ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- monovaryOn_inv_right
@[to_additive (attr := simp)]
lemma monovaryOn_inv_right : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s
-- monovaryOn_neg_right
@[to_additive (attr := simp)]
lemma monovaryOn_inv_right : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn f (-g) s ↔ AntivaryOn f g s
-- antivaryOn_inv_right
@[to_additive (attr := simp)]
lemma antivaryOn_inv_right : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s
-- antivaryOn_neg_right
@[to_additive (attr := simp)]
lemma antivaryOn_inv_right : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn f (-g) s ↔ MonovaryOn f g s
-- monovaryOn_inv
@[to_additive] lemma monovaryOn_inv : MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s
-- monovaryOn_neg
@[to_additive] lemma monovaryOn_inv : MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn (-f) (-g) s ↔ MonovaryOn f g s
-- antivaryOn_inv
@[to_additive] lemma antivaryOn_inv : AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s
-- antivaryOn_neg
@[to_additive] lemma antivaryOn_inv : AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn (-f) (-g) s ↔ AntivaryOn f g s
-- monovary_neg_left
@[to_additive (attr := simp)] lemma monovary_inv_left : Monovary f⁻¹ g ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary (-f) g ↔ Antivary f g
-- monovary_inv_left
@[to_additive (attr := simp)] lemma monovary_inv_left : Monovary f⁻¹ g ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary f⁻¹ g ↔ Antivary f g
-- antivary_neg_left
@[to_additive (attr := simp)] lemma antivary_inv_left : Antivary f⁻¹ g ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary (-f) g ↔ Monovary f g
-- antivary_inv_left
@[to_additive (attr := simp)] lemma antivary_inv_left : Antivary f⁻¹ g ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary f⁻¹ g ↔ Monovary f g
-- monovary_neg_right
@[to_additive (attr := simp)] lemma monovary_inv_right : Monovary f g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary f (-g) ↔ Antivary f g
-- monovary_inv_right
@[to_additive (attr := simp)] lemma monovary_inv_right : Monovary f g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary f g⁻¹ ↔ Antivary f g
-- antivary_neg_right
@[to_additive (attr := simp)] lemma antivary_inv_right : Antivary f g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary f (-g) ↔ Monovary f g
-- antivary_inv_right
@[to_additive (attr := simp)] lemma antivary_inv_right : Antivary f g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary f g⁻¹ ↔ Monovary f g
-- monovary_inv
@[to_additive] lemma monovary_inv : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary f⁻¹ g⁻¹ ↔ Monovary f g
-- monovary_neg
@[to_additive] lemma monovary_inv : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Monovary (-f) (-g) ↔ Monovary f g
-- antivary_neg
@[to_additive] lemma antivary_inv : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary (-f) (-g) ↔ Antivary f g
-- antivary_inv
@[to_additive] lemma antivary_inv : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β}
Goal: Antivary f⁻¹ g⁻¹ ↔ Antivary f g
-- MonovaryOn.mul_left
@[to_additive] lemma MonovaryOn.mul_left (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    MonovaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s)
Goal: MonovaryOn (f₁ * f₂) g s
-- MonovaryOn.add_left
@[to_additive] lemma MonovaryOn.mul_left (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    MonovaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s)
Goal: MonovaryOn (f₁ + f₂) g s
-- AntivaryOn.add_left
@[to_additive] lemma AntivaryOn.mul_left (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    AntivaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s)
Goal: AntivaryOn (f₁ + f₂) g s
-- AntivaryOn.mul_left
@[to_additive] lemma AntivaryOn.mul_left (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    AntivaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s)
Goal: AntivaryOn (f₁ * f₂) g s
-- MonovaryOn.div_left
@[to_additive] lemma MonovaryOn.div_left (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    MonovaryOn (f₁ / f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s)
Goal: MonovaryOn (f₁ / f₂) g s
-- MonovaryOn.sub_left
@[to_additive] lemma MonovaryOn.div_left (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    MonovaryOn (f₁ / f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s)
Goal: MonovaryOn (f₁ - f₂) g s
-- AntivaryOn.sub_left
@[to_additive] lemma AntivaryOn.div_left (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    AntivaryOn (f₁ / f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s)
Goal: AntivaryOn (f₁ - f₂) g s
-- AntivaryOn.div_left
@[to_additive] lemma AntivaryOn.div_left (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    AntivaryOn (f₁ / f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s)
Goal: AntivaryOn (f₁ / f₂) g s
-- MonovaryOn.pow_left
@[to_additive] lemma MonovaryOn.pow_left (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn (f ^ n) g s
-- MonovaryOn.nsmul_left
@[to_additive] lemma MonovaryOn.pow_left (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn (n • f) g s
-- AntivaryOn.nsmul_left
@[to_additive] lemma AntivaryOn.pow_left (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn (n • f) g s
-- AntivaryOn.pow_left
@[to_additive] lemma AntivaryOn.pow_left (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn (f ^ n) g s
-- Monovary.mul_left
@[to_additive]
lemma Monovary.mul_left (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) : Monovary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g)
Goal: Monovary (f₁ * f₂) g
-- Monovary.add_left
@[to_additive]
lemma Monovary.mul_left (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) : Monovary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g)
Goal: Monovary (f₁ + f₂) g
-- Antivary.mul_left
@[to_additive]
lemma Antivary.mul_left (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) : Antivary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g)
Goal: Antivary (f₁ * f₂) g
-- Antivary.add_left
@[to_additive]
lemma Antivary.mul_left (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) : Antivary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g)
Goal: Antivary (f₁ + f₂) g
-- Monovary.div_left
@[to_additive]
lemma Monovary.div_left (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g) : Monovary (f₁ / f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g)
Goal: Monovary (f₁ / f₂) g
-- Monovary.sub_left
@[to_additive]
lemma Monovary.div_left (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g) : Monovary (f₁ / f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g)
Goal: Monovary (f₁ - f₂) g
-- Antivary.sub_left
@[to_additive]
lemma Antivary.div_left (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g) : Antivary (f₁ / f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g)
Goal: Antivary (f₁ - f₂) g
-- Antivary.div_left
@[to_additive]
lemma Antivary.div_left (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g) : Antivary (f₁ / f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f₁ f₂ : ι → α} {g : ι → β} (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g)
Goal: Antivary (f₁ / f₂) g
-- Monovary.nsmul_left
@[to_additive] lemma Monovary.pow_left (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β} (hfg : Monovary f g) (n : ℕ)
Goal: Monovary (n • f) g
-- Monovary.pow_left
@[to_additive] lemma Monovary.pow_left (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β} (hfg : Monovary f g) (n : ℕ)
Goal: Monovary (f ^ n) g
-- Antivary.pow_left
@[to_additive] lemma Antivary.pow_left (hfg : Antivary f g) (n : ℕ) : Antivary (f ^ n) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {f : ι → α} {g : ι → β} (hfg : Antivary f g) (n : ℕ)
Goal: Antivary (f ^ n) g
-- Antivary.nsmul_left
@[to_additive] lemma Antivary.pow_left (hfg : Antivary f g) (n : ℕ) : Antivary (f ^ n) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {f : ι → α} {g : ι → β} (hfg : Antivary f g) (n : ℕ)
Goal: Antivary (n • f) g
-- MonovaryOn.add_right
@[to_additive] lemma MonovaryOn.mul_right (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    MonovaryOn f (g₁ * g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: MonovaryOn f (g₁ + g₂) s
-- MonovaryOn.mul_right
@[to_additive] lemma MonovaryOn.mul_right (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    MonovaryOn f (g₁ * g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: MonovaryOn f (g₁ * g₂) s
-- AntivaryOn.mul_right
@[to_additive] lemma AntivaryOn.mul_right (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    AntivaryOn f (g₁ * g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: AntivaryOn f (g₁ * g₂) s
-- AntivaryOn.add_right
@[to_additive] lemma AntivaryOn.mul_right (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    AntivaryOn f (g₁ * g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: AntivaryOn f (g₁ + g₂) s
-- MonovaryOn.div_right
@[to_additive] lemma MonovaryOn.div_right (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    MonovaryOn f (g₁ / g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: MonovaryOn f (g₁ / g₂) s
-- MonovaryOn.sub_right
@[to_additive] lemma MonovaryOn.div_right (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    MonovaryOn f (g₁ / g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: MonovaryOn f (g₁ - g₂) s
-- AntivaryOn.div_right
@[to_additive] lemma AntivaryOn.div_right (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    AntivaryOn f (g₁ / g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: AntivaryOn f (g₁ / g₂) s
-- AntivaryOn.sub_right
@[to_additive] lemma AntivaryOn.div_right (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    AntivaryOn f (g₁ / g₂) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: AntivaryOn f (g₁ - g₂) s
-- MonovaryOn.nsmul_right
@[to_additive] lemma MonovaryOn.pow_right (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn f (g ^ n) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn f (n • g) s
-- MonovaryOn.pow_right
@[to_additive] lemma MonovaryOn.pow_right (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn f (g ^ n) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn f (g ^ n) s
-- AntivaryOn.pow_right
@[to_additive] lemma AntivaryOn.pow_right (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn f (g ^ n) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn f (g ^ n) s
-- AntivaryOn.nsmul_right
@[to_additive] lemma AntivaryOn.pow_right (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn f (g ^ n) s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn f (n • g) s
-- Monovary.mul_right
@[to_additive] lemma Monovary.mul_right (h₁ : Monovary f g₁) (h₂ : Monovary f g₂) :
    Monovary f (g₁ * g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Monovary f g₁) (h₂ : Monovary f g₂)
Goal: Monovary f (g₁ * g₂)
-- Monovary.add_right
@[to_additive] lemma Monovary.mul_right (h₁ : Monovary f g₁) (h₂ : Monovary f g₂) :
    Monovary f (g₁ * g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Monovary f g₁) (h₂ : Monovary f g₂)
Goal: Monovary f (g₁ + g₂)
-- Antivary.mul_right
@[to_additive] lemma Antivary.mul_right (h₁ : Antivary f g₁) (h₂ : Antivary f g₂) :
    Antivary f (g₁ * g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Antivary f g₁) (h₂ : Antivary f g₂)
Goal: Antivary f (g₁ * g₂)
-- Antivary.add_right
@[to_additive] lemma Antivary.mul_right (h₁ : Antivary f g₁) (h₂ : Antivary f g₂) :
    Antivary f (g₁ * g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Antivary f g₁) (h₂ : Antivary f g₂)
Goal: Antivary f (g₁ + g₂)
-- Monovary.sub_right
@[to_additive] lemma Monovary.div_right (h₁ : Monovary f g₁) (h₂ : Antivary f g₂) :
    Monovary f (g₁ / g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Monovary f g₁) (h₂ : Antivary f g₂)
Goal: Monovary f (g₁ - g₂)
-- Monovary.div_right
@[to_additive] lemma Monovary.div_right (h₁ : Monovary f g₁) (h₂ : Antivary f g₂) :
    Monovary f (g₁ / g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Monovary f g₁) (h₂ : Antivary f g₂)
Goal: Monovary f (g₁ / g₂)
-- Antivary.div_right
@[to_additive] lemma Antivary.div_right (h₁ : Antivary f g₁) (h₂ : Monovary f g₂) :
    Antivary f (g₁ / g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Antivary f g₁) (h₂ : Monovary f g₂)
Goal: Antivary f (g₁ / g₂)
-- Antivary.sub_right
@[to_additive] lemma Antivary.div_right (h₁ : Antivary f g₁) (h₂ : Monovary f g₂) :
    Antivary f (g₁ / g₂) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g₁ g₂ : ι → β} (h₁ : Antivary f g₁) (h₂ : Monovary f g₂)
Goal: Antivary f (g₁ - g₂)
-- Monovary.nsmul_right
@[to_additive] lemma Monovary.pow_right (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g : ι → β} (hfg : Monovary f g) (n : ℕ)
Goal: Monovary f (n • g)
-- Monovary.pow_right
@[to_additive] lemma Monovary.pow_right (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g : ι → β} (hfg : Monovary f g) (n : ℕ)
Goal: Monovary f (g ^ n)
-- Antivary.pow_right
@[to_additive] lemma Antivary.pow_right (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [LinearOrderedCommGroup β] {f : ι → α} {g : ι → β} (hfg : Antivary f g) (n : ℕ)
Goal: Antivary f (g ^ n)
-- Antivary.nsmul_right
@[to_additive] lemma Antivary.pow_right (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [LinearOrderedAddCommGroup β] {f : ι → α} {g : ι → β} (hfg : Antivary f g) (n : ℕ)
Goal: Antivary f (n • g)
-- MonovaryOn.mul_left₀
lemma MonovaryOn.mul_left₀ (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 ≤ f₂ i)
    (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) : MonovaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 ≤ f₂ i) (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s)
Goal: MonovaryOn (f₁ * f₂) g s
-- AntivaryOn.mul_left₀
lemma AntivaryOn.mul_left₀ (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 ≤ f₂ i)
    (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) : AntivaryOn (f₁ * f₂) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {s : Set ι} {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 ≤ f₂ i) (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s)
Goal: AntivaryOn (f₁ * f₂) g s
-- MonovaryOn.pow_left₀
lemma MonovaryOn.pow_left₀ (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn (f ^ n) g s
-- AntivaryOn.pow_left₀
lemma AntivaryOn.pow_left₀ (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn (f ^ n) g s
-- Monovary.mul_left₀
lemma Monovary.mul_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) :
    Monovary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g)
Goal: Monovary (f₁ * f₂) g
-- Antivary.mul_left₀
lemma Antivary.mul_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) :
    Antivary (f₁ * f₂) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g)
Goal: Antivary (f₁ * f₂) g
-- Monovary.pow_left₀
lemma Monovary.pow_left₀ (hf : 0 ≤ f) (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedSemiring α] [OrderedSemiring β] {f : ι → α} {g : ι → β} (hf : 0 ≤ f) (hfg : Monovary f g) (n : ℕ)
Goal: Monovary (f ^ n) g