|
|
import Mathlib.GroupTheory.GroupAction.Basic
|
|
|
|
|
|
namespace Rubin
|
|
|
|
|
|
variable {G α β : Type _} [Group G]
|
|
|
variable [MulAction G α]
|
|
|
|
|
|
theorem smul_congr (g : G) {x y : α} (h : x = y) : g • x = g • y :=
|
|
|
congr_arg ((· • ·) g) h
|
|
|
#align smul_congr Rubin.smul_congr
|
|
|
|
|
|
theorem smul_eq_iff_inv_smul_eq {x : α} {g : G} : g • x = x ↔ g⁻¹ • x = x :=
|
|
|
⟨fun h => (smul_congr g⁻¹ h).symm.trans (inv_smul_smul g x), fun h =>
|
|
|
(smul_congr g h).symm.trans (smul_inv_smul g x)⟩
|
|
|
#align smul_eq_iff_inv_smul_eq Rubin.smul_eq_iff_inv_smul_eq
|
|
|
|
|
|
theorem smul_pow_eq_of_smul_eq {x : α} {g : G} (n : ℕ) :
|
|
|
g • x = x → g ^ n • x = x := by
|
|
|
induction n with
|
|
|
| zero => simp only [pow_zero, one_smul, eq_self_iff_true, imp_true_iff]
|
|
|
| succ n n_ih =>
|
|
|
intro h
|
|
|
nth_rw 2 [← (smul_congr g (n_ih h)).trans h]
|
|
|
rw [← mul_smul, ← pow_succ]
|
|
|
#align smul_pow_eq_of_smul_eq Rubin.smul_pow_eq_of_smul_eq
|
|
|
|
|
|
theorem smul_zpow_eq_of_smul_eq {x : α} {g : G} (n : ℤ) :
|
|
|
g • x = x → g ^ n • x = x := by
|
|
|
intro h
|
|
|
cases n with
|
|
|
| ofNat n => let res := smul_pow_eq_of_smul_eq n h; simp; exact res
|
|
|
| negSucc n =>
|
|
|
let res :=
|
|
|
smul_eq_iff_inv_smul_eq.mp (smul_pow_eq_of_smul_eq (1 + n) h);
|
|
|
simp
|
|
|
rw [add_comm]
|
|
|
exact res
|
|
|
#align smul_zpow_eq_of_smul_eq Rubin.smul_zpow_eq_of_smul_eq
|
|
|
|
|
|
lemma disjoint_not_mem {α : Type _} {U V : Set α} (disj: Disjoint U V) :
|
|
|
∀ {x : α}, x ∈ U → x ∉ V :=
|
|
|
by
|
|
|
intro x x_in_U x_in_V
|
|
|
apply disj <;> try simp
|
|
|
· exact Set.singleton_subset_iff.mpr x_in_U
|
|
|
· rw [Set.singleton_subset_iff]
|
|
|
exact x_in_V
|
|
|
· rfl
|
|
|
|
|
|
lemma disjoint_not_mem₂ {α : Type _} {U V : Set α} (disj: Disjoint U V) :
|
|
|
∀ {x : α}, x ∈ V → x ∉ U := disjoint_not_mem disj.symm
|
|
|
|
|
|
lemma fixes_inv {G α : Type _} [Group G] [MulAction G α] {g : G} {x : α}:
|
|
|
g • x = x ↔ g⁻¹ • x = x :=
|
|
|
by
|
|
|
constructor
|
|
|
{
|
|
|
intro h
|
|
|
nth_rw 1 [<-h]
|
|
|
rw [<-mul_smul, mul_left_inv, one_smul]
|
|
|
}
|
|
|
{
|
|
|
intro h
|
|
|
nth_rw 1 [<-h]
|
|
|
rw [<-mul_smul, mul_right_inv, one_smul]
|
|
|
}
|
|
|
|
|
|
lemma exists_smul_ne {G : Type _} (α : Type _) [Group G] [MulAction G α] [h_f : FaithfulSMul G α]
|
|
|
{f g : G} (f_ne_g : f ≠ g) : ∃ (x : α), f • x ≠ g • x :=
|
|
|
by
|
|
|
by_contra h
|
|
|
rw [Classical.not_exists_not] at h
|
|
|
apply f_ne_g
|
|
|
apply h_f.eq_of_smul_eq_smul
|
|
|
exact h
|
|
|
|
|
|
@[simp]
|
|
|
theorem orbit_bot {G : Type _} [Group G] {H : Subgroup G} (H_eq_bot : H = ⊥):
|
|
|
∀ (g : G), MulAction.orbit H g = {g} :=
|
|
|
by
|
|
|
intro g
|
|
|
ext x
|
|
|
rw [MulAction.mem_orbit_iff]
|
|
|
simp
|
|
|
rw [H_eq_bot]
|
|
|
simp
|
|
|
constructor <;> tauto
|
|
|
|
|
|
@[simp]
|
|
|
theorem orbit_bot₂ {G : Type _} [Group G] {α : Type _} [MulAction G α] (H : Subgroup G) (H_eq_bot : H = ⊥):
|
|
|
∀ (x : α), MulAction.orbit H x = {x} :=
|
|
|
by
|
|
|
intro g
|
|
|
ext x
|
|
|
rw [MulAction.mem_orbit_iff]
|
|
|
simp
|
|
|
rw [H_eq_bot]
|
|
|
simp
|
|
|
constructor <;> tauto
|
|
|
|
|
|
end Rubin
|