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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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import Rubin.MulActionExt
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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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-- 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`.
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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 rigidStabilizer_support {g : G} {U : Set α} :
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g ∈ RigidStabilizer G U ↔ Support α g ⊆ U :=
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⟨
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fun h x x_in_support =>
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by_contradiction (x_in_support ∘ h x),
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by
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intro support_sub
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rw [<-Subgroup.mem_carrier]
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unfold RigidStabilizer; simp
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intro x x_notin_U
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by_contra h
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exact x_notin_U (support_sub h)
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⟩
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#align rist_supported_in_set Rubin.rigidStabilizer_support
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theorem rigidStabilizer_mono {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.rigidStabilizer_mono
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theorem monotone_rigidStabilizer : Monotone (RigidStabilizer (α := α) G) := fun _ _ => rigidStabilizer_mono
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theorem rigidStabilizer_to_subgroup_closure {g : G} {U : Set α} :
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g ∈ RigidStabilizer G U → g ∈ Subgroup.closure { g : G | Support α g ⊆ U } :=
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by
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rw [rigidStabilizer_support]
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intro h
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rw [Subgroup.mem_closure]
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intro V orbit_subset_V
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apply orbit_subset_V
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simp
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exact h
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theorem commute_if_rigidStabilizer_and_disjoint {g h : G} {U : Set α} [FaithfulSMul G α] :
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g ∈ RigidStabilizer G U → Disjoint U (Support α h) → Commute g h :=
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by
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intro g_in_rist U_disj
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unfold Commute
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unfold SemiconjBy
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apply eq_of_smul_eq_smul (α := α)
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intro x
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by_cases x_in_U?: x ∈ U
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{
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rw [rigidStabilizer_support] at g_in_rist
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have x_notin_support : x ∉ Support α h := disjoint_not_mem U_disj x_in_U?
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rw [mul_smul]
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rw [not_mem_support.mp x_notin_support]
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rw [mul_smul]
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by_cases gx_in_U?: g • x ∈ U
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{
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symm
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apply not_mem_support.mp
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apply disjoint_not_mem U_disj
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exact gx_in_U?
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}
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{
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have gx_notin_support : g • x ∉ Support α g := by
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intro h
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exact gx_in_U? (g_in_rist h)
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rw [<-support_inv] at gx_notin_support
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rw [not_mem_support] at gx_notin_support
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simp at gx_notin_support
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symm at gx_notin_support
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rw [fixes_inv] at gx_notin_support
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rw [<-gx_notin_support]
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group_action
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rw [not_mem_support.mp x_notin_support]
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}
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}
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{
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have x_fixed : g • x = x := g_in_rist _ x_in_U?
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repeat rw [mul_smul]
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rw [x_fixed]
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by_cases hx_in_U?: h • x ∈ U
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{
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have hx_notin_support := disjoint_not_mem U_disj hx_in_U?
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rw [<-support_inv] at hx_notin_support
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rw [not_mem_support] at hx_notin_support
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group_action at hx_notin_support
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rw [<-hx_notin_support]
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exact x_fixed
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}
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{
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rw [g_in_rist _ hx_in_U?]
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}
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}
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end Rubin
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