ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DistribMulActionHom.congr_fun
protected theorem congr_fun {f g : A →ₑ+[φ] B} (h : f = g) (x : A) : f x = g x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f g : A →ₑ+[φ] B} (h : f = g) (x : A)
Goal: f x = g x
-- DistribMulActionHom.toMulActionHom_injective
theorem toMulActionHom_injective {f g : A →ₑ+[φ] B} (h : (f : A →ₑ[φ] B) = (g : A →ₑ[φ] B)) :
    f = g 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f g : A →ₑ+[φ] B} (h : ↑f = ↑g)
Goal: f = g
-- DistribMulActionHom.toAddMonoidHom_injective
theorem toAddMonoidHom_injective {f g : A →ₑ+[φ] B} (h : (f : A →+ B) = (g : A →+ B)) : f = g 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {f g : A →ₑ+[φ] B} (h : ↑f = ↑g)
Goal: f = g
-- DistribMulActionHom.map_zero
protected theorem map_zero (f : A →ₑ+[φ] B) : f 0 = 0 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B)
Goal: f 0 = 0
-- DistribMulActionHom.map_add
protected theorem map_add (f : A →ₑ+[φ] B) (x y : A) : f (x + y) = f x + f y 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) (x y : A)
Goal: f (x + y) = f x + f y
-- DistribMulActionHom.map_neg
protected theorem map_neg (f : A' →ₑ+[φ] B') (x : A') : f (-x) = -f x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (A' : Type u_8) [AddGroup A'] [DistribMulAction M A'] (B' : Type u_9) [AddGroup B'] [DistribMulAction N B'] (f : A' →ₑ+[φ] B') (x : A')
Goal: f (-x) = -f x
-- DistribMulActionHom.map_sub
protected theorem map_sub (f : A' →ₑ+[φ] B') (x y : A') : f (x - y) = f x - f y 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (A' : Type u_8) [AddGroup A'] [DistribMulAction M A'] (B' : Type u_9) [AddGroup B'] [DistribMulAction N B'] (f : A' →ₑ+[φ] B') (x y : A')
Goal: f (x - y) = f x - f y
-- DistribMulActionHom.map_smulₑ
protected theorem map_smulₑ (f : A →ₑ+[φ] B) (m : M) (x : A) : f (m • x) = (φ m) • f x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B) (m : M) (x : A)
Goal: f (m • x) = φ m • f x
-- DistribMulActionHom.id_apply
@[simp]
theorem id_apply (x : A) : DistribMulActionHom.id M x = x 
Arguments: (M : Type u_1) [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] (x : A)
Goal: (DistribMulActionHom.id M) x = x
-- DistribMulActionHom.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : A →ₑ+[φ] B) = 0 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B]
Goal: ⇑0 = 0
-- DistribMulActionHom.coe_one
@[simp]
theorem coe_one : ⇑(1 : A →+[M] A) = id 
Arguments: {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A]
Goal: ⇑1 = id
-- DistribMulActionHom.zero_apply
theorem zero_apply (a : A) : (0 : A →ₑ+[φ] B) a = 0 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (a : A)
Goal: 0 a = 0
-- DistribMulActionHom.one_apply
theorem one_apply (a : A) : (1 : A →+[M] A) a = a 
Arguments: {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] (a : A)
Goal: 1 a = a
-- DistribMulActionHom.comp_apply
@[simp]
theorem comp_apply
    (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [MonoidHom.CompTriple φ ψ χ] (x : A) : g.comp f x = g (f x) 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {C : Type u_7} [AddMonoid C] [DistribMulAction P C] (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [φ.CompTriple ψ χ] (x : A)
Goal: (g.comp f) x = g (f x)
-- DistribMulActionHom.id_comp
@[simp]
theorem id_comp (f : A →ₑ+[φ] B) : comp (DistribMulActionHom.id N) f = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B)
Goal: (DistribMulActionHom.id N).comp f = f
-- DistribMulActionHom.comp_id
@[simp]
theorem comp_id (f : A →ₑ+[φ] B) : f.comp (DistribMulActionHom.id M) = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] (f : A →ₑ+[φ] B)
Goal: f.comp (DistribMulActionHom.id M) = f
-- DistribMulActionHom.comp_assoc
@[simp]
theorem comp_assoc {Q D : Type*} [Monoid Q] [AddMonoid D] [DistribMulAction Q D]
    {η : P →* Q} {θ : M →* Q} {ζ : N →* Q}
    (h : C →ₑ+[η] D) (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B)
    [MonoidHom.CompTriple φ ψ χ] [MonoidHom.CompTriple χ η θ]
    [MonoidHom.CompTriple ψ η ζ] [MonoidHom.CompTriple φ ζ θ] :
    h.comp (g.comp f) = (h.comp g).comp f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B : Type u_5} [AddMonoid B] [DistribMulAction N B] {C : Type u_7} [AddMonoid C] [DistribMulAction P C] {Q : Type u_11} {D : Type u_12} [Monoid Q] [AddMonoid D] [DistribMulAction Q D] {η : P →* Q} {θ : M →* Q} {ζ : N →* Q} (h : C →ₑ+[η] D) (g : B →ₑ+[ψ] C) (f : A →ₑ+[φ] B) [φ.CompTriple ψ χ] [χ.CompTriple η θ] [ψ.CompTriple η ζ] [φ.CompTriple ζ θ]
Goal: h.comp (g.comp f) = (h.comp g).comp f
-- DistribMulActionHom.inverse_toFun
@[simps]
def inverse (f : A →+[M] B₁) (g : B₁ → A) (h₁ : Function.LeftInverse g f)
    (h₂ : Function.RightInverse g f) : B₁ →+[M] A 
Arguments: {M : Type u_1} [Monoid M] {A : Type u_4} [AddMonoid A] [DistribMulAction M A] {B₁ : Type u_6} [AddMonoid B₁] [DistribMulAction M B₁] (f : A →+[M] B₁) (g : B₁ → A) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f)
Goal: ∀ (a : B₁), (f.inverse g h₁ h₂) a = g a
-- DistribMulActionHom.ext_ring
@[ext]
theorem ext_ring {f g : R →ₑ+[σ] N'} (h : f 1 = g 1) : f = g 
Arguments: {R : Type u_11} [Semiring R] {S : Type u_13} [Semiring S] {N' : Type u_16} [AddMonoid N'] [DistribMulAction S N'] {σ : R →* S} {f g : R →ₑ+[σ] N'} (h : f 1 = g 1)
Goal: f = g
-- DistribMulActionHom.ext_ring_iff
theorem ext_ring_iff {f g : R →ₑ+[σ] N'} : f = g ↔ f 1 = g 1 
Arguments: {R : Type u_11} [Semiring R] {S : Type u_13} [Semiring S] {N' : Type u_16} [AddMonoid N'] [DistribMulAction S N'] {σ : R →* S} {f g : R →ₑ+[σ] N'}
Goal: f = g ↔ f 1 = g 1
-- MulSemiringActionHom.coe_fn_coe
@[norm_cast]
theorem coe_fn_coe (f : R →ₑ+*[φ] S) : ⇑(f : R →+* S) = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: ⇑↑f = ⇑f
-- MulSemiringActionHom.coe_fn_coe'
@[norm_cast]
theorem coe_fn_coe' (f : R →ₑ+*[φ] S) : ⇑(f : R →ₑ+[φ] S) = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: ⇑↑f = ⇑f
-- MulSemiringActionHom.ext
@[ext]
theorem ext {f g : R →ₑ+*[φ] S} : (∀ x, f x = g x) → f = g 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {f g : R →ₑ+*[φ] S}
Goal: (∀ (x : R), f x = g x) → f = g
-- MulSemiringActionHom.ext_iff
theorem ext_iff {f g : R →ₑ+*[φ] S} : f = g ↔ ∀ x, f x = g x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {f g : R →ₑ+*[φ] S}
Goal: f = g ↔ ∀ (x : R), f x = g x
-- MulSemiringActionHom.map_zero
protected theorem map_zero (f : R →ₑ+*[φ] S) : f 0 = 0 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: f 0 = 0
-- MulSemiringActionHom.map_add
protected theorem map_add (f : R →ₑ+*[φ] S) (x y : R) : f (x + y) = f x + f y 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (x y : R)
Goal: f (x + y) = f x + f y
-- MulSemiringActionHom.map_neg
protected theorem map_neg (f : R' →ₑ+*[φ] S') (x : R') : f (-x) = -f x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (R' : Type u_11) [Ring R'] [MulSemiringAction M R'] (S' : Type u_13) [Ring S'] [MulSemiringAction N S'] (f : R' →ₑ+*[φ] S') (x : R')
Goal: f (-x) = -f x
-- MulSemiringActionHom.map_sub
protected theorem map_sub (f : R' →ₑ+*[φ] S') (x y : R') : f (x - y) = f x - f y 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} (R' : Type u_11) [Ring R'] [MulSemiringAction M R'] (S' : Type u_13) [Ring S'] [MulSemiringAction N S'] (f : R' →ₑ+*[φ] S') (x y : R')
Goal: f (x - y) = f x - f y
-- MulSemiringActionHom.map_one
protected theorem map_one (f : R →ₑ+*[φ] S) : f 1 = 1 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: f 1 = 1
-- MulSemiringActionHom.map_mul
protected theorem map_mul (f : R →ₑ+*[φ] S) (x y : R) : f (x * y) = f x * f y 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (x y : R)
Goal: f (x * y) = f x * f y
-- MulSemiringActionHom.map_smulₛₗ
protected theorem map_smulₛₗ (f : R →ₑ+*[φ] S) (m : M) (x : R) : f (m • x) = φ m • f x 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (m : M) (x : R)
Goal: f (m • x) = φ m • f x
-- MulSemiringActionHom.map_smul
protected theorem map_smul [MulSemiringAction M S] (f : R →+*[M] S) (m : M) (x : R) :
    f (m • x) = m • f x 
Arguments: {M : Type u_1} [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction M S] (f : R →+*[M] S) (m : M) (x : R)
Goal: f (m • x) = m • f x
-- MulSemiringActionHom.id_apply
@[simp]
theorem id_apply (x : R) : MulSemiringActionHom.id M x = x 
Arguments: (M : Type u_1) [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] (x : R)
Goal: (MulSemiringActionHom.id M) x = x
-- MulSemiringActionHom.comp_apply
@[simp]
theorem comp_apply (g : S →ₑ+*[ψ] T) (f : R →ₑ+*[φ] S) [MonoidHom.CompTriple φ ψ χ] (x : R) :
    g.comp f x = g (f x) 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {P : Type u_3} [Monoid P] {φ : M →* N} {ψ : N →* P} {χ : M →* P} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] {T : Type u_14} [Semiring T] [MulSemiringAction P T] (g : S →ₑ+*[ψ] T) (f : R →ₑ+*[φ] S) [φ.CompTriple ψ χ] (x : R)
Goal: (g.comp f) x = g (f x)
-- MulSemiringActionHom.id_comp
@[simp]
theorem id_comp (f : R →ₑ+*[φ] S) : (MulSemiringActionHom.id N).comp f = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: (MulSemiringActionHom.id N).comp f = f
-- MulSemiringActionHom.comp_id
@[simp]
theorem comp_id (f : R →ₑ+*[φ] S) : f.comp (MulSemiringActionHom.id M) = f 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S)
Goal: f.comp (MulSemiringActionHom.id M) = f
-- MulSemiringActionHom.inverse'_toFun
@[simps]
def inverse' (f : R →ₑ+*[φ] S) (g : S → R) (k : Function.RightInverse φ' φ)
    (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    S →ₑ+*[φ'] R 
Arguments: {M : Type u_1} [Monoid M] {N : Type u_2} [Monoid N] {φ : M →* N} {φ' : N →* M} {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S : Type u_12} [Semiring S] [MulSemiringAction N S] (f : R →ₑ+*[φ] S) (g : S → R) (k : Function.RightInverse ⇑φ' ⇑φ) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f)
Goal: ∀ (a : S), (f.inverse' g k h₁ h₂) a = g a
-- MulSemiringActionHom.inverse_toFun
@[simps]
def inverse {S₁ : Type*} [Semiring S₁] [MulSemiringAction M S₁]
    (f : R →+*[M] S₁) (g : S₁ → R)
    (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    S₁ →+*[M] R 
Arguments: {M : Type u_1} [Monoid M] {R : Type u_10} [Semiring R] [MulSemiringAction M R] {S₁ : Type u_15} [Semiring S₁] [MulSemiringAction M S₁] (f : R →+*[M] S₁) (g : S₁ → R) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f)
Goal: ∀ (a : S₁), (f.inverse g h₁ h₂) a = g a
-- isPeriodicPt_add_iff_nsmul_eq_zero
@[to_additive]
theorem isPeriodicPt_mul_iff_pow_eq_one (x : G) : IsPeriodicPt (x * ·) n 1 ↔ x ^ n = 1 
Arguments: {G : Type u_1} [AddMonoid G] {n : ℕ} (x : G)
Goal: Function.IsPeriodicPt (fun x_1 => x + x_1) n 0 ↔ n • x = 0
-- isPeriodicPt_mul_iff_pow_eq_one
@[to_additive]
theorem isPeriodicPt_mul_iff_pow_eq_one (x : G) : IsPeriodicPt (x * ·) n 1 ↔ x ^ n = 1 
Arguments: {G : Type u_1} [Monoid G] {n : ℕ} (x : G)
Goal: Function.IsPeriodicPt (fun x_1 => x * x_1) n 1 ↔ x ^ n = 1
-- isOfFinAddOrder_ofMul_iff
theorem isOfFinAddOrder_ofMul_iff : IsOfFinAddOrder (Additive.ofMul x) ↔ IsOfFinOrder x 
Arguments: {G : Type u_1} [Monoid G] {x : G}
Goal: IsOfFinAddOrder (Additive.ofMul x) ↔ IsOfFinOrder x
-- isOfFinOrder_ofAdd_iff
theorem isOfFinOrder_ofAdd_iff {α : Type*} [AddMonoid α] {x : α} :
    IsOfFinOrder (Multiplicative.ofAdd x) ↔ IsOfFinAddOrder x 
Arguments: {α : Type u_6} [AddMonoid α] {x : α}
Goal: IsOfFinOrder (Multiplicative.ofAdd x) ↔ IsOfFinAddOrder x
-- isOfFinAddOrder_iff_nsmul_eq_zero
@[to_additive]
theorem isOfFinOrder_iff_pow_eq_one : IsOfFinOrder x ↔ ∃ n, 0 < n ∧ x ^ n = 1 
Arguments: {G : Type u_1} [AddMonoid G] {x : G}
Goal: IsOfFinAddOrder x ↔ ∃ n, 0 < n ∧ n • x = 0
-- isOfFinOrder_iff_pow_eq_one
@[to_additive]
theorem isOfFinOrder_iff_pow_eq_one : IsOfFinOrder x ↔ ∃ n, 0 < n ∧ x ^ n = 1 
Arguments: {G : Type u_1} [Monoid G] {x : G}
Goal: IsOfFinOrder x ↔ ∃ n, 0 < n ∧ x ^ n = 1
-- isOfFinAddOrder_iff_zsmul_eq_zero
@[to_additive]
lemma isOfFinOrder_iff_zpow_eq_one {G} [Group G] {x : G} :
    IsOfFinOrder x ↔ ∃ (n : ℤ), n ≠ 0 ∧ x ^ n = 1 
Arguments: {G : Type u_6} [AddGroup G] {x : G}
Goal: IsOfFinAddOrder x ↔ ∃ n, n ≠ 0 ∧ n • x = 0
-- isOfFinOrder_iff_zpow_eq_one
@[to_additive]
lemma isOfFinOrder_iff_zpow_eq_one {G} [Group G] {x : G} :
    IsOfFinOrder x ↔ ∃ (n : ℤ), n ≠ 0 ∧ x ^ n = 1 
Arguments: {G : Type u_6} [Group G] {x : G}
Goal: IsOfFinOrder x ↔ ∃ n, n ≠ 0 ∧ x ^ n = 1
-- not_isOfFinOrder_of_injective_pow
@[to_additive "See also `injective_nsmul_iff_not_isOfFinAddOrder`."]
theorem not_isOfFinOrder_of_injective_pow {x : G} (h : Injective fun n : ℕ => x ^ n) :
    ¬IsOfFinOrder x 
Arguments: {G : Type u_1} [Monoid G] {x : G} (h : Function.Injective fun n => x ^ n)
Goal: ¬IsOfFinOrder x
-- not_isOfFinAddOrder_of_injective_nsmul
@[to_additive "See also `injective_nsmul_iff_not_isOfFinAddOrder`."]
theorem not_isOfFinOrder_of_injective_pow {x : G} (h : Injective fun n : ℕ => x ^ n) :
    ¬IsOfFinOrder x 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} (h : Function.Injective fun n => n • x)
Goal: ¬IsOfFinAddOrder x
-- IsOfFinOrder.pow
lemma IsOfFinOrder.pow {n : ℕ} : IsOfFinOrder a → IsOfFinOrder (a ^ n) 
Arguments: {G : Type u_1} [Monoid G] {a : G} {n : ℕ}
Goal: IsOfFinOrder a → IsOfFinOrder (a ^ n)
-- AddSubmonoid.isOfFinAddOrder_coe
@[to_additive "Elements of finite order are of finite order in submonoids."]
theorem Submonoid.isOfFinOrder_coe {H : Submonoid G} {x : H} :
    IsOfFinOrder (x : G) ↔ IsOfFinOrder x 
Arguments: {G : Type u_1} [AddMonoid G] {H : AddSubmonoid G} {x : ↥H}
Goal: IsOfFinAddOrder ↑x ↔ IsOfFinAddOrder x
-- Submonoid.isOfFinOrder_coe
@[to_additive "Elements of finite order are of finite order in submonoids."]
theorem Submonoid.isOfFinOrder_coe {H : Submonoid G} {x : H} :
    IsOfFinOrder (x : G) ↔ IsOfFinOrder x 
Arguments: {G : Type u_1} [Monoid G] {H : Submonoid G} {x : ↥H}
Goal: IsOfFinOrder ↑x ↔ IsOfFinOrder x
-- AddMonoidHom.isOfFinAddOrder
@[to_additive "The image of an element of finite additive order has finite additive order."]
theorem MonoidHom.isOfFinOrder [Monoid H] (f : G →* H) {x : G} (h : IsOfFinOrder x) :
    IsOfFinOrder <| f x 
Arguments: {G : Type u_1} {H : Type u_2} [AddMonoid G] [AddMonoid H] (f : G →+ H) {x : G} (h : IsOfFinAddOrder x)
Goal: IsOfFinAddOrder (f x)
-- MonoidHom.isOfFinOrder
@[to_additive "The image of an element of finite additive order has finite additive order."]
theorem MonoidHom.isOfFinOrder [Monoid H] (f : G →* H) {x : G} (h : IsOfFinOrder x) :
    IsOfFinOrder <| f x 
Arguments: {G : Type u_1} {H : Type u_2} [Monoid G] [Monoid H] (f : G →* H) {x : G} (h : IsOfFinOrder x)
Goal: IsOfFinOrder (f x)
-- IsOfFinAddOrder.apply
@[to_additive "If a direct product has finite additive order then so does each component."]
theorem IsOfFinOrder.apply {η : Type*} {Gs : η → Type*} [∀ i, Monoid (Gs i)] {x : ∀ i, Gs i}
    (h : IsOfFinOrder x) : ∀ i, IsOfFinOrder (x i) 
Arguments: {η : Type u_6} {Gs : η → Type u_7} [(i : η) → AddMonoid (Gs i)] {x : (i : η) → Gs i} (h : IsOfFinAddOrder x) (i : η)
Goal: IsOfFinAddOrder (x i)
-- IsOfFinOrder.apply
@[to_additive "If a direct product has finite additive order then so does each component."]
theorem IsOfFinOrder.apply {η : Type*} {Gs : η → Type*} [∀ i, Monoid (Gs i)] {x : ∀ i, Gs i}
    (h : IsOfFinOrder x) : ∀ i, IsOfFinOrder (x i) 
Arguments: {η : Type u_6} {Gs : η → Type u_7} [(i : η) → Monoid (Gs i)] {x : (i : η) → Gs i} (h : IsOfFinOrder x) (i : η)
Goal: IsOfFinOrder (x i)
-- isOfFinAddOrder_zero
@[to_additive "0 is of finite order in any additive monoid."]
theorem isOfFinOrder_one : IsOfFinOrder (1 : G) 
Arguments: {G : Type u_1} [AddMonoid G]
Goal: IsOfFinAddOrder 0
-- isOfFinOrder_one
@[to_additive "0 is of finite order in any additive monoid."]
theorem isOfFinOrder_one : IsOfFinOrder (1 : G) 
Arguments: {G : Type u_1} [Monoid G]
Goal: IsOfFinOrder 1
-- addOrderOf_ofMul_eq_orderOf
@[simp]
theorem addOrderOf_ofMul_eq_orderOf (x : G) : addOrderOf (Additive.ofMul x) = orderOf x 
Arguments: {G : Type u_1} [Monoid G] (x : G)
Goal: addOrderOf (Additive.ofMul x) = orderOf x
-- orderOf_ofAdd_eq_addOrderOf
@[simp]
lemma orderOf_ofAdd_eq_addOrderOf {α : Type*} [AddMonoid α] (a : α) :
    orderOf (Multiplicative.ofAdd a) = addOrderOf a 
Arguments: {α : Type u_6} [AddMonoid α] (a : α)
Goal: orderOf (Multiplicative.ofAdd a) = addOrderOf a
-- IsOfFinAddOrder.addOrderOf_pos
@[to_additive]
protected lemma IsOfFinOrder.orderOf_pos (h : IsOfFinOrder x) : 0 < orderOf x 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} (h : IsOfFinAddOrder x)
Goal: 0 < addOrderOf x
-- IsOfFinOrder.orderOf_pos
@[to_additive]
protected lemma IsOfFinOrder.orderOf_pos (h : IsOfFinOrder x) : 0 < orderOf x 
Arguments: {G : Type u_1} [Monoid G] {x : G} (h : IsOfFinOrder x)
Goal: 0 < orderOf x
-- pow_orderOf_eq_one
@[to_additive addOrderOf_nsmul_eq_zero]
theorem pow_orderOf_eq_one (x : G) : x ^ orderOf x = 1 
Arguments: {G : Type u_1} [Monoid G] (x : G)
Goal: x ^ orderOf x = 1
-- addOrderOf_nsmul_eq_zero
@[to_additive addOrderOf_nsmul_eq_zero]
theorem pow_orderOf_eq_one (x : G) : x ^ orderOf x = 1 
Arguments: {G : Type u_1} [AddMonoid G] (x : G)
Goal: addOrderOf x • x = 0
-- orderOf_eq_zero
@[to_additive]
theorem orderOf_eq_zero (h : ¬IsOfFinOrder x) : orderOf x = 0 
Arguments: {G : Type u_1} [Monoid G] {x : G} (h : ¬IsOfFinOrder x)
Goal: orderOf x = 0
-- addOrderOf_eq_zero
@[to_additive]
theorem orderOf_eq_zero (h : ¬IsOfFinOrder x) : orderOf x = 0 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} (h : ¬IsOfFinAddOrder x)
Goal: addOrderOf x = 0
-- orderOf_eq_zero_iff
@[to_additive]
theorem orderOf_eq_zero_iff : orderOf x = 0 ↔ ¬IsOfFinOrder x 
Arguments: {G : Type u_1} [Monoid G] {x : G}
Goal: orderOf x = 0 ↔ ¬IsOfFinOrder x
-- addOrderOf_eq_zero_iff
@[to_additive]
theorem orderOf_eq_zero_iff : orderOf x = 0 ↔ ¬IsOfFinOrder x 
Arguments: {G : Type u_1} [AddMonoid G] {x : G}
Goal: addOrderOf x = 0 ↔ ¬IsOfFinAddOrder x
-- orderOf_eq_zero_iff'
@[to_additive]
theorem orderOf_eq_zero_iff' : orderOf x = 0 ↔ ∀ n : ℕ, 0 < n → x ^ n ≠ 1 
Arguments: {G : Type u_1} [Monoid G] {x : G}
Goal: orderOf x = 0 ↔ ∀ (n : ℕ), 0 < n → x ^ n ≠ 1
-- addOrderOf_eq_zero_iff'
@[to_additive]
theorem orderOf_eq_zero_iff' : orderOf x = 0 ↔ ∀ n : ℕ, 0 < n → x ^ n ≠ 1 
Arguments: {G : Type u_1} [AddMonoid G] {x : G}
Goal: addOrderOf x = 0 ↔ ∀ (n : ℕ), 0 < n → n • x ≠ 0
-- addOrderOf_eq_iff
@[to_additive]
theorem orderOf_eq_iff {n} (h : 0 < n) :
    orderOf x = n ↔ x ^ n = 1 ∧ ∀ m, m < n → 0 < m → x ^ m ≠ 1 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} {n : ℕ} (h : 0 < n)
Goal: addOrderOf x = n ↔ n • x = 0 ∧ ∀ m < n, 0 < m → m • x ≠ 0
-- orderOf_eq_iff
@[to_additive]
theorem orderOf_eq_iff {n} (h : 0 < n) :
    orderOf x = n ↔ x ^ n = 1 ∧ ∀ m, m < n → 0 < m → x ^ m ≠ 1 
Arguments: {G : Type u_1} [Monoid G] {x : G} {n : ℕ} (h : 0 < n)
Goal: orderOf x = n ↔ x ^ n = 1 ∧ ∀ m < n, 0 < m → x ^ m ≠ 1
-- addOrderOf_pos_iff
@[to_additive
      "A group element has finite additive order iff its order is positive."]
theorem orderOf_pos_iff : 0 < orderOf x ↔ IsOfFinOrder x 
Arguments: {G : Type u_1} [AddMonoid G] {x : G}
Goal: 0 < addOrderOf x ↔ IsOfFinAddOrder x
-- orderOf_pos_iff
@[to_additive
      "A group element has finite additive order iff its order is positive."]
theorem orderOf_pos_iff : 0 < orderOf x ↔ IsOfFinOrder x 
Arguments: {G : Type u_1} [Monoid G] {x : G}
Goal: 0 < orderOf x ↔ IsOfFinOrder x
-- IsOfFinOrder.mono
@[to_additive]
theorem IsOfFinOrder.mono [Monoid β] {y : β} (hx : IsOfFinOrder x) (h : orderOf y ∣ orderOf x) :
    IsOfFinOrder y 
Arguments: {G : Type u_1} {β : Type u_5} [Monoid G] {x : G} [Monoid β] {y : β} (hx : IsOfFinOrder x) (h : orderOf y ∣ orderOf x)
Goal: IsOfFinOrder y
-- IsOfFinAddOrder.mono
@[to_additive]
theorem IsOfFinOrder.mono [Monoid β] {y : β} (hx : IsOfFinOrder x) (h : orderOf y ∣ orderOf x) :
    IsOfFinOrder y 
Arguments: {G : Type u_1} {β : Type u_5} [AddMonoid G] {x : G} [AddMonoid β] {y : β} (hx : IsOfFinAddOrder x) (h : addOrderOf y ∣ addOrderOf x)
Goal: IsOfFinAddOrder y
-- nsmul_ne_zero_of_lt_addOrderOf
@[to_additive]
theorem pow_ne_one_of_lt_orderOf (n0 : n ≠ 0) (h : n < orderOf x) : x ^ n ≠ 1 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} {n : ℕ} (n0 : n ≠ 0) (h : n < addOrderOf x)
Goal: n • x ≠ 0
-- pow_ne_one_of_lt_orderOf
@[to_additive]
theorem pow_ne_one_of_lt_orderOf (n0 : n ≠ 0) (h : n < orderOf x) : x ^ n ≠ 1 
Arguments: {G : Type u_1} [Monoid G] {x : G} {n : ℕ} (n0 : n ≠ 0) (h : n < orderOf x)
Goal: x ^ n ≠ 1
-- addOrderOf_le_of_nsmul_eq_zero
@[to_additive]
theorem orderOf_le_of_pow_eq_one (hn : 0 < n) (h : x ^ n = 1) : orderOf x ≤ n 
Arguments: {G : Type u_1} [AddMonoid G] {x : G} {n : ℕ} (hn : 0 < n) (h : n • x = 0)
Goal: addOrderOf x ≤ n
-- orderOf_le_of_pow_eq_one
@[to_additive]
theorem orderOf_le_of_pow_eq_one (hn : 0 < n) (h : x ^ n = 1) : orderOf x ≤ n 
Arguments: {G : Type u_1} [Monoid G] {x : G} {n : ℕ} (hn : 0 < n) (h : x ^ n = 1)
Goal: orderOf x ≤ n
-- orderOf_one
@[to_additive (attr := simp)]
theorem orderOf_one : orderOf (1 : G) = 1 
Arguments: {G : Type u_1} [Monoid G]
Goal: orderOf 1 = 1