ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MulAction.smul_zpow_movedBy_eq_of_commute
@[to_additive "If `g` and `h` commute, then `g` moves `(j • h) +ᵥ x` iff `g` moves `x`."]
theorem smul_zpow_movedBy_eq_of_commute {g h : G} (comm : Commute g h) (j : ℤ) :
    h ^ j • (fixedBy α g)ᶜ = (fixedBy α g)ᶜ 
Arguments: {α : Type u_1} {G : Type u_2} [Group G] [MulAction G α] {g h : G} (comm : Commute g h) (j : ℤ)
Goal: h ^ j • (MulAction.fixedBy α g)ᶜ = (MulAction.fixedBy α g)ᶜ
-- AddAction.vadd_zsmul_movedBy_eq_of_addCommute
@[to_additive "If `g` and `h` commute, then `g` moves `(j • h) +ᵥ x` iff `g` moves `x`."]
theorem smul_zpow_movedBy_eq_of_commute {g h : G} (comm : Commute g h) (j : ℤ) :
    h ^ j • (fixedBy α g)ᶜ = (fixedBy α g)ᶜ 
Arguments: {α : Type u_1} {G : Type u_2} [AddGroup G] [AddAction G α] {g h : G} (comm : AddCommute g h) (j : ℤ)
Goal: j • h +ᵥ (AddAction.fixedBy α g)ᶜ = (AddAction.fixedBy α g)ᶜ
-- AddAction.fixedBy_eq_univ_iff_eq_zero
@[to_additive "If the additive action of `M` on `α` is faithful,
then `fixedBy α m = Set.univ` implies that `m = 1`."]
theorem fixedBy_eq_univ_iff_eq_one {m : M} : fixedBy α m = Set.univ ↔ m = 1 
Arguments: {α : Type u_1} {M : Type u_3} [AddMonoid M] [AddAction M α] [FaithfulVAdd M α] {m : M}
Goal: AddAction.fixedBy α m = Set.univ ↔ m = 0
-- MulAction.fixedBy_eq_univ_iff_eq_one
@[to_additive "If the additive action of `M` on `α` is faithful,
then `fixedBy α m = Set.univ` implies that `m = 1`."]
theorem fixedBy_eq_univ_iff_eq_one {m : M} : fixedBy α m = Set.univ ↔ m = 1 
Arguments: {α : Type u_1} {M : Type u_3} [Monoid M] [MulAction M α] [FaithfulSMul M α] {m : M}
Goal: MulAction.fixedBy α m = Set.univ ↔ m = 1
-- MulAction.not_commute_of_disjoint_movedBy_preimage
@[to_additive "If the image of the `(fixedBy α g)ᶜ` set by the pointwise action of `h: G`
is disjoint from `(fixedBy α g)ᶜ`, then `g` and `h` cannot commute."]
theorem not_commute_of_disjoint_movedBy_preimage {g h : G} (ne_one : g ≠ 1)
    (disjoint : Disjoint (fixedBy α g)ᶜ (h • (fixedBy α g)ᶜ)) : ¬Commute g h 
Arguments: {α : Type u_1} {G : Type u_2} [Group G] [MulAction G α] [FaithfulSMul G α] {g h : G} (ne_one : g ≠ 1) (disjoint : Disjoint (MulAction.fixedBy α g)ᶜ (h • (MulAction.fixedBy α g)ᶜ))
Goal: ¬Commute g h
-- AddAction.not_addCommute_of_disjoint_movedBy_preimage
@[to_additive "If the image of the `(fixedBy α g)ᶜ` set by the pointwise action of `h: G`
is disjoint from `(fixedBy α g)ᶜ`, then `g` and `h` cannot commute."]
theorem not_commute_of_disjoint_movedBy_preimage {g h : G} (ne_one : g ≠ 1)
    (disjoint : Disjoint (fixedBy α g)ᶜ (h • (fixedBy α g)ᶜ)) : ¬Commute g h 
Arguments: {α : Type u_1} {G : Type u_2} [AddGroup G] [AddAction G α] [FaithfulVAdd G α] {g h : G} (ne_one : g ≠ 0) (disjoint : Disjoint (AddAction.fixedBy α g)ᶜ (h +ᵥ (AddAction.fixedBy α g)ᶜ))
Goal: ¬AddCommute g h
-- card_dvd_exponent_pow_rank
@[to_additive]
theorem card_dvd_exponent_pow_rank : Nat.card G ∣ Monoid.exponent G ^ Group.rank G 
Arguments: (G : Type u_1) [CommGroup G] [Group.FG G]
Goal: Nat.card G ∣ Monoid.exponent G ^ Group.rank G
-- card_dvd_exponent_nsmul_rank
@[to_additive]
theorem card_dvd_exponent_pow_rank : Nat.card G ∣ Monoid.exponent G ^ Group.rank G 
Arguments: (G : Type u_1) [AddCommGroup G] [AddGroup.FG G]
Goal: Nat.card G ∣ AddMonoid.exponent G ^ AddGroup.rank G
-- card_dvd_exponent_pow_rank'
@[to_additive]
theorem card_dvd_exponent_pow_rank' {n : ℕ} (hG : ∀ g : G, g ^ n = 1) :
    Nat.card G ∣ n ^ Group.rank G 
Arguments: (G : Type u_1) [CommGroup G] [Group.FG G] {n : ℕ} (hG : ∀ (g : G), g ^ n = 1)
Goal: Nat.card G ∣ n ^ Group.rank G
-- card_dvd_exponent_nsmul_rank'
@[to_additive]
theorem card_dvd_exponent_pow_rank' {n : ℕ} (hG : ∀ g : G, g ^ n = 1) :
    Nat.card G ∣ n ^ Group.rank G 
Arguments: (G : Type u_1) [AddCommGroup G] [AddGroup.FG G] {n : ℕ} (hG : ∀ (g : G), n • g = 0)
Goal: Nat.card G ∣ n ^ AddGroup.rank G
-- Subgroup.closure_mul_image_mul_eq_top
theorem closure_mul_image_mul_eq_top
    (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R) (hS : closure S = ⊤) :
    (closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹)) * R = ⊤ 
Arguments: {G : Type u_1} [Group G] {H : Subgroup G} {R S : Set G} (hR : R ∈ Subgroup.rightTransversals ↑H) (hR1 : 1 ∈ R) (hS : Subgroup.closure S = ⊤)
Goal: ↑(Subgroup.closure ((fun g => g * (↑(Subgroup.MemRightTransversals.toFun hR g))⁻¹) '' (R * S))) * R = ⊤
-- Subgroup.closure_mul_image_eq
theorem closure_mul_image_eq (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
    (hS : closure S = ⊤) : closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹) = H 
Arguments: {G : Type u_1} [Group G] {H : Subgroup G} {R S : Set G} (hR : R ∈ Subgroup.rightTransversals ↑H) (hR1 : 1 ∈ R) (hS : Subgroup.closure S = ⊤)
Goal: Subgroup.closure ((fun g => g * (↑(Subgroup.MemRightTransversals.toFun hR g))⁻¹) '' (R * S)) = H
-- Subgroup.closure_mul_image_eq_top
theorem closure_mul_image_eq_top (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
    (hS : closure S = ⊤) : closure ((R * S).image fun g =>
      ⟨g * (toFun hR g : G)⁻¹, mul_inv_toFun_mem hR g⟩ : Set H) = ⊤ 
Arguments: {G : Type u_1} [Group G] {H : Subgroup G} {R S : Set G} (hR : R ∈ Subgroup.rightTransversals ↑H) (hR1 : 1 ∈ R) (hS : Subgroup.closure S = ⊤)
Goal: Subgroup.closure ((fun g => ⟨g * (↑(Subgroup.MemRightTransversals.toFun hR g))⁻¹, ⋯⟩) '' (R * S)) = ⊤
-- Subgroup.closure_mul_image_eq_top'
theorem closure_mul_image_eq_top' [DecidableEq G] {R S : Finset G}
    (hR : (R : Set G) ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
    (hS : closure (S : Set G) = ⊤) :
    closure (((R * S).image fun g => ⟨_, mul_inv_toFun_mem hR g⟩ : Finset H) : Set H) = ⊤ 
Arguments: {G : Type u_1} [Group G] {H : Subgroup G} [DecidableEq G] {R S : Finset G} (hR : ↑R ∈ Subgroup.rightTransversals ↑H) (hR1 : 1 ∈ R) (hS : Subgroup.closure ↑S = ⊤)
Goal: Subgroup.closure ↑(Finset.image (fun g => ⟨g * (↑(Subgroup.MemRightTransversals.toFun hR g))⁻¹, ⋯⟩) (R * S)) = ⊤
-- Subgroup.exists_finset_card_le_mul
theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
    ∃ T : Finset H, T.card ≤ H.index * S.card ∧ closure (T : Set H) = ⊤ 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] {S : Finset G} (hS : Subgroup.closure ↑S = ⊤)
Goal: ∃ T, T.card ≤ H.index * S.card ∧ Subgroup.closure ↑T = ⊤
-- Subgroup.rank_le_index_mul_rank
theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
    Group.rank H ≤ H.index * Group.rank G 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G) [hG : Group.FG G] [H.FiniteIndex]
Goal: Group.rank ↥H ≤ H.index * Group.rank G
-- Subgroup.card_commutator_dvd_index_center_pow
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) 
Arguments: (G : Type u_1) [Group G] [Finite ↑(commutatorSet G)]
Goal: Nat.card ↥(commutator G) ∣ (Subgroup.center G).index ^ ((Subgroup.center G).index * Nat.card ↑(commutatorSet G) + 1)
-- Subgroup.card_commutator_le_of_finite_commutatorSet
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) 
Arguments: (G : Type u_1) [Group G] [Finite ↑(commutatorSet G)]
Goal: Nat.card ↥(commutator G) ≤ Subgroup.cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- MulAction.smul_bijective_of_is_unit
theorem MulAction.smul_bijective_of_is_unit
    {M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
    Function.Bijective (fun (a : α) ↦ m • a) 
Arguments: {M : Type u_1} [Monoid M] {α : Type u_2} [MulAction M α] {m : M} (hm : IsUnit m)
Goal: Function.Bijective fun a => m • a
-- image_smul_setₛₗ
theorem image_smul_setₛₗ :
    h '' (c • s) = σ c • h '' s 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] (σ : R → S) [MulAction R M] [MulAction S N] {F : Type u_7} (h : F) [FunLike F M N] [MulActionSemiHomClass F σ M N] (c : R) (s : Set M)
Goal: ⇑h '' (c • s) = σ c • ⇑h '' s
-- smul_preimage_set_leₛₗ
theorem smul_preimage_set_leₛₗ :
    c • h ⁻¹' t ⊆ h ⁻¹' (σ c • t) 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] (σ : R → S) [MulAction R M] [MulAction S N] {F : Type u_7} (h : F) [FunLike F M N] [MulActionSemiHomClass F σ M N] (c : R) (t : Set N)
Goal: c • ⇑h ⁻¹' t ⊆ ⇑h ⁻¹' (σ c • t)
-- preimage_smul_setₛₗ'
theorem preimage_smul_setₛₗ'
    (hc : Function.Surjective (fun (m : M) ↦ c • m))
    (hc' : Function.Injective (fun (n : N) ↦ σ c • n)) :
    h ⁻¹' (σ c • t) = c • h ⁻¹' t 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] (σ : R → S) [MulAction R M] [MulAction S N] {F : Type u_7} (h : F) [FunLike F M N] [MulActionSemiHomClass F σ M N] {c : R} (t : Set N) (hc : Function.Surjective fun m => c • m) (hc' : Function.Injective fun n => σ c • n)
Goal: ⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t
-- preimage_smul_setₛₗ_of_units
theorem preimage_smul_setₛₗ_of_units (hc : IsUnit c) (hc' : IsUnit (σ c)) :
    h ⁻¹' (σ c • t) = c • h ⁻¹' t 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] (σ : R → S) [MulAction R M] [MulAction S N] {F : Type u_7} (h : F) [FunLike F M N] [MulActionSemiHomClass F σ M N] {c : R} (t : Set N) (hc : IsUnit c) (hc' : IsUnit (σ c))
Goal: ⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t
-- MonoidHom.preimage_smul_setₛₗ
theorem MonoidHom.preimage_smul_setₛₗ (σ : R →* S)
    {F : Type*} [FunLike F M N] [MulActionSemiHomClass F ⇑σ M N] (h : F)
    {c : R} (hc : IsUnit c) (t : Set N) :
    h ⁻¹' (σ c • t) = c • h ⁻¹' t 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] [MulAction R M] [MulAction S N] (σ : R →* S) {F : Type u_8} [FunLike F M N] [MulActionSemiHomClass F (⇑σ) M N] (h : F) {c : R} (hc : IsUnit c) (t : Set N)
Goal: ⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t
-- preimage_smul_setₛₗ
theorem preimage_smul_setₛₗ
    {G : Type*} [FunLike G R S] [MonoidHomClass G R S] (σ : G)
    {F : Type*} [FunLike F M N] [MulActionSemiHomClass F σ M N] (h : F)
    {c : R} (hc : IsUnit c) (t : Set N) :
    h ⁻¹' (σ c • t) = c • h ⁻¹' t 
Arguments: {R : Type u_1} {S : Type u_2} (M : Type u_3) (N : Type u_6) [Monoid R] [Monoid S] [MulAction R M] [MulAction S N] {G : Type u_8} [FunLike G R S] [MonoidHomClass G R S] (σ : G) {F : Type u_9} [FunLike F M N] [MulActionSemiHomClass F (⇑σ) M N] (h : F) {c : R} (hc : IsUnit c) (t : Set N)
Goal: ⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t
-- Group.preimage_smul_setₛₗ
theorem Group.preimage_smul_setₛₗ
    {R S : Type*} [Group R] [Group S] (σ : R → S)
    [MulAction R M] [MulAction S N]
    {F : Type*} [FunLike F M N] [MulActionSemiHomClass F σ M N] (h : F)
    (c : R) (t : Set N) :
    h ⁻¹' (σ c • t) = c • h ⁻¹' t 
Arguments: (M : Type u_3) (N : Type u_6) {R : Type u_8} {S : Type u_9} [Group R] [Group S] (σ : R → S) [MulAction R M] [MulAction S N] {F : Type u_10} [FunLike F M N] [MulActionSemiHomClass F σ M N] (h : F) (c : R) (t : Set N)
Goal: ⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t
-- image_smul_set
@[simp] -- This can be safely removed as a `@[simp]` lemma if `image_smul_setₛₗ` is readded.
theorem image_smul_set :
    h '' (c • s) = c • h '' s 
Arguments: (R : Type u_1) (M₁ : Type u_4) (M₂ : Type u_5) [Monoid R] [MulAction R M₁] [MulAction R M₂] {F : Type u_7} (h : F) [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] (c : R) (s : Set M₁)
Goal: ⇑h '' (c • s) = c • ⇑h '' s
-- smul_preimage_set_le
theorem smul_preimage_set_le :
    c • h ⁻¹' t ⊆ h ⁻¹' (c • t) 
Arguments: (R : Type u_1) (M₁ : Type u_4) (M₂ : Type u_5) [Monoid R] [MulAction R M₁] [MulAction R M₂] {F : Type u_7} (h : F) [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] (c : R) (t : Set M₂)
Goal: c • ⇑h ⁻¹' t ⊆ ⇑h ⁻¹' (c • t)
-- preimage_smul_set
theorem preimage_smul_set (hc : IsUnit c) :
    h ⁻¹' (c • t) = c • h ⁻¹' t 
Arguments: (R : Type u_1) (M₁ : Type u_4) (M₂ : Type u_5) [Monoid R] [MulAction R M₁] [MulAction R M₂] {F : Type u_7} (h : F) [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] {c : R} (t : Set M₂) (hc : IsUnit c)
Goal: ⇑h ⁻¹' (c • t) = c • ⇑h ⁻¹' t
-- Group.preimage_smul_set
theorem Group.preimage_smul_set
    {R : Type*} [Group R] (M₁ M₂ : Type*)
    [MulAction R M₁] [MulAction R M₂]
    {F : Type*} [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] (h : F)
    (c : R) (t : Set M₂) :
    h ⁻¹' (c • t) = c • h ⁻¹' t 
Arguments: {R : Type u_8} [Group R] (M₁ : Type u_9) (M₂ : Type u_10) [MulAction R M₁] [MulAction R M₂] {F : Type u_11} [FunLike F M₁ M₂] [MulActionHomClass F R M₁ M₂] (h : F) (c : R) (t : Set M₂)
Goal: ⇑h ⁻¹' (c • t) = c • ⇑h ⁻¹' t
-- Commensurable.refl
@[refl]
protected theorem refl (H : Subgroup G) : Commensurable H H 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G)
Goal: Commensurable H H
-- Commensurable.comm
theorem comm {H K : Subgroup G} : Commensurable H K ↔ Commensurable K H 
Arguments: {G : Type u_1} [Group G] {H K : Subgroup G}
Goal: Commensurable H K ↔ Commensurable K H
-- Commensurable.symm
@[symm]
theorem symm {H K : Subgroup G} : Commensurable H K → Commensurable K H 
Arguments: {G : Type u_1} [Group G] {H K : Subgroup G}
Goal: Commensurable H K → Commensurable K H
-- Commensurable.trans
@[trans]
theorem trans {H K L : Subgroup G} (hhk : Commensurable H K) (hkl : Commensurable K L) :
    Commensurable H L 
Arguments: {G : Type u_1} [Group G] {H K L : Subgroup G} (hhk : Commensurable H K) (hkl : Commensurable K L)
Goal: Commensurable H L
-- Commensurable.equivalence
theorem equivalence : Equivalence (@Commensurable G _) 
Arguments: {G : Type u_1} [Group G]
Goal: Equivalence Commensurable
-- Commensurable.commensurable_conj
theorem commensurable_conj {H K : Subgroup G} (g : ConjAct G) :
    Commensurable H K ↔ Commensurable (g • H) (g • K) 
Arguments: {G : Type u_1} [Group G] {H K : Subgroup G} (g : ConjAct G)
Goal: Commensurable H K ↔ Commensurable (g • H) (g • K)
-- Commensurable.commensurable_inv
theorem commensurable_inv (H : Subgroup G) (g : ConjAct G) :
    Commensurable (g • H) H ↔ Commensurable H (g⁻¹ • H) 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G) (g : ConjAct G)
Goal: Commensurable (g • H) H ↔ Commensurable H (g⁻¹ • H)
-- Commensurable.commensurator'_mem_iff
@[simp]
theorem commensurator'_mem_iff (H : Subgroup G) (g : ConjAct G) :
    g ∈ commensurator' H ↔ Commensurable (g • H) H 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G) (g : ConjAct G)
Goal: g ∈ Commensurable.commensurator' H ↔ Commensurable (g • H) H
-- Commensurable.commensurator_mem_iff
@[simp]
theorem commensurator_mem_iff (H : Subgroup G) (g : G) :
    g ∈ commensurator H ↔ Commensurable (ConjAct.toConjAct g • H) H 
Arguments: {G : Type u_1} [Group G] (H : Subgroup G) (g : G)
Goal: g ∈ Commensurable.commensurator H ↔ Commensurable (ConjAct.toConjAct g • H) H
-- Commensurable.eq
theorem eq {H K : Subgroup G} (hk : Commensurable H K) : commensurator H = commensurator K 
Arguments: {G : Type u_1} [Group G] {H K : Subgroup G} (hk : Commensurable H K)
Goal: Commensurable.commensurator H = Commensurable.commensurator K
-- ConjAct.card
@[simp]
theorem card [Fintype G] : Fintype.card (ConjAct G) = Fintype.card G 
Arguments: {G : Type u_3} [Fintype G]
Goal: Fintype.card (ConjAct G) = Fintype.card G
-- ConjAct.forall
@[simp]
theorem «forall» (p : ConjAct G → Prop) : (∀ x : ConjAct G, p x) ↔ ∀ x : G, p (toConjAct x) 
Arguments: {G : Type u_3} [DivInvMonoid G] (p : ConjAct G → Prop)
Goal: (∀ (x : ConjAct G), p x) ↔ ∀ (x : G), p (ConjAct.toConjAct x)
-- ConjAct.of_mul_symm_eq
@[simp]
theorem of_mul_symm_eq : (@ofConjAct G _).symm = toConjAct 
Arguments: {G : Type u_3} [DivInvMonoid G]
Goal: ConjAct.ofConjAct.symm = ConjAct.toConjAct
-- ConjAct.to_mul_symm_eq
@[simp]
theorem to_mul_symm_eq : (@toConjAct G _).symm = ofConjAct 
Arguments: {G : Type u_3} [DivInvMonoid G]
Goal: ConjAct.toConjAct.symm = ConjAct.ofConjAct
-- ConjAct.toConjAct_ofConjAct
@[simp]
theorem toConjAct_ofConjAct (x : ConjAct G) : toConjAct (ofConjAct x) = x 
Arguments: {G : Type u_3} [DivInvMonoid G] (x : ConjAct G)
Goal: ConjAct.toConjAct (ConjAct.ofConjAct x) = x
-- ConjAct.ofConjAct_toConjAct
@[simp]
theorem ofConjAct_toConjAct (x : G) : ofConjAct (toConjAct x) = x 
Arguments: {G : Type u_3} [DivInvMonoid G] (x : G)
Goal: ConjAct.ofConjAct (ConjAct.toConjAct x) = x
-- ConjAct.ofConjAct_one
theorem ofConjAct_one : ofConjAct (1 : ConjAct G) = 1 
Arguments: {G : Type u_3} [DivInvMonoid G]
Goal: ConjAct.ofConjAct 1 = 1
-- ConjAct.toConjAct_one
theorem toConjAct_one : toConjAct (1 : G) = 1 
Arguments: {G : Type u_3} [DivInvMonoid G]
Goal: ConjAct.toConjAct 1 = 1
-- ConjAct.ofConjAct_inv
@[simp]
theorem ofConjAct_inv (x : ConjAct G) : ofConjAct x⁻¹ = (ofConjAct x)⁻¹ 
Arguments: {G : Type u_3} [DivInvMonoid G] (x : ConjAct G)
Goal: ConjAct.ofConjAct x⁻¹ = (ConjAct.ofConjAct x)⁻¹
-- ConjAct.toConjAct_inv
@[simp]
theorem toConjAct_inv (x : G) : toConjAct x⁻¹ = (toConjAct x)⁻¹ 
Arguments: {G : Type u_3} [DivInvMonoid G] (x : G)
Goal: ConjAct.toConjAct x⁻¹ = (ConjAct.toConjAct x)⁻¹
-- ConjAct.ofConjAct_mul
theorem ofConjAct_mul (x y : ConjAct G) : ofConjAct (x * y) = ofConjAct x * ofConjAct y 
Arguments: {G : Type u_3} [DivInvMonoid G] (x y : ConjAct G)
Goal: ConjAct.ofConjAct (x * y) = ConjAct.ofConjAct x * ConjAct.ofConjAct y
-- ConjAct.toConjAct_mul
theorem toConjAct_mul (x y : G) : toConjAct (x * y) = toConjAct x * toConjAct y 
Arguments: {G : Type u_3} [DivInvMonoid G] (x y : G)
Goal: ConjAct.toConjAct (x * y) = ConjAct.toConjAct x * ConjAct.toConjAct y
-- ConjAct.smul_def
theorem smul_def (g : ConjAct G) (h : G) : g • h = ofConjAct g * h * (ofConjAct g)⁻¹ 
Arguments: {G : Type u_3} [DivInvMonoid G] (g : ConjAct G) (h : G)
Goal: g • h = ConjAct.ofConjAct g * h * (ConjAct.ofConjAct g)⁻¹
-- ConjAct.toConjAct_smul
theorem toConjAct_smul (g h : G) : toConjAct g • h = g * h * g⁻¹ 
Arguments: {G : Type u_3} [DivInvMonoid G] (g h : G)
Goal: ConjAct.toConjAct g • h = g * h * g⁻¹
-- ConjAct.units_smul_def
theorem units_smul_def (g : ConjAct Mˣ) (h : M) : g • h = ofConjAct g * h * ↑(ofConjAct g)⁻¹ 
Arguments: {M : Type u_2} [Monoid M] (g : ConjAct Mˣ) (h : M)
Goal: g • h = ↑(ConjAct.ofConjAct g) * h * ↑(ConjAct.ofConjAct g)⁻¹
-- ConjAct.ofConjAct_zero
theorem ofConjAct_zero : ofConjAct (0 : ConjAct G₀) = 0 
Arguments: {G₀ : Type u_4} [GroupWithZero G₀]
Goal: ConjAct.ofConjAct 0 = 0
-- ConjAct.toConjAct_zero
theorem toConjAct_zero : toConjAct (0 : G₀) = 0 
Arguments: {G₀ : Type u_4} [GroupWithZero G₀]
Goal: ConjAct.toConjAct 0 = 0
-- ConjAct.smul_eq_mulAut_conj
theorem smul_eq_mulAut_conj (g : ConjAct G) (h : G) : g • h = MulAut.conj (ofConjAct g) h 
Arguments: {G : Type u_3} [Group G] (g : ConjAct G) (h : G)
Goal: g • h = (MulAut.conj (ConjAct.ofConjAct g)) h
-- ConjAct.fixedPoints_eq_center
theorem fixedPoints_eq_center : fixedPoints (ConjAct G) G = center G 
Arguments: {G : Type u_3} [Group G]
Goal: MulAction.fixedPoints (ConjAct G) G = ↑(Subgroup.center G)
-- ConjAct.mem_orbit_conjAct
@[simp]
theorem mem_orbit_conjAct {g h : G} : g ∈ orbit (ConjAct G) h ↔ IsConj g h 
Arguments: {G : Type u_3} [Group G] {g h : G}
Goal: g ∈ MulAction.orbit (ConjAct G) h ↔ IsConj g h
-- ConjAct.orbitRel_conjAct
theorem orbitRel_conjAct : (orbitRel (ConjAct G) G).Rel = IsConj 
Arguments: {G : Type u_3} [Group G]
Goal: (MulAction.orbitRel (ConjAct G) G).Rel = IsConj
-- ConjAct.orbit_eq_carrier_conjClasses
theorem orbit_eq_carrier_conjClasses (g : G) :
    orbit (ConjAct G) g = (ConjClasses.mk g).carrier 
Arguments: {G : Type u_3} [Group G] (g : G)
Goal: MulAction.orbit (ConjAct G) g = (ConjClasses.mk g).carrier
-- ConjAct.stabilizer_eq_centralizer
theorem stabilizer_eq_centralizer (g : G) :
    stabilizer (ConjAct G) g = centralizer (zpowers (toConjAct g) : Set (ConjAct G)) 
Arguments: {G : Type u_3} [Group G] (g : G)
Goal: MulAction.stabilizer (ConjAct G) g = Subgroup.centralizer ↑(Subgroup.zpowers (ConjAct.toConjAct g))
-- ConjAct.Subgroup.val_conj_smul
theorem Subgroup.val_conj_smul {H : Subgroup G} [H.Normal] (g : ConjAct G) (h : H) :
    ↑(g • h) = g • (h : G) 
Arguments: {G : Type u_3} [Group G] {H : Subgroup G} [H.Normal] (g : ConjAct G) (h : ↥H)
Goal: ↑(g • h) = g • ↑h
-- MulAut.conjNormal_apply
@[simp]
theorem _root_.MulAut.conjNormal_apply {H : Subgroup G} [H.Normal] (g : G) (h : H) :
    ↑(MulAut.conjNormal g h) = g * h * g⁻¹ 
Arguments: {G : Type u_3} [Group G] {H : Subgroup G} [H.Normal] (g : G) (h : ↥H)
Goal: ↑((MulAut.conjNormal g) h) = g * ↑h * g⁻¹
-- MulAut.conjNormal_symm_apply
@[simp]
theorem _root_.MulAut.conjNormal_symm_apply {H : Subgroup G} [H.Normal] (g : G) (h : H) :
    ↑((MulAut.conjNormal g).symm h) = g⁻¹ * h * g 
Arguments: {G : Type u_3} [Group G] {H : Subgroup G} [H.Normal] (g : G) (h : ↥H)
Goal: ↑((MulEquiv.symm (MulAut.conjNormal g)) h) = g⁻¹ * ↑h * g
-- MulAut.conjNormal_inv_apply
@[simp]
theorem _root_.MulAut.conjNormal_inv_apply {H : Subgroup G} [H.Normal] (g : G) (h : H) :
    ↑((MulAut.conjNormal g)⁻¹ h) = g⁻¹ * h * g 
Arguments: {G : Type u_3} [Group G] {H : Subgroup G} [H.Normal] (g : G) (h : ↥H)
Goal: ↑((MulAut.conjNormal g)⁻¹ h) = g⁻¹ * ↑h * g
-- MulAut.conjNormal_val
theorem _root_.MulAut.conjNormal_val {H : Subgroup G} [H.Normal] {h : H} :
    MulAut.conjNormal ↑h = MulAut.conj h 
Arguments: {G : Type u_3} [Group G] {H : Subgroup G} [H.Normal] {h : ↥H}
Goal: MulAut.conjNormal ↑h = MulAut.conj h
-- val_unitsCentralizerEquiv_symm_apply_coe
@[simps! apply_coe_val symm_apply_val_coe]
def unitsCentralizerEquiv (x : Mˣ) :
    (Submonoid.centralizer ({↑x} : Set M))ˣ ≃* MulAction.stabilizer (ConjAct Mˣ) x 
Arguments: (M : Type u_2) [Monoid M] (x : Mˣ)
Goal: ∀ (a : ↥(MulAction.stabilizer (ConjAct Mˣ) x)), ↑↑((unitsCentralizerEquiv M x).symm a) = ↑(ConjAct.ofConjAct ↑a)
-- val_unitsCentralizerEquiv_apply_coe
@[simps! apply_coe_val symm_apply_val_coe]
def unitsCentralizerEquiv (x : Mˣ) :
    (Submonoid.centralizer ({↑x} : Set M))ˣ ≃* MulAction.stabilizer (ConjAct Mˣ) x 
Arguments: (M : Type u_2) [Monoid M] (x : Mˣ)
Goal: ∀ (a : (↥(Submonoid.centralizer {↑x}))ˣ), ↑↑((unitsCentralizerEquiv M x) a) = ↑↑a
-- MulAction.QuotientAction.inv_mul_mem
class QuotientAction : Prop where
    inv_mul_mem : ∀ (b : β) {a a' : α}, a⁻¹ * a' ∈ H → (b • a)⁻¹ * b • a' ∈ H
Arguments: {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] {H : Subgroup α} [self : MulAction.QuotientAction β H] (b : β) {a a' : α}
Goal: a⁻¹ * a' ∈ H → (b • a)⁻¹ * b • a' ∈ H
-- AddAction.QuotientAction.inv_mul_mem
class _root_.AddAction.QuotientAction {α : Type u} (β : Type v) [AddGroup α] [AddMonoid β]
  [AddAction β α] (H : AddSubgroup α) : Prop where
    inv_mul_mem : ∀ (b : β) {a a' : α}, -a + a' ∈ H → -(b +ᵥ a) + (b +ᵥ a') ∈ H
Arguments: {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] {H : AddSubgroup α} [self : AddAction.QuotientAction β H] (b : β) {a a' : α}
Goal: -a + a' ∈ H → -(b +ᵥ a) + (b +ᵥ a') ∈ H
-- MulAction.Quotient.smul_mk
@[to_additive (attr := simp)]
theorem Quotient.smul_mk [QuotientAction β H] (b : β) (a : α) :
    (b • QuotientGroup.mk a : α ⧸ H) = QuotientGroup.mk (b • a) 
Arguments: {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (a : α)
Goal: b • ↑a = ↑(b • a)
-- AddAction.Quotient.vadd_mk
@[to_additive (attr := simp)]
theorem Quotient.smul_mk [QuotientAction β H] (b : β) (a : α) :
    (b • QuotientGroup.mk a : α ⧸ H) = QuotientGroup.mk (b • a) 
Arguments: {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (a : α)
Goal: b +ᵥ ↑a = ↑(b +ᵥ a)
-- MulAction.Quotient.smul_coe
@[to_additive (attr := simp)]
theorem Quotient.smul_coe [QuotientAction β H] (b : β) (a : α) :
    b • (a : α ⧸ H) = (↑(b • a) : α ⧸ H) 
Arguments: {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (a : α)
Goal: b • ↑a = ↑(b • a)
-- AddAction.Quotient.vadd_coe
@[to_additive (attr := simp)]
theorem Quotient.smul_coe [QuotientAction β H] (b : β) (a : α) :
    b • (a : α ⧸ H) = (↑(b • a) : α ⧸ H) 
Arguments: {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (a : α)
Goal: b +ᵥ ↑a = ↑(b +ᵥ a)