ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Function.Periodic.comp_addHom
theorem Periodic.comp_addHom [Add α] [Add γ] (h : Periodic f c) (g : AddHom γ α) (g_inv : α → γ)
    (hg : RightInverse g_inv g) : Periodic (f ∘ g) (g_inv c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [Add α] [Add γ] (h : Function.Periodic f c) (g : AddHom γ α) (g_inv : α → γ) (hg : Function.RightInverse g_inv ⇑g)
Goal: Function.Periodic (f ∘ ⇑g) (g_inv c)
-- Function.Periodic.add
@[to_additive]
protected theorem Periodic.mul [Add α] [Mul β] (hf : Periodic f c) (hg : Periodic g c) :
    Periodic (f * g) c 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {c : α} [Add α] [Add β] (hf : Function.Periodic f c) (hg : Function.Periodic g c)
Goal: Function.Periodic (f + g) c
-- Function.Periodic.mul
@[to_additive]
protected theorem Periodic.mul [Add α] [Mul β] (hf : Periodic f c) (hg : Periodic g c) :
    Periodic (f * g) c 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {c : α} [Add α] [Mul β] (hf : Function.Periodic f c) (hg : Function.Periodic g c)
Goal: Function.Periodic (f * g) c
-- Function.Periodic.sub
@[to_additive]
protected theorem Periodic.div [Add α] [Div β] (hf : Periodic f c) (hg : Periodic g c) :
    Periodic (f / g) c 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {c : α} [Add α] [Sub β] (hf : Function.Periodic f c) (hg : Function.Periodic g c)
Goal: Function.Periodic (f - g) c
-- Function.Periodic.div
@[to_additive]
protected theorem Periodic.div [Add α] [Div β] (hf : Periodic f c) (hg : Periodic g c) :
    Periodic (f / g) c 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {c : α} [Add α] [Div β] (hf : Function.Periodic f c) (hg : Function.Periodic g c)
Goal: Function.Periodic (f / g) c
-- List.periodic_prod
@[to_additive]
theorem _root_.List.periodic_prod [Add α] [Monoid β] (l : List (α → β))
    (hl : ∀ f ∈ l, Periodic f c) : Periodic l.prod c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [Monoid β] (l : List (α → β)) (hl : ∀ f ∈ l, Function.Periodic f c)
Goal: Function.Periodic l.prod c
-- List.periodic_sum
@[to_additive]
theorem _root_.List.periodic_prod [Add α] [Monoid β] (l : List (α → β))
    (hl : ∀ f ∈ l, Periodic f c) : Periodic l.prod c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [AddMonoid β] (l : List (α → β)) (hl : ∀ f ∈ l, Function.Periodic f c)
Goal: Function.Periodic l.sum c
-- Multiset.periodic_sum
@[to_additive]
theorem _root_.Multiset.periodic_prod [Add α] [CommMonoid β] (s : Multiset (α → β))
    (hs : ∀ f ∈ s, Periodic f c) : Periodic s.prod c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [AddCommMonoid β] (s : Multiset (α → β)) (hs : ∀ f ∈ s, Function.Periodic f c)
Goal: Function.Periodic s.sum c
-- Multiset.periodic_prod
@[to_additive]
theorem _root_.Multiset.periodic_prod [Add α] [CommMonoid β] (s : Multiset (α → β))
    (hs : ∀ f ∈ s, Periodic f c) : Periodic s.prod c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [CommMonoid β] (s : Multiset (α → β)) (hs : ∀ f ∈ s, Function.Periodic f c)
Goal: Function.Periodic s.prod c
-- Finset.periodic_sum
@[to_additive]
theorem _root_.Finset.periodic_prod [Add α] [CommMonoid β] {ι : Type*} {f : ι → α → β}
    (s : Finset ι) (hs : ∀ i ∈ s, Periodic (f i) c) : Periodic (∏ i ∈ s, f i) c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [AddCommMonoid β] {ι : Type u_4} {f : ι → α → β} (s : Finset ι) (hs : ∀ i ∈ s, Function.Periodic (f i) c)
Goal: Function.Periodic (∑ i ∈ s, f i) c
-- Finset.periodic_prod
@[to_additive]
theorem _root_.Finset.periodic_prod [Add α] [CommMonoid β] {ι : Type*} {f : ι → α → β}
    (s : Finset ι) (hs : ∀ i ∈ s, Periodic (f i) c) : Periodic (∏ i ∈ s, f i) c 
Arguments: {α : Type u_1} {β : Type u_2} {c : α} [Add α] [CommMonoid β] {ι : Type u_4} {f : ι → α → β} (s : Finset ι) (hs : ∀ i ∈ s, Function.Periodic (f i) c)
Goal: Function.Periodic (∏ i ∈ s, f i) c
-- Function.Periodic.smul
@[to_additive]
protected theorem Periodic.smul [Add α] [SMul γ β] (h : Periodic f c) (a : γ) :
    Periodic (a • f) c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [Add α] [SMul γ β] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (a • f) c
-- Function.Periodic.vadd
@[to_additive]
protected theorem Periodic.smul [Add α] [SMul γ β] (h : Periodic f c) (a : γ) :
    Periodic (a • f) c 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [Add α] [VAdd γ β] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (a +ᵥ f) c
-- Function.Periodic.const_smul
protected theorem Periodic.const_smul [AddMonoid α] [Group γ] [DistribMulAction γ α]
    (h : Periodic f c) (a : γ) : Periodic (fun x => f (a • x)) (a⁻¹ • c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [AddMonoid α] [Group γ] [DistribMulAction γ α] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (fun x => f (a • x)) (a⁻¹ • c)
-- Function.Periodic.const_smul₀
protected theorem Periodic.const_smul₀ [AddCommMonoid α] [DivisionSemiring γ] [Module γ α]
    (h : Periodic f c) (a : γ) : Periodic (fun x => f (a • x)) (a⁻¹ • c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [AddCommMonoid α] [DivisionSemiring γ] [Module γ α] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (fun x => f (a • x)) (a⁻¹ • c)
-- Function.Periodic.const_mul
protected theorem Periodic.const_mul [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (a * x)) (a⁻¹ * c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (a * x)) (a⁻¹ * c)
-- Function.Periodic.const_inv_smul
theorem Periodic.const_inv_smul [AddMonoid α] [Group γ] [DistribMulAction γ α] (h : Periodic f c)
    (a : γ) : Periodic (fun x => f (a⁻¹ • x)) (a • c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [AddMonoid α] [Group γ] [DistribMulAction γ α] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (fun x => f (a⁻¹ • x)) (a • c)
-- Function.Periodic.const_inv_smul₀
theorem Periodic.const_inv_smul₀ [AddCommMonoid α] [DivisionSemiring γ] [Module γ α]
    (h : Periodic f c) (a : γ) : Periodic (fun x => f (a⁻¹ • x)) (a • c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {c : α} [AddCommMonoid α] [DivisionSemiring γ] [Module γ α] (h : Function.Periodic f c) (a : γ)
Goal: Function.Periodic (fun x => f (a⁻¹ • x)) (a • c)
-- Function.Periodic.const_inv_mul
theorem Periodic.const_inv_mul [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (a⁻¹ * x)) (a * c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (a⁻¹ * x)) (a * c)
-- Function.Periodic.mul_const
theorem Periodic.mul_const [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x * a)) (c * a⁻¹) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x * a)) (c * a⁻¹)
-- Function.Periodic.mul_const'
theorem Periodic.mul_const' [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x * a)) (c / a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x * a)) (c / a)
-- Function.Periodic.mul_const_inv
theorem Periodic.mul_const_inv [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x * a⁻¹)) (c * a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x * a⁻¹)) (c * a)
-- Function.Periodic.div_const
theorem Periodic.div_const [DivisionSemiring α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x / a)) (c * a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [DivisionSemiring α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x / a)) (c * a)
-- Function.Periodic.add_period
theorem Periodic.add_period [AddSemigroup α] (h1 : Periodic f c₁) (h2 : Periodic f c₂) :
    Periodic f (c₁ + c₂) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c₁ c₂ : α} [AddSemigroup α] (h1 : Function.Periodic f c₁) (h2 : Function.Periodic f c₂)
Goal: Function.Periodic f (c₁ + c₂)
-- Function.Periodic.sub_eq
theorem Periodic.sub_eq [AddGroup α] (h : Periodic f c) (x : α) : f (x - c) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c) (x : α)
Goal: f (x - c) = f x
-- Function.Periodic.sub_eq'
theorem Periodic.sub_eq' [AddCommGroup α] (h : Periodic f c) : f (c - x) = f (-x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [AddCommGroup α] (h : Function.Periodic f c)
Goal: f (c - x) = f (-x)
-- Function.Periodic.neg
protected theorem Periodic.neg [AddGroup α] (h : Periodic f c) : Periodic f (-c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c)
Goal: Function.Periodic f (-c)
-- Function.Periodic.sub_period
theorem Periodic.sub_period [AddGroup α] (h1 : Periodic f c₁) (h2 : Periodic f c₂) :
    Periodic f (c₁ - c₂) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c₁ c₂ : α} [AddGroup α] (h1 : Function.Periodic f c₁) (h2 : Function.Periodic f c₂)
Goal: Function.Periodic f (c₁ - c₂)
-- Function.Periodic.const_add
theorem Periodic.const_add [AddSemigroup α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (a + x)) c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddSemigroup α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (a + x)) c
-- Function.Periodic.add_const
theorem Periodic.add_const [AddCommSemigroup α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x + a)) c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddCommSemigroup α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x + a)) c
-- Function.Periodic.const_sub
theorem Periodic.const_sub [AddCommGroup α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (a - x)) c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddCommGroup α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (a - x)) c
-- Function.Periodic.sub_const
theorem Periodic.sub_const [AddCommGroup α] (h : Periodic f c) (a : α) :
    Periodic (fun x => f (x - a)) c 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddCommGroup α] (h : Function.Periodic f c) (a : α)
Goal: Function.Periodic (fun x => f (x - a)) c
-- Function.Periodic.nsmul
theorem Periodic.nsmul [AddMonoid α] (h : Periodic f c) (n : ℕ) : Periodic f (n • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddMonoid α] (h : Function.Periodic f c) (n : ℕ)
Goal: Function.Periodic f (n • c)
-- Function.Periodic.nat_mul
theorem Periodic.nat_mul [Semiring α] (h : Periodic f c) (n : ℕ) : Periodic f (n * c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Semiring α] (h : Function.Periodic f c) (n : ℕ)
Goal: Function.Periodic f (↑n * c)
-- Function.Periodic.neg_nsmul
theorem Periodic.neg_nsmul [AddGroup α] (h : Periodic f c) (n : ℕ) : Periodic f (-(n • c)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c) (n : ℕ)
Goal: Function.Periodic f (-(n • c))
-- Function.Periodic.neg_nat_mul
theorem Periodic.neg_nat_mul [Ring α] (h : Periodic f c) (n : ℕ) : Periodic f (-(n * c)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Ring α] (h : Function.Periodic f c) (n : ℕ)
Goal: Function.Periodic f (-(↑n * c))
-- Function.Periodic.sub_nsmul_eq
theorem Periodic.sub_nsmul_eq [AddGroup α] (h : Periodic f c) (n : ℕ) : f (x - n • c) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [AddGroup α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (x - n • c) = f x
-- Function.Periodic.sub_nat_mul_eq
theorem Periodic.sub_nat_mul_eq [Ring α] (h : Periodic f c) (n : ℕ) : f (x - n * c) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [Ring α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (x - ↑n * c) = f x
-- Function.Periodic.nsmul_sub_eq
theorem Periodic.nsmul_sub_eq [AddCommGroup α] (h : Periodic f c) (n : ℕ) :
    f (n • c - x) = f (-x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [AddCommGroup α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (n • c - x) = f (-x)
-- Function.Periodic.nat_mul_sub_eq
theorem Periodic.nat_mul_sub_eq [Ring α] (h : Periodic f c) (n : ℕ) : f (n * c - x) = f (-x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [Ring α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (↑n * c - x) = f (-x)
-- Function.Periodic.zsmul
protected theorem Periodic.zsmul [AddGroup α] (h : Periodic f c) (n : ℤ) : Periodic f (n • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c) (n : ℤ)
Goal: Function.Periodic f (n • c)
-- Function.Periodic.int_mul
protected theorem Periodic.int_mul [Ring α] (h : Periodic f c) (n : ℤ) : Periodic f (n * c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Ring α] (h : Function.Periodic f c) (n : ℤ)
Goal: Function.Periodic f (↑n * c)
-- Function.Periodic.sub_zsmul_eq
theorem Periodic.sub_zsmul_eq [AddGroup α] (h : Periodic f c) (n : ℤ) : f (x - n • c) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [AddGroup α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (x - n • c) = f x
-- Function.Periodic.sub_int_mul_eq
theorem Periodic.sub_int_mul_eq [Ring α] (h : Periodic f c) (n : ℤ) : f (x - n * c) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [Ring α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (x - ↑n * c) = f x
-- Function.Periodic.zsmul_sub_eq
theorem Periodic.zsmul_sub_eq [AddCommGroup α] (h : Periodic f c) (n : ℤ) :
    f (n • c - x) = f (-x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [AddCommGroup α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (n • c - x) = f (-x)
-- Function.Periodic.int_mul_sub_eq
theorem Periodic.int_mul_sub_eq [Ring α] (h : Periodic f c) (n : ℤ) : f (n * c - x) = f (-x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c x : α} [Ring α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (↑n * c - x) = f (-x)
-- Function.Periodic.eq
protected theorem Periodic.eq [AddZeroClass α] (h : Periodic f c) : f c = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddZeroClass α] (h : Function.Periodic f c)
Goal: f c = f 0
-- Function.Periodic.neg_eq
protected theorem Periodic.neg_eq [AddGroup α] (h : Periodic f c) : f (-c) = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c)
Goal: f (-c) = f 0
-- Function.Periodic.nsmul_eq
protected theorem Periodic.nsmul_eq [AddMonoid α] (h : Periodic f c) (n : ℕ) : f (n • c) = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddMonoid α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (n • c) = f 0
-- Function.Periodic.nat_mul_eq
theorem Periodic.nat_mul_eq [Semiring α] (h : Periodic f c) (n : ℕ) : f (n * c) = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Semiring α] (h : Function.Periodic f c) (n : ℕ)
Goal: f (↑n * c) = f 0
-- Function.Periodic.zsmul_eq
theorem Periodic.zsmul_eq [AddGroup α] (h : Periodic f c) (n : ℤ) : f (n • c) = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (n • c) = f 0
-- Function.Periodic.int_mul_eq
theorem Periodic.int_mul_eq [Ring α] (h : Periodic f c) (n : ℤ) : f (n * c) = f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Ring α] (h : Function.Periodic f c) (n : ℤ)
Goal: f (↑n * c) = f 0
-- Function.Periodic.exists_mem_Ico₀
theorem Periodic.exists_mem_Ico₀ [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : 0 < c) (x) : ∃ y ∈ Ico 0 c, f x = f y 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : 0 < c) (x : α)
Goal: ∃ y ∈ Set.Ico 0 c, f x = f y
-- Function.Periodic.exists_mem_Ico
theorem Periodic.exists_mem_Ico [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : 0 < c) (x a) : ∃ y ∈ Ico a (a + c), f x = f y 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : 0 < c) (x a : α)
Goal: ∃ y ∈ Set.Ico a (a + c), f x = f y
-- Function.Periodic.exists_mem_Ioc
theorem Periodic.exists_mem_Ioc [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : 0 < c) (x a) : ∃ y ∈ Ioc a (a + c), f x = f y 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : 0 < c) (x a : α)
Goal: ∃ y ∈ Set.Ioc a (a + c), f x = f y
-- Function.Periodic.image_Ioc
theorem Periodic.image_Ioc [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : 0 < c) (a : α) : f '' Ioc a (a + c) = range f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : 0 < c) (a : α)
Goal: f '' Set.Ioc a (a + c) = Set.range f
-- Function.Periodic.image_Icc
theorem Periodic.image_Icc [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : 0 < c) (a : α) : f '' Icc a (a + c) = range f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : 0 < c) (a : α)
Goal: f '' Set.Icc a (a + c) = Set.range f
-- Function.Periodic.image_uIcc
theorem Periodic.image_uIcc [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
    (hc : c ≠ 0) (a : α) : f '' uIcc a (a + c) = range f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [LinearOrderedAddCommGroup α] [Archimedean α] (h : Function.Periodic f c) (hc : c ≠ 0) (a : α)
Goal: f '' Set.uIcc a (a + c) = Set.range f
-- Function.periodic_with_period_zero
theorem periodic_with_period_zero [AddZeroClass α] (f : α → β) : Periodic f 0 
Arguments: {α : Type u_1} {β : Type u_2} [AddZeroClass α] (f : α → β)
Goal: Function.Periodic f 0
-- Function.Periodic.map_vadd_zmultiples
theorem Periodic.map_vadd_zmultiples [AddCommGroup α] (hf : Periodic f c)
    (a : AddSubgroup.zmultiples c) (x : α) : f (a +ᵥ x) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddCommGroup α] (hf : Function.Periodic f c) (a : ↥(AddSubgroup.zmultiples c)) (x : α)
Goal: f (a +ᵥ x) = f x
-- Function.Periodic.map_vadd_multiples
theorem Periodic.map_vadd_multiples [AddCommMonoid α] (hf : Periodic f c)
    (a : AddSubmonoid.multiples c) (x : α) : f (a +ᵥ x) = f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddCommMonoid α] (hf : Function.Periodic f c) (a : ↥(AddSubmonoid.multiples c)) (x : α)
Goal: f (a +ᵥ x) = f x
-- Function.Periodic.lift_coe
@[simp]
theorem Periodic.lift_coe [AddGroup α] (h : Periodic f c) (a : α) :
    h.lift (a : α ⧸ AddSubgroup.zmultiples c) = f a 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] (h : Function.Periodic f c) (a : α)
Goal: h.lift ↑a = f a
-- Function.Periodic.not_injective
lemma Periodic.not_injective {R X : Type*} [AddZeroClass R] {f : R → X} {c : R}
    (hf : Periodic f c) (hc : c ≠ 0) : ¬ Injective f 
Arguments: {R : Type u_4} {X : Type u_5} [AddZeroClass R] {f : R → X} {c : R} (hf : Function.Periodic f c) (hc : c ≠ 0)
Goal: ¬Function.Injective f
-- Function.Antiperiodic.funext
protected theorem Antiperiodic.funext [Add α] [Neg β] (h : Antiperiodic f c) :
    (fun x => f (x + c)) = -f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Add α] [Neg β] (h : Function.Antiperiodic f c)
Goal: (fun x => f (x + c)) = -f
-- Function.Antiperiodic.funext'
protected theorem Antiperiodic.funext' [Add α] [InvolutiveNeg β] (h : Antiperiodic f c) :
    (fun x => -f (x + c)) = f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Add α] [InvolutiveNeg β] (h : Function.Antiperiodic f c)
Goal: (fun x => -f (x + c)) = f
-- Function.Antiperiodic.periodic
protected theorem Antiperiodic.periodic [AddMonoid α] [InvolutiveNeg β]
    (h : Antiperiodic f c) : Periodic f (2 • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddMonoid α] [InvolutiveNeg β] (h : Function.Antiperiodic f c)
Goal: Function.Periodic f (2 • c)
-- Function.Antiperiodic.periodic_two_mul
protected theorem Antiperiodic.periodic_two_mul [Semiring α] [InvolutiveNeg β]
    (h : Antiperiodic f c) : Periodic f (2 * c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Semiring α] [InvolutiveNeg β] (h : Function.Antiperiodic f c)
Goal: Function.Periodic f (2 * c)
-- Function.Antiperiodic.eq
protected theorem Antiperiodic.eq [AddZeroClass α] [Neg β] (h : Antiperiodic f c) : f c = -f 0 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddZeroClass α] [Neg β] (h : Function.Antiperiodic f c)
Goal: f c = -f 0
-- Function.Antiperiodic.even_nsmul_periodic
theorem Antiperiodic.even_nsmul_periodic [AddMonoid α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℕ) : Periodic f ((2 * n) • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddMonoid α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℕ)
Goal: Function.Periodic f ((2 * n) • c)
-- Function.Antiperiodic.nat_even_mul_periodic
theorem Antiperiodic.nat_even_mul_periodic [Semiring α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℕ) : Periodic f (n * (2 * c)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Semiring α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℕ)
Goal: Function.Periodic f (↑n * (2 * c))
-- Function.Antiperiodic.odd_nsmul_antiperiodic
theorem Antiperiodic.odd_nsmul_antiperiodic [AddMonoid α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℕ) : Antiperiodic f ((2 * n + 1) • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddMonoid α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℕ)
Goal: Function.Antiperiodic f ((2 * n + 1) • c)
-- Function.Antiperiodic.nat_odd_mul_antiperiodic
theorem Antiperiodic.nat_odd_mul_antiperiodic [Semiring α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℕ) : Antiperiodic f (n * (2 * c) + c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Semiring α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℕ)
Goal: Function.Antiperiodic f (↑n * (2 * c) + c)
-- Function.Antiperiodic.even_zsmul_periodic
theorem Antiperiodic.even_zsmul_periodic [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℤ) : Periodic f ((2 * n) • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℤ)
Goal: Function.Periodic f ((2 * n) • c)
-- Function.Antiperiodic.int_even_mul_periodic
theorem Antiperiodic.int_even_mul_periodic [Ring α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℤ) : Periodic f (n * (2 * c)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Ring α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℤ)
Goal: Function.Periodic f (↑n * (2 * c))
-- Function.Antiperiodic.odd_zsmul_antiperiodic
theorem Antiperiodic.odd_zsmul_antiperiodic [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℤ) : Antiperiodic f ((2 * n + 1) • c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℤ)
Goal: Function.Antiperiodic f ((2 * n + 1) • c)
-- Function.Antiperiodic.int_odd_mul_antiperiodic
theorem Antiperiodic.int_odd_mul_antiperiodic [Ring α] [InvolutiveNeg β] (h : Antiperiodic f c)
    (n : ℤ) : Antiperiodic f (n * (2 * c) + c) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [Ring α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (n : ℤ)
Goal: Function.Antiperiodic f (↑n * (2 * c) + c)
-- Function.Antiperiodic.sub_eq
theorem Antiperiodic.sub_eq [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c) (x : α) :
    f (x - c) = -f x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [AddGroup α] [InvolutiveNeg β] (h : Function.Antiperiodic f c) (x : α)
Goal: f (x - c) = -f x