ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Equiv.Perm.ofSubtype_apply_of_not_mem
theorem ofSubtype_apply_of_not_mem (f : Perm (Subtype p)) (ha : ¬p a) : ofSubtype f a = a 
Arguments: {α : Type u} {p : α → Prop} [DecidablePred p] {a : α} (f : Equiv.Perm (Subtype p)) (ha : ¬p a)
Goal: (Equiv.Perm.ofSubtype f) a = a
-- Equiv.Perm.mem_iff_ofSubtype_apply_mem
theorem mem_iff_ofSubtype_apply_mem (f : Perm (Subtype p)) (x : α) :
    p x ↔ p ((ofSubtype f : α → α) x) 
Arguments: {α : Type u} {p : α → Prop} [DecidablePred p] (f : Equiv.Perm (Subtype p)) (x : α)
Goal: p x ↔ p ((Equiv.Perm.ofSubtype f) x)
-- Equiv.Perm.subtypePerm_ofSubtype
@[simp]
theorem subtypePerm_ofSubtype (f : Perm (Subtype p)) :
    subtypePerm (ofSubtype f) (mem_iff_ofSubtype_apply_mem f) = f 
Arguments: {α : Type u} {p : α → Prop} [DecidablePred p] (f : Equiv.Perm (Subtype p))
Goal: (Equiv.Perm.ofSubtype f).subtypePerm ⋯ = f
-- Equiv.Perm.subtypeEquivSubtypePerm_apply_coe
@[simps]
protected def subtypeEquivSubtypePerm (p : α → Prop) [DecidablePred p] :
    Perm (Subtype p) ≃ { f : Perm α // ∀ a, ¬p a → f a = a } 
Arguments: {α : Type u} (p : α → Prop) [DecidablePred p] (f : Equiv.Perm (Subtype p))
Goal: ↑((Equiv.Perm.subtypeEquivSubtypePerm p) f) = Equiv.Perm.ofSubtype f
-- Equiv.Perm.subtypeEquivSubtypePerm_symm_apply
@[simps]
protected def subtypeEquivSubtypePerm (p : α → Prop) [DecidablePred p] :
    Perm (Subtype p) ≃ { f : Perm α // ∀ a, ¬p a → f a = a } 
Arguments: {α : Type u} (p : α → Prop) [DecidablePred p] (f : { f // ∀ (a : α), ¬p a → f a = a })
Goal: (Equiv.Perm.subtypeEquivSubtypePerm p).symm f = (↑f).subtypePerm ⋯
-- Equiv.Perm.subtypeEquivSubtypePerm_apply_of_mem
theorem subtypeEquivSubtypePerm_apply_of_mem (f : Perm (Subtype p)) (h : p a) :
    -- Porting note: was `Perm.subtypeEquivSubtypePerm p f a`
    ((Perm.subtypeEquivSubtypePerm p).toFun f).1 a = f ⟨a, h⟩ 
Arguments: {α : Type u} {p : α → Prop} [DecidablePred p] {a : α} (f : Equiv.Perm (Subtype p)) (h : p a)
Goal: ↑((Equiv.Perm.subtypeEquivSubtypePerm p).toFun f) a = ↑(f ⟨a, h⟩)
-- Equiv.Perm.subtypeEquivSubtypePerm_apply_of_not_mem
theorem subtypeEquivSubtypePerm_apply_of_not_mem (f : Perm (Subtype p)) (h : ¬p a) :
    -- Porting note: was `Perm.subtypeEquivSubtypePerm p f a`
    ((Perm.subtypeEquivSubtypePerm p).toFun f).1 a = a 
Arguments: {α : Type u} {p : α → Prop} [DecidablePred p] {a : α} (f : Equiv.Perm (Subtype p)) (h : ¬p a)
Goal: ↑((Equiv.Perm.subtypeEquivSubtypePerm p).toFun f) a = a
-- Equiv.swap_inv
@[simp]
theorem swap_inv (x y : α) : (swap x y)⁻¹ = swap x y 
Arguments: {α : Type u} [DecidableEq α] (x y : α)
Goal: (Equiv.swap x y)⁻¹ = Equiv.swap x y
-- Equiv.swap_mul_self
@[simp]
theorem swap_mul_self (i j : α) : swap i j * swap i j = 1 
Arguments: {α : Type u} [DecidableEq α] (i j : α)
Goal: Equiv.swap i j * Equiv.swap i j = 1
-- Equiv.swap_mul_eq_mul_swap
theorem swap_mul_eq_mul_swap (f : Perm α) (x y : α) : swap x y * f = f * swap (f⁻¹ x) (f⁻¹ y) 
Arguments: {α : Type u} [DecidableEq α] (f : Equiv.Perm α) (x y : α)
Goal: Equiv.swap x y * f = f * Equiv.swap (f⁻¹ x) (f⁻¹ y)
-- Equiv.mul_swap_eq_swap_mul
theorem mul_swap_eq_swap_mul (f : Perm α) (x y : α) : f * swap x y = swap (f x) (f y) * f 
Arguments: {α : Type u} [DecidableEq α] (f : Equiv.Perm α) (x y : α)
Goal: f * Equiv.swap x y = Equiv.swap (f x) (f y) * f
-- Equiv.swap_apply_apply
theorem swap_apply_apply (f : Perm α) (x y : α) : swap (f x) (f y) = f * swap x y * f⁻¹ 
Arguments: {α : Type u} [DecidableEq α] (f : Equiv.Perm α) (x y : α)
Goal: Equiv.swap (f x) (f y) = f * Equiv.swap x y * f⁻¹
-- Equiv.swap_smul_self_smul
@[simp]
theorem swap_smul_self_smul [MulAction (Perm α) β] (i j : α) (x : β) :
    swap i j • swap i j • x = x 
Arguments: {α : Type u} {β : Type v} [DecidableEq α] [MulAction (Equiv.Perm α) β] (i j : α) (x : β)
Goal: Equiv.swap i j • Equiv.swap i j • x = x
-- Equiv.swap_smul_involutive
theorem swap_smul_involutive [MulAction (Perm α) β] (i j : α) :
    Function.Involutive (swap i j • · : β → β) 
Arguments: {α : Type u} {β : Type v} [DecidableEq α] [MulAction (Equiv.Perm α) β] (i j : α)
Goal: Function.Involutive fun x => Equiv.swap i j • x
-- Equiv.swap_mul_self_mul
@[simp]
theorem swap_mul_self_mul (i j : α) (σ : Perm α) : Equiv.swap i j * (Equiv.swap i j * σ) = σ 
Arguments: {α : Type u} [DecidableEq α] (i j : α) (σ : Equiv.Perm α)
Goal: Equiv.swap i j * (Equiv.swap i j * σ) = σ
-- Equiv.mul_swap_mul_self
@[simp]
theorem mul_swap_mul_self (i j : α) (σ : Perm α) : σ * Equiv.swap i j * Equiv.swap i j = σ 
Arguments: {α : Type u} [DecidableEq α] (i j : α) (σ : Equiv.Perm α)
Goal: σ * Equiv.swap i j * Equiv.swap i j = σ
-- Equiv.swap_mul_involutive
@[simp]
theorem swap_mul_involutive (i j : α) : Function.Involutive (Equiv.swap i j * ·) 
Arguments: {α : Type u} [DecidableEq α] (i j : α)
Goal: Function.Involutive fun x => Equiv.swap i j * x
-- Equiv.mul_swap_involutive
@[simp]
theorem mul_swap_involutive (i j : α) : Function.Involutive (· * Equiv.swap i j) 
Arguments: {α : Type u} [DecidableEq α] (i j : α)
Goal: Function.Involutive fun x => x * Equiv.swap i j
-- Equiv.swap_eq_one_iff
@[simp]
theorem swap_eq_one_iff {i j : α} : swap i j = (1 : Perm α) ↔ i = j 
Arguments: {α : Type u} [DecidableEq α] {i j : α}
Goal: Equiv.swap i j = 1 ↔ i = j
-- Equiv.swap_mul_eq_iff
theorem swap_mul_eq_iff {i j : α} {σ : Perm α} : swap i j * σ = σ ↔ i = j 
Arguments: {α : Type u} [DecidableEq α] {i j : α} {σ : Equiv.Perm α}
Goal: Equiv.swap i j * σ = σ ↔ i = j
-- Equiv.mul_swap_eq_iff
theorem mul_swap_eq_iff {i j : α} {σ : Perm α} : σ * swap i j = σ ↔ i = j 
Arguments: {α : Type u} [DecidableEq α] {i j : α} {σ : Equiv.Perm α}
Goal: σ * Equiv.swap i j = σ ↔ i = j
-- Equiv.swap_mul_swap_mul_swap
theorem swap_mul_swap_mul_swap {x y z : α} (hxy : x ≠ y) (hxz : x ≠ z) :
    swap y z * swap x y * swap y z = swap z x 
Arguments: {α : Type u} [DecidableEq α] {x y z : α} (hxy : x ≠ y) (hxz : x ≠ z)
Goal: Equiv.swap y z * Equiv.swap x y * Equiv.swap y z = Equiv.swap z x
-- Equiv.addLeft_zero
@[simp] lemma addLeft_zero : Equiv.addLeft (0 : α) = 1 
Arguments: {α : Type u} [AddGroup α]
Goal: Equiv.addLeft 0 = 1
-- Equiv.addRight_zero
@[simp] lemma addRight_zero : Equiv.addRight (0 : α) = 1 
Arguments: {α : Type u} [AddGroup α]
Goal: Equiv.addRight 0 = 1
-- Equiv.addLeft_add
@[simp] lemma addLeft_add : Equiv.addLeft (a + b) = Equiv.addLeft a * Equiv.addLeft b 
Arguments: {α : Type u} [AddGroup α] (a b : α)
Goal: Equiv.addLeft (a + b) = Equiv.addLeft a * Equiv.addLeft b
-- Equiv.addRight_add
@[simp] lemma addRight_add : Equiv.addRight (a + b) = Equiv.addRight b * Equiv.addRight a 
Arguments: {α : Type u} [AddGroup α] (a b : α)
Goal: Equiv.addRight (a + b) = Equiv.addRight b * Equiv.addRight a
-- Equiv.inv_addLeft
@[simp] lemma inv_addLeft : (Equiv.addLeft a)⁻¹ = Equiv.addLeft (-a) 
Arguments: {α : Type u} [AddGroup α] (a : α)
Goal: (Equiv.addLeft a)⁻¹ = Equiv.addLeft (-a)
-- Equiv.inv_addRight
@[simp] lemma inv_addRight : (Equiv.addRight a)⁻¹ = Equiv.addRight (-a) 
Arguments: {α : Type u} [AddGroup α] (a : α)
Goal: (Equiv.addRight a)⁻¹ = Equiv.addRight (-a)
-- Equiv.pow_addLeft
@[simp] lemma pow_addLeft (n : ℕ) : Equiv.addLeft a ^ n = Equiv.addLeft (n • a) 
Arguments: {α : Type u} [AddGroup α] (a : α) (n : ℕ)
Goal: Equiv.addLeft a ^ n = Equiv.addLeft (n • a)
-- Equiv.pow_addRight
@[simp] lemma pow_addRight (n : ℕ) : Equiv.addRight a ^ n = Equiv.addRight (n • a) 
Arguments: {α : Type u} [AddGroup α] (a : α) (n : ℕ)
Goal: Equiv.addRight a ^ n = Equiv.addRight (n • a)
-- Equiv.zpow_addLeft
@[simp] lemma zpow_addLeft (n : ℤ) : Equiv.addLeft a ^ n = Equiv.addLeft (n • a) 
Arguments: {α : Type u} [AddGroup α] (a : α) (n : ℤ)
Goal: Equiv.addLeft a ^ n = Equiv.addLeft (n • a)
-- Equiv.zpow_addRight
@[simp] lemma zpow_addRight : ∀ (n : ℤ), Equiv.addRight a ^ n = Equiv.addRight (n • a)
  
Arguments: {α : Type u} [AddGroup α] (a : α) (n : ℤ)
Goal: Equiv.addRight a ^ n = Equiv.addRight (n • a)
-- Equiv.mulLeft_one
@[to_additive existing (attr := simp)]
lemma mulLeft_one : Equiv.mulLeft (1 : α) = 1 
Arguments: {α : Type u} [Group α]
Goal: Equiv.mulLeft 1 = 1
-- Equiv.mulRight_one
@[to_additive existing (attr := simp)]
lemma mulRight_one : Equiv.mulRight (1 : α) = 1 
Arguments: {α : Type u} [Group α]
Goal: Equiv.mulRight 1 = 1
-- Equiv.mulLeft_mul
@[to_additive existing (attr := simp)]
lemma mulLeft_mul : Equiv.mulLeft (a * b) = Equiv.mulLeft a * Equiv.mulLeft b 
Arguments: {α : Type u} [Group α] (a b : α)
Goal: Equiv.mulLeft (a * b) = Equiv.mulLeft a * Equiv.mulLeft b
-- Equiv.mulRight_mul
@[to_additive existing (attr := simp)]
lemma mulRight_mul : Equiv.mulRight (a * b) = Equiv.mulRight b * Equiv.mulRight a 
Arguments: {α : Type u} [Group α] (a b : α)
Goal: Equiv.mulRight (a * b) = Equiv.mulRight b * Equiv.mulRight a
-- Equiv.inv_mulLeft
@[to_additive existing (attr := simp) inv_addLeft]
lemma inv_mulLeft : (Equiv.mulLeft a)⁻¹ = Equiv.mulLeft a⁻¹ 
Arguments: {α : Type u} [Group α] (a : α)
Goal: (Equiv.mulLeft a)⁻¹ = Equiv.mulLeft a⁻¹
-- Equiv.inv_mulRight
@[to_additive existing (attr := simp) inv_addRight]
lemma inv_mulRight : (Equiv.mulRight a)⁻¹ = Equiv.mulRight a⁻¹ 
Arguments: {α : Type u} [Group α] (a : α)
Goal: (Equiv.mulRight a)⁻¹ = Equiv.mulRight a⁻¹
-- Equiv.pow_mulLeft
@[to_additive existing (attr := simp) pow_addLeft]
lemma pow_mulLeft (n : ℕ) : Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n) 
Arguments: {α : Type u} [Group α] (a : α) (n : ℕ)
Goal: Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n)
-- Equiv.pow_mulRight
@[to_additive existing (attr := simp) pow_addRight]
lemma pow_mulRight (n : ℕ) : Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n) 
Arguments: {α : Type u} [Group α] (a : α) (n : ℕ)
Goal: Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n)
-- Equiv.zpow_mulLeft
@[to_additive existing (attr := simp) zpow_addLeft]
lemma zpow_mulLeft (n : ℤ) : Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n) 
Arguments: {α : Type u} [Group α] (a : α) (n : ℤ)
Goal: Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n)
-- Equiv.zpow_mulRight
@[to_additive existing (attr := simp) zpow_addRight]
lemma zpow_mulRight : ∀ n : ℤ, Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n)
  
Arguments: {α : Type u} [Group α] (a : α) (n : ℤ)
Goal: Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n)
-- Set.BijOn.perm_inv
lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s 
Arguments: {α : Type u_1} {f : Equiv.Perm α} {s : Set α} (hf : Set.BijOn (⇑f) s s)
Goal: Set.BijOn (⇑f⁻¹) s s
-- Set.MapsTo.perm_pow
lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s 
Arguments: {α : Type u_1} {f : Equiv.Perm α} {s : Set α}
Goal: Set.MapsTo (⇑f) s s → ∀ (n : ℕ), Set.MapsTo (⇑(f ^ n)) s s
-- Set.SurjOn.perm_pow
lemma SurjOn.perm_pow : SurjOn f s s → ∀ n : ℕ, SurjOn (f ^ n) s s 
Arguments: {α : Type u_1} {f : Equiv.Perm α} {s : Set α}
Goal: Set.SurjOn (⇑f) s s → ∀ (n : ℕ), Set.SurjOn (⇑(f ^ n)) s s
-- Set.BijOn.perm_pow
lemma BijOn.perm_pow : BijOn f s s → ∀ n : ℕ, BijOn (f ^ n) s s 
Arguments: {α : Type u_1} {f : Equiv.Perm α} {s : Set α}
Goal: Set.BijOn (⇑f) s s → ∀ (n : ℕ), Set.BijOn (⇑(f ^ n)) s s
-- Set.BijOn.perm_zpow
lemma BijOn.perm_zpow (hf : BijOn f s s) : ∀ n : ℤ, BijOn (f ^ n) s s
  
Arguments: {α : Type u_1} {f : Equiv.Perm α} {s : Set α} (hf : Set.BijOn (⇑f) s s) (n : ℤ)
Goal: Set.BijOn (⇑(f ^ n)) s s
-- IsAddCyclic.exists_generator
class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
  exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
Arguments: {α : Type u} [AddGroup α] [self : IsAddCyclic α]
Goal: ∃ g, ∀ (x : α), x ∈ AddSubgroup.zmultiples g
-- IsCyclic.exists_generator
@[to_additive]
class IsCyclic (α : Type u) [Group α] : Prop where
  exists_generator : ∃ g : α, ∀ x, x ∈ zpowers g
Arguments: {α : Type u} [Group α] [self : IsCyclic α]
Goal: ∃ g, ∀ (x : α), x ∈ Subgroup.zpowers g
-- isCyclic_multiplicative_iff
@[simp]
theorem isCyclic_multiplicative_iff [AddGroup α] : IsCyclic (Multiplicative α) ↔ IsAddCyclic α 
Arguments: {α : Type u} [AddGroup α]
Goal: IsCyclic (Multiplicative α) ↔ IsAddCyclic α
-- isAddCyclic_additive_iff
@[simp]
theorem isAddCyclic_additive_iff [Group α] : IsAddCyclic (Additive α) ↔ IsCyclic α 
Arguments: {α : Type u} [Group α]
Goal: IsAddCyclic (Additive α) ↔ IsCyclic α
-- Nontrivial.of_not_isCyclic
@[to_additive "A non-cyclic additive group is non-trivial."]
theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α 
Arguments: {α : Type u} [Group α] (nc : ¬IsCyclic α)
Goal: Nontrivial α
-- Nontrivial.of_not_isAddCyclic
@[to_additive "A non-cyclic additive group is non-trivial."]
theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α 
Arguments: {α : Type u} [AddGroup α] (nc : ¬IsAddCyclic α)
Goal: Nontrivial α
-- MonoidHom.map_cyclic
@[to_additive]
theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
    ∃ m : ℤ, ∀ g : G, σ g = g ^ m 
Arguments: {G : Type u_1} [Group G] [h : IsCyclic G] (σ : G →* G)
Goal: ∃ m, ∀ (g : G), σ g = g ^ m
-- AddMonoidHom.map_addCyclic
@[to_additive]
theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
    ∃ m : ℤ, ∀ g : G, σ g = g ^ m 
Arguments: {G : Type u_1} [AddGroup G] [h : IsAddCyclic G] (σ : G →+ G)
Goal: ∃ m, ∀ (g : G), σ g = m • g
-- isCyclic_of_orderOf_eq_card
@[to_additive]
theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
    IsCyclic α 
Arguments: {α : Type u} [Group α] [Fintype α] (x : α) (hx : orderOf x = Fintype.card α)
Goal: IsCyclic α
-- isAddCyclic_of_addOrderOf_eq_card
@[to_additive]
theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
    IsCyclic α 
Arguments: {α : Type u} [AddGroup α] [Fintype α] (x : α) (hx : addOrderOf x = Fintype.card α)
Goal: IsAddCyclic α
-- Subgroup.eq_bot_or_eq_top_of_prime_card
@[to_additive]
theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
    (H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ 
Arguments: {G : Type u_1} [Group G]
Goal: ∀ {x : Fintype G} (H : Subgroup G) [hp : Fact (Nat.Prime (Fintype.card G))], H = ⊥ ∨ H = ⊤
-- AddSubgroup.eq_bot_or_eq_top_of_prime_card
@[to_additive]
theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
    (H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ 
Arguments: {G : Type u_1} [AddGroup G]
Goal: ∀ {x : Fintype G} (H : AddSubgroup G) [hp : Fact (Nat.Prime (Fintype.card G))], H = ⊥ ∨ H = ⊤
-- zmultiples_eq_top_of_prime_card
@[to_additive "Any non-identity element of a finite group of prime order generates the group."]
theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : zpowers g = ⊤ 
Arguments: {G : Type u_1} [AddGroup G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)],
  Fintype.card G = p → ∀ {g : G}, g ≠ 0 → AddSubgroup.zmultiples g = ⊤
-- zpowers_eq_top_of_prime_card
@[to_additive "Any non-identity element of a finite group of prime order generates the group."]
theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : zpowers g = ⊤ 
Arguments: {G : Type u_1} [Group G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)], Fintype.card G = p → ∀ {g : G}, g ≠ 1 → Subgroup.zpowers g = ⊤
-- mem_zpowers_of_prime_card
@[to_additive]
theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g 
Arguments: {G : Type u_1} [Group G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)], Fintype.card G = p → ∀ {g g' : G}, g ≠ 1 → g' ∈ Subgroup.zpowers g
-- mem_zmultiples_of_prime_card
@[to_additive]
theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g 
Arguments: {G : Type u_1} [AddGroup G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)],
  Fintype.card G = p → ∀ {g g' : G}, g ≠ 0 → g' ∈ AddSubgroup.zmultiples g
-- mem_powers_of_prime_card
@[to_additive]
theorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g 
Arguments: {G : Type u_1} [Group G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)], Fintype.card G = p → ∀ {g g' : G}, g ≠ 1 → g' ∈ Submonoid.powers g
-- mem_multiples_of_prime_card
@[to_additive]
theorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g 
Arguments: {G : Type u_1} [AddGroup G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)],
  Fintype.card G = p → ∀ {g g' : G}, g ≠ 0 → g' ∈ AddSubmonoid.multiples g
-- multiples_eq_top_of_prime_card
@[to_additive]
theorem powers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : Submonoid.powers g = ⊤ 
Arguments: {G : Type u_1} [AddGroup G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)],
  Fintype.card G = p → ∀ {g : G}, g ≠ 0 → AddSubmonoid.multiples g = ⊤
-- powers_eq_top_of_prime_card
@[to_additive]
theorem powers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : Submonoid.powers g = ⊤ 
Arguments: {G : Type u_1} [Group G]
Goal: ∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)], Fintype.card G = p → ∀ {g : G}, g ≠ 1 → Submonoid.powers g = ⊤
-- isAddCyclic_of_prime_card
@[to_additive "A finite group of prime order is cyclic."]
theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card α = p) : IsCyclic α 
Arguments: {α : Type u} [AddGroup α] [Fintype α] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : Fintype.card α = p)
Goal: IsAddCyclic α
-- isCyclic_of_prime_card
@[to_additive "A finite group of prime order is cyclic."]
theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
    (h : Fintype.card α = p) : IsCyclic α 
Arguments: {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : Fintype.card α = p)
Goal: IsCyclic α
-- isAddCyclic_of_card_dvd_prime
@[to_additive "A finite group of order dividing a prime is cyclic."]
theorem isCyclic_of_card_dvd_prime {α : Type u} [Group α] {p : ℕ} [hp : Fact p.Prime]
    (h : Nat.card α ∣ p) : IsCyclic α 
Arguments: {α : Type u} [AddGroup α] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : Nat.card α ∣ p)
Goal: IsAddCyclic α
-- isCyclic_of_card_dvd_prime
@[to_additive "A finite group of order dividing a prime is cyclic."]
theorem isCyclic_of_card_dvd_prime {α : Type u} [Group α] {p : ℕ} [hp : Fact p.Prime]
    (h : Nat.card α ∣ p) : IsCyclic α 
Arguments: {α : Type u} [Group α] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : Nat.card α ∣ p)
Goal: IsCyclic α
-- isCyclic_of_surjective
@[to_additive]
theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) :
    IsCyclic G 
Arguments: {H : Type u_1} {G : Type u_2} {F : Type u_3} [Group H] [Group G] [hH : IsCyclic H] [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective ⇑f)
Goal: IsCyclic G
-- isAddCyclic_of_surjective
@[to_additive]
theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) :
    IsCyclic G 
Arguments: {H : Type u_1} {G : Type u_2} {F : Type u_3} [AddGroup H] [AddGroup G] [hH : IsAddCyclic H] [FunLike F H G] [AddMonoidHomClass F H G] (f : F) (hf : Function.Surjective ⇑f)
Goal: IsAddCyclic G
-- addOrderOf_eq_card_of_forall_mem_zmultiples
@[to_additive]
theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
    orderOf g = Fintype.card α 
Arguments: {α : Type u} [AddGroup α] [Fintype α] {g : α} (hx : ∀ (x : α), x ∈ AddSubgroup.zmultiples g)
Goal: addOrderOf g = Fintype.card α
-- orderOf_eq_card_of_forall_mem_zpowers
@[to_additive]
theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
    orderOf g = Fintype.card α 
Arguments: {α : Type u} [Group α] [Fintype α] {g : α} (hx : ∀ (x : α), x ∈ Subgroup.zpowers g)
Goal: orderOf g = Fintype.card α
-- orderOf_generator_eq_natCard
@[to_additive]
lemma orderOf_generator_eq_natCard (h : ∀ x, x ∈ Subgroup.zpowers a) : orderOf a = Nat.card α 
Arguments: {α : Type u} {a : α} [Group α] (h : ∀ (x : α), x ∈ Subgroup.zpowers a)
Goal: orderOf a = Nat.card α
-- addOrderOf_generator_eq_natCard
@[to_additive]
lemma orderOf_generator_eq_natCard (h : ∀ x, x ∈ Subgroup.zpowers a) : orderOf a = Nat.card α 
Arguments: {α : Type u} {a : α} [AddGroup α] (h : ∀ (x : α), x ∈ AddSubgroup.zmultiples a)
Goal: addOrderOf a = Nat.card α
-- exists_nsmul_ne_zero_of_isAddCyclic
@[to_additive]
theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]
    {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 
Arguments: {G : Type u_1} [AddGroup G] [Fintype G] [G_cyclic : IsAddCyclic G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G)
Goal: ∃ a, k • a ≠ 0
-- exists_pow_ne_one_of_isCyclic
@[to_additive]
theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]
    {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 
Arguments: {G : Type u_1} [Group G] [Fintype G] [G_cyclic : IsCyclic G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G)
Goal: ∃ a, a ^ k ≠ 1
-- Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
@[to_additive]
theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 
Arguments: {α : Type u} [AddGroup α] [Infinite α] {g : α} (h : ∀ (x : α), x ∈ AddSubgroup.zmultiples g)
Goal: addOrderOf g = 0
-- Infinite.orderOf_eq_zero_of_forall_mem_zpowers
@[to_additive]
theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 
Arguments: {α : Type u} [Group α] [Infinite α] {g : α} (h : ∀ (x : α), x ∈ Subgroup.zpowers g)
Goal: orderOf g = 0
-- IsAddCyclic.card_nsmul_eq_zero_le
@[to_additive IsAddCyclic.card_nsmul_eq_zero_le]
theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
    (univ.filter fun a : α => a ^ n = 1).card ≤ n 
Arguments: {α : Type u} [AddGroup α] [DecidableEq α] [Fintype α] [IsAddCyclic α] {n : ℕ} (hn0 : 0 < n)
Goal: (Finset.filter (fun a => n • a = 0) Finset.univ).card ≤ n
-- IsCyclic.card_pow_eq_one_le
@[to_additive IsAddCyclic.card_nsmul_eq_zero_le]
theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
    (univ.filter fun a : α => a ^ n = 1).card ≤ n 
Arguments: {α : Type u} [Group α] [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n)
Goal: (Finset.filter (fun a => a ^ n = 1) Finset.univ).card ≤ n
-- IsCyclic.exists_monoid_generator
@[to_additive]
theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
    ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x 
Arguments: {α : Type u} [Group α] [Finite α] [IsCyclic α]
Goal: ∃ x, ∀ (y : α), y ∈ Submonoid.powers x
-- IsAddCyclic.exists_addMonoid_generator
@[to_additive]
theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
    ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x 
Arguments: {α : Type u} [AddGroup α] [Finite α] [IsAddCyclic α]
Goal: ∃ x, ∀ (y : α), y ∈ AddSubmonoid.multiples x
-- IsCyclic.exists_ofOrder_eq_natCard
@[to_additive]
lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf g = Nat.card α 
Arguments: {α : Type u} [Group α] [h : IsCyclic α]
Goal: ∃ g, orderOf g = Nat.card α
-- IsAddCyclic.exists_ofOrder_eq_natCard
@[to_additive]
lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf g = Nat.card α 
Arguments: {α : Type u} [AddGroup α] [h : IsAddCyclic α]
Goal: ∃ g, addOrderOf g = Nat.card α