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rubin-lean4/Rubin/SmulImage.lean

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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
theorem mem_smulImage' {x : α} (g : G) {U : Set α} : x ∈ U ↔ g • x ∈ g •'' U :=
by
rw [mem_smulImage]
rw [<-mul_smul, mul_left_inv, one_smul]
@[simp]
theorem smulImage_mul (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.smulImage_mul
@[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 smulImage_disjoint (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.smulImage_disjoint
theorem SmulImage.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
theorem SmulImage.inv_congr (g: G) {U V : Set α} : g •'' U = g •'' V → U = V :=
by
intro h
rw [<-one_smulImage (G := G) U]
rw [<-one_smulImage (G := G) V]
rw [<-mul_left_inv g]
repeat rw [<-smulImage_mul]
exact SmulImage.congr g⁻¹ h
theorem smulImage_inv (g: G) (U V : Set α) : g •'' U = V ↔ U = g⁻¹ •'' V := by
nth_rw 2 [<-one_smulImage (G := G) U]
rw [<-mul_left_inv g, <-smulImage_mul]
constructor
· intro h
rw [SmulImage.congr]
exact h
· intro h
apply SmulImage.inv_congr at h
exact h
-- TODO: rename to smulImage_mono
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
def smulImage_mono (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V := smulImage_subset g
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_sUnion (g : G) {S : Set (Set α)} : g •'' (⋃₀ S) = ⋃₀ {g •'' T | T ∈ S} :=
by
ext x
constructor
· intro h
rw [mem_smulImage, Set.mem_sUnion] at h
rw [Set.mem_sUnion]
let ⟨T, ⟨T_in_S, ginv_x_in_T⟩⟩ := h
simp
use T
constructor; trivial
rw [mem_smulImage]
exact ginv_x_in_T
· intro h
rw [Set.mem_sUnion] at h
rw [mem_smulImage, Set.mem_sUnion]
simp at h
let ⟨T, ⟨T_in_S, x_in_gT⟩⟩ := h
use T
constructor; trivial
rw [<-mem_smulImage]
exact x_in_gT
theorem smulImage_sInter (g : G) {S : Set (Set α)} : g •'' (⋂₀ S) = ⋂₀ {g •'' T | T ∈ S} :=
by
ext x
constructor
· intro h
rw [mem_smulImage, Set.mem_sInter] at h
rw [Set.mem_sInter]
simp
intro T T_in_S
rw [mem_smulImage]
exact h T T_in_S
· intro h
rw [Set.mem_sInter] at h
rw [mem_smulImage, Set.mem_sInter]
intro T T_in_S
rw [<-mem_smulImage]
simp at h
exact h T T_in_S
@[simp]
theorem smulImage_compl (g : G) (U : Set α) : (g •'' U)ᶜ = g •'' Uᶜ :=
by
ext x
rw [Set.mem_compl_iff]
repeat rw [mem_smulImage]
rw [Set.mem_compl_iff]
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
theorem smulImage_subset_inv {G α : Type _} [Group G] [MulAction G α]
(f : G) (U V : Set α) :
f •'' U ⊆ V ↔ U ⊆ f⁻¹ •'' V :=
by
constructor
· intro h
apply smulImage_subset f⁻¹ at h
rw [smulImage_mul] at h
rw [mul_left_inv, one_smulImage] at h
exact h
· intro h
apply smulImage_subset f at h
rw [smulImage_mul] at h
rw [mul_right_inv, one_smulImage] at h
exact h
theorem smulImage_subset_inv' {G α : Type _} [Group G] [MulAction G α]
(f : G) (U V : Set α) :
f⁻¹ •'' U ⊆ V ↔ U ⊆ f •'' V :=
by
nth_rewrite 2 [<-inv_inv f]
exact smulImage_subset_inv f⁻¹ U V
theorem smulImage_disjoint_mul {G α : Type _} [Group G] [MulAction G α]
(f g : G) (U V : Set α) :
Disjoint (f •'' U) (g •'' V) ↔ Disjoint U ((f⁻¹ * g) •'' V) := by
constructor
· intro h
apply smulImage_disjoint f⁻¹ at h
repeat rw [smulImage_mul] at h
rw [mul_left_inv, one_smulImage] at h
exact h
· intro h
apply smulImage_disjoint f at h
rw [smulImage_mul] at h
rw [<-mul_assoc] at h
rw [mul_right_inv, one_mul] at h
exact h
theorem smulImage_disjoint_inv_pow {G α : Type _} [Group G] [MulAction G α]
(g : G) (i j : ) (U V : Set α) :
Disjoint (g^i •'' U) (g^j •'' V) ↔ Disjoint (g^(-j) •'' U) (g^(-i) •'' V) :=
by
rw [smulImage_disjoint_mul]
rw [<-zpow_neg, <-zpow_add, add_comm, zpow_add, zpow_neg]
rw [<-inv_inv (g^j)]
rw [<-smulImage_disjoint_mul]
simp
theorem smulImage_disjoint_subset {G α : Type _} [Group G] [MulAction G α]
{f g : G} {U V : Set α} (h_sub: U ⊆ V):
Disjoint (f •'' V) (g •'' V) → Disjoint (f •'' U) (g •'' U) :=
by
apply Set.disjoint_of_subset (smulImage_subset _ h_sub) (smulImage_subset _ h_sub)
-- States that if `g^i •'' V` and `g^j •'' V` are disjoint for any `i ≠ j` and `x ∈ V`
-- then `g^i • x` will always lie outside of `V`.
lemma smulImage_distinct_of_disjoint_pow {G α : Type _} [Group G] [MulAction G α] {g : G} {V : Set α} {n : }
(n_pos : 0 < n)
(h_disj : ∀ (i j : Fin n), i ≠ j → Disjoint (g ^ (i : ) •'' V) (g ^ (j : ) •'' V)) :
∀ (x : α) (_hx : x ∈ V) (i : Fin n), 0 < (i : ) → g ^ (i : ) • (x : α) ∉ V :=
by
intro x hx i i_pos
have i_ne_zero : i ≠ (⟨ 0, n_pos ⟩ : Fin n) := by
intro h
rw [h] at i_pos
simp at i_pos
have h_contra : g ^ (i : ) • (x : α) ∈ g ^ (i : ) •'' V := by use x
have h_notin_V := Set.disjoint_left.mp (h_disj i (⟨0, n_pos⟩ : Fin n) i_ne_zero) h_contra
simp only [pow_zero, one_smulImage] at h_notin_V
exact h_notin_V
#align distinct_images_from_disjoint Rubin.smulImage_distinct_of_disjoint_pow
end Rubin