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import Mathlib.Data.Finset.Basic
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import Mathlib.GroupTheory.GroupAction.Basic
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import Rubin.Support
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namespace Rubin
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-- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup
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def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G :=
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{g : G | ∀ x : α, g • x = x ∨ x ∈ U}
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#align rigid_stabilizer' Rubin.rigidStabilizer'
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def RigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G
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where
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carrier := {g : G | ∀ (x) (_ : x ∉ U), g • x = x}
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mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U]
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inv_mem' hg x x_notin_U := smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U)
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one_mem' x _ := one_smul G x
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#align rigid_stabilizer Rubin.RigidStabilizer
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variable {G α: Type _}
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variable [Group G]
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variable [MulAction G α]
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theorem rist_supported_in_set {g : G} {U : Set α} :
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g ∈ RigidStabilizer G U → Support α g ⊆ U := fun h x x_in_support =>
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by_contradiction (x_in_support ∘ h x)
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#align rist_supported_in_set Rubin.rist_supported_in_set
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theorem rist_ss_rist {U V : Set α} (V_ss_U : V ⊆ U) :
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(RigidStabilizer G V : Set G) ⊆ (RigidStabilizer G U : Set G) :=
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by
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intro g g_in_ristV x x_notin_U
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have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V)
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exact g_in_ristV x x_notin_V
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#align rist_ss_rist Rubin.rist_ss_rist
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end Rubin
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