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rubin-lean4/Rubin/RigidStabilizer.lean

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import Mathlib.Data.Finset.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Rubin.Support
namespace Rubin
-- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup
def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G :=
{g : G | ∀ x : α, g • x = x x ∈ U}
#align rigid_stabilizer' Rubin.rigidStabilizer'
-- A subgroup of G for which `Support α g ⊆ U`, or in other words, all elements of `G` that don't move points outside of `U`.
def RigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G
where
carrier := {g : G | ∀ (x) (_ : x ∉ U), g • x = x}
mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U]
inv_mem' hg x x_notin_U := smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U)
one_mem' x _ := one_smul G x
#align rigid_stabilizer Rubin.RigidStabilizer
variable {G α: Type _}
variable [Group G]
variable [MulAction G α]
theorem rigidStabilizer_support {g : G} {U : Set α} :
g ∈ RigidStabilizer G U ↔ Support α g ⊆ U :=
fun h x x_in_support =>
by_contradiction (x_in_support ∘ h x),
by
intro support_sub
rw [<-Subgroup.mem_carrier]
unfold RigidStabilizer; simp
intro x x_notin_U
by_contra h
exact x_notin_U (support_sub h)
#align rist_supported_in_set Rubin.rigidStabilizer_support
theorem rigidStabilizer_mono {U V : Set α} (V_ss_U : V ⊆ U) :
(RigidStabilizer G V : Set G) ⊆ (RigidStabilizer G U : Set G) :=
by
intro g g_in_ristV x x_notin_U
have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V)
exact g_in_ristV x x_notin_V
#align rist_ss_rist Rubin.rigidStabilizer_mono
theorem monotone_rigidStabilizer : Monotone (RigidStabilizer (α := α) G) := fun _ _ => rigidStabilizer_mono
theorem rigidStabilizer_to_subgroup_closure {g : G} {U : Set α} :
g ∈ RigidStabilizer G U → g ∈ Subgroup.closure { g : G | Support α g ⊆ U } :=
by
rw [rigidStabilizer_support]
intro h
rw [Subgroup.mem_closure]
intro V orbit_subset_V
apply orbit_subset_V
simp
exact h
end Rubin