group_theory.group_action.basic ⟷ Mathlib.GroupTheory.GroupAction.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -7,7 +7,7 @@ import Data.Fintype.Card
 import GroupTheory.GroupAction.Defs
 import GroupTheory.GroupAction.Group
 import Data.Setoid.Basic
-import Data.Set.Pointwise.Smul
+import Data.Set.Pointwise.SMul
 import GroupTheory.Subgroup.Basic
 
 #align_import group_theory.group_action.basic from "leanprover-community/mathlib"@"d30d31261cdb4d2f5e612eabc3c4bf45556350d5"

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -360,7 +360,7 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
     rw [Set.mem_iUnion]
     exact ⟨a⁻¹, a • x, hy, inv_smul_smul a x⟩
   · intro hx
-    rw [Set.mem_iUnion] at hx 
+    rw [Set.mem_iUnion] at hx
     obtain ⟨a, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
     refine' ⟨a⁻¹ • x, _, by simp only [Quotient.eq'] <;> use a⁻¹⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -186,25 +186,26 @@ theorem mem_fixedPoints' {b : β} : b ∈ fixedPoints α β ↔ ∀ b', b' ∈ o
 
 variable (α) {β}
 
-#print MulAction.Stabilizer.submonoid /-
+#print MulAction.stabilizerSubmonoid /-
 /-- The stabilizer of a point `b` as a submonoid of `α`. -/
 @[to_additive "The stabilizer of a point `b` as an additive submonoid of `α`."]
-def Stabilizer.submonoid (b : β) : Submonoid α
+def MulAction.stabilizerSubmonoid (b : β) : Submonoid α
     where
   carrier := {a | a • b = b}
   one_mem' := one_smul _ b
   hMul_mem' a a' (ha : a • b = b) (hb : a' • b = b) :=
     show (a * a') • b = b by rw [← smul_smul, hb, ha]
-#align mul_action.stabilizer.submonoid MulAction.Stabilizer.submonoid
-#align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid
+#align mul_action.stabilizer.submonoid MulAction.stabilizerSubmonoid
+#align add_action.stabilizer.add_submonoid AddAction.stabilizerAddSubmonoid
 -/
 
-#print MulAction.mem_stabilizer_submonoid_iff /-
+#print MulAction.mem_stabilizerSubmonoid_iff /-
 @[simp, to_additive]
-theorem mem_stabilizer_submonoid_iff {b : β} {a : α} : a ∈ Stabilizer.submonoid α b ↔ a • b = b :=
+theorem mem_stabilizerSubmonoid_iff {b : β} {a : α} :
+    a ∈ MulAction.stabilizerSubmonoid α b ↔ a • b = b :=
   Iff.rfl
-#align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizer_submonoid_iff
-#align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizer_addSubmonoid_iff
+#align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizerSubmonoid_iff
+#align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizerAddSubmonoid_iff
 -/
 
 #print MulAction.orbit_eq_univ /-
@@ -248,7 +249,7 @@ A subgroup. -/
 @[to_additive
       "The stabilizer of an element under an action, i.e. what sends the element to itself.\nAn additive subgroup."]
 def stabilizer (b : β) : Subgroup α :=
-  { Stabilizer.submonoid α b with
+  { MulAction.stabilizerSubmonoid α b with
     inv_mem' := fun a (ha : a • b = b) => show a⁻¹ • b = b by rw [inv_smul_eq_iff, ha] }
 #align mul_action.stabilizer MulAction.stabilizer
 #align add_action.stabilizer AddAction.stabilizer

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,12 +3,12 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
-import Mathbin.Data.Fintype.Card
-import Mathbin.GroupTheory.GroupAction.Defs
-import Mathbin.GroupTheory.GroupAction.Group
-import Mathbin.Data.Setoid.Basic
-import Mathbin.Data.Set.Pointwise.Smul
-import Mathbin.GroupTheory.Subgroup.Basic
+import Data.Fintype.Card
+import GroupTheory.GroupAction.Defs
+import GroupTheory.GroupAction.Group
+import Data.Setoid.Basic
+import Data.Set.Pointwise.Smul
+import GroupTheory.Subgroup.Basic
 
 #align_import group_theory.group_action.basic from "leanprover-community/mathlib"@"d30d31261cdb4d2f5e612eabc3c4bf45556350d5"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -231,7 +231,7 @@ theorem mem_fixedPoints_iff_card_orbit_eq_one {a : β} [Fintype (orbit α a)] :
       x • a = z := Subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
       _ = a := (Subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm
 #align mul_action.mem_fixed_points_iff_card_orbit_eq_one MulAction.mem_fixedPoints_iff_card_orbit_eq_one
-#align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_zero
+#align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_one
 -/
 
 end MulAction

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -90,7 +90,7 @@ theorem orbit_nonempty (b : β) : Set.Nonempty (orbit α b) :=
 #print MulAction.mapsTo_smul_orbit /-
 @[to_additive]
 theorem mapsTo_smul_orbit (a : α) (b : β) : Set.MapsTo ((· • ·) a) (orbit α b) (orbit α b) :=
-  Set.range_subset_iff.2 fun a' => ⟨a * a', mul_smul _ _ _⟩
+  Set.range_subset_iff.2 fun a' => ⟨a * a', hMul_smul _ _ _⟩
 #align mul_action.maps_to_smul_orbit MulAction.mapsTo_smul_orbit
 #align add_action.maps_to_vadd_orbit AddAction.mapsTo_vadd_orbit
 -/
@@ -106,7 +106,7 @@ theorem smul_orbit_subset (a : α) (b : β) : a • orbit α b ⊆ orbit α b :=
 #print MulAction.orbit_smul_subset /-
 @[to_additive]
 theorem orbit_smul_subset (a : α) (b : β) : orbit α (a • b) ⊆ orbit α b :=
-  Set.range_subset_iff.2 fun a' => mul_smul a' a b ▸ mem_orbit _ _
+  Set.range_subset_iff.2 fun a' => hMul_smul a' a b ▸ mem_orbit _ _
 #align mul_action.orbit_smul_subset MulAction.orbit_smul_subset
 #align add_action.orbit_vadd_subset AddAction.orbit_vadd_subset
 -/
@@ -116,7 +116,7 @@ instance {b : β} : MulAction α (orbit α b)
     where
   smul a := (mapsTo_smul_orbit a b).restrict _ _ _
   one_smul a := Subtype.ext (one_smul α a)
-  mul_smul a a' b' := Subtype.ext (mul_smul a a' b')
+  hMul_smul a a' b' := Subtype.ext (hMul_smul a a' b')
 
 #print MulAction.orbit.coe_smul /-
 @[simp, to_additive]
@@ -193,7 +193,7 @@ def Stabilizer.submonoid (b : β) : Submonoid α
     where
   carrier := {a | a • b = b}
   one_mem' := one_smul _ b
-  mul_mem' a a' (ha : a • b = b) (hb : a' • b = b) :=
+  hMul_mem' a a' (ha : a • b = b) (hb : a' • b = b) :=
     show (a * a') • b = b by rw [← smul_smul, hb, ha]
 #align mul_action.stabilizer.submonoid MulAction.Stabilizer.submonoid
 #align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
-
-! This file was ported from Lean 3 source module group_theory.group_action.basic
-! leanprover-community/mathlib commit d30d31261cdb4d2f5e612eabc3c4bf45556350d5
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Fintype.Card
 import Mathbin.GroupTheory.GroupAction.Defs
@@ -15,6 +10,8 @@ import Mathbin.Data.Setoid.Basic
 import Mathbin.Data.Set.Pointwise.Smul
 import Mathbin.GroupTheory.Subgroup.Basic
 
+#align_import group_theory.group_action.basic from "leanprover-community/mathlib"@"d30d31261cdb4d2f5e612eabc3c4bf45556350d5"
+
 /-!
 # Basic properties of group actions
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

@@ -338,11 +338,13 @@ attribute [local instance] orbit_rel
 
 variable {α} {β}
 
+#print MulAction.orbitRel_apply /-
 @[to_additive]
 theorem orbitRel_apply {x y : β} : (orbitRel α β).Rel x y ↔ x ∈ orbit α y :=
   Iff.rfl
 #align mul_action.orbit_rel_apply MulAction.orbitRel_apply
 #align add_action.orbit_rel_apply AddAction.orbitRel_apply
+-/
 
 #print MulAction.quotient_preimage_image_eq_union_mul /-
 /-- When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the union

mathlib commit https://github.com/leanprover-community/mathlib/commit/d30d31261cdb4d2f5e612eabc3c4bf45556350d5

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 
 ! This file was ported from Lean 3 source module group_theory.group_action.basic
-! leanprover-community/mathlib commit a11f9106a169dd302a285019e5165f8ab32ff433
+! leanprover-community/mathlib commit d30d31261cdb4d2f5e612eabc3c4bf45556350d5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -338,6 +338,12 @@ attribute [local instance] orbit_rel
 
 variable {α} {β}
 
+@[to_additive]
+theorem orbitRel_apply {x y : β} : (orbitRel α β).Rel x y ↔ x ∈ orbit α y :=
+  Iff.rfl
+#align mul_action.orbit_rel_apply MulAction.orbitRel_apply
+#align add_action.orbit_rel_apply AddAction.orbitRel_apply
+
 #print MulAction.quotient_preimage_image_eq_union_mul /-
 /-- When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the union
 of the orbit of `U` under `α`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -121,11 +121,13 @@ instance {b : β} : MulAction α (orbit α b)
   one_smul a := Subtype.ext (one_smul α a)
   mul_smul a a' b' := Subtype.ext (mul_smul a a' b')
 
+#print MulAction.orbit.coe_smul /-
 @[simp, to_additive]
 theorem orbit.coe_smul {b : β} {a : α} {b' : orbit α b} : ↑(a • b') = a • (b' : β) :=
   rfl
 #align mul_action.orbit.coe_smul MulAction.orbit.coe_smul
 #align add_action.orbit.coe_vadd AddAction.orbit.coe_vadd
+-/
 
 variable (α) (β)
 
@@ -200,11 +202,13 @@ def Stabilizer.submonoid (b : β) : Submonoid α
 #align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid
 -/
 
+#print MulAction.mem_stabilizer_submonoid_iff /-
 @[simp, to_additive]
 theorem mem_stabilizer_submonoid_iff {b : β} {a : α} : a ∈ Stabilizer.submonoid α b ↔ a • b = b :=
   Iff.rfl
 #align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizer_submonoid_iff
 #align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizer_addSubmonoid_iff
+-/
 
 #print MulAction.orbit_eq_univ /-
 @[to_additive]
@@ -255,11 +259,13 @@ def stabilizer (b : β) : Subgroup α :=
 
 variable {α} {β}
 
+#print MulAction.mem_stabilizer_iff /-
 @[simp, to_additive]
 theorem mem_stabilizer_iff {b : β} {a : α} : a ∈ stabilizer α b ↔ a • b = b :=
   Iff.rfl
 #align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
 #align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
+-/
 
 #print MulAction.smul_orbit /-
 @[simp, to_additive]
@@ -359,6 +365,7 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
 #align add_action.quotient_preimage_image_eq_union_add AddAction.quotient_preimage_image_eq_union_add
 -/
 
+#print MulAction.disjoint_image_image_iff /-
 @[to_additive]
 theorem disjoint_image_image_iff {U V : Set β} :
     Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
@@ -375,13 +382,16 @@ theorem disjoint_image_image_iff {U V : Set β} :
     exact h y hy₁ a hz₁
 #align mul_action.disjoint_image_image_iff MulAction.disjoint_image_image_iff
 #align add_action.disjoint_image_image_iff AddAction.disjoint_image_image_iff
+-/
 
+#print MulAction.image_inter_image_iff /-
 @[to_additive]
 theorem image_inter_image_iff (U V : Set β) :
     Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
   Set.disjoint_iff_inter_eq_empty.symm.trans disjoint_image_image_iff
 #align mul_action.image_inter_image_iff MulAction.image_inter_image_iff
 #align add_action.image_inter_image_iff AddAction.image_inter_image_iff
+-/
 
 variable (α β)
 
@@ -440,7 +450,6 @@ theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient α β)
 
 variable (α) (β)
 
--- mathport name: exprΩ
 local notation "Ω" => orbitRel.Quotient α β
 
 #print MulAction.selfEquivSigmaOrbits' /-
@@ -474,6 +483,7 @@ def selfEquivSigmaOrbits : β ≃ Σ ω : Ω, orbit α ω.out' :=
 
 variable {α β}
 
+#print MulAction.stabilizer_smul_eq_stabilizer_map_conj /-
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
     stabilizer α (g • x) = (stabilizer α x).map (MulAut.conj g).toMonoidHom :=
@@ -482,7 +492,9 @@ theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
   rw [mem_stabilizer_iff, ← smul_left_cancel_iff g⁻¹, smul_smul, smul_smul, smul_smul, mul_left_inv,
     one_smul, ← mem_stabilizer_iff, Subgroup.mem_map_equiv, MulAut.conj_symm_apply]
 #align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conj
+-/
 
+#print MulAction.stabilizerEquivStabilizerOfOrbitRel /-
 /-- A bijection between the stabilizers of two elements in the same orbit. -/
 noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
     stabilizer α x ≃* stabilizer α y :=
@@ -492,6 +504,7 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel
     rw [← hg, stabilizer_smul_eq_stabilizer_map_conj]
   (MulEquiv.subgroupCongr this).trans ((MulAut.conj g).subgroupMap <| stabilizer α y).symm
 #align mul_action.stabilizer_equiv_stabilizer_of_orbit_rel MulAction.stabilizerEquivStabilizerOfOrbitRel
+-/
 
 end MulAction
 
@@ -499,6 +512,7 @@ namespace AddAction
 
 variable [AddGroup α] [AddAction α β]
 
+#print AddAction.stabilizer_vadd_eq_stabilizer_map_conj /-
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
     stabilizer α (g +ᵥ x) = (stabilizer α x).map (AddAut.conj g).toAddMonoidHom :=
@@ -508,7 +522,9 @@ theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
     add_left_neg, zero_vadd, ← mem_stabilizer_iff, AddSubgroup.mem_map_equiv,
     AddAut.conj_symm_apply]
 #align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conj
+-/
 
+#print AddAction.stabilizerEquivStabilizerOfOrbitRel /-
 /-- A bijection between the stabilizers of two elements in the same orbit. -/
 noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
     stabilizer α x ≃+ stabilizer α y :=
@@ -518,9 +534,11 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel
     rw [← hg, stabilizer_vadd_eq_stabilizer_map_conj]
   (AddEquiv.addSubgroupCongr this).trans ((AddAut.conj g).addSubgroupMap <| stabilizer α y).symm
 #align add_action.stabilizer_equiv_stabilizer_of_orbit_rel AddAction.stabilizerEquivStabilizerOfOrbitRel
+-/
 
 end AddAction
 
+#print smul_cancel_of_non_zero_divisor /-
 /-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
 The general theory of such `k` is elaborated by `is_smul_regular`.
 The typeclass that restricts all terms of `M` to have this property is `no_zero_smul_divisors`. -/
@@ -531,4 +549,5 @@ theorem smul_cancel_of_non_zero_divisor {M R : Type _} [Monoid M] [NonUnitalNonA
   refine' h _ _
   rw [smul_sub, h', sub_self]
 #align smul_cancel_of_non_zero_divisor smul_cancel_of_non_zero_divisor
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -229,7 +229,6 @@ theorem mem_fixedPoints_iff_card_orbit_eq_one {a : β} [Fintype (orbit α a)] :
     calc
       x • a = z := Subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
       _ = a := (Subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm
-      
 #align mul_action.mem_fixed_points_iff_card_orbit_eq_one MulAction.mem_fixedPoints_iff_card_orbit_eq_one
 #align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_zero
 -/
@@ -269,7 +268,6 @@ theorem smul_orbit (a : α) (b : β) : a • orbit α b = orbit α b :=
     calc
       orbit α b = a • a⁻¹ • orbit α b := (smul_inv_smul _ _).symm
       _ ⊆ a • orbit α b := Set.image_subset _ (smul_orbit_subset _ _)
-      
 #align mul_action.smul_orbit MulAction.smul_orbit
 #align add_action.vadd_orbit AddAction.vadd_orbit
 -/
@@ -281,7 +279,6 @@ theorem orbit_smul (a : α) (b : β) : orbit α (a • b) = orbit α b :=
     calc
       orbit α b = orbit α (a⁻¹ • a • b) := by rw [inv_smul_smul]
       _ ⊆ orbit α (a • b) := orbit_smul_subset _ _
-      
 #align mul_action.orbit_smul MulAction.orbit_smul
 #align add_action.orbit_vadd AddAction.orbit_vadd
 -/
@@ -459,7 +456,6 @@ def selfEquivSigmaOrbits' : β ≃ Σ ω : Ω, ω.orbit :=
     _ ≃ Σ ω : Ω, ω.orbit :=
       Equiv.sigmaCongrRight fun ω =>
         Equiv.subtypeEquivRight fun x => orbitRel.Quotient.mem_orbit.symm
-    
 #align mul_action.self_equiv_sigma_orbits' MulAction.selfEquivSigmaOrbits'
 #align add_action.self_equiv_sigma_orbits' AddAction.selfEquivSigmaOrbits'
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -133,7 +133,7 @@ variable (α) (β)
 /-- The set of elements fixed under the whole action. -/
 @[to_additive "The set of elements fixed under the whole action."]
 def fixedPoints : Set β :=
-  { b : β | ∀ x : α, x • b = b }
+  {b : β | ∀ x : α, x • b = b}
 #align mul_action.fixed_points MulAction.fixedPoints
 #align add_action.fixed_points AddAction.fixedPoints
 -/
@@ -142,7 +142,7 @@ def fixedPoints : Set β :=
 /-- `fixed_by g` is the subfield of elements fixed by `g`. -/
 @[to_additive "`fixed_by g` is the subfield of elements fixed by `g`."]
 def fixedBy (g : α) : Set β :=
-  { x | g • x = x }
+  {x | g • x = x}
 #align mul_action.fixed_by MulAction.fixedBy
 #align add_action.fixed_by AddAction.fixedBy
 -/
@@ -192,7 +192,7 @@ variable (α) {β}
 @[to_additive "The stabilizer of a point `b` as an additive submonoid of `α`."]
 def Stabilizer.submonoid (b : β) : Submonoid α
     where
-  carrier := { a | a • b = b }
+  carrier := {a | a • b = b}
   one_mem' := one_smul _ b
   mul_mem' a a' (ha : a • b = b) (hb : a' • b = b) :=
     show (a * a') • b = b by rw [← smul_smul, hb, ha]

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -351,7 +351,7 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
     rw [Set.mem_iUnion]
     exact ⟨a⁻¹, a • x, hy, inv_smul_smul a x⟩
   · intro hx
-    rw [Set.mem_iUnion] at hx
+    rw [Set.mem_iUnion] at hx 
     obtain ⟨a, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
     refine' ⟨a⁻¹ • x, _, by simp only [Quotient.eq'] <;> use a⁻¹⟩
@@ -453,10 +453,10 @@ This version is expressed in terms of `mul_action.orbit_rel.quotient.orbit` inst
 `mul_action.orbit`, to avoid mentioning `quotient.out'`. -/
 @[to_additive
       "Decomposition of a type `X` as a disjoint union of its orbits under an additive group\naction.\n\nThis version is expressed in terms of `add_action.orbit_rel.quotient.orbit` instead of\n`add_action.orbit`, to avoid mentioning `quotient.out'`. "]
-def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
+def selfEquivSigmaOrbits' : β ≃ Σ ω : Ω, ω.orbit :=
   calc
-    β ≃ Σω : Ω, { b // Quotient.mk'' b = ω } := (Equiv.sigmaFiberEquiv Quotient.mk'').symm
-    _ ≃ Σω : Ω, ω.orbit :=
+    β ≃ Σ ω : Ω, { b // Quotient.mk'' b = ω } := (Equiv.sigmaFiberEquiv Quotient.mk'').symm
+    _ ≃ Σ ω : Ω, ω.orbit :=
       Equiv.sigmaCongrRight fun ω =>
         Equiv.subtypeEquivRight fun x => orbitRel.Quotient.mem_orbit.symm
     
@@ -468,7 +468,7 @@ def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
 /-- Decomposition of a type `X` as a disjoint union of its orbits under a group action. -/
 @[to_additive
       "Decomposition of a type `X` as a disjoint union of its orbits under an additive group\naction."]
-def selfEquivSigmaOrbits : β ≃ Σω : Ω, orbit α ω.out' :=
+def selfEquivSigmaOrbits : β ≃ Σ ω : Ω, orbit α ω.out' :=
   (selfEquivSigmaOrbits' α β).trans <|
     Equiv.sigmaCongrRight fun i =>
       Equiv.Set.ofEq <| orbitRel.Quotient.orbit_eq_orbit_out _ Quotient.out_eq'

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -39,7 +39,7 @@ universe u v w
 
 variable {α : Type u} {β : Type v} {γ : Type w}
 
-open Pointwise
+open scoped Pointwise
 
 open Function
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -121,12 +121,6 @@ instance {b : β} : MulAction α (orbit α b)
   one_smul a := Subtype.ext (one_smul α a)
   mul_smul a a' b' := Subtype.ext (mul_smul a a' b')
 
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 @[simp, to_additive]
 theorem orbit.coe_smul {b : β} {a : α} {b' : orbit α b} : ↑(a • b') = a • (b' : β) :=
   rfl
@@ -206,12 +200,6 @@ def Stabilizer.submonoid (b : β) : Submonoid α
 #align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid
 -/
 
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 @[simp, to_additive]
 theorem mem_stabilizer_submonoid_iff {b : β} {a : α} : a ∈ Stabilizer.submonoid α b ↔ a • b = b :=
   Iff.rfl
@@ -268,12 +256,6 @@ def stabilizer (b : β) : Subgroup α :=
 
 variable {α} {β}
 
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 @[simp, to_additive]
 theorem mem_stabilizer_iff {b : β} {a : α} : a ∈ stabilizer α b ↔ a • b = b :=
   Iff.rfl
@@ -380,12 +362,6 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
 #align add_action.quotient_preimage_image_eq_union_add AddAction.quotient_preimage_image_eq_union_add
 -/
 
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 @[to_additive]
 theorem disjoint_image_image_iff {U V : Set β} :
     Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
@@ -403,12 +379,6 @@ theorem disjoint_image_image_iff {U V : Set β} :
 #align mul_action.disjoint_image_image_iff MulAction.disjoint_image_image_iff
 #align add_action.disjoint_image_image_iff AddAction.disjoint_image_image_iff
 
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 @[to_additive]
 theorem image_inter_image_iff (U V : Set β) :
     Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
@@ -508,9 +478,6 @@ def selfEquivSigmaOrbits : β ≃ Σω : Ω, orbit α ω.out' :=
 
 variable {α β}
 
-/- warning: mul_action.stabilizer_smul_eq_stabilizer_map_conj -> MulAction.stabilizer_smul_eq_stabilizer_map_conj is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
     stabilizer α (g • x) = (stabilizer α x).map (MulAut.conj g).toMonoidHom :=
@@ -520,12 +487,6 @@ theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
     one_smul, ← mem_stabilizer_iff, Subgroup.mem_map_equiv, MulAut.conj_symm_apply]
 #align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conj
 
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 /-- A bijection between the stabilizers of two elements in the same orbit. -/
 noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
     stabilizer α x ≃* stabilizer α y :=
@@ -542,9 +503,6 @@ namespace AddAction
 
 variable [AddGroup α] [AddAction α β]
 
-/- warning: add_action.stabilizer_vadd_eq_stabilizer_map_conj -> AddAction.stabilizer_vadd_eq_stabilizer_map_conj is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
     stabilizer α (g +ᵥ x) = (stabilizer α x).map (AddAut.conj g).toAddMonoidHom :=
@@ -555,12 +513,6 @@ theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
     AddAut.conj_symm_apply]
 #align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conj
 
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 /-- A bijection between the stabilizers of two elements in the same orbit. -/
 noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
     stabilizer α x ≃+ stabilizer α y :=
@@ -573,12 +525,6 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel
 
 end AddAction
 
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 /-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
 The general theory of such `k` is elaborated by `is_smul_regular`.
 The typeclass that restricts all terms of `M` to have this property is `no_zero_smul_divisors`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -307,11 +307,7 @@ theorem orbit_smul (a : α) (b : β) : orbit α (a • b) = orbit α b :=
 /-- The action of a group on an orbit is transitive. -/
 @[to_additive "The action of an additive group on an orbit is transitive."]
 instance (x : β) : IsPretransitive α (orbit α x) :=
-  ⟨by
-    rintro ⟨_, a, rfl⟩ ⟨_, b, rfl⟩
-    use b * a⁻¹
-    ext1
-    simp [mul_smul]⟩
+  ⟨by rintro ⟨_, a, rfl⟩ ⟨_, b, rfl⟩; use b * a⁻¹; ext1; simp [mul_smul]⟩
 
 #print MulAction.orbit_eq_iff /-
 @[to_additive]
@@ -455,11 +451,8 @@ theorem orbitRel.Quotient.orbit_mk (b : β) :
 #print MulAction.orbitRel.Quotient.mem_orbit /-
 @[to_additive]
 theorem orbitRel.Quotient.mem_orbit {b : β} {x : orbitRel.Quotient α β} :
-    b ∈ x.orbit ↔ Quotient.mk'' b = x :=
-  by
-  induction x using Quotient.inductionOn'
-  rw [Quotient.eq'']
-  rfl
+    b ∈ x.orbit ↔ Quotient.mk'' b = x := by induction x using Quotient.inductionOn';
+  rw [Quotient.eq'']; rfl
 #align mul_action.orbit_rel.quotient.mem_orbit MulAction.orbitRel.Quotient.mem_orbit
 #align add_action.orbit_rel.quotient.mem_orbit AddAction.orbitRel.Quotient.mem_orbit
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -516,10 +516,7 @@ def selfEquivSigmaOrbits : β ≃ Σω : Ω, orbit α ω.out' :=
 variable {α β}
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
@@ -553,10 +550,7 @@ namespace AddAction
 variable [AddGroup α] [AddAction α β]
 
 /- warning: add_action.stabilizer_vadd_eq_stabilizer_map_conj -> AddAction.stabilizer_vadd_eq_stabilizer_map_conj is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -519,7 +519,7 @@ variable {α β}
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (SMul.smul.{u1, u2} α β (MulAction.toHasSmul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (fun (_x : MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) => α -> (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 Case conversion may be inaccurate. Consider using '#align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -153,13 +153,13 @@ def fixedBy (g : α) : Set β :=
 #align add_action.fixed_by AddAction.fixedBy
 -/
 
-#print MulAction.fixed_eq_interᵢ_fixedBy /-
+#print MulAction.fixed_eq_iInter_fixedBy /-
 @[to_additive]
-theorem fixed_eq_interᵢ_fixedBy : fixedPoints α β = ⋂ g : α, fixedBy α β g :=
+theorem fixed_eq_iInter_fixedBy : fixedPoints α β = ⋂ g : α, fixedBy α β g :=
   Set.ext fun x =>
-    ⟨fun hx => Set.mem_interᵢ.2 fun g => hx g, fun hx g => (Set.mem_interᵢ.1 hx g : _)⟩
-#align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_interᵢ_fixedBy
-#align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_interᵢ_fixedBy
+    ⟨fun hx => Set.mem_iInter.2 fun g => hx g, fun hx g => (Set.mem_iInter.1 hx g : _)⟩
+#align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_iInter_fixedBy
+#align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_iInter_fixedBy
 -/
 
 variable {α} (β)
@@ -370,10 +370,10 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
   constructor
   · rintro ⟨y, hy, hxy⟩
     obtain ⟨a, rfl⟩ := Quotient.exact hxy
-    rw [Set.mem_unionᵢ]
+    rw [Set.mem_iUnion]
     exact ⟨a⁻¹, a • x, hy, inv_smul_smul a x⟩
   · intro hx
-    rw [Set.mem_unionᵢ] at hx
+    rw [Set.mem_iUnion] at hx
     obtain ⟨a, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
     refine' ⟨a⁻¹ • x, _, by simp only [Quotient.eq'] <;> use a⁻¹⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -519,7 +519,7 @@ variable {α β}
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (SMul.smul.{u1, u2} α β (MulAction.toHasSmul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (fun (_x : MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) => α -> (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 Case conversion may be inaccurate. Consider using '#align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
@@ -556,7 +556,7 @@ variable [AddGroup α] [AddAction α β]
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : AddGroup.{u1} α] [_inst_2 : AddAction.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (AddSubgroup.{u1} α _inst_1) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (VAdd.vadd.{u1, u2} α β (AddAction.toHasVadd.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)) _inst_2) g x)) (AddSubgroup.map.{u1, u1} α _inst_1 α _inst_1 (AddEquiv.toAddMonoidHom.{u1, u1} α α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) (fun (_x : AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) => α -> (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))))) (AddMonoidHom.hasCoeToFun.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toHasAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) (AddAut.conj.{u1} α _inst_1) g)) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : AddGroup.{u1} α] [_inst_2 : AddAction.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (AddSubgroup.{u1} α _inst_1) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HVAdd.hVAdd.{u1, u2, u2} α β β (instHVAdd.{u1, u2} α β (AddAction.toVAdd.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)) _inst_2)) g x)) (AddSubgroup.map.{u1, u1} α _inst_1 α _inst_1 (AddEquiv.toAddMonoidHom.{u1, u1} α α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : α) => Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) _x) (AddHomClass.toFunLike.{u1, u1, u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))) (AddZeroClass.toAdd.{u1} (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) (AddMonoidHomClass.toAddHomClass.{u1, u1, u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))))))) (AddMonoidHom.addMonoidHomClass.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))))) (AddAut.conj.{u1} α _inst_1) g)) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : AddGroup.{u1} α] [_inst_2 : AddAction.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (AddSubgroup.{u1} α _inst_1) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HVAdd.hVAdd.{u1, u2, u2} α β β (instHVAdd.{u1, u2} α β (AddAction.toVAdd.{u1, u2} α β (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)) _inst_2)) g x)) (AddSubgroup.map.{u1, u1} α _inst_1 α _inst_1 (AddEquiv.toAddMonoidHom.{u1, u1} α α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : α) => Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) _x) (AddHomClass.toFunLike.{u1, u1, u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))) (AddZeroClass.toAdd.{u1} (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) (AddMonoidHomClass.toAddHomClass.{u1, u1, u1} (AddMonoidHom.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))) α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))))))) (AddMonoidHom.addMonoidHomClass.{u1, u1} α (Additive.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))) (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))) (Additive.addZeroClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (AddAut.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1))))) (AddAut.group.{u1} α (AddZeroClass.toAdd.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_1)))))))))))) (AddAut.conj.{u1} α _inst_1) g)) (AddAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 Case conversion may be inaccurate. Consider using '#align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -519,7 +519,7 @@ variable {α β}
 lean 3 declaration is
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (SMul.smul.{u1, u2} α β (MulAction.toHasSmul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (fun (_x : MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) => α -> (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))) (MonoidHom.hasCoeToFun.{u1, u1} α (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.group.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Group.{u1} α] [_inst_2 : MulAction.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))] (g : α) (x : β), Eq.{succ u1} (Subgroup.{u1} α _inst_1) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β (MulAction.toSMul.{u1, u2} α β (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) _inst_2)) g x)) (Subgroup.map.{u1, u1} α _inst_1 α _inst_1 (MulEquiv.toMonoidHom.{u1, u1} α α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (fun (_x : α) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : α) => MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (MulOneClass.toMul.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))) α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (MonoidHom.monoidHomClass.{u1, u1} α (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))) (Monoid.toMulOneClass.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (DivInvMonoid.toMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (Group.toDivInvMonoid.{u1} (MulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) (MulAut.instGroupMulAut.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))))))))) (MulAut.conj.{u1} α _inst_1) g)) (MulAction.stabilizer.{u1, u2} α β _inst_1 _inst_2 x))
 Case conversion may be inaccurate. Consider using '#align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conjₓ'. -/
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Follow-up to #10816.

Remaining places containing such lemmas are

• Option.bex_ne_none and Option.ball_ne_none: defined in Lean core
• Nat.decidableBallLT and Nat.decidableBallLE: defined in Lean core
• bef_def is still used in a number of places and could be renamed
• BAll.imp_{left,right}, BEx.imp_{left,right}, BEx.intro and BEx.elim

I only audited the first ~150 lemmas mentioning "ball"; too many lemmas named after Metric.ball/openBall/closedBall.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -434,7 +434,7 @@ theorem quotient_preimage_image_eq_union_mul (U : Set α) :
   · intro hx
     rw [Set.mem_iUnion] at hx
     obtain ⟨g, u, hu₁, hu₂⟩ := hx
-    rw [Set.mem_preimage, Set.mem_image_iff_bex]
+    rw [Set.mem_preimage, Set.mem_image]
     refine' ⟨g⁻¹ • a, _, by simp only [f, Quotient.eq']; use g⁻¹⟩
     rw [← hu₂]
     convert hu₁

Quoting [@digama0](https://github.com/digama0):

These were actually never meant to go in the file, they are basically debugging information and only useful on significantly broken mathport files. You can safely remove all of them.

@@ -535,7 +535,6 @@ theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient G α)
 
 variable (G) (α)
 
--- mathport name: exprΩ
 local notation "Ω" => orbitRel.Quotient G α
 
 /-- Decomposition of a type `X` as a disjoint union of its orbits under a group action.

Add various miscellaneous lemmas about orbits, mainly about orbits under the subgroup / submonoid action.

From AperiodicMonotilesLean.

@@ -106,6 +106,16 @@ theorem orbit.coe_smul {a : α} {m : M} {a' : orbit M a} : ↑(m • a') = m •
 #align mul_action.orbit.coe_smul MulAction.orbit.coe_smul
 #align add_action.orbit.coe_vadd AddAction.orbit.coe_vadd
 
+@[to_additive]
+lemma orbit_submonoid_subset (S : Submonoid M) (a : α) : orbit S a ⊆ orbit M a := by
+  rintro b ⟨g, rfl⟩
+  exact mem_orbit _ _
+
+@[to_additive]
+lemma mem_orbit_of_mem_orbit_submonoid {S : Submonoid M} {a b : α} (h : a ∈ orbit S b) :
+    a ∈ orbit M b :=
+  orbit_submonoid_subset S _ h
+
 variable (M)
 
 @[to_additive]
@@ -355,6 +365,32 @@ theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) :
 #align mul_action.smul_mem_orbit_smul MulAction.smul_mem_orbit_smul
 #align add_action.vadd_mem_orbit_vadd AddAction.vadd_mem_orbit_vadd
 
+@[to_additive]
+lemma orbit_subgroup_subset (H : Subgroup G) (a : α) : orbit H a ⊆ orbit G a :=
+  orbit_submonoid_subset H.toSubmonoid a
+
+@[to_additive]
+lemma mem_orbit_of_mem_orbit_subgroup {H : Subgroup G} {a b : α} (h : a ∈ orbit H b) :
+    a ∈ orbit G b :=
+  orbit_subgroup_subset H _ h
+
+@[to_additive]
+lemma mem_orbit_symm {a₁ a₂ : α} : a₁ ∈ orbit G a₂ ↔ a₂ ∈ orbit G a₁ := by
+  simp_rw [← orbit_eq_iff, eq_comm]
+
+@[to_additive]
+lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
+    a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) := by
+  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
+  · rcases h with ⟨g, rfl⟩
+    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, orbit.coe_smul, ← Submonoid.smul_def]
+    exact MulAction.mem_orbit _ g
+  · rcases h with ⟨g, h⟩
+    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, ← orbit.coe_smul,
+             ← Submonoid.smul_def, ← Subtype.ext_iff] at h
+    subst h
+    exact MulAction.mem_orbit _ g
+
 variable (G α)
 
 /-- The relation 'in the same orbit'. -/
@@ -375,6 +411,10 @@ theorem orbitRel_apply {a b : α} : (orbitRel G α).Rel a b ↔ a ∈ orbit G b
 #align mul_action.orbit_rel_apply MulAction.orbitRel_apply
 #align add_action.orbit_rel_apply AddAction.orbitRel_apply
 
+@[to_additive]
+lemma orbitRel_subgroup_le (H : Subgroup G) : orbitRel H α ≤ orbitRel G α :=
+  Setoid.le_def.2 mem_orbit_of_mem_orbit_subgroup
+
 /-- When you take a set `U` in `α`, push it down to the quotient, and pull back, you get the union
 of the orbit of `U` under `G`. -/
 @[to_additive

I see no reason for the omission of @[to_additive] on these two pretransitive_iff_... lemmas, so add it here.

From AperiodicMonotilesLean.

@@ -440,6 +440,8 @@ def orbitRel.Quotient : Type _ :=
 
 /-- An action is pretransitive if and only if the quotient by `MulAction.orbitRel` is a
 subsingleton. -/
+@[to_additive "An additive action is pretransitive if and only if the quotient by
+`AddAction.orbitRel` is a subsingleton."]
 theorem pretransitive_iff_subsingleton_quotient :
     IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α) := by
   refine ⟨fun _ ↦ ⟨fun a b ↦ ?_⟩, fun _ ↦ ⟨fun a b ↦ ?_⟩⟩
@@ -450,6 +452,8 @@ theorem pretransitive_iff_subsingleton_quotient :
 
 /-- If `α` is non-empty, an action is pretransitive if and only if the quotient has exactly one
 element. -/
+@[to_additive "If `α` is non-empty, an additive action is pretransitive if and only if the
+quotient has exactly one element."]
 theorem pretransitive_iff_unique_quotient_of_nonempty [Nonempty α] :
     IsPretransitive G α ↔ Nonempty (Unique <| orbitRel.Quotient G α) := by
   rw [unique_iff_subsingleton_and_nonempty, pretransitive_iff_subsingleton_quotient, iff_self_and]

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -395,7 +395,7 @@ theorem quotient_preimage_image_eq_union_mul (U : Set α) :
     rw [Set.mem_iUnion] at hx
     obtain ⟨g, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
-    refine' ⟨g⁻¹ • a, _, by simp only [Quotient.eq']; use g⁻¹⟩
+    refine' ⟨g⁻¹ • a, _, by simp only [f, Quotient.eq']; use g⁻¹⟩
     rw [← hu₂]
     convert hu₁
     simp only [inv_smul_smul]

Defines Galois objects in a Galois category in a fibre functor independent way, also gives an equivalent characterisation in terms of a fibre functor.

To allow for a definition that only depends on C, contrary to what was said earlier, we introduce a GaloisCategory typeclass extending PreGaloisCategory that additionally asserts the existence of a fibre functor.

Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>

@@ -438,6 +438,23 @@ def orbitRel.Quotient : Type _ :=
 #align mul_action.orbit_rel.quotient MulAction.orbitRel.Quotient
 #align add_action.orbit_rel.quotient AddAction.orbitRel.Quotient
 
+/-- An action is pretransitive if and only if the quotient by `MulAction.orbitRel` is a
+subsingleton. -/
+theorem pretransitive_iff_subsingleton_quotient :
+    IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α) := by
+  refine ⟨fun _ ↦ ⟨fun a b ↦ ?_⟩, fun _ ↦ ⟨fun a b ↦ ?_⟩⟩
+  · refine Quot.inductionOn a (fun x ↦ ?_)
+    exact Quot.inductionOn b (fun y ↦ Quot.sound <| exists_smul_eq G y x)
+  · have h : Quotient.mk (orbitRel G α) b = ⟦a⟧ := Subsingleton.elim _ _
+    exact Quotient.eq_rel.mp h
+
+/-- If `α` is non-empty, an action is pretransitive if and only if the quotient has exactly one
+element. -/
+theorem pretransitive_iff_unique_quotient_of_nonempty [Nonempty α] :
+    IsPretransitive G α ↔ Nonempty (Unique <| orbitRel.Quotient G α) := by
+  rw [unique_iff_subsingleton_and_nonempty, pretransitive_iff_subsingleton_quotient, iff_self_and]
+  exact fun _ ↦ (nonempty_quotient_iff _).mpr inferInstance
+
 variable {G α}
 
 /-- The orbit corresponding to an element of the quotient by `MulAction.orbitRel` -/

@@ -212,6 +212,88 @@ end Stabilizers
 
 end MulAction
 
+section FixedPoints
+
+variable (M : Type u) (α : Type v) [Monoid M]
+
+section Monoid
+
+variable [Monoid α] [MulDistribMulAction M α]
+
+/-- The submonoid of elements fixed under the whole action. -/
+def FixedPoints.submonoid : Submonoid α where
+  carrier := MulAction.fixedPoints M α
+  one_mem' := smul_one
+  mul_mem' ha hb _ := by rw [smul_mul', ha, hb]
+
+@[simp]
+lemma FixedPoints.mem_submonoid (a : α) : a ∈ submonoid M α ↔ ∀ m : M, m • a = a :=
+  Iff.rfl
+
+end Monoid
+
+section Group
+
+variable [Group α] [MulDistribMulAction M α]
+
+/-- The subgroup of elements fixed under the whole action. -/
+def FixedPoints.subgroup : Subgroup α where
+  __ := submonoid M α
+  inv_mem' ha _ := by rw [smul_inv', ha]
+
+/-- The notation for `FixedPoints.subgroup`, chosen to resemble `αᴹ`. -/
+notation α "^*" M:51 => FixedPoints.subgroup M α
+
+@[simp]
+lemma FixedPoints.mem_subgroup (a : α) : a ∈ α^*M ↔ ∀ m : M, m • a = a :=
+  Iff.rfl
+
+@[simp]
+lemma FixedPoints.subgroup_toSubmonoid : (α^*M).toSubmonoid = submonoid M α :=
+  rfl
+
+end Group
+
+section AddMonoid
+
+variable [AddMonoid α] [DistribMulAction M α]
+
+/-- The additive submonoid of elements fixed under the whole action. -/
+def FixedPoints.addSubmonoid : AddSubmonoid α where
+  carrier := MulAction.fixedPoints M α
+  zero_mem' := smul_zero
+  add_mem' ha hb _ := by rw [smul_add, ha, hb]
+
+@[simp]
+lemma FixedPoints.mem_addSubmonoid (a : α) : a ∈ addSubmonoid M α ↔ ∀ m : M, m • a = a :=
+  Iff.rfl
+
+end AddMonoid
+
+section AddGroup
+
+variable [AddGroup α] [DistribMulAction M α]
+
+/-- The additive subgroup of elements fixed under the whole action. -/
+def FixedPoints.addSubgroup : AddSubgroup α where
+  __ := addSubmonoid M α
+  neg_mem' ha _ := by rw [smul_neg, ha]
+
+/-- The notation for `FixedPoints.addSubgroup`, chosen to resemble `αᴹ`. -/
+notation α "^+" M:51 => FixedPoints.addSubgroup M α
+
+@[simp]
+lemma FixedPoints.mem_addSubgroup (a : α) : a ∈ α^+M ↔ ∀ m : M, m • a = a :=
+  Iff.rfl
+
+@[simp]
+lemma FixedPoints.addSubgroup_toAddSubmonoid : (α^+M).toAddSubmonoid = addSubmonoid M α :=
+  rfl
+
+end AddGroup
+
+end FixedPoints
+
 /-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
 The general theory of such `k` is elaborated by `IsSMulRegular`.
 The typeclass that restricts all terms of `M` to have this property is `NoZeroSMulDivisors`. -/

@@ -531,3 +531,9 @@ end AddAction
 
 attribute [to_additive existing] MulAction.stabilizer_smul_eq_stabilizer_map_conj
 attribute [to_additive existing] MulAction.stabilizerEquivStabilizerOfOrbitRel
+
+theorem Equiv.swap_mem_stabilizer {α : Type*} [DecidableEq α] {S : Set α} {a b : α} :
+    Equiv.swap a b ∈ MulAction.stabilizer (Equiv.Perm α) S ↔ (a ∈ S ↔ b ∈ S) := by
+  rw [MulAction.mem_stabilizer_iff, Set.ext_iff, ← swap_inv]
+  simp_rw [Set.mem_inv_smul_set_iff, Perm.smul_def, swap_apply_def]
+  exact ⟨fun h ↦ by simpa [Iff.comm] using h a, by intros; split_ifs <;> simp [*]⟩

See Zulip thread for the discussion.

@@ -472,12 +472,12 @@ lemma stabilizer_smul_eq_left [SMul α β] [IsScalarTower G α β] (a : α) (b :
 @[to_additive (attr := simp)]
 lemma stabilizer_smul_eq_right [Group α] [MulAction α β] [SMulCommClass G α β] (a : α) (b : β) :
     stabilizer G (a • b) = stabilizer G b :=
-  (le_stabilizer_smul_right _ _).antisymm' $ (le_stabilizer_smul_right a⁻¹ _).trans_eq $ by
+  (le_stabilizer_smul_right _ _).antisymm' <| (le_stabilizer_smul_right a⁻¹ _).trans_eq <| by
     rw [inv_smul_smul]
 
 @[to_additive (attr := simp)]
 lemma stabilizer_mul_eq_left [Group α] [IsScalarTower G α α] (a b : α)  :
-    stabilizer G (a * b) = stabilizer G a := stabilizer_smul_eq_left a _ $ mul_left_injective _
+    stabilizer G (a * b) = stabilizer G a := stabilizer_smul_eq_left a _ <| mul_left_injective _
 
 @[to_additive (attr := simp)]
 lemma stabilizer_mul_eq_right [Group α] [SMulCommClass G α α] (a b : α) :

Prove a bunch of results about stabilizer G s where Group G, MulAction G α, s : Set α.

@@ -224,13 +224,10 @@ theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAs
 #align smul_cancel_of_non_zero_divisor smul_cancel_of_non_zero_divisor
 
 namespace MulAction
-
-variable (G : Type u) [Group G] (α : Type v) [MulAction G α]
+variable {G α β : Type*} [Group G] [MulAction G α] [MulAction G β]
 
 section Orbit
 
-variable {G α}
-
 @[to_additive (attr := simp)]
 theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a :=
   (smul_orbit_subset g a).antisymm <|
@@ -431,8 +428,7 @@ def selfEquivSigmaOrbits : α ≃ Σω : Ω, orbit G ω.out' :=
 end Orbit
 
 section Stabilizer
-
-variable {α}
+variable (G)
 
 /-- The stabilizer of an element under an action, i.e. what sends the element to itself.
 A subgroup. -/
@@ -457,6 +453,36 @@ theorem mem_stabilizer_iff {a : α} {g : G} : g ∈ stabilizer G a ↔ g • a =
 #align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
 #align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
 
+@[to_additive]
+lemma le_stabilizer_smul_left [SMul α β] [IsScalarTower G α β] (a : α) (b : β) :
+    stabilizer G a ≤ stabilizer G (a • b) := by
+  simp_rw [SetLike.le_def, mem_stabilizer_iff, ← smul_assoc]; rintro a h; rw [h]
+
+@[to_additive]
+lemma le_stabilizer_smul_right [SMul α β] [SMulCommClass G α β] (a : α) (b : β) :
+    stabilizer G b ≤ stabilizer G (a • b) := by
+  simp_rw [SetLike.le_def, mem_stabilizer_iff, smul_comm]; rintro a h; rw [h]
+
+@[to_additive (attr := simp)]
+lemma stabilizer_smul_eq_left [SMul α β] [IsScalarTower G α β] (a : α) (b : β)
+    (h : Injective (· • b : α → β)) : stabilizer G (a • b) = stabilizer G a := by
+  refine' (le_stabilizer_smul_left _ _).antisymm' fun a ha ↦ _
+  simpa only [mem_stabilizer_iff, ← smul_assoc, h.eq_iff] using ha
+
+@[to_additive (attr := simp)]
+lemma stabilizer_smul_eq_right [Group α] [MulAction α β] [SMulCommClass G α β] (a : α) (b : β) :
+    stabilizer G (a • b) = stabilizer G b :=
+  (le_stabilizer_smul_right _ _).antisymm' $ (le_stabilizer_smul_right a⁻¹ _).trans_eq $ by
+    rw [inv_smul_smul]
+
+@[to_additive (attr := simp)]
+lemma stabilizer_mul_eq_left [Group α] [IsScalarTower G α α] (a b : α)  :
+    stabilizer G (a * b) = stabilizer G a := stabilizer_smul_eq_left a _ $ mul_left_injective _
+
+@[to_additive (attr := simp)]
+lemma stabilizer_mul_eq_right [Group α] [SMulCommClass G α α] (a b : α) :
+    stabilizer G (a * b) = stabilizer G b := stabilizer_smul_eq_right a _
+
 /-- If the stabilizer of `a` is `S`, then the stabilizer of `g • a` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : G) (a : α) :
     stabilizer G (g • a) = (stabilizer G a).map (MulAut.conj g).toMonoidHom := by
@@ -480,8 +506,7 @@ end Stabilizer
 end MulAction
 
 namespace AddAction
-
-variable (G : Type u) [AddGroup G] (α : Type v) [AddAction G α]
+variable {G α : Type*} [AddGroup G] [AddAction G α]
 
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : G) (a : α) :
@@ -503,3 +528,6 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {a b : α} (h : (orbitRel
 #align add_action.stabilizer_equiv_stabilizer_of_orbit_rel AddAction.stabilizerEquivStabilizerOfOrbitRel
 
 end AddAction
+
+attribute [to_additive existing] MulAction.stabilizer_smul_eq_stabilizer_map_conj
+attribute [to_additive existing] MulAction.stabilizerEquivStabilizerOfOrbitRel

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -77,7 +77,7 @@ theorem orbit_nonempty (a : α) : Set.Nonempty (orbit M a) :=
 #align add_action.orbit_nonempty AddAction.orbit_nonempty
 
 @[to_additive]
-theorem mapsTo_smul_orbit (m : M) (a : α) : Set.MapsTo ((· • ·) m) (orbit M a) (orbit M a) :=
+theorem mapsTo_smul_orbit (m : M) (a : α) : Set.MapsTo (m • ·) (orbit M a) (orbit M a) :=
   Set.range_subset_iff.2 fun m' => ⟨m * m', mul_smul _ _ _⟩
 #align mul_action.maps_to_smul_orbit MulAction.mapsTo_smul_orbit
 #align add_action.maps_to_vadd_orbit AddAction.mapsTo_vadd_orbit
@@ -303,7 +303,7 @@ of the orbit of `U` under `G`. -/
       union of the orbit of `U` under `G`."]
 theorem quotient_preimage_image_eq_union_mul (U : Set α) :
     letI := orbitRel G α
-    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (· • ·) g '' U := by
+    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (g • ·) '' U := by
   letI := orbitRel G α
   set f : α → Quotient (MulAction.orbitRel G α) := Quotient.mk'
   ext a

@@ -385,7 +385,7 @@ theorem orbitRel.Quotient.mem_orbit {a : α} {x : orbitRel.Quotient G α} :
 #align add_action.orbit_rel.quotient.mem_orbit AddAction.orbitRel.Quotient.mem_orbit
 
 /-- Note that `hφ = Quotient.out_eq'` is a useful choice here. -/
-@[to_additive "Note that `hφ = quotient.out_eq'` is m useful choice here."]
+@[to_additive "Note that `hφ = Quotient.out_eq'` is a useful choice here."]
 theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient G α)
     {φ : orbitRel.Quotient G α → α} (hφ : letI := orbitRel G α; RightInverse φ Quotient.mk') :
     orbitRel.Quotient.orbit x = MulAction.orbit G (φ x) := by

• Re-organise the namespace and section structure of GroupTheory/GroupAction/Basic.lean.
• Remove the namespaces MulAction.Stabilizer and AddAction.Stabilizer, renaming MulAction.Stabilizer.submonoid to MulAction.stabilizerSubmonoid.
• Make variables for the monoid/group/set implicit when an element or subset is used in the statement.

@@ -37,7 +37,11 @@ open Function
 
 namespace MulAction
 
-variable (M : Type u) {α : Type v} [Monoid M] [MulAction M α]
+variable (M : Type u) [Monoid M] (α : Type v) [MulAction M α]
+
+section Orbit
+
+variable {α}
 
 /-- The orbit of an element under an action. -/
 @[to_additive "The orbit of an element under an action."]
@@ -55,8 +59,8 @@ theorem mem_orbit_iff {a₁ a₂ : α} : a₂ ∈ orbit M a₁ ↔ ∃ x : M, x
 #align add_action.mem_orbit_iff AddAction.mem_orbit_iff
 
 @[to_additive (attr := simp)]
-theorem mem_orbit (a : α) (x : M) : x • a ∈ orbit M a :=
-  ⟨x, rfl⟩
+theorem mem_orbit (a : α) (m : M) : m • a ∈ orbit M a :=
+  ⟨m, rfl⟩
 #align mul_action.mem_orbit MulAction.mem_orbit
 #align add_action.mem_orbit AddAction.mem_orbit
 
@@ -102,7 +106,17 @@ theorem orbit.coe_smul {a : α} {m : M} {a' : orbit M a} : ↑(m • a') = m •
 #align mul_action.orbit.coe_smul MulAction.orbit.coe_smul
 #align add_action.orbit.coe_vadd AddAction.orbit.coe_vadd
 
-variable (M) (α)
+variable (M)
+
+@[to_additive]
+theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ :=
+  (surjective_smul M a).range_eq
+#align mul_action.orbit_eq_univ MulAction.orbit_eq_univ
+#align add_action.orbit_eq_univ AddAction.orbit_eq_univ
+
+end Orbit
+
+section FixedPoints
 
 /-- The set of elements fixed under the whole action. -/
 @[to_additive "The set of elements fixed under the whole action."]
@@ -111,6 +125,8 @@ def fixedPoints : Set α :=
 #align mul_action.fixed_points MulAction.fixedPoints
 #align add_action.fixed_points AddAction.fixedPoints
 
+variable {M}
+
 /-- `fixedBy m` is the set of elements fixed by `m`. -/
 @[to_additive "`fixedBy m` is the set of elements fixed by `m`."]
 def fixedBy (m : M) : Set α :=
@@ -118,14 +134,16 @@ def fixedBy (m : M) : Set α :=
 #align mul_action.fixed_by MulAction.fixedBy
 #align add_action.fixed_by AddAction.fixedBy
 
+variable (M)
+
 @[to_additive]
-theorem fixed_eq_iInter_fixedBy : fixedPoints M α = ⋂ m : M, fixedBy M α m :=
+theorem fixed_eq_iInter_fixedBy : fixedPoints M α = ⋂ m : M, fixedBy α m :=
   Set.ext fun _ =>
     ⟨fun hx => Set.mem_iInter.2 fun m => hx m, fun hx m => (Set.mem_iInter.1 hx m : _)⟩
 #align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_iInter_fixedBy
 #align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_iInter_fixedBy
 
-variable {M}
+variable {M α}
 
 @[to_additive (attr := simp)]
 theorem mem_fixedPoints {a : α} : a ∈ fixedPoints M α ↔ ∀ m : M, m • a = a :=
@@ -134,7 +152,7 @@ theorem mem_fixedPoints {a : α} : a ∈ fixedPoints M α ↔ ∀ m : M, m • a
 #align add_action.mem_fixed_points AddAction.mem_fixedPoints
 
 @[to_additive (attr := simp)]
-theorem mem_fixedBy {m : M} {a : α} : a ∈ fixedBy M α m ↔ m • a = a :=
+theorem mem_fixedBy {m : M} {a : α} : a ∈ fixedBy α m ↔ m • a = a :=
   Iff.rfl
 #align mul_action.mem_fixed_by MulAction.mem_fixedBy
 #align add_action.mem_fixed_by AddAction.mem_fixedBy
@@ -148,36 +166,6 @@ theorem mem_fixedPoints' {a : α} : a ∈ fixedPoints M α ↔ ∀ a', a' ∈ or
 #align mul_action.mem_fixed_points' MulAction.mem_fixedPoints'
 #align add_action.mem_fixed_points' AddAction.mem_fixedPoints'
 
-variable (M) {α}
-
-/-- The stabilizer of a point `a` as a submonoid of `M`. -/
-@[to_additive "The stabilizer of m point `a` as an additive submonoid of `M`."]
-def Stabilizer.submonoid (a : α) : Submonoid M where
-  carrier := { m | m • a = a }
-  one_mem' := one_smul _ a
-  mul_mem' {m m'} (ha : m • a = a) (hb : m' • a = a) :=
-    show (m * m') • a = a by rw [← smul_smul, hb, ha]
-#align mul_action.stabilizer.submonoid MulAction.Stabilizer.submonoid
-#align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid
-
-@[to_additive (attr := simp)]
-theorem mem_stabilizer_submonoid_iff {a : α} {m : M} : m ∈ Stabilizer.submonoid M a ↔ m • a = a :=
-  Iff.rfl
-#align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizer_submonoid_iff
-#align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizer_addSubmonoid_iff
-
-@[to_additive]
-instance [DecidableEq α] (a : α) : DecidablePred (· ∈ Stabilizer.submonoid M a) :=
-  fun _ => inferInstanceAs <| Decidable (_ = _)
-
-@[to_additive]
-theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ :=
-  (surjective_smul M a).range_eq
-#align mul_action.orbit_eq_univ MulAction.orbit_eq_univ
-#align add_action.orbit_eq_univ AddAction.orbit_eq_univ
-
-variable {M}
-
 @[to_additive mem_fixedPoints_iff_card_orbit_eq_one]
 theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
     a ∈ fixedPoints M α ↔ Fintype.card (orbit M a) = 1 := by
@@ -192,34 +180,56 @@ theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
 #align mul_action.mem_fixed_points_iff_card_orbit_eq_one MulAction.mem_fixedPoints_iff_card_orbit_eq_one
 #align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_one
 
-end MulAction
+end FixedPoints
 
-namespace MulAction
+section Stabilizers
 
-variable (G : Type u) {α : Type v} [Group G] [MulAction G α]
+variable {α}
 
-/-- The stabilizer of an element under an action, i.e. what sends the element to itself.
-A subgroup. -/
-@[to_additive
-      "The stabilizer of an element under an action, i.e. what sends the element to itself.
-      An additive subgroup."]
-def stabilizer (a : α) : Subgroup G :=
-  { Stabilizer.submonoid G a with
-    inv_mem' := fun {m} (ha : m • a = a) => show m⁻¹ • a = a by rw [inv_smul_eq_iff, ha] }
-#align mul_action.stabilizer MulAction.stabilizer
-#align add_action.stabilizer AddAction.stabilizer
+/-- The stabilizer of a point `a` as a submonoid of `M`. -/
+@[to_additive "The stabilizer of a point `a` as an additive submonoid of `M`."]
+def stabilizerSubmonoid (a : α) : Submonoid M where
+  carrier := { m | m • a = a }
+  one_mem' := one_smul _ a
+  mul_mem' {m m'} (ha : m • a = a) (hb : m' • a = a) :=
+    show (m * m') • a = a by rw [← smul_smul, hb, ha]
+#align mul_action.stabilizer.submonoid MulAction.stabilizerSubmonoid
+#align add_action.stabilizer.add_submonoid AddAction.stabilizerAddSubmonoid
 
-variable {G}
+variable {M}
+
+@[to_additive]
+instance [DecidableEq α] (a : α) : DecidablePred (· ∈ stabilizerSubmonoid M a) :=
+  fun _ => inferInstanceAs <| Decidable (_ = _)
 
 @[to_additive (attr := simp)]
-theorem mem_stabilizer_iff {g : G} {a : α} : g ∈ stabilizer G a ↔ g • a = a :=
+theorem mem_stabilizerSubmonoid_iff {a : α} {m : M} : m ∈ stabilizerSubmonoid M a ↔ m • a = a :=
   Iff.rfl
-#align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
-#align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
+#align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizerSubmonoid_iff
+#align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizerAddSubmonoid_iff
 
-@[to_additive]
-instance [DecidableEq α] (a : α) : DecidablePred (· ∈ stabilizer G a) :=
-  fun _ => inferInstanceAs <| Decidable (_ = _)
+end Stabilizers
+
+end MulAction
+
+/-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
+The general theory of such `k` is elaborated by `IsSMulRegular`.
+The typeclass that restricts all terms of `M` to have this property is `NoZeroSMulDivisors`. -/
+theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
+    [DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
+    a = b := by
+  rw [← sub_eq_zero]
+  refine' h _ _
+  rw [smul_sub, h', sub_self]
+#align smul_cancel_of_non_zero_divisor smul_cancel_of_non_zero_divisor
+
+namespace MulAction
+
+variable (G : Type u) [Group G] (α : Type v) [MulAction G α]
+
+section Orbit
+
+variable {G α}
 
 @[to_additive (attr := simp)]
 theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a :=
@@ -254,8 +264,6 @@ theorem orbit_eq_iff {a b : α} : orbit G a = orbit G b ↔ a ∈ orbit G b :=
 #align mul_action.orbit_eq_iff MulAction.orbit_eq_iff
 #align add_action.orbit_eq_iff AddAction.orbit_eq_iff
 
-variable (G)
-
 @[to_additive]
 theorem mem_orbit_smul (g : G) (a : α) : a ∈ orbit G (g • a) := by
   simp only [orbit_smul, mem_orbit_self]
@@ -268,7 +276,7 @@ theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) :
 #align mul_action.smul_mem_orbit_smul MulAction.smul_mem_orbit_smul
 #align add_action.vadd_mem_orbit_vadd AddAction.vadd_mem_orbit_vadd
 
-variable (α)
+variable (G α)
 
 /-- The relation 'in the same orbit'. -/
 @[to_additive "The relation 'in the same orbit'."]
@@ -420,7 +428,34 @@ def selfEquivSigmaOrbits : α ≃ Σω : Ω, orbit G ω.out' :=
 #align mul_action.self_equiv_sigma_orbits MulAction.selfEquivSigmaOrbits
 #align add_action.self_equiv_sigma_orbits AddAction.selfEquivSigmaOrbits
 
-variable {G α}
+end Orbit
+
+section Stabilizer
+
+variable {α}
+
+/-- The stabilizer of an element under an action, i.e. what sends the element to itself.
+A subgroup. -/
+@[to_additive
+      "The stabilizer of an element under an action, i.e. what sends the element to itself.
+      An additive subgroup."]
+def stabilizer (a : α) : Subgroup G :=
+  { stabilizerSubmonoid G a with
+    inv_mem' := fun {m} (ha : m • a = a) => show m⁻¹ • a = a by rw [inv_smul_eq_iff, ha] }
+#align mul_action.stabilizer MulAction.stabilizer
+#align add_action.stabilizer AddAction.stabilizer
+
+variable {G}
+
+@[to_additive]
+instance [DecidableEq α] (a : α) : DecidablePred (· ∈ stabilizer G a) :=
+  fun _ => inferInstanceAs <| Decidable (_ = _)
+
+@[to_additive (attr := simp)]
+theorem mem_stabilizer_iff {a : α} {g : G} : g ∈ stabilizer G a ↔ g • a = a :=
+  Iff.rfl
+#align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
+#align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
 
 /-- If the stabilizer of `a` is `S`, then the stabilizer of `g • a` is `gSg⁻¹`. -/
 theorem stabilizer_smul_eq_stabilizer_map_conj (g : G) (a : α) :
@@ -440,11 +475,13 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {a b : α} (h : (orbitRel
   (MulEquiv.subgroupCongr this).trans ((MulAut.conj g).subgroupMap <| stabilizer G b).symm
 #align mul_action.stabilizer_equiv_stabilizer_of_orbit_rel MulAction.stabilizerEquivStabilizerOfOrbitRel
 
+end Stabilizer
+
 end MulAction
 
 namespace AddAction
 
-variable (G : Type u) (α : Type v) [AddGroup G] [AddAction G α]
+variable (G : Type u) [AddGroup G] (α : Type v) [AddAction G α]
 
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
 theorem stabilizer_vadd_eq_stabilizer_map_conj (g : G) (a : α) :
@@ -466,14 +503,3 @@ noncomputable def stabilizerEquivStabilizerOfOrbitRel {a b : α} (h : (orbitRel
 #align add_action.stabilizer_equiv_stabilizer_of_orbit_rel AddAction.stabilizerEquivStabilizerOfOrbitRel
 
 end AddAction
-
-/-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
-The general theory of such `k` is elaborated by `IsSMulRegular`.
-The typeclass that restricts all terms of `M` to have this property is `NoZeroSMulDivisors`. -/
-theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
-    [DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
-    a = b := by
-  rw [← sub_eq_zero]
-  refine' h _ _
-  rw [smul_sub, h', sub_self]
-#align smul_cancel_of_non_zero_divisor smul_cancel_of_non_zero_divisor

@@ -166,6 +166,10 @@ theorem mem_stabilizer_submonoid_iff {a : α} {m : M} : m ∈ Stabilizer.submono
 #align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizer_submonoid_iff
 #align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizer_addSubmonoid_iff
 
+@[to_additive]
+instance [DecidableEq α] (a : α) : DecidablePred (· ∈ Stabilizer.submonoid M a) :=
+  fun _ => inferInstanceAs <| Decidable (_ = _)
+
 @[to_additive]
 theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ :=
   (surjective_smul M a).range_eq
@@ -213,6 +217,10 @@ theorem mem_stabilizer_iff {g : G} {a : α} : g ∈ stabilizer G a ↔ g • a =
 #align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
 #align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
 
+@[to_additive]
+instance [DecidableEq α] (a : α) : DecidablePred (· ∈ stabilizer G a) :=
+  fun _ => inferInstanceAs <| Decidable (_ = _)
+
 @[to_additive (attr := simp)]
 theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a :=
   (smul_orbit_subset g a).antisymm <|

And fix some names in comments where this revealed issues

@@ -346,7 +346,7 @@ def orbitRel.Quotient : Type _ :=
 variable {G α}
 
 /-- The orbit corresponding to an element of the quotient by `MulAction.orbitRel` -/
-@[to_additive "The orbit corresponding to an element of the quotient by `add_action.orbit_rel`"]
+@[to_additive "The orbit corresponding to an element of the quotient by `AddAction.orbitRel`"]
 nonrec def orbitRel.Quotient.orbit (x : orbitRel.Quotient G α) : Set α :=
   Quotient.liftOn' x (orbit G) fun _ _ => MulAction.orbit_eq_iff.2
 #align mul_action.orbit_rel.quotient.orbit MulAction.orbitRel.Quotient.orbit

correct AddAction.mem_fixedPoints_iff_card_orbit_eq_zero to AddAction.mem_fixedPoints_iff_card_orbit_eq_one, as it was wrongly chosen from MulAction.mem_fixedPoints_iff_card_orbit_eq_one by to_additive tag

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

@@ -174,7 +174,7 @@ theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ :=
 
 variable {M}
 
-@[to_additive]
+@[to_additive mem_fixedPoints_iff_card_orbit_eq_one]
 theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
     a ∈ fixedPoints M α ↔ Fintype.card (orbit M a) = 1 := by
   rw [Fintype.card_eq_one_iff, mem_fixedPoints]
@@ -186,7 +186,7 @@ theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
       x • a = z := Subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
       _ = a := (Subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm
 #align mul_action.mem_fixed_points_iff_card_orbit_eq_one MulAction.mem_fixedPoints_iff_card_orbit_eq_one
-#align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_zero
+#align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_one
 
 end MulAction
 

Renaming variable names in basic definitions, theorems, and proofs regarding MulAction and AddAction, so that the choice of letter is

1. consistent with sibling files (e.g. GroupTheory/GroupAction/Defs.lean) and
2. easy to parse.

Specifically,

• M (for "monoid") or G (for "group") is the type that acts;
• α is the type that is acted on;
• terms of M are m etc and terms of G are g etc;
• terms of α are a etc.

Miswording in the docstrings for MulAction.fixedBy and AddAction.fixedby is also fixed.

@@ -29,9 +29,7 @@ of `•` belong elsewhere.
 -/
 
 
-universe u v w
-
-variable {α : Type u} {β : Type v} {γ : Type w}
+universe u v
 
 open Pointwise
 
@@ -39,149 +37,149 @@ open Function
 
 namespace MulAction
 
-variable (α) [Monoid α] [MulAction α β]
+variable (M : Type u) {α : Type v} [Monoid M] [MulAction M α]
 
 /-- The orbit of an element under an action. -/
 @[to_additive "The orbit of an element under an action."]
-def orbit (b : β) :=
-  Set.range fun x : α => x • b
+def orbit (a : α) :=
+  Set.range fun m : M => m • a
 #align mul_action.orbit MulAction.orbit
 #align add_action.orbit AddAction.orbit
 
-variable {α}
+variable {M}
 
 @[to_additive]
-theorem mem_orbit_iff {b₁ b₂ : β} : b₂ ∈ orbit α b₁ ↔ ∃ x : α, x • b₁ = b₂ :=
+theorem mem_orbit_iff {a₁ a₂ : α} : a₂ ∈ orbit M a₁ ↔ ∃ x : M, x • a₁ = a₂ :=
   Iff.rfl
 #align mul_action.mem_orbit_iff MulAction.mem_orbit_iff
 #align add_action.mem_orbit_iff AddAction.mem_orbit_iff
 
 @[to_additive (attr := simp)]
-theorem mem_orbit (b : β) (x : α) : x • b ∈ orbit α b :=
+theorem mem_orbit (a : α) (x : M) : x • a ∈ orbit M a :=
   ⟨x, rfl⟩
 #align mul_action.mem_orbit MulAction.mem_orbit
 #align add_action.mem_orbit AddAction.mem_orbit
 
 @[to_additive (attr := simp)]
-theorem mem_orbit_self (b : β) : b ∈ orbit α b :=
+theorem mem_orbit_self (a : α) : a ∈ orbit M a :=
   ⟨1, by simp [MulAction.one_smul]⟩
 #align mul_action.mem_orbit_self MulAction.mem_orbit_self
 #align add_action.mem_orbit_self AddAction.mem_orbit_self
 
 @[to_additive]
-theorem orbit_nonempty (b : β) : Set.Nonempty (orbit α b) :=
+theorem orbit_nonempty (a : α) : Set.Nonempty (orbit M a) :=
   Set.range_nonempty _
 #align mul_action.orbit_nonempty MulAction.orbit_nonempty
 #align add_action.orbit_nonempty AddAction.orbit_nonempty
 
 @[to_additive]
-theorem mapsTo_smul_orbit (a : α) (b : β) : Set.MapsTo ((· • ·) a) (orbit α b) (orbit α b) :=
-  Set.range_subset_iff.2 fun a' => ⟨a * a', mul_smul _ _ _⟩
+theorem mapsTo_smul_orbit (m : M) (a : α) : Set.MapsTo ((· • ·) m) (orbit M a) (orbit M a) :=
+  Set.range_subset_iff.2 fun m' => ⟨m * m', mul_smul _ _ _⟩
 #align mul_action.maps_to_smul_orbit MulAction.mapsTo_smul_orbit
 #align add_action.maps_to_vadd_orbit AddAction.mapsTo_vadd_orbit
 
 @[to_additive]
-theorem smul_orbit_subset (a : α) (b : β) : a • orbit α b ⊆ orbit α b :=
-  (mapsTo_smul_orbit a b).image_subset
+theorem smul_orbit_subset (m : M) (a : α) : m • orbit M a ⊆ orbit M a :=
+  (mapsTo_smul_orbit m a).image_subset
 #align mul_action.smul_orbit_subset MulAction.smul_orbit_subset
 #align add_action.vadd_orbit_subset AddAction.vadd_orbit_subset
 
 @[to_additive]
-theorem orbit_smul_subset (a : α) (b : β) : orbit α (a • b) ⊆ orbit α b :=
-  Set.range_subset_iff.2 fun a' => mul_smul a' a b ▸ mem_orbit _ _
+theorem orbit_smul_subset (m : M) (a : α) : orbit M (m • a) ⊆ orbit M a :=
+  Set.range_subset_iff.2 fun m' => mul_smul m' m a ▸ mem_orbit _ _
 #align mul_action.orbit_smul_subset MulAction.orbit_smul_subset
 #align add_action.orbit_vadd_subset AddAction.orbit_vadd_subset
 
 @[to_additive]
-instance {b : β} : MulAction α (orbit α b) where
-  smul a := (mapsTo_smul_orbit a b).restrict _ _ _
-  one_smul a := Subtype.ext (one_smul α (a : β))
-  mul_smul a a' b' := Subtype.ext (mul_smul a a' (b' : β))
+instance {a : α} : MulAction M (orbit M a) where
+  smul m := (mapsTo_smul_orbit m a).restrict _ _ _
+  one_smul m := Subtype.ext (one_smul M (m : α))
+  mul_smul m m' a' := Subtype.ext (mul_smul m m' (a' : α))
 
 @[to_additive (attr := simp)]
-theorem orbit.coe_smul {b : β} {a : α} {b' : orbit α b} : ↑(a • b') = a • (b' : β) :=
+theorem orbit.coe_smul {a : α} {m : M} {a' : orbit M a} : ↑(m • a') = m • (a' : α) :=
   rfl
 #align mul_action.orbit.coe_smul MulAction.orbit.coe_smul
 #align add_action.orbit.coe_vadd AddAction.orbit.coe_vadd
 
-variable (α) (β)
+variable (M) (α)
 
 /-- The set of elements fixed under the whole action. -/
 @[to_additive "The set of elements fixed under the whole action."]
-def fixedPoints : Set β :=
-  { b : β | ∀ x : α, x • b = b }
+def fixedPoints : Set α :=
+  { a : α | ∀ m : M, m • a = a }
 #align mul_action.fixed_points MulAction.fixedPoints
 #align add_action.fixed_points AddAction.fixedPoints
 
-/-- `fixedBy g` is the subfield of elements fixed by `g`. -/
-@[to_additive "`fixedBy g` is the subfield of elements fixed by `g`."]
-def fixedBy (g : α) : Set β :=
-  { x | g • x = x }
+/-- `fixedBy m` is the set of elements fixed by `m`. -/
+@[to_additive "`fixedBy m` is the set of elements fixed by `m`."]
+def fixedBy (m : M) : Set α :=
+  { x | m • x = x }
 #align mul_action.fixed_by MulAction.fixedBy
 #align add_action.fixed_by AddAction.fixedBy
 
 @[to_additive]
-theorem fixed_eq_iInter_fixedBy : fixedPoints α β = ⋂ g : α, fixedBy α β g :=
+theorem fixed_eq_iInter_fixedBy : fixedPoints M α = ⋂ m : M, fixedBy M α m :=
   Set.ext fun _ =>
-    ⟨fun hx => Set.mem_iInter.2 fun g => hx g, fun hx g => (Set.mem_iInter.1 hx g : _)⟩
+    ⟨fun hx => Set.mem_iInter.2 fun m => hx m, fun hx m => (Set.mem_iInter.1 hx m : _)⟩
 #align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_iInter_fixedBy
 #align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_iInter_fixedBy
 
-variable {α}
+variable {M}
 
 @[to_additive (attr := simp)]
-theorem mem_fixedPoints {b : β} : b ∈ fixedPoints α β ↔ ∀ x : α, x • b = b :=
+theorem mem_fixedPoints {a : α} : a ∈ fixedPoints M α ↔ ∀ m : M, m • a = a :=
   Iff.rfl
 #align mul_action.mem_fixed_points MulAction.mem_fixedPoints
 #align add_action.mem_fixed_points AddAction.mem_fixedPoints
 
 @[to_additive (attr := simp)]
-theorem mem_fixedBy {g : α} {b : β} : b ∈ fixedBy α β g ↔ g • b = b :=
+theorem mem_fixedBy {m : M} {a : α} : a ∈ fixedBy M α m ↔ m • a = a :=
   Iff.rfl
 #align mul_action.mem_fixed_by MulAction.mem_fixedBy
 #align add_action.mem_fixed_by AddAction.mem_fixedBy
 
 @[to_additive]
-theorem mem_fixedPoints' {b : β} : b ∈ fixedPoints α β ↔ ∀ b', b' ∈ orbit α b → b' = b :=
+theorem mem_fixedPoints' {a : α} : a ∈ fixedPoints M α ↔ ∀ a', a' ∈ orbit M a → a' = a :=
   ⟨fun h _ h₁ =>
-    let ⟨x, hx⟩ := mem_orbit_iff.1 h₁
-    hx ▸ h x,
+    let ⟨m, hm⟩ := mem_orbit_iff.1 h₁
+    hm ▸ h m,
     fun h _ => h _ (mem_orbit _ _)⟩
 #align mul_action.mem_fixed_points' MulAction.mem_fixedPoints'
 #align add_action.mem_fixed_points' AddAction.mem_fixedPoints'
 
-variable (α) {β}
+variable (M) {α}
 
-/-- The stabilizer of a point `b` as a submonoid of `α`. -/
-@[to_additive "The stabilizer of a point `b` as an additive submonoid of `α`."]
-def Stabilizer.submonoid (b : β) : Submonoid α where
-  carrier := { a | a • b = b }
-  one_mem' := one_smul _ b
-  mul_mem' {a a'} (ha : a • b = b) (hb : a' • b = b) :=
-    show (a * a') • b = b by rw [← smul_smul, hb, ha]
+/-- The stabilizer of a point `a` as a submonoid of `M`. -/
+@[to_additive "The stabilizer of m point `a` as an additive submonoid of `M`."]
+def Stabilizer.submonoid (a : α) : Submonoid M where
+  carrier := { m | m • a = a }
+  one_mem' := one_smul _ a
+  mul_mem' {m m'} (ha : m • a = a) (hb : m' • a = a) :=
+    show (m * m') • a = a by rw [← smul_smul, hb, ha]
 #align mul_action.stabilizer.submonoid MulAction.Stabilizer.submonoid
 #align add_action.stabilizer.add_submonoid AddAction.Stabilizer.addSubmonoid
 
 @[to_additive (attr := simp)]
-theorem mem_stabilizer_submonoid_iff {b : β} {a : α} : a ∈ Stabilizer.submonoid α b ↔ a • b = b :=
+theorem mem_stabilizer_submonoid_iff {a : α} {m : M} : m ∈ Stabilizer.submonoid M a ↔ m • a = a :=
   Iff.rfl
 #align mul_action.mem_stabilizer_submonoid_iff MulAction.mem_stabilizer_submonoid_iff
 #align add_action.mem_stabilizer_add_submonoid_iff AddAction.mem_stabilizer_addSubmonoid_iff
 
 @[to_additive]
-theorem orbit_eq_univ [IsPretransitive α β] (x : β) : orbit α x = Set.univ :=
-  (surjective_smul α x).range_eq
+theorem orbit_eq_univ [IsPretransitive M α] (a : α) : orbit M a = Set.univ :=
+  (surjective_smul M a).range_eq
 #align mul_action.orbit_eq_univ MulAction.orbit_eq_univ
 #align add_action.orbit_eq_univ AddAction.orbit_eq_univ
 
-variable {α}
+variable {M}
 
 @[to_additive]
-theorem mem_fixedPoints_iff_card_orbit_eq_one {a : β} [Fintype (orbit α a)] :
-    a ∈ fixedPoints α β ↔ Fintype.card (orbit α a) = 1 := by
+theorem mem_fixedPoints_iff_card_orbit_eq_one {a : α} [Fintype (orbit M a)] :
+    a ∈ fixedPoints M α ↔ Fintype.card (orbit M a) = 1 := by
   rw [Fintype.card_eq_one_iff, mem_fixedPoints]
   constructor
-  · exact fun h => ⟨⟨a, mem_orbit_self _⟩, fun ⟨b, ⟨x, hx⟩⟩ => Subtype.eq <| by simp [h x, hx.symm]⟩
+  · exact fun h => ⟨⟨a, mem_orbit_self _⟩, fun ⟨a, ⟨x, hx⟩⟩ => Subtype.eq <| by simp [h x, hx.symm]⟩
   · intro h x
     rcases h with ⟨⟨z, hz⟩, hz₁⟩
     calc
@@ -194,117 +192,115 @@ end MulAction
 
 namespace MulAction
 
-variable (α)
-
-variable [Group α] [MulAction α β]
+variable (G : Type u) {α : Type v} [Group G] [MulAction G α]
 
 /-- The stabilizer of an element under an action, i.e. what sends the element to itself.
 A subgroup. -/
 @[to_additive
       "The stabilizer of an element under an action, i.e. what sends the element to itself.
       An additive subgroup."]
-def stabilizer (b : β) : Subgroup α :=
-  { Stabilizer.submonoid α b with
-    inv_mem' := fun {a} (ha : a • b = b) => show a⁻¹ • b = b by rw [inv_smul_eq_iff, ha] }
+def stabilizer (a : α) : Subgroup G :=
+  { Stabilizer.submonoid G a with
+    inv_mem' := fun {m} (ha : m • a = a) => show m⁻¹ • a = a by rw [inv_smul_eq_iff, ha] }
 #align mul_action.stabilizer MulAction.stabilizer
 #align add_action.stabilizer AddAction.stabilizer
 
-variable {α}
+variable {G}
 
 @[to_additive (attr := simp)]
-theorem mem_stabilizer_iff {b : β} {a : α} : a ∈ stabilizer α b ↔ a • b = b :=
+theorem mem_stabilizer_iff {g : G} {a : α} : g ∈ stabilizer G a ↔ g • a = a :=
   Iff.rfl
 #align mul_action.mem_stabilizer_iff MulAction.mem_stabilizer_iff
 #align add_action.mem_stabilizer_iff AddAction.mem_stabilizer_iff
 
 @[to_additive (attr := simp)]
-theorem smul_orbit (a : α) (b : β) : a • orbit α b = orbit α b :=
-  (smul_orbit_subset a b).antisymm <|
+theorem smul_orbit (g : G) (a : α) : g • orbit G a = orbit G a :=
+  (smul_orbit_subset g a).antisymm <|
     calc
-      orbit α b = a • a⁻¹ • orbit α b := (smul_inv_smul _ _).symm
-      _ ⊆ a • orbit α b := Set.image_subset _ (smul_orbit_subset _ _)
+      orbit G a = g • g⁻¹ • orbit G a := (smul_inv_smul _ _).symm
+      _ ⊆ g • orbit G a := Set.image_subset _ (smul_orbit_subset _ _)
 #align mul_action.smul_orbit MulAction.smul_orbit
 #align add_action.vadd_orbit AddAction.vadd_orbit
 
 @[to_additive (attr := simp)]
-theorem orbit_smul (a : α) (b : β) : orbit α (a • b) = orbit α b :=
-  (orbit_smul_subset a b).antisymm <|
+theorem orbit_smul (g : G) (a : α) : orbit G (g • a) = orbit G a :=
+  (orbit_smul_subset g a).antisymm <|
     calc
-      orbit α b = orbit α (a⁻¹ • a • b) := by rw [inv_smul_smul]
-      _ ⊆ orbit α (a • b) := orbit_smul_subset _ _
+      orbit G a = orbit G (g⁻¹ • g • a) := by rw [inv_smul_smul]
+      _ ⊆ orbit G (g • a) := orbit_smul_subset _ _
 #align mul_action.orbit_smul MulAction.orbit_smul
 #align add_action.orbit_vadd AddAction.orbit_vadd
 
 /-- The action of a group on an orbit is transitive. -/
 @[to_additive "The action of an additive group on an orbit is transitive."]
-instance (x : β) : IsPretransitive α (orbit α x) :=
+instance (a : α) : IsPretransitive G (orbit G a) :=
   ⟨by
-    rintro ⟨_, a, rfl⟩ ⟨_, b, rfl⟩
-    use b * a⁻¹
+    rintro ⟨_, g, rfl⟩ ⟨_, h, rfl⟩
+    use h * g⁻¹
     ext1
     simp [mul_smul]⟩
 
 @[to_additive]
-theorem orbit_eq_iff {a b : β} : orbit α a = orbit α b ↔ a ∈ orbit α b :=
+theorem orbit_eq_iff {a b : α} : orbit G a = orbit G b ↔ a ∈ orbit G b :=
   ⟨fun h => h ▸ mem_orbit_self _, fun ⟨_, hc⟩ => hc ▸ orbit_smul _ _⟩
 #align mul_action.orbit_eq_iff MulAction.orbit_eq_iff
 #align add_action.orbit_eq_iff AddAction.orbit_eq_iff
 
-variable (α)
+variable (G)
 
 @[to_additive]
-theorem mem_orbit_smul (g : α) (a : β) : a ∈ orbit α (g • a) := by
+theorem mem_orbit_smul (g : G) (a : α) : a ∈ orbit G (g • a) := by
   simp only [orbit_smul, mem_orbit_self]
 #align mul_action.mem_orbit_smul MulAction.mem_orbit_smul
 #align add_action.mem_orbit_vadd AddAction.mem_orbit_vadd
 
 @[to_additive]
-theorem smul_mem_orbit_smul (g h : α) (a : β) : g • a ∈ orbit α (h • a) := by
+theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) := by
   simp only [orbit_smul, mem_orbit]
 #align mul_action.smul_mem_orbit_smul MulAction.smul_mem_orbit_smul
 #align add_action.vadd_mem_orbit_vadd AddAction.vadd_mem_orbit_vadd
 
-variable (β)
+variable (α)
 
 /-- The relation 'in the same orbit'. -/
 @[to_additive "The relation 'in the same orbit'."]
-def orbitRel : Setoid β where
-  r a b := a ∈ orbit α b
+def orbitRel : Setoid α where
+  r a b := a ∈ orbit G b
   iseqv :=
     ⟨mem_orbit_self, fun {a b} => by simp [orbit_eq_iff.symm, eq_comm], fun {a b} => by
       simp (config := { contextual := true }) [orbit_eq_iff.symm, eq_comm]⟩
 #align mul_action.orbit_rel MulAction.orbitRel
 #align add_action.orbit_rel AddAction.orbitRel
 
-variable {α} {β}
+variable {G α}
 
 @[to_additive]
-theorem orbitRel_apply {x y : β} : (orbitRel α β).Rel x y ↔ x ∈ orbit α y :=
+theorem orbitRel_apply {a b : α} : (orbitRel G α).Rel a b ↔ a ∈ orbit G b :=
   Iff.rfl
 #align mul_action.orbit_rel_apply MulAction.orbitRel_apply
 #align add_action.orbit_rel_apply AddAction.orbitRel_apply
 
-/-- When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the union
-of the orbit of `U` under `α`. -/
+/-- When you take a set `U` in `α`, push it down to the quotient, and pull back, you get the union
+of the orbit of `U` under `G`. -/
 @[to_additive
-      "When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the
-      union of the orbit of `U` under `α`."]
-theorem quotient_preimage_image_eq_union_mul (U : Set β) :
-    letI := orbitRel α β
-    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ a : α, (· • ·) a '' U := by
-  letI := orbitRel α β
-  set f : β → Quotient (MulAction.orbitRel α β) := Quotient.mk'
-  ext x
+      "When you take a set `U` in `α`, push it down to the quotient, and pull back, you get the
+      union of the orbit of `U` under `G`."]
+theorem quotient_preimage_image_eq_union_mul (U : Set α) :
+    letI := orbitRel G α
+    Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (· • ·) g '' U := by
+  letI := orbitRel G α
+  set f : α → Quotient (MulAction.orbitRel G α) := Quotient.mk'
+  ext a
   constructor
-  · rintro ⟨y, hy, hxy⟩
-    obtain ⟨a, rfl⟩ := Quotient.exact hxy
+  · rintro ⟨b, hb, hab⟩
+    obtain ⟨g, rfl⟩ := Quotient.exact hab
     rw [Set.mem_iUnion]
-    exact ⟨a⁻¹, a • x, hy, inv_smul_smul a x⟩
+    exact ⟨g⁻¹, g • a, hb, inv_smul_smul g a⟩
   · intro hx
     rw [Set.mem_iUnion] at hx
-    obtain ⟨a, u, hu₁, hu₂⟩ := hx
+    obtain ⟨g, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
-    refine' ⟨a⁻¹ • x, _, by simp only [Quotient.eq']; use a⁻¹⟩
+    refine' ⟨g⁻¹ • a, _, by simp only [Quotient.eq']; use g⁻¹⟩
     rw [← hu₂]
     convert hu₁
     simp only [inv_smul_smul]
@@ -312,60 +308,60 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
 #align add_action.quotient_preimage_image_eq_union_add AddAction.quotient_preimage_image_eq_union_add
 
 @[to_additive]
-theorem disjoint_image_image_iff {U V : Set β} :
-    letI := orbitRel α β
-    Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V := by
-  letI := orbitRel α β
-  set f : β → Quotient (MulAction.orbitRel α β) := Quotient.mk'
+theorem disjoint_image_image_iff {U V : Set α} :
+    letI := orbitRel G α
+    Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V := by
+  letI := orbitRel G α
+  set f : α → Quotient (MulAction.orbitRel G α) := Quotient.mk'
   refine'
-    ⟨fun h x x_in_U a a_in_V =>
-      h.le_bot ⟨⟨x, x_in_U, Quotient.sound ⟨a⁻¹, _⟩⟩, ⟨a • x, a_in_V, rfl⟩⟩, _⟩
+    ⟨fun h a a_in_U g g_in_V =>
+      h.le_bot ⟨⟨a, a_in_U, Quotient.sound ⟨g⁻¹, _⟩⟩, ⟨g • a, g_in_V, rfl⟩⟩, _⟩
   · simp
   · intro h
     rw [Set.disjoint_left]
-    rintro x ⟨y, hy₁, hy₂⟩ ⟨z, hz₁, hz₂⟩
-    obtain ⟨a, rfl⟩ := Quotient.exact (hz₂.trans hy₂.symm)
-    exact h y hy₁ a hz₁
+    rintro _ ⟨b, hb₁, hb₂⟩ ⟨c, hc₁, hc₂⟩
+    obtain ⟨g, rfl⟩ := Quotient.exact (hc₂.trans hb₂.symm)
+    exact h b hb₁ g hc₁
 #align mul_action.disjoint_image_image_iff MulAction.disjoint_image_image_iff
 #align add_action.disjoint_image_image_iff AddAction.disjoint_image_image_iff
 
 @[to_additive]
-theorem image_inter_image_iff (U V : Set β) :
-    letI := orbitRel α β
-    Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
+theorem image_inter_image_iff (U V : Set α) :
+    letI := orbitRel G α
+    Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ g : G, g • x ∉ V :=
   Set.disjoint_iff_inter_eq_empty.symm.trans disjoint_image_image_iff
 #align mul_action.image_inter_image_iff MulAction.image_inter_image_iff
 #align add_action.image_inter_image_iff AddAction.image_inter_image_iff
 
-variable (α β)
+variable (G α)
 
 /-- The quotient by `MulAction.orbitRel`, given a name to enable dot notation. -/
 @[to_additive (attr := reducible)
     "The quotient by `AddAction.orbitRel`, given a name to enable dot notation."]
 def orbitRel.Quotient : Type _ :=
-  _root_.Quotient <| orbitRel α β
+  _root_.Quotient <| orbitRel G α
 #align mul_action.orbit_rel.quotient MulAction.orbitRel.Quotient
 #align add_action.orbit_rel.quotient AddAction.orbitRel.Quotient
 
-variable {α β}
+variable {G α}
 
 /-- The orbit corresponding to an element of the quotient by `MulAction.orbitRel` -/
 @[to_additive "The orbit corresponding to an element of the quotient by `add_action.orbit_rel`"]
-nonrec def orbitRel.Quotient.orbit (x : orbitRel.Quotient α β) : Set β :=
-  Quotient.liftOn' x (orbit α) fun _ _ => MulAction.orbit_eq_iff.2
+nonrec def orbitRel.Quotient.orbit (x : orbitRel.Quotient G α) : Set α :=
+  Quotient.liftOn' x (orbit G) fun _ _ => MulAction.orbit_eq_iff.2
 #align mul_action.orbit_rel.quotient.orbit MulAction.orbitRel.Quotient.orbit
 #align add_action.orbit_rel.quotient.orbit AddAction.orbitRel.Quotient.orbit
 
 @[to_additive (attr := simp)]
-theorem orbitRel.Quotient.orbit_mk (b : β) :
-    orbitRel.Quotient.orbit (Quotient.mk'' b : orbitRel.Quotient α β) = MulAction.orbit α b :=
+theorem orbitRel.Quotient.orbit_mk (a : α) :
+    orbitRel.Quotient.orbit (Quotient.mk'' a : orbitRel.Quotient G α) = MulAction.orbit G a :=
   rfl
 #align mul_action.orbit_rel.quotient.orbit_mk MulAction.orbitRel.Quotient.orbit_mk
 #align add_action.orbit_rel.quotient.orbit_mk AddAction.orbitRel.Quotient.orbit_mk
 
 @[to_additive]
-theorem orbitRel.Quotient.mem_orbit {b : β} {x : orbitRel.Quotient α β} :
-    b ∈ x.orbit ↔ Quotient.mk'' b = x := by
+theorem orbitRel.Quotient.mem_orbit {a : α} {x : orbitRel.Quotient G α} :
+    a ∈ x.orbit ↔ Quotient.mk'' a = x := by
   induction x using Quotient.inductionOn'
   rw [Quotient.eq'']
   rfl
@@ -373,18 +369,18 @@ theorem orbitRel.Quotient.mem_orbit {b : β} {x : orbitRel.Quotient α β} :
 #align add_action.orbit_rel.quotient.mem_orbit AddAction.orbitRel.Quotient.mem_orbit
 
 /-- Note that `hφ = Quotient.out_eq'` is a useful choice here. -/
-@[to_additive "Note that `hφ = quotient.out_eq'` is a useful choice here."]
-theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient α β)
-    {φ : orbitRel.Quotient α β → β} (hφ : letI := orbitRel α β; RightInverse φ Quotient.mk') :
-    orbitRel.Quotient.orbit x = MulAction.orbit α (φ x) := by
+@[to_additive "Note that `hφ = quotient.out_eq'` is m useful choice here."]
+theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient G α)
+    {φ : orbitRel.Quotient G α → α} (hφ : letI := orbitRel G α; RightInverse φ Quotient.mk') :
+    orbitRel.Quotient.orbit x = MulAction.orbit G (φ x) := by
   conv_lhs => rw [← hφ x]
 #align mul_action.orbit_rel.quotient.orbit_eq_orbit_out MulAction.orbitRel.Quotient.orbit_eq_orbit_out
 #align add_action.orbit_rel.quotient.orbit_eq_orbit_out AddAction.orbitRel.Quotient.orbit_eq_orbit_out
 
-variable (α) (β)
+variable (G) (α)
 
 -- mathport name: exprΩ
-local notation "Ω" => orbitRel.Quotient α β
+local notation "Ω" => orbitRel.Quotient G α
 
 /-- Decomposition of a type `X` as a disjoint union of its orbits under a group action.
 
@@ -395,10 +391,10 @@ This version is expressed in terms of `MulAction.orbitRel.Quotient.orbit` instea
 
       This version is expressed in terms of `AddAction.orbitRel.Quotient.orbit` instead of
       `AddAction.orbit`, to avoid mentioning `Quotient.out'`. "]
-def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
-  letI := orbitRel α β
+def selfEquivSigmaOrbits' : α ≃ Σω : Ω, ω.orbit :=
+  letI := orbitRel G α
   calc
-    β ≃ Σω : Ω, { b // Quotient.mk' b = ω } := (Equiv.sigmaFiberEquiv Quotient.mk').symm
+    α ≃ Σω : Ω, { a // Quotient.mk' a = ω } := (Equiv.sigmaFiberEquiv Quotient.mk').symm
     _ ≃ Σω : Ω, ω.orbit :=
       Equiv.sigmaCongrRight fun _ =>
         Equiv.subtypeEquivRight fun _ => orbitRel.Quotient.mem_orbit.symm
@@ -409,42 +405,42 @@ def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
 @[to_additive
       "Decomposition of a type `X` as a disjoint union of its orbits under an additive group
       action."]
-def selfEquivSigmaOrbits : β ≃ Σω : Ω, orbit α ω.out' :=
-  (selfEquivSigmaOrbits' α β).trans <|
+def selfEquivSigmaOrbits : α ≃ Σω : Ω, orbit G ω.out' :=
+  (selfEquivSigmaOrbits' G α).trans <|
     Equiv.sigmaCongrRight fun _ =>
       Equiv.Set.ofEq <| orbitRel.Quotient.orbit_eq_orbit_out _ Quotient.out_eq'
 #align mul_action.self_equiv_sigma_orbits MulAction.selfEquivSigmaOrbits
 #align add_action.self_equiv_sigma_orbits AddAction.selfEquivSigmaOrbits
 
-variable {α β}
+variable {G α}
 
-/-- If the stabilizer of `x` is `S`, then the stabilizer of `g • x` is `gSg⁻¹`. -/
-theorem stabilizer_smul_eq_stabilizer_map_conj (g : α) (x : β) :
-    stabilizer α (g • x) = (stabilizer α x).map (MulAut.conj g).toMonoidHom := by
+/-- If the stabilizer of `a` is `S`, then the stabilizer of `g • a` is `gSg⁻¹`. -/
+theorem stabilizer_smul_eq_stabilizer_map_conj (g : G) (a : α) :
+    stabilizer G (g • a) = (stabilizer G a).map (MulAut.conj g).toMonoidHom := by
   ext h
   rw [mem_stabilizer_iff, ← smul_left_cancel_iff g⁻¹, smul_smul, smul_smul, smul_smul, mul_left_inv,
     one_smul, ← mem_stabilizer_iff, Subgroup.mem_map_equiv, MulAut.conj_symm_apply]
 #align mul_action.stabilizer_smul_eq_stabilizer_map_conj MulAction.stabilizer_smul_eq_stabilizer_map_conj
 
 /-- A bijection between the stabilizers of two elements in the same orbit. -/
-noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
-    stabilizer α x ≃* stabilizer α y :=
-  let g : α := Classical.choose h
-  have hg : g • y = x := Classical.choose_spec h
-  have this : stabilizer α x = (stabilizer α y).map (MulAut.conj g).toMonoidHom := by
+noncomputable def stabilizerEquivStabilizerOfOrbitRel {a b : α} (h : (orbitRel G α).Rel a b) :
+    stabilizer G a ≃* stabilizer G b :=
+  let g : G := Classical.choose h
+  have hg : g • b = a := Classical.choose_spec h
+  have this : stabilizer G a = (stabilizer G b).map (MulAut.conj g).toMonoidHom := by
     rw [← hg, stabilizer_smul_eq_stabilizer_map_conj]
-  (MulEquiv.subgroupCongr this).trans ((MulAut.conj g).subgroupMap <| stabilizer α y).symm
+  (MulEquiv.subgroupCongr this).trans ((MulAut.conj g).subgroupMap <| stabilizer G b).symm
 #align mul_action.stabilizer_equiv_stabilizer_of_orbit_rel MulAction.stabilizerEquivStabilizerOfOrbitRel
 
 end MulAction
 
 namespace AddAction
 
-variable [AddGroup α] [AddAction α β]
+variable (G : Type u) (α : Type v) [AddGroup G] [AddAction G α]
 
 /-- If the stabilizer of `x` is `S`, then the stabilizer of `g +ᵥ x` is `g + S + (-g)`. -/
-theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
-    stabilizer α (g +ᵥ x) = (stabilizer α x).map (AddAut.conj g).toAddMonoidHom := by
+theorem stabilizer_vadd_eq_stabilizer_map_conj (g : G) (a : α) :
+    stabilizer G (g +ᵥ a) = (stabilizer G a).map (AddAut.conj g).toAddMonoidHom := by
   ext h
   rw [mem_stabilizer_iff, ← vadd_left_cancel_iff (-g), vadd_vadd, vadd_vadd, vadd_vadd,
     add_left_neg, zero_vadd, ← mem_stabilizer_iff, AddSubgroup.mem_map_equiv,
@@ -452,13 +448,13 @@ theorem stabilizer_vadd_eq_stabilizer_map_conj (g : α) (x : β) :
 #align add_action.stabilizer_vadd_eq_stabilizer_map_conj AddAction.stabilizer_vadd_eq_stabilizer_map_conj
 
 /-- A bijection between the stabilizers of two elements in the same orbit. -/
-noncomputable def stabilizerEquivStabilizerOfOrbitRel {x y : β} (h : (orbitRel α β).Rel x y) :
-    stabilizer α x ≃+ stabilizer α y :=
-  let g : α := Classical.choose h
-  have hg : g +ᵥ y = x := Classical.choose_spec h
-  have this : stabilizer α x = (stabilizer α y).map (AddAut.conj g).toAddMonoidHom := by
+noncomputable def stabilizerEquivStabilizerOfOrbitRel {a b : α} (h : (orbitRel G α).Rel a b) :
+    stabilizer G a ≃+ stabilizer G b :=
+  let g : G := Classical.choose h
+  have hg : g +ᵥ b = a := Classical.choose_spec h
+  have this : stabilizer G a = (stabilizer G b).map (AddAut.conj g).toAddMonoidHom := by
     rw [← hg, stabilizer_vadd_eq_stabilizer_map_conj]
-  (AddEquiv.addSubgroupCongr this).trans ((AddAut.conj g).addSubgroupMap <| stabilizer α y).symm
+  (AddEquiv.addSubgroupCongr this).trans ((AddAut.conj g).addSubgroupMap <| stabilizer G b).symm
 #align add_action.stabilizer_equiv_stabilizer_of_orbit_rel AddAction.stabilizerEquivStabilizerOfOrbitRel
 
 end AddAction

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -466,7 +466,7 @@ end AddAction
 /-- `smul` by a `k : M` over a ring is injective, if `k` is not a zero divisor.
 The general theory of such `k` is elaborated by `IsSMulRegular`.
 The typeclass that restricts all terms of `M` to have this property is `NoZeroSMulDivisors`. -/
-theorem smul_cancel_of_non_zero_divisor {M R : Type _} [Monoid M] [NonUnitalNonAssocRing R]
+theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
     [DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
     a = b := by
   rw [← sub_eq_zero]

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
-
-! This file was ported from Lean 3 source module group_theory.group_action.basic
-! leanprover-community/mathlib commit d30d31261cdb4d2f5e612eabc3c4bf45556350d5
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Fintype.Card
 import Mathlib.GroupTheory.GroupAction.Defs
@@ -15,6 +10,8 @@ import Mathlib.Data.Setoid.Basic
 import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.GroupTheory.Subgroup.Basic
 
+#align_import group_theory.group_action.basic from "leanprover-community/mathlib"@"d30d31261cdb4d2f5e612eabc3c4bf45556350d5"
+
 /-!
 # Basic properties of group actions
 

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 
 ! This file was ported from Lean 3 source module group_theory.group_action.basic
-! leanprover-community/mathlib commit f93c11933efbc3c2f0299e47b8ff83e9b539cbf6
+! leanprover-community/mathlib commit d30d31261cdb4d2f5e612eabc3c4bf45556350d5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -281,6 +281,12 @@ def orbitRel : Setoid β where
 
 variable {α} {β}
 
+@[to_additive]
+theorem orbitRel_apply {x y : β} : (orbitRel α β).Rel x y ↔ x ∈ orbit α y :=
+  Iff.rfl
+#align mul_action.orbit_rel_apply MulAction.orbitRel_apply
+#align add_action.orbit_rel_apply AddAction.orbitRel_apply
+
 /-- When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the union
 of the orbit of `U` under `α`. -/
 @[to_additive

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -124,11 +124,11 @@ def fixedBy (g : α) : Set β :=
 #align add_action.fixed_by AddAction.fixedBy
 
 @[to_additive]
-theorem fixed_eq_interᵢ_fixedBy : fixedPoints α β = ⋂ g : α, fixedBy α β g :=
+theorem fixed_eq_iInter_fixedBy : fixedPoints α β = ⋂ g : α, fixedBy α β g :=
   Set.ext fun _ =>
-    ⟨fun hx => Set.mem_interᵢ.2 fun g => hx g, fun hx g => (Set.mem_interᵢ.1 hx g : _)⟩
-#align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_interᵢ_fixedBy
-#align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_interᵢ_fixedBy
+    ⟨fun hx => Set.mem_iInter.2 fun g => hx g, fun hx g => (Set.mem_iInter.1 hx g : _)⟩
+#align mul_action.fixed_eq_Inter_fixed_by MulAction.fixed_eq_iInter_fixedBy
+#align add_action.fixed_eq_Inter_fixed_by AddAction.fixed_eq_iInter_fixedBy
 
 variable {α}
 
@@ -295,10 +295,10 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
   constructor
   · rintro ⟨y, hy, hxy⟩
     obtain ⟨a, rfl⟩ := Quotient.exact hxy
-    rw [Set.mem_unionᵢ]
+    rw [Set.mem_iUnion]
     exact ⟨a⁻¹, a • x, hy, inv_smul_smul a x⟩
   · intro hx
-    rw [Set.mem_unionᵢ] at hx
+    rw [Set.mem_iUnion] at hx
     obtain ⟨a, u, hu₁, hu₂⟩ := hx
     rw [Set.mem_preimage, Set.mem_image_iff_bex]
     refine' ⟨a⁻¹ • x, _, by simp only [Quotient.eq']; use a⁻¹⟩

This PR fixes two things:

• Most align statements for definitions and theorems and instances that are separated by two newlines from the relevant declaration (s/\n\n#align/\n#align). This is often seen in the mathport output after ending calc blocks.
• All remaining more-than-one-line #align statements. (This was needed for a script I wrote for #3630.)

@@ -190,7 +190,6 @@ theorem mem_fixedPoints_iff_card_orbit_eq_one {a : β} [Fintype (orbit α a)] :
     calc
       x • a = z := Subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
       _ = a := (Subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm
-
 #align mul_action.mem_fixed_points_iff_card_orbit_eq_one MulAction.mem_fixedPoints_iff_card_orbit_eq_one
 #align add_action.mem_fixed_points_iff_card_orbit_eq_zero AddAction.mem_fixedPoints_iff_card_orbit_eq_zero
 
@@ -227,7 +226,6 @@ theorem smul_orbit (a : α) (b : β) : a • orbit α b = orbit α b :=
     calc
       orbit α b = a • a⁻¹ • orbit α b := (smul_inv_smul _ _).symm
       _ ⊆ a • orbit α b := Set.image_subset _ (smul_orbit_subset _ _)
-
 #align mul_action.smul_orbit MulAction.smul_orbit
 #align add_action.vadd_orbit AddAction.vadd_orbit
 
@@ -237,7 +235,6 @@ theorem orbit_smul (a : α) (b : β) : orbit α (a • b) = orbit α b :=
     calc
       orbit α b = orbit α (a⁻¹ • a • b) := by rw [inv_smul_smul]
       _ ⊆ orbit α (a • b) := orbit_smul_subset _ _
-
 #align mul_action.orbit_smul MulAction.orbit_smul
 #align add_action.orbit_vadd AddAction.orbit_vadd
 
@@ -402,7 +399,6 @@ def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
     _ ≃ Σω : Ω, ω.orbit :=
       Equiv.sigmaCongrRight fun _ =>
         Equiv.subtypeEquivRight fun _ => orbitRel.Quotient.mem_orbit.symm
-
 #align mul_action.self_equiv_sigma_orbits' MulAction.selfEquivSigmaOrbits'
 #align add_action.self_equiv_sigma_orbits' AddAction.selfEquivSigmaOrbits'
 

@@ -282,8 +282,6 @@ def orbitRel : Setoid β where
 #align mul_action.orbit_rel MulAction.orbitRel
 #align add_action.orbit_rel AddAction.orbitRel
 
-attribute [local instance] orbitRel
-
 variable {α} {β}
 
 /-- When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the union
@@ -292,7 +290,9 @@ of the orbit of `U` under `α`. -/
       "When you take a set `U` in `β`, push it down to the quotient, and pull back, you get the
       union of the orbit of `U` under `α`."]
 theorem quotient_preimage_image_eq_union_mul (U : Set β) :
+    letI := orbitRel α β
     Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ a : α, (· • ·) a '' U := by
+  letI := orbitRel α β
   set f : β → Quotient (MulAction.orbitRel α β) := Quotient.mk'
   ext x
   constructor
@@ -313,7 +313,9 @@ theorem quotient_preimage_image_eq_union_mul (U : Set β) :
 
 @[to_additive]
 theorem disjoint_image_image_iff {U V : Set β} :
+    letI := orbitRel α β
     Disjoint (Quotient.mk' '' U) (Quotient.mk' '' V) ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V := by
+  letI := orbitRel α β
   set f : β → Quotient (MulAction.orbitRel α β) := Quotient.mk'
   refine'
     ⟨fun h x x_in_U a a_in_V =>
@@ -329,6 +331,7 @@ theorem disjoint_image_image_iff {U V : Set β} :
 
 @[to_additive]
 theorem image_inter_image_iff (U V : Set β) :
+    letI := orbitRel α β
     Quotient.mk' '' U ∩ Quotient.mk' '' V = ∅ ↔ ∀ x ∈ U, ∀ a : α, a • x ∉ V :=
   Set.disjoint_iff_inter_eq_empty.symm.trans disjoint_image_image_iff
 #align mul_action.image_inter_image_iff MulAction.image_inter_image_iff
@@ -372,7 +375,7 @@ theorem orbitRel.Quotient.mem_orbit {b : β} {x : orbitRel.Quotient α β} :
 /-- Note that `hφ = Quotient.out_eq'` is a useful choice here. -/
 @[to_additive "Note that `hφ = quotient.out_eq'` is a useful choice here."]
 theorem orbitRel.Quotient.orbit_eq_orbit_out (x : orbitRel.Quotient α β)
-    {φ : orbitRel.Quotient α β → β} (hφ : RightInverse φ Quotient.mk') :
+    {φ : orbitRel.Quotient α β → β} (hφ : letI := orbitRel α β; RightInverse φ Quotient.mk') :
     orbitRel.Quotient.orbit x = MulAction.orbit α (φ x) := by
   conv_lhs => rw [← hφ x]
 #align mul_action.orbit_rel.quotient.orbit_eq_orbit_out MulAction.orbitRel.Quotient.orbit_eq_orbit_out
@@ -393,6 +396,7 @@ This version is expressed in terms of `MulAction.orbitRel.Quotient.orbit` instea
       This version is expressed in terms of `AddAction.orbitRel.Quotient.orbit` instead of
       `AddAction.orbit`, to avoid mentioning `Quotient.out'`. "]
 def selfEquivSigmaOrbits' : β ≃ Σω : Ω, ω.orbit :=
+  letI := orbitRel α β
   calc
     β ≃ Σω : Ω, { b // Quotient.mk' b = ω } := (Equiv.sigmaFiberEquiv Quotient.mk').symm
     _ ≃ Σω : Ω, ω.orbit :=

@@ -337,8 +337,8 @@ theorem image_inter_image_iff (U V : Set β) :
 variable (α β)
 
 /-- The quotient by `MulAction.orbitRel`, given a name to enable dot notation. -/
-@[reducible,
-  to_additive "The quotient by `AddAction.orbitRel`, given a name to enable dot notation."]
+@[to_additive (attr := reducible)
+    "The quotient by `AddAction.orbitRel`, given a name to enable dot notation."]
 def orbitRel.Quotient : Type _ :=
   _root_.Quotient <| orbitRel α β
 #align mul_action.orbit_rel.quotient MulAction.orbitRel.Quotient