open_open g⁻¹ V V_open) W_open
  }
  constructor
  {
    simp
    rw [mem_inv_smulImage]
    trivial
  }
  {
    apply Set.disjoint_of_subset
    · apply Set.inter_subset_right
    · intro y hy; show y ∈ V

      rw [<-smul_inv_smul g y]
      rw [<-mem_inv_smulImage]

      rw [mem_smulImage] at hy
      simp at hy

      exact hy.left
    · exact disjoint_V_W.symm
  }

lemma disjoint_nbhd_in [T2Space α] {g : G} {x : α} {V : Set α}
  (V_open : IsOpen V) (x_in_V : x ∈ V) (x_moved : g • x ≠ x) :
  ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ U ⊆ V ∧ Disjoint U (g •'' U) :=
by
  have ⟨W, W_open, x_in_W, disjoint_W_img⟩ := disjoint_nbhd x_moved
  use W ∩ V
  simp
  constructor
  {
    apply IsOpen.inter <;> assumption
  }
  constructor
  {
    constructor <;> assumption
  }
  show Disjoint (W ∩ V) (g •'' W ∩ V)
  apply Set.disjoint_of_subset
  · exact Set.inter_subset_left W V
  · show g •'' W ∩ V ⊆ g •'' W
    rewrite [smulImage_inter]
    exact Set.inter_subset_left _ _
  · exact disjoint_W_img

-- Kind of a boring lemma but okay
lemma rewrite_Union (f : Fin 2 × Fin 2 → Set α) :
  (⋃(i : Fin 2 × Fin 2), f i) = (f (0,0) ∪ f (0,1)) ∪ (f (1,0) ∪ f (1,1)) :=
by
  ext x
  simp only [Set.mem_iUnion, Set.mem_union]
  constructor
  · rewrite [forall_exists_index]
    intro i
    fin_cases i
      <;> simp only [Fin.zero_eta, Fin.mk_one]
      <;> intro h
      <;> simp only [h, true_or, or_true]
  · rintro ((h|h)|(h|h)) <;> exact ⟨_, h⟩

-- TODO: move to Rubin.lean
-- TODO: modify the proof to be less "let everything"-y, especially the first half
-- TODO: use the new class thingy to write a cleaner proof?
lemma proposition_1_1_1 [h_lm : LocallyMoving G α] [T2Space α] (f g : G) (supp_disjoint : Disjoint (Support α f) (Support α g)) : AlgebraicallyDisjoint f g := by
  apply AlgebraicallyDisjoint_mk
  intros h h_not_commute
  -- h is not the identity on `Support α f`
  have f_h_not_disjoint := (mt (disjoint_commute (G := G) (α := α)) h_not_commute)
  have ⟨x, ⟨x_in_supp_f, x_in_supp_h⟩⟩ := Set.not_disjoint_iff.mp f_h_not_disjoint
  have hx_ne_x := mem_support.mp x_in_supp_h

  -- Since α is Hausdoff, there is a nonempty V ⊆ Support α f, with h •'' V disjoint from V
  have ⟨V, V_open, x_in_V, V_in_support, disjoint_img_V⟩ := disjoint_nbhd_in (support_open f) x_in_supp_f hx_ne_x

  -- let f₂ be a nontrivial element of the RigidStabilizer G V
  let ⟨f₂, f₂_in_rist_V, f₂_ne_one⟩ := h_lm.get_nontrivial_rist_elem V_open (Set.nonempty_of_mem x_in_V)

  -- Re-use the Hausdoff property of α again, this time yielding W ⊆ V
  let ⟨y, y_moved⟩ := faithful_moves_point' α f₂_ne_one
  have y_in_V := (rigidStabilizer_support.mp f₂_in_rist_V) (mem_support.mpr y_moved)
  let ⟨W, W_open, y_in_W, W_in_V, disjoint_img_W⟩ := disjoint_nbhd_in V_open y_in_V y_moved

  -- Let f₁ be a nontrivial element of RigidStabilizer G W
  let ⟨f₁, f₁_in_rist_W, f₁_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open (Set.nonempty_of_mem y_in_W)

  use f₁
  use f₂
  constructor <;> try constructor
  · apply disjoint_commute (α := α)
    apply Set.disjoint_of_subset_left _ supp_disjoint
    calc
      Support α f₁ ⊆ W := rigidStabilizer_support.mp f₁_in_rist_W
      W ⊆ V := W_in_V
      V ⊆ Support α f := V_in_support
  · apply disjoint_commute (α := α)
    apply Set.disjoint_of_subset_left _ supp_disjoint
    calc
      Support α f₂ ⊆ V := rigidStabilizer_support.mp f₂_in_rist_V
      V ⊆ Support α f := V_in_support

  -- We claim that [f₁, [f₂, h]] is a nontrivial elelement of Centralizer G g
  let k := ⁅f₂, h⁆
  have h₂ : ∀ z ∈ W, f₂ • z = k • z := by
    intro z z_in_W
    simp
    symm
    apply disjoint_support_comm f₂ h _ disjoint_img_V
    · exact W_in_V z_in_W
    · exact rigidStabilizer_support.mp f₂_in_rist_V

  constructor
  · -- then `k*f₁⁻¹*k⁻¹` is supported on k W = f₂ W,
    -- so [f₁,k] is supported on W ∪ f₂ W ⊆ V ⊆ support f, so commutes with g.
    apply disjoint_commute (α := α)
    apply Set.disjoint_of_subset_left _ supp_disjoint
    have supp_f₁_subset_W := (rigidStabilizer_support.mp f₁_in_rist_W)

    show Support α ⁅f₁, ⁅f₂, h⁆⁆ ⊆ Support α f
    calc
      Support α ⁅f₁, k⁆ = Support α ⁅k, f₁⁆ := by rw [<-commutatorElement_inv, support_inv]
      _ ⊆ Support α f₁ ∪ (k •'' Support α f₁) := support_comm α k f₁
      _ ⊆ W ∪ (k •'' Support α f₁) := Set.union_subset_union_left _ supp_f₁_subset_W
      _ ⊆ W ∪ (k •'' W) := by
        apply Set.union_subset_union_right
        exact (smulImage_subset k supp_f₁_subset_W)
      _ = W ∪ (f₂ •'' W) := by rw [<-smulImage_eq_of_smul_eq h₂]
      _ ⊆ V ∪ (f₂ •'' W) := Set.union_subset_union_left _ W_in_V
      _ ⊆ V ∪ V := by
        apply Set.union_subset_union_right
        apply smulImage_subset_in_support f₂ W V W_in_V
        exact rigidStabilizer_support.mp f₂_in_rist_V
      _ ⊆ V := by rw [Set.union_self]
      _ ⊆ Support α f := V_in_support

  · -- finally, [f₁,k] agrees with f₁ on W, so is not the identity.
    have h₄: ∀ z ∈ W, ⁅f₁, k⁆ • z = f₁ • z := by
      apply disjoint_support_comm f₁ k
      exact rigidStabilizer_support.mp f₁_in_rist_W
      rw [<-smulImage_eq_of_smul_eq h₂]
      exact disjoint_img_W
    let ⟨z, z_in_W, z_moved⟩ := faithful_rigid_stabilizer_moves_point f₁_in_rist_W f₁_ne_one

    by_contra h₅
    rw [<-h₄ z z_in_W] at z_moved
    have h₆ : ⁅f₁, ⁅f₂, h⁆⁆ • z = z := by rw [h₅, one_smul]
    exact z_moved h₆
#align proposition_1_1_1 Rubin.proposition_1_1_1

@[simp] lemma smulImage_mul {g h : G} {U : Set α} : g •'' (h •'' U) = (g*h) •'' U :=
  (mul_smulImage g h U)

lemma smul_inj_moves {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j)
  (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) :
  ((f j)⁻¹ * f i) • x ≠ x := by
    by_contra h
    apply i_ne_j
    apply f_smul_inj
    group_action
    group_action at h
    exact h

def smul_inj_nbhd {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j)
  (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) :
  Set α :=
  (disjoint_nbhd (smul_inj_moves i_ne_j f_smul_inj)).choose

lemma smul_inj_nbhd_open {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j)
  (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) :
  IsOpen (smul_inj_nbhd i_ne_j f_smul_inj) :=
by
  exact (disjoint_nbhd (smul_inj_moves i_ne_j f_smul_inj)).choose_spec.1

lemma smul_inj_nbhd_mem {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j)
  (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) :
  x ∈ (smul_inj_nbhd i_ne_j f_smul_inj) :=
by
  exact (disjoint_nbhd (smul_inj_moves i_ne_j f_smul_inj)).choose_spec.2.1

lemma smul_inj_nbhd_disjoint {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} {i j : ι} (i_ne_j : i ≠ j)
  (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)) :
  Disjoint
    (smul_inj_nbhd i_ne_j f_smul_inj)
    ((f j)⁻¹ * f i •'' (smul_inj_nbhd i_ne_j f_smul_inj)) :=
by
  exact (disjoint_nbhd (smul_inj_moves i_ne_j f_smul_inj)).choose_spec.2.2


lemma disjoint_nbhd_fin {ι : Type*} [Fintype ι] [T2Space α]
  {f : ι → G} {x : α} (f_smul_inj : Function.Injective (fun i : ι => (f i) • x)):
  ∃ U : Set α,
  IsOpen U ∧ x ∈ U ∧ (∀ (i j : ι), i ≠ j → Disjoint (f i •'' U) (f j •'' U)) :=
by
  let ι₂ := { p : ι × ι | p.1 ≠ p.2 }
  let U := ⋂(p : ι₂), smul_inj_nbhd p.prop f_smul_inj
  use U

  -- The notations provided afterwards tend to be quite ugly because we used `Exists.choose`,
  -- but the idea is that this all boils down to applying `Exists.choose_spec`, except in the disjointness case,
  -- where we transform `Disjoint (f i •'' U) (f j •'' U)` into `Disjoint U ((f i)⁻¹ * f j •'' U)`
  -- and transform both instances of `U` into `N`, the neighborhood of the chosen `(i, j) ∈ ι₂`
  repeat' constructor
  · apply isOpen_iInter_of_finite
    intro ⟨⟨i, j⟩, i_ne_j⟩
    apply smul_inj_nbhd_open
  · apply Set.mem_iInter.mpr
    intro ⟨⟨i, j⟩, i_ne_j⟩
    apply smul_inj_nbhd_mem
  · intro i j i_ne_j
    let N := smul_inj_nbhd i_ne_j f_smul_inj
    have U_subset_N : U ⊆ N := Set.iInter_subset
      (fun (⟨⟨i, j⟩, i_ne_j⟩ : ι₂) => (smul_inj_nbhd i_ne_j f_smul_inj))
      ⟨⟨i, j⟩, i_ne_j⟩

    rw [disjoint_comm, smulImage_disjoint_mul]

    apply Set.disjoint_of_subset U_subset_N
    · apply smulImage_subset
      exact U_subset_N
    · exact smul_inj_nbhd_disjoint i_ne_j f_smul_inj

lemma moves_inj {g : G} {x : α} {n : ℕ} (period_ge_n : ∀ (k : ℤ), 1 ≤ k → k < n → g^k • x ≠ x) :
  Function.Injective (fun (i : Fin n) => g^(i : ℤ) • x) :=
by
  intro a b same_img
  by_contra a_ne_b

  let abs_diff := |(a : ℤ) - (b : ℤ)|

  apply period_ge_n abs_diff _ _ _
  {
    show 1 ≤ abs_diff
    unfold_let
    rw [<-zero_add 1, Int.add_one_le_iff]
    apply abs_pos.mpr
    apply sub_ne_zero.mpr
    simp
    apply Fin.vne_of_ne
    apply a_ne_b
  }
  {
    show abs_diff < (n : ℤ)
    apply abs_lt.mpr
    constructor
    · rw [<-zero_sub]
      apply Int.sub_lt_sub_of_le_of_lt <;> simp
    · rw [<-sub_zero (n : ℤ)]
      apply Int.sub_lt_sub_of_lt_of_le <;> simp
  }
  {
    show g^abs_diff • x = x

    simp at same_img
    group_action at same_img
    rw [neg_add_eq_sub] at same_img

    cases abs_cases ((a : ℤ) - (b : ℤ)) with
    | inl h =>
      unfold_let
      rw [h.1]
      exact same_img
    | inr h =>
      unfold_let
      rw [h.1]
      rw [smul_eq_iff_eq_inv_smul]
      group_action
      symm
      exact same_img
  }

-- Note: this can be strengthened to `k ≥ 0`
lemma natAbs_eq_of_pos' (k : ℤ) (k_ge_one : k ≥ 1) : k = k.natAbs := by
  cases Int.natAbs_eq k with
  | inl _ => assumption
  | inr h =>
    exfalso
    have k_lt_one : k < 1 := by
      calc
        k ≤ 0 := by
          rw [h]
          apply nonpos_of_neg_nonneg
          rw [neg_neg]
          apply Int.ofNat_nonneg
        _ < 1 := by norm_num
    exact ((lt_iff_not_ge _ _).mp k_lt_one) k_ge_one

lemma period_ge_n_cast {g : G} {x : α} {n : ℕ} :
  (∀ (k : ℕ), 1 ≤ k → k < n → g ^ k • x ≠ x) →
  (∀ (k : ℤ), 1 ≤ k → k < n → g ^ k • x ≠ x) :=
by
  intro period_ge_n'
  intro k one_le_k k_lt_n

  have one_le_abs_k : 1 ≤ k.natAbs := by
    rw [<-Nat.cast_le (α := ℤ)]
    norm_num
    calc
      1 ≤ k := one_le_k
      _ ≤ |k| := le_abs_self k
  have abs_k_lt_n : k.natAbs < n := by
    rw [<-Nat.cast_lt (α := ℤ)]
    norm_num
    calc
      |k| = k := abs_of_pos one_le_k
      _ < n := k_lt_n
  have res := period_ge_n' k.natAbs one_le_abs_k abs_k_lt_n

  rw [<-zpow_ofNat, Int.coe_natAbs, abs_of_pos _] at res
  exact res
  exact one_le_k

instance {g : G} {x : α} {n : ℕ} :
  Coe
    (∀ (k : ℕ), 1 ≤ k → k < n → g ^ k • x ≠ x)
    (∀ (k : ℤ), 1 ≤ k → k < n → g ^ k • x ≠ x)
where
  coe := period_ge_n_cast

lemma moves_1234_of_moves_12 {g : G} {x : α} (g12_moves : g^12 • x ≠ x) :
  Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) :=
by
  apply moves_inj
  intros k k_ge_1 k_lt_5
  simp at k_lt_5

  by_contra x_fixed
  have k_div_12 : k ∣ 12 := by
    -- Note: norm_num does not support ℤ.dvd yet, nor ℤ.mod, nor Int.natAbs, nor Int.div, etc.
    have h: (12 : ℤ) = (12 : ℕ) := by norm_num
    rw [h, Int.ofNat_dvd_right]
    apply Nat.dvd_of_mod_eq_zero

    interval_cases k
    all_goals unfold Int.natAbs
    all_goals norm_num

  have g12_fixed : g^12 • x = x := by
    rw [<-zpow_ofNat]
    simp
    rw [<-Int.mul_ediv_cancel' k_div_12]
    have res := smul_zpow_eq_of_smul_eq (12/k) x_fixed
    group_action at res
    exact res

  exact g12_moves g12_fixed

lemma proposition_1_1_2 [T2Space α] [h_lm : LocallyMoving G α]
  (f g : G) (h_disj : AlgebraicallyDisjoint f g) : Disjoint (Support α f) (Support α (g^12)) :=
by
  by_contra not_disjoint
  let U := Support α f ∩ Support α (g^12)
  have U_nonempty : U.Nonempty := by
    apply Set.not_disjoint_iff_nonempty_inter.mp
    exact not_disjoint

  -- Since G is Hausdorff, we can find a nonempty set V ⊆ such that f(V) is disjoint from V and the sets {g^i(V): i=0..4} are pairwise disjoint
  let x := U_nonempty.some
  have x_in_U : x ∈ U := Set.Nonempty.some_mem U_nonempty
  have fx_moves : f • x ≠ x := Set.inter_subset_left _ _ x_in_U

  have five_points : Function.Injective (fun i : Fin 5 => g^(i : ℤ) • x) := by
    apply moves_1234_of_moves_12
    exact (Set.inter_subset_right _ _ x_in_U)
  have U_open: IsOpen U := (IsOpen.inter (support_open f) (support_open (g^12)))

  let ⟨V₀, V₀_open, x_in_V₀, V₀_in_support, disjoint_img_V₀⟩ := disjoint_nbhd_in U_open x_in_U fx_moves
  let ⟨V₁, V₁_open, x_in_V₁, disjoint_img_V₁⟩ := disjoint_nbhd_fin five_points

  let V := V₀ ∩ V₁
  -- Let h be a nontrivial element of the RigidStabilizer G V
  let ⟨h, ⟨h_in_ristV, h_ne_one⟩⟩ := h_lm.get_nontrivial_rist_elem (IsOpen.inter V₀_open V₁_open) (Set.nonempty_of_mem ⟨x_in_V₀, x_in_V₁⟩)

  have V_disjoint_smulImage: Disjoint V (f •'' V) := by
    apply Set.disjoint_of_subset
    · exact Set.inter_subset_left _ _
    · apply smulImage_subset
      exact Set.inter_subset_left _ _
    · exact disjoint_img_V₀

  have comm_non_trivial : ¬Commute f h := by
    by_contra comm_trivial
    let ⟨z, z_in_V, z_moved⟩ := faithful_rigid_stabilizer_moves_point h_in_ristV h_ne_one
    apply z_moved

    nth_rewrite 2 [<-one_smul G z]
    rw [<-commutatorElement_eq_one_iff_commute.mpr comm_trivial.symm]
    symm

    apply disjoint_support_comm h f
    · exact rigidStabilizer_support.mp h_in_ristV
    · exact V_disjoint_smulImage
    · exact z_in_V

  -- Since g is algebraically disjoint from f, there exist f₁,f₂ ∈ C_G(g) so that the commutator h' = [f1,[f2,h]] is a nontrivial element of C_G(g)
  let alg_disj_elem := h_disj h comm_non_trivial
  let f₁ := alg_disj_elem.fst
  let f₂ := alg_disj_elem.snd
  let h' := alg_disj_elem.comm_elem
  have f₁_commutes : Commute f₁ g := alg_disj_elem.fst_commute
  have f₂_commutes : Commute f₂ g := alg_disj_elem.snd_commute
  have h'_commutes : Commute h' g := alg_disj_elem.comm_elem_commute
  have h'_nontrivial : h' ≠ 1 := alg_disj_elem.comm_elem_nontrivial

  have support_f₂_h : Support α ⁅f₂,h⁆ ⊆ V ∪ (f₂ •'' V) := by
    calc
      Support α ⁅f₂, h⁆ ⊆ Support α h ∪ (f₂ •'' Support α h) := support_comm α f₂ h
      _ ⊆ V ∪ (f₂ •'' Support α h) := by
        apply Set.union_subset_union_left
        exact rigidStabilizer_support.mp h_in_ristV
      _ ⊆ V ∪ (f₂ •'' V) := by
        apply Set.union_subset_union_right
        apply smulImage_subset
        exact rigidStabilizer_support.mp h_in_ristV
  have support_h' : Support α h' ⊆ ⋃(i : Fin 2 × Fin 2), (f₁^(i.1.val) * f₂^(i.2.val)) •'' V := by
    rw [rewrite_Union]
    simp (config := {zeta := false})
    rw [<-smulImage_mul, <-smulImage_union]
    calc
      Support α h' ⊆ Support α ⁅f₂,h⁆ ∪ (f₁ •'' Support α ⁅f₂, h⁆) := support_comm α f₁ ⁅f₂,h⁆
      _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' Support α ⁅f₂, h⁆) := by
        apply Set.union_subset_union_left
        exact support_f₂_h
      _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' V ∪ (f₂ •'' V)) := by
        apply Set.union_subset_union_right
        apply smulImage_subset f₁
        exact support_f₂_h

  -- Since h' is nontrivial, it has at least one point p in its support
  let ⟨p, p_moves⟩ := faithful_moves_point' α h'_nontrivial
  -- Since g commutes with h', all five of the points {gi(p):i=0..4} lie in supp(h')
  have gi_in_support : ∀ (i: Fin 5), g^(i.val) • p ∈ Support α h' := by
    intro i
    rw [mem_support]
    by_contra p_fixed
    rw [<-mul_smul, h'_commutes.pow_right, mul_smul] at p_fixed
    group_action at p_fixed
    exact p_moves p_fixed

  -- The next section gets tricky, so let us clear away some stuff first :3
  clear h'_commutes h'_nontrivial
  clear V₀_open x_in_V₀ V₀_in_support disjoint_img_V₀
  clear V₁_open x_in_V₁
  clear five_points h_in_ristV h_ne_one V_disjoint_smulImage
  clear support_f₂_h

  -- by the pigeonhole principle, one of the four sets V, f₁(V), f₂(V), f₁f₂(V) must contain two of these points,
  -- say g^i(p),g^j(p) ∈ k(V) for some 0 ≤ i < j ≤ 4 and k ∈ {1,f₁,f₂,f₁f₂}
  let pigeonhole : Fintype.card (Fin 5) > Fintype.card (Fin 2 × Fin 2) := by trivial
  let choice_pred := fun (i : Fin 5) => (Set.mem_iUnion.mp (support_h' (gi_in_support i)))
  let choice := fun (i : Fin 5) => (choice_pred i).choose
  let ⟨i, _, j, _, i_ne_j, same_choice⟩ := Finset.exists_ne_map_eq_of_card_lt_of_maps_to
    pigeonhole
    (fun (i : Fin 5) _ => Finset.mem_univ (choice i))

  let k := f₁^(choice i).1.val * f₂^(choice i).2.val
  have same_k : f₁^(choice j).1.val * f₂^(choice j).2.val = k := by rw [<-same_choice]
  have gi : g^i.val • p ∈ k •'' V := (choice_pred i).choose_spec
  have gk : g^j.val • p ∈ k •'' V := by
    have gk' := (choice_pred j).choose_spec
    rw [same_k] at gk'
    exact gk'

  -- Since g^(j-i)(V) is disjoint from V and k commutes with g,
  -- we know that g^(j−i)k(V) is disjoint from k(V),
  -- which leads to a contradiction since g^i(p) and g^j(p) both lie in k(V).

  have g_disjoint : Disjoint ((g^i.val)⁻¹ •'' V) ((g^j.val)⁻¹ •'' V) := by
    apply smulImage_disjoint_subset (Set.inter_subset_right V₀ V₁)
    group
    rw [smulImage_disjoint_inv_pow]
    group
    apply disjoint_img_V₁
    symm; exact i_ne_j

  have k_commutes: Commute k g := by
    apply Commute.mul_left
    · exact f₁_commutes.pow_left _
    · exact f₂_commutes.pow_left _
  clear f₁_commutes f₂_commutes

  have g_k_disjoint : Disjoint ((g^i.val)⁻¹ •'' (k •'' V)) ((g^j.val)⁻¹ •'' (k •'' V)) := by
    repeat rw [mul_smulImage]
    repeat rw [<-inv_pow]
    repeat rw [k_commutes.symm.inv_left.pow_left]
    repeat rw [<-mul_smulImage k]
    repeat rw [inv_pow]
    exact disjoint_smulImage k g_disjoint

  apply Set.disjoint_left.mp g_k_disjoint
  · rw [mem_inv_smulImage]
    exact gi
  · rw [mem_inv_smulImage]
    exact gk

lemma remark_1_2 (f g : G) (h_disj : AlgebraicallyDisjoint f g): Commute f g := by
  by_contra non_commute
  let disj_elem := h_disj g non_commute
  let nontrivial := disj_elem.comm_elem_nontrivial

  rw [commutatorElement_eq_one_iff_commute.mpr disj_elem.snd_commute] at nontrivial
  rw [commutatorElement_one_right] at nontrivial

  tauto

end Rubin