ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AddAction.pretransitive_iff_unique_quotient_of_nonempty
@[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) 
Arguments: (G : Type u_1) (α : Type u_2) [AddGroup G] [AddAction G α] [Nonempty α]
Goal: AddAction.IsPretransitive G α ↔ Nonempty (Unique (AddAction.orbitRel.Quotient G α))
-- MulAction.orbitRel.Quotient.orbit_mk
@[to_additive (attr := simp)]
theorem orbitRel.Quotient.orbit_mk (a : α) :
    orbitRel.Quotient.orbit (Quotient.mk'' a : orbitRel.Quotient G α) = MulAction.orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (a : α)
Goal: MulAction.orbitRel.Quotient.orbit (Quotient.mk'' a) = MulAction.orbit G a
-- AddAction.orbitRel.Quotient.orbit_mk
@[to_additive (attr := simp)]
theorem orbitRel.Quotient.orbit_mk (a : α) :
    orbitRel.Quotient.orbit (Quotient.mk'' a : orbitRel.Quotient G α) = MulAction.orbit G a 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (a : α)
Goal: AddAction.orbitRel.Quotient.orbit (Quotient.mk'' a) = AddAction.orbit G a
-- MulAction.orbitRel.Quotient.mem_orbit
@[to_additive]
theorem orbitRel.Quotient.mem_orbit {a : α} {x : orbitRel.Quotient G α} :
    a ∈ x.orbit ↔ Quotient.mk'' a = x 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a : α} {x : MulAction.orbitRel.Quotient G α}
Goal: a ∈ x.orbit ↔ Quotient.mk'' a = x
-- AddAction.orbitRel.Quotient.mem_orbit
@[to_additive]
theorem orbitRel.Quotient.mem_orbit {a : α} {x : orbitRel.Quotient G α} :
    a ∈ x.orbit ↔ Quotient.mk'' a = x 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a : α} {x : AddAction.orbitRel.Quotient G α}
Goal: a ∈ x.orbit ↔ Quotient.mk'' a = x
-- MulAction.orbitRel.Quotient.orbit_eq_orbit_out
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (x : MulAction.orbitRel.Quotient G α) {φ : MulAction.orbitRel.Quotient G α → α} (hφ : Function.RightInverse φ Quotient.mk')
Goal: x.orbit = MulAction.orbit G (φ x)
-- AddAction.orbitRel.Quotient.orbit_eq_orbit_out
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (x : AddAction.orbitRel.Quotient G α) {φ : AddAction.orbitRel.Quotient G α → α} (hφ : Function.RightInverse φ Quotient.mk')
Goal: x.orbit = AddAction.orbit G (φ x)
-- AddAction.orbitRel.Quotient.orbit_injective
@[to_additive]
lemma orbitRel.Quotient.orbit_injective :
    Injective (orbitRel.Quotient.orbit : orbitRel.Quotient G α → Set α) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α]
Goal: Function.Injective AddAction.orbitRel.Quotient.orbit
-- MulAction.orbitRel.Quotient.orbit_injective
@[to_additive]
lemma orbitRel.Quotient.orbit_injective :
    Injective (orbitRel.Quotient.orbit : orbitRel.Quotient G α → Set α) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α]
Goal: Function.Injective MulAction.orbitRel.Quotient.orbit
-- AddAction.orbitRel.Quotient.orbit_inj
@[to_additive (attr := simp)]
lemma orbitRel.Quotient.orbit_inj {x y : orbitRel.Quotient G α} : x.orbit = y.orbit ↔ x = y 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {x y : AddAction.orbitRel.Quotient G α}
Goal: x.orbit = y.orbit ↔ x = y
-- MulAction.orbitRel.Quotient.orbit_inj
@[to_additive (attr := simp)]
lemma orbitRel.Quotient.orbit_inj {x y : orbitRel.Quotient G α} : x.orbit = y.orbit ↔ x = y 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {x y : MulAction.orbitRel.Quotient G α}
Goal: x.orbit = y.orbit ↔ x = y
-- AddAction.orbitRel.quotient_eq_of_quotient_addSubgroup_eq
@[to_additive]
lemma orbitRel.quotient_eq_of_quotient_subgroup_eq {H : Subgroup G} {a b : α}
    (h : (⟦a⟧ : orbitRel.Quotient H α) = ⟦b⟧) : (⟦a⟧ : orbitRel.Quotient G α) = ⟦b⟧ 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {a b : α} (h : ⟦a⟧ = ⟦b⟧)
Goal: ⟦a⟧ = ⟦b⟧
-- MulAction.orbitRel.quotient_eq_of_quotient_subgroup_eq
@[to_additive]
lemma orbitRel.quotient_eq_of_quotient_subgroup_eq {H : Subgroup G} {a b : α}
    (h : (⟦a⟧ : orbitRel.Quotient H α) = ⟦b⟧) : (⟦a⟧ : orbitRel.Quotient G α) = ⟦b⟧ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {a b : α} (h : ⟦a⟧ = ⟦b⟧)
Goal: ⟦a⟧ = ⟦b⟧
-- MulAction.orbitRel.quotient_eq_of_quotient_subgroup_eq'
@[to_additive]
lemma orbitRel.quotient_eq_of_quotient_subgroup_eq' {H : Subgroup G} {a b : α}
    (h : (Quotient.mk'' a : orbitRel.Quotient H α) = Quotient.mk'' b) :
    (Quotient.mk'' a : orbitRel.Quotient G α) = Quotient.mk'' b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {a b : α} (h : Quotient.mk'' a = Quotient.mk'' b)
Goal: Quotient.mk'' a = Quotient.mk'' b
-- AddAction.orbitRel.quotient_eq_of_quotient_addSubgroup_eq'
@[to_additive]
lemma orbitRel.quotient_eq_of_quotient_subgroup_eq' {H : Subgroup G} {a b : α}
    (h : (Quotient.mk'' a : orbitRel.Quotient H α) = Quotient.mk'' b) :
    (Quotient.mk'' a : orbitRel.Quotient G α) = Quotient.mk'' b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {a b : α} (h : Quotient.mk'' a = Quotient.mk'' b)
Goal: Quotient.mk'' a = Quotient.mk'' b
-- AddAction.orbitRel.Quotient.orbit_nonempty
@[to_additive]
nonrec lemma orbitRel.Quotient.orbit_nonempty (x : orbitRel.Quotient G α) :
    Set.Nonempty x.orbit 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (x : AddAction.orbitRel.Quotient G α)
Goal: x.orbit.Nonempty
-- MulAction.orbitRel.Quotient.orbit_nonempty
@[to_additive]
nonrec lemma orbitRel.Quotient.orbit_nonempty (x : orbitRel.Quotient G α) :
    Set.Nonempty x.orbit 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (x : MulAction.orbitRel.Quotient G α)
Goal: x.orbit.Nonempty
-- MulAction.orbitRel.Quotient.mapsTo_smul_orbit
@[to_additive]
nonrec lemma orbitRel.Quotient.mapsTo_smul_orbit (g : G) (x : orbitRel.Quotient G α) :
    Set.MapsTo (g • ·) x.orbit x.orbit 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (x : MulAction.orbitRel.Quotient G α)
Goal: Set.MapsTo (fun x => g • x) x.orbit x.orbit
-- AddAction.orbitRel.Quotient.mapsTo_vadd_orbit
@[to_additive]
nonrec lemma orbitRel.Quotient.mapsTo_smul_orbit (g : G) (x : orbitRel.Quotient G α) :
    Set.MapsTo (g • ·) x.orbit x.orbit 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (x : AddAction.orbitRel.Quotient G α)
Goal: Set.MapsTo (fun x => g +ᵥ x) x.orbit x.orbit
-- MulAction.orbitRel.Quotient.orbit.coe_smul
@[to_additive (attr := simp)]
lemma orbitRel.Quotient.orbit.coe_smul {g : G} {x : orbitRel.Quotient G α} {a : x.orbit} :
    ↑(g • a) = g • (a : α) 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {g : G} {x : MulAction.orbitRel.Quotient G α} {a : ↑x.orbit}
Goal: ↑(g • a) = g • ↑a
-- AddAction.orbitRel.Quotient.orbit.coe_vadd
@[to_additive (attr := simp)]
lemma orbitRel.Quotient.orbit.coe_smul {g : G} {x : orbitRel.Quotient G α} {a : x.orbit} :
    ↑(g • a) = g • (a : α) 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {g : G} {x : AddAction.orbitRel.Quotient G α} {a : ↑x.orbit}
Goal: ↑(g +ᵥ a) = g +ᵥ ↑a
-- AddAction.orbitRel.Quotient.mem_addSubgroup_orbit_iff
@[to_additive (attr := norm_cast, simp)]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} : (a : α) ∈ MulAction.orbit H (b : α) ↔ a ∈ MulAction.orbit H b 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {x : AddAction.orbitRel.Quotient G α} {a b : ↑x.orbit}
Goal: ↑a ∈ AddAction.orbit ↥H ↑b ↔ a ∈ AddAction.orbit (↥H) b
-- MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff
@[to_additive (attr := norm_cast, simp)]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} : (a : α) ∈ MulAction.orbit H (b : α) ↔ a ∈ MulAction.orbit H b 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {x : MulAction.orbitRel.Quotient G α} {a b : ↑x.orbit}
Goal: ↑a ∈ MulAction.orbit ↥H ↑b ↔ a ∈ MulAction.orbit (↥H) b
-- AddAction.orbitRel.Quotient.addSubgroup_quotient_eq_iff
@[to_additive]
lemma orbitRel.Quotient.subgroup_quotient_eq_iff {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} : (⟦a⟧ : orbitRel.Quotient H x.orbit) = ⟦b⟧ ↔
      (⟦↑a⟧ : orbitRel.Quotient H α) = ⟦↑b⟧ 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {x : AddAction.orbitRel.Quotient G α} {a b : ↑x.orbit}
Goal: ⟦a⟧ = ⟦b⟧ ↔ ⟦↑a⟧ = ⟦↑b⟧
-- MulAction.orbitRel.Quotient.subgroup_quotient_eq_iff
@[to_additive]
lemma orbitRel.Quotient.subgroup_quotient_eq_iff {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} : (⟦a⟧ : orbitRel.Quotient H x.orbit) = ⟦b⟧ ↔
      (⟦↑a⟧ : orbitRel.Quotient H α) = ⟦↑b⟧ 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {x : MulAction.orbitRel.Quotient G α} {a b : ↑x.orbit}
Goal: ⟦a⟧ = ⟦b⟧ ↔ ⟦↑a⟧ = ⟦↑b⟧
-- AddAction.orbitRel.Quotient.mem_addSubgroup_orbit_iff'
@[to_additive]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff' {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} {c : α} (h : (⟦a⟧ : orbitRel.Quotient H x.orbit) = ⟦b⟧) :
    (a : α) ∈ MulAction.orbit H c ↔ (b : α) ∈ MulAction.orbit H c 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {H : AddSubgroup G} {x : AddAction.orbitRel.Quotient G α} {a b : ↑x.orbit} {c : α} (h : ⟦a⟧ = ⟦b⟧)
Goal: ↑a ∈ AddAction.orbit (↥H) c ↔ ↑b ∈ AddAction.orbit (↥H) c
-- MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff'
@[to_additive]
lemma orbitRel.Quotient.mem_subgroup_orbit_iff' {H : Subgroup G} {x : orbitRel.Quotient G α}
    {a b : x.orbit} {c : α} (h : (⟦a⟧ : orbitRel.Quotient H x.orbit) = ⟦b⟧) :
    (a : α) ∈ MulAction.orbit H c ↔ (b : α) ∈ MulAction.orbit H c 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {H : Subgroup G} {x : MulAction.orbitRel.Quotient G α} {a b : ↑x.orbit} {c : α} (h : ⟦a⟧ = ⟦b⟧)
Goal: ↑a ∈ MulAction.orbit (↥H) c ↔ ↑b ∈ MulAction.orbit (↥H) c
-- MulAction.orbitRel_le_fst
@[to_additive]
lemma orbitRel_le_fst :
    orbitRel G (α × β) ≤ (orbitRel G α).comap Prod.fst 
Arguments: (G : Type u_1) (α : Type u_2) (β : Type u_3) [Group G] [MulAction G α] [MulAction G β]
Goal: MulAction.orbitRel G (α × β) ≤ Setoid.comap Prod.fst (MulAction.orbitRel G α)
-- AddAction.orbitRel_le_fst
@[to_additive]
lemma orbitRel_le_fst :
    orbitRel G (α × β) ≤ (orbitRel G α).comap Prod.fst 
Arguments: (G : Type u_1) (α : Type u_2) (β : Type u_3) [AddGroup G] [AddAction G α] [AddAction G β]
Goal: AddAction.orbitRel G (α × β) ≤ Setoid.comap Prod.fst (AddAction.orbitRel G α)
-- MulAction.orbitRel_le_snd
@[to_additive]
lemma orbitRel_le_snd :
    orbitRel G (α × β) ≤ (orbitRel G β).comap Prod.snd 
Arguments: (G : Type u_1) (α : Type u_2) (β : Type u_3) [Group G] [MulAction G α] [MulAction G β]
Goal: MulAction.orbitRel G (α × β) ≤ Setoid.comap Prod.snd (MulAction.orbitRel G β)
-- AddAction.orbitRel_le_snd
@[to_additive]
lemma orbitRel_le_snd :
    orbitRel G (α × β) ≤ (orbitRel G β).comap Prod.snd 
Arguments: (G : Type u_1) (α : Type u_2) (β : Type u_3) [AddGroup G] [AddAction G α] [AddAction G β]
Goal: AddAction.orbitRel G (α × β) ≤ Setoid.comap Prod.snd (AddAction.orbitRel G β)
-- MulAction.mem_stabilizer_iff
@[to_additive (attr := simp)]
theorem mem_stabilizer_iff {a : α} {g : G} : g ∈ stabilizer G a ↔ g • a = a 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a : α} {g : G}
Goal: g ∈ MulAction.stabilizer G a ↔ g • a = a
-- AddAction.mem_stabilizer_iff
@[to_additive (attr := simp)]
theorem mem_stabilizer_iff {a : α} {g : G} : g ∈ stabilizer G a ↔ g • a = a 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a : α} {g : G}
Goal: g ∈ AddAction.stabilizer G a ↔ g +ᵥ a = a
-- AddAction.le_stabilizer_vadd_left
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [AddAction G α] [AddAction G β] [VAdd α β] [VAddAssocClass G α β] (a : α) (b : β)
Goal: AddAction.stabilizer G a ≤ AddAction.stabilizer G (a +ᵥ b)
-- MulAction.le_stabilizer_smul_left
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G α] [MulAction G β] [SMul α β] [IsScalarTower G α β] (a : α) (b : β)
Goal: MulAction.stabilizer G a ≤ MulAction.stabilizer G (a • b)
-- AddAction.le_stabilizer_vadd_right
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [AddAction G β] [VAdd α β] [VAddCommClass G α β] (a : α) (b : β)
Goal: AddAction.stabilizer G b ≤ AddAction.stabilizer G (a +ᵥ b)
-- MulAction.le_stabilizer_smul_right
@[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) 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G β] [SMul α β] [SMulCommClass G α β] (a : α) (b : β)
Goal: MulAction.stabilizer G b ≤ MulAction.stabilizer G (a • b)
-- AddAction.stabilizer_vadd_eq_left
@[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 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [AddAction G α] [AddAction G β] [VAdd α β] [VAddAssocClass G α β] (a : α) (b : β) (h : Function.Injective fun x => x +ᵥ b)
Goal: AddAction.stabilizer G (a +ᵥ b) = AddAction.stabilizer G a
-- MulAction.stabilizer_smul_eq_left
@[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 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G α] [MulAction G β] [SMul α β] [IsScalarTower G α β] (a : α) (b : β) (h : Function.Injective fun x => x • b)
Goal: MulAction.stabilizer G (a • b) = MulAction.stabilizer G a
-- MulAction.stabilizer_smul_eq_right
@[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 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G β] [Group α] [MulAction α β] [SMulCommClass G α β] (a : α) (b : β)
Goal: MulAction.stabilizer G (a • b) = MulAction.stabilizer G b
-- AddAction.stabilizer_vadd_eq_right
@[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 
Arguments: {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [AddAction G β] [AddGroup α] [AddAction α β] [VAddCommClass G α β] (a : α) (b : β)
Goal: AddAction.stabilizer G (a +ᵥ b) = AddAction.stabilizer G b
-- MulAction.stabilizer_mul_eq_left
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [Group α] [IsScalarTower G α α] (a b : α)
Goal: MulAction.stabilizer G (a * b) = MulAction.stabilizer G a
-- AddAction.stabilizer_add_eq_left
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] [AddGroup α] [VAddAssocClass G α α] (a b : α)
Goal: AddAction.stabilizer G (a + b) = AddAction.stabilizer G a
-- MulAction.stabilizer_mul_eq_right
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [Group α] [SMulCommClass G α α] (a b : α)
Goal: MulAction.stabilizer G (a * b) = MulAction.stabilizer G b
-- AddAction.stabilizer_add_eq_right
@[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 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] [AddGroup α] [VAddCommClass G α α] (a b : α)
Goal: AddAction.stabilizer G (a + b) = AddAction.stabilizer G b
-- MulAction.stabilizer_smul_eq_stabilizer_map_conj
theorem stabilizer_smul_eq_stabilizer_map_conj (g : G) (a : α) :
    stabilizer G (g • a) = (stabilizer G a).map (MulAut.conj g).toMonoidHom 
Arguments: {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α)
Goal: MulAction.stabilizer G (g • a) = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (MulAction.stabilizer G a)
-- AddAction.stabilizer_vadd_eq_stabilizer_map_conj
theorem stabilizer_vadd_eq_stabilizer_map_conj (g : G) (a : α) :
    stabilizer G (g +ᵥ a) = (stabilizer G a).map (AddAut.conj g).toAddMonoidHom 
Arguments: {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α)
Goal: AddAction.stabilizer G (g +ᵥ a) = AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (AddAction.stabilizer G a)
-- Equiv.swap_mem_stabilizer
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) 
Arguments: {α : Type u_1} [DecidableEq α] {S : Set α} {a b : α}
Goal: Equiv.swap a b ∈ MulAction.stabilizer (Equiv.Perm α) S ↔ (a ∈ S ↔ b ∈ S)
-- MulAction.le_stabilizer_iff_smul_le
theorem le_stabilizer_iff_smul_le (s : Set α) (H : Subgroup G) :
    H ≤ stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s 
Arguments: {G : Type u_1} [Group G] {α : Type u_2} [MulAction G α] (s : Set α) (H : Subgroup G)
Goal: H ≤ MulAction.stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s
-- MulAction.mem_stabilizer_of_finite_iff_smul_le
theorem mem_stabilizer_of_finite_iff_smul_le (s : Set α) (hs : s.Finite) (g : G) :
    g ∈ stabilizer G s ↔ g • s ⊆ s 
Arguments: {G : Type u_1} [Group G] {α : Type u_2} [MulAction G α] (s : Set α) (hs : s.Finite) (g : G)
Goal: g ∈ MulAction.stabilizer G s ↔ g • s ⊆ s
-- MulAction.mem_stabilizer_of_finite_iff_le_smul
theorem mem_stabilizer_of_finite_iff_le_smul (s : Set α) (hs : s.Finite) (g : G) :
    g ∈ stabilizer G s ↔ s ⊆ g • s 
Arguments: {G : Type u_1} [Group G] {α : Type u_2} [MulAction G α] (s : Set α) (hs : s.Finite) (g : G)
Goal: g ∈ MulAction.stabilizer G s ↔ s ⊆ g • s
-- Equiv.Perm.sumCongrHom.card_range
@[simp]
theorem sumCongrHom.card_range {α β : Type*} [Fintype (sumCongrHom α β).range]
    [Fintype (Perm α × Perm β)] :
    Fintype.card (sumCongrHom α β).range = Fintype.card (Perm α × Perm β) 
Arguments: {α : Type u_1} {β : Type u_2} [Fintype ↥(Equiv.Perm.sumCongrHom α β).range] [Fintype (Equiv.Perm α × Equiv.Perm β)]
Goal: Fintype.card ↥(Equiv.Perm.sumCongrHom α β).range = Fintype.card (Equiv.Perm α × Equiv.Perm β)
-- Equiv.Perm.sigmaCongrRightHom.card_range
@[simp]
theorem sigmaCongrRightHom.card_range {α : Type*} {β : α → Type*}
    [Fintype (sigmaCongrRightHom β).range] [Fintype (∀ a, Perm (β a))] :
    Fintype.card (sigmaCongrRightHom β).range = Fintype.card (∀ a, Perm (β a)) 
Arguments: {α : Type u_1} {β : α → Type u_2} [Fintype ↥(Equiv.Perm.sigmaCongrRightHom β).range] [Fintype ((a : α) → Equiv.Perm (β a))]
Goal: Fintype.card ↥(Equiv.Perm.sigmaCongrRightHom β).range = Fintype.card ((a : α) → Equiv.Perm (β a))
-- Equiv.Perm.subtypeCongrHom.card_range
@[simp]
theorem subtypeCongrHom.card_range {α : Type*} (p : α → Prop) [DecidablePred p]
    [Fintype (subtypeCongrHom p).range] [Fintype (Perm { a // p a } × Perm { a // ¬p a })] :
    Fintype.card (subtypeCongrHom p).range =
      Fintype.card (Perm { a // p a } × Perm { a // ¬p a }) 
Arguments: {α : Type u_1} (p : α → Prop) [DecidablePred p] [Fintype ↥(Equiv.Perm.subtypeCongrHom p).range] [Fintype (Equiv.Perm { a // p a } × Equiv.Perm { a // ¬p a })]
Goal: Fintype.card ↥(Equiv.Perm.subtypeCongrHom p).range = Fintype.card (Equiv.Perm { a // p a } × Equiv.Perm { a // ¬p a })
-- MulAction.orbit.eq_or_disjoint
theorem orbit.eq_or_disjoint (a b : X) :
    orbit G a = orbit G b ∨ Disjoint (orbit G a) (orbit G b) 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] (a b : X)
Goal: MulAction.orbit G a = MulAction.orbit G b ∨ Disjoint (MulAction.orbit G a) (MulAction.orbit G b)
-- MulAction.orbit.pairwiseDisjoint
theorem orbit.pairwiseDisjoint :
    (Set.range fun x : X => orbit G x).PairwiseDisjoint id 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X]
Goal: (Set.range fun x => MulAction.orbit G x).PairwiseDisjoint id
-- MulAction.IsPartition.of_orbits
theorem IsPartition.of_orbits :
    Setoid.IsPartition (Set.range fun a : X => orbit G a) 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X]
Goal: Setoid.IsPartition (Set.range fun a => MulAction.orbit G a)
-- MulAction.IsBlock.def
theorem IsBlock.def {B : Set X} :
    IsBlock G B ↔ ∀ g g' : G, g • B = g' • B ∨ Disjoint (g • B) (g' • B) 
Arguments: {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X}
Goal: MulAction.IsBlock G B ↔ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
-- MulAction.IsBlock.mk_notempty
theorem IsBlock.mk_notempty {B : Set X} :
    IsBlock G B ↔ ∀ g g' : G, g • B ∩ g' • B ≠ ∅ → g • B = g' • B 
Arguments: {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X}
Goal: MulAction.IsBlock G B ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B
-- MulAction.IsFixedBlock.isBlock
theorem IsFixedBlock.isBlock {B : Set X} (hfB : IsFixedBlock G B) :
    IsBlock G B 
Arguments: {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X} (hfB : MulAction.IsFixedBlock G B)
Goal: MulAction.IsBlock G B
-- MulAction.isBlock_empty
theorem isBlock_empty : IsBlock G (⊥ : Set X) 
Arguments: {G : Type u_1} (X : Type u_2) [SMul G X]
Goal: MulAction.IsBlock G ⊥
-- MulAction.isBlock_singleton
theorem isBlock_singleton (a : X) : IsBlock G ({a} : Set X) 
Arguments: {G : Type u_1} {X : Type u_2} [SMul G X] (a : X)
Goal: MulAction.IsBlock G {a}
-- MulAction.isBlock_subsingleton
theorem isBlock_subsingleton {B : Set X} (hB : B.Subsingleton) :
    IsBlock G B 
Arguments: {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X} (hB : B.Subsingleton)
Goal: MulAction.IsBlock G B
-- MulAction.IsBlock.smul_eq_or_disjoint
theorem IsBlock.smul_eq_or_disjoint {B : Set X} (hB : IsBlock G B) (g : G) :
    g • B = B ∨ Disjoint (g • B) B 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] {B : Set X} (hB : MulAction.IsBlock G B) (g : G)
Goal: g • B = B ∨ Disjoint (g • B) B
-- MulAction.IsBlock.def_one
theorem IsBlock.def_one {B : Set X} :
    IsBlock G B ↔ ∀ g : G, g • B = B ∨ Disjoint (g • B) B 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] {B : Set X}
Goal: MulAction.IsBlock G B ↔ ∀ (g : G), g • B = B ∨ Disjoint (g • B) B
-- MulAction.IsBlock.mk_notempty_one
theorem IsBlock.mk_notempty_one {B : Set X} :
    IsBlock G B ↔ ∀ g : G, g • B ∩ B ≠ ∅ → g • B = B 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] {B : Set X}
Goal: MulAction.IsBlock G B ↔ ∀ (g : G), g • B ∩ B ≠ ∅ → g • B = B
-- MulAction.IsBlock.mk_mem
theorem IsBlock.mk_mem {B : Set X} :
    IsBlock G B ↔ ∀ (g : G) (a : X) (_ : a ∈ B) (_ : g • a ∈ B), g • B = B 
Arguments: {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] {B : Set X}
Goal: MulAction.IsBlock G B ↔ ∀ (g : G), ∀ a ∈ B, g • a ∈ B → g • B = B