rubin-lean4/Rubin/SmulImage.lean

import Mathlib.Data.Finset.Basic
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.GroupAction.Basic

import Rubin.MulActionExt

namespace Rubin

/--
The image of a group action (here generalized to any pair `(G, α)` implementing `SMul`)
is the image of the elements of `U` under the `g • u` operation.

An alternative definition (which is available through the [`mem_smulImage`] theorem and the [`smulImage_set`] equality) would be:
`SmulImage g U = {x | g⁻¹ • x ∈ U}`.

The notation used for this operator is `g •'' U`.
-/
def SmulImage {G α : Type _} [SMul G α] (g : G) (U : Set α) :=
  (g • ·) '' U
#align subset_img Rubin.SmulImage

infixl:60 " •'' " => Rubin.SmulImage

/--
The pre-image of a group action (here generalized to any pair `(G, α)` implementing `SMul`)
is the set of values `x: α` such that `g • x ∈ U`.

Unlike [`SmulImage`], no notation is defined for this operator.
--/
def SmulPreImage {G α : Type _} [SMul G α] (g : G) (U : Set α) :=
  {x | g • x ∈ U}
#align subset_preimg' Rubin.SmulPreImage

variable {G α : Type _}
variable [Group G]
variable [MulAction G α]

theorem smulImage_def {g : G} {U : Set α} : g •'' U = (· • ·) g '' U :=
  rfl
#align subset_img_def Rubin.smulImage_def

theorem mem_smulImage {x : α} {g : G} {U : Set α} : x ∈ g •'' U ↔ g⁻¹ • x ∈ U :=
  by
  rw [Rubin.smulImage_def, Set.mem_image (g • ·) U x]
  constructor
  · rintro ⟨y, yU, gyx⟩
    let ygx : y = g⁻¹ • x := inv_smul_smul g y ▸ Rubin.smul_congr g⁻¹ gyx
    exact ygx ▸ yU
  · intro h
    exact ⟨g⁻¹ • x, ⟨Set.mem_preimage.mp h, smul_inv_smul g x⟩⟩
#align mem_smul'' Rubin.mem_smulImage

-- Provides a way to express a [`SmulImage`] as a `Set`;
-- this is simply [`mem_smulImage`] paired with set extensionality.
theorem smulImage_set {g: G} {U: Set α} : g •'' U = {x | g⁻¹ • x ∈ U} := Set.ext (fun _x => mem_smulImage)

theorem mem_inv_smulImage {x : α} {g : G} {U : Set α} : x ∈ g⁻¹ •'' U ↔ g • x ∈ U :=
  by
  let msi := @Rubin.mem_smulImage _ _ _ _ x g⁻¹ U
  rw [inv_inv] at msi
  exact msi
#align mem_inv_smul'' Rubin.mem_inv_smulImage

@[simp]
theorem mul_smulImage (g h : G) (U : Set α) : g •'' (h •'' U) = (g * h) •'' U :=
  by
  ext
  rw [Rubin.mem_smulImage, Rubin.mem_smulImage, Rubin.mem_smulImage, ←
    mul_smul, mul_inv_rev]
#align mul_smul'' Rubin.mul_smulImage

@[simp]
theorem one_smulImage (U : Set α) : (1 : G) •'' U = U :=
  by
  ext
  rw [Rubin.mem_smulImage, inv_one, one_smul]
#align one_smul'' Rubin.one_smulImage

theorem disjoint_smulImage (g : G) {U V : Set α} :
    Disjoint U V → Disjoint (g •'' U) (g •'' V) :=
  by
  intro disjoint_U_V
  rw [Set.disjoint_left]
  rw [Set.disjoint_left] at disjoint_U_V
  intro x x_in_gU
  by_contra h
  exact (disjoint_U_V (mem_smulImage.mp x_in_gU)) (mem_smulImage.mp h)
#align disjoint_smul'' Rubin.disjoint_smulImage

namespace SmulImage
theorem congr (g : G) {U V : Set α} : U = V → g •'' U = g •'' V :=
  congr_arg fun W : Set α => g •'' W
#align smul''_congr Rubin.SmulImage.congr
end SmulImage

theorem smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V :=
  by
  intro h1 x
  rw [Rubin.mem_smulImage, Rubin.mem_smulImage]
  exact fun h2 => h1 h2
#align smul''_subset Rubin.smulImage_subset

theorem smulImage_union (g : G) {U V : Set α} : g •'' U ∪ V = (g •'' U) ∪ (g •'' V) :=
  by
  ext
  rw [Rubin.mem_smulImage, Set.mem_union, Set.mem_union, Rubin.mem_smulImage,
    Rubin.mem_smulImage]
#align smul''_union Rubin.smulImage_union

theorem smulImage_inter (g : G) {U V : Set α} : g •'' U ∩ V = (g •'' U) ∩ (g •'' V) :=
  by
  ext
  rw [Set.mem_inter_iff, Rubin.mem_smulImage, Rubin.mem_smulImage,
    Rubin.mem_smulImage, Set.mem_inter_iff]
#align smul''_inter Rubin.smulImage_inter

theorem smulImage_eq_inv_preimage {g : G} {U : Set α} : g •'' U = (· • ·) g⁻¹ ⁻¹' U :=
  by
  ext
  constructor
  · intro h; rw [Set.mem_preimage]; exact mem_smulImage.mp h
  · intro h; rw [Rubin.mem_smulImage]; exact Set.mem_preimage.mp h
#align smul''_eq_inv_preimage Rubin.smulImage_eq_inv_preimage

theorem smulImage_eq_of_smul_eq {g h : G} {U : Set α} :
    (∀ x ∈ U, g • x = h • x) → g •'' U = h •'' U :=
  by
  intro hU
  ext x
  rw [Rubin.mem_smulImage, Rubin.mem_smulImage]
  constructor
  · intro k; let a := congr_arg (h⁻¹ • ·) (hU (g⁻¹ • x) k);
    simp only [smul_inv_smul, inv_smul_smul] at a ; exact Set.mem_of_eq_of_mem a k
  · intro k; let a := congr_arg (g⁻¹ • ·) (hU (h⁻¹ • x) k);
    simp only [smul_inv_smul, inv_smul_smul] at a ; exact Set.mem_of_eq_of_mem a.symm k
#align smul''_eq_of_smul_eq Rubin.smulImage_eq_of_smul_eq

end Rubin