File: Mathlib/Algebra/Order/Monovary.lean

full_name: monovaryOn_inv_left
code: @[to_additive (attr := simp)]
lemma monovaryOn_inv_left : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
goal: MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s

full_name: monovaryOn_neg_left
code: @[to_additive (attr := simp)]
lemma monovaryOn_inv_left : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
goal: MonovaryOn (-f) g s ↔ AntivaryOn f g s

full_name: antivaryOn_inv_left
code: @[to_additive (attr := simp)]
lemma antivaryOn_inv_left : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
goal: AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s

full_name: antivaryOn_neg_left
code: @[to_additive (attr := simp)]
lemma antivaryOn_inv_left : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
goal: AntivaryOn (-f) g s ↔ MonovaryOn f g s

full_name: monovaryOn_inv_right
code: @[to_additive (attr := simp)]
lemma monovaryOn_inv_right : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
goal: MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s

full_name: monovaryOn_neg_right
code: @[to_additive (attr := simp)]
lemma monovaryOn_inv_right : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
goal: MonovaryOn f (-g) s ↔ AntivaryOn f g s

full_name: antivaryOn_inv_right
code: @[to_additive (attr := simp)]
lemma antivaryOn_inv_right : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
goal: AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s

full_name: antivaryOn_neg_right
code: @[to_additive (attr := simp)]
lemma antivaryOn_inv_right : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
goal: AntivaryOn f (-g) s ↔ MonovaryOn f g s

full_name: monovaryOn_inv
code: @[to_additive] lemma monovaryOn_inv : MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s 
goal: MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s

full_name: monovaryOn_neg
code: @[to_additive] lemma monovaryOn_inv : MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s 
goal: MonovaryOn (-f) (-g) s ↔ MonovaryOn f g s

full_name: antivaryOn_inv
code: @[to_additive] lemma antivaryOn_inv : AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s 
goal: AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s

full_name: antivaryOn_neg
code: @[to_additive] lemma antivaryOn_inv : AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s 
goal: AntivaryOn (-f) (-g) s ↔ AntivaryOn f g s

full_name: monovary_neg_left
code: @[to_additive (attr := simp)] lemma monovary_inv_left : Monovary f⁻¹ g ↔ Antivary f g 
goal: Monovary (-f) g ↔ Antivary f g

full_name: monovary_inv_left
code: @[to_additive (attr := simp)] lemma monovary_inv_left : Monovary f⁻¹ g ↔ Antivary f g 
goal: Monovary f⁻¹ g ↔ Antivary f g

full_name: antivary_neg_left
code: @[to_additive (attr := simp)] lemma antivary_inv_left : Antivary f⁻¹ g ↔ Monovary f g 
goal: Antivary (-f) g ↔ Monovary f g

full_name: antivary_inv_left
code: @[to_additive (attr := simp)] lemma antivary_inv_left : Antivary f⁻¹ g ↔ Monovary f g 
goal: Antivary f⁻¹ g ↔ Monovary f g

full_name: monovary_neg_right
code: @[to_additive (attr := simp)] lemma monovary_inv_right : Monovary f g⁻¹ ↔ Antivary f g 
goal: Monovary f (-g) ↔ Antivary f g

full_name: monovary_inv_right
code: @[to_additive (attr := simp)] lemma monovary_inv_right : Monovary f g⁻¹ ↔ Antivary f g 
goal: Monovary f g⁻¹ ↔ Antivary f g

full_name: antivary_neg_right
code: @[to_additive (attr := simp)] lemma antivary_inv_right : Antivary f g⁻¹ ↔ Monovary f g 
goal: Antivary f (-g) ↔ Monovary f g

full_name: antivary_inv_right
code: @[to_additive (attr := simp)] lemma antivary_inv_right : Antivary f g⁻¹ ↔ Monovary f g 
goal: Antivary f g⁻¹ ↔ Monovary f g

full_name: monovary_inv
code: @[to_additive] lemma monovary_inv : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
goal: Monovary f⁻¹ g⁻¹ ↔ Monovary f g

full_name: monovary_neg
code: @[to_additive] lemma monovary_inv : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
goal: Monovary (-f) (-g) ↔ Monovary f g

full_name: antivary_neg
code: @[to_additive] lemma antivary_inv : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
goal: Antivary (-f) (-g) ↔ Antivary f g

full_name: antivary_inv
code: @[to_additive] lemma antivary_inv : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
goal: Antivary f⁻¹ g⁻¹ ↔ Antivary f g

full_name: MonovaryOn.mul_left
code: @[to_additive] lemma MonovaryOn.mul_left (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    MonovaryOn (f₁ * f₂) g s 
goal: MonovaryOn (f₁ * f₂) g s

full_name: MonovaryOn.add_left
code: @[to_additive] lemma MonovaryOn.mul_left (h₁ : MonovaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    MonovaryOn (f₁ * f₂) g s 
goal: MonovaryOn (f₁ + f₂) g s

full_name: AntivaryOn.add_left
code: @[to_additive] lemma AntivaryOn.mul_left (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    AntivaryOn (f₁ * f₂) g s 
goal: AntivaryOn (f₁ + f₂) g s

full_name: AntivaryOn.mul_left
code: @[to_additive] lemma AntivaryOn.mul_left (h₁ : AntivaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    AntivaryOn (f₁ * f₂) g s 
goal: AntivaryOn (f₁ * f₂) g s

full_name: MonovaryOn.div_left
code: @[to_additive] lemma MonovaryOn.div_left (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    MonovaryOn (f₁ / f₂) g s 
goal: MonovaryOn (f₁ / f₂) g s

full_name: MonovaryOn.sub_left
code: @[to_additive] lemma MonovaryOn.div_left (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) :
    MonovaryOn (f₁ / f₂) g s 
goal: MonovaryOn (f₁ - f₂) g s

full_name: AntivaryOn.sub_left
code: @[to_additive] lemma AntivaryOn.div_left (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    AntivaryOn (f₁ / f₂) g s 
goal: AntivaryOn (f₁ - f₂) g s

full_name: AntivaryOn.div_left
code: @[to_additive] lemma AntivaryOn.div_left (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) :
    AntivaryOn (f₁ / f₂) g s 
goal: AntivaryOn (f₁ / f₂) g s

full_name: MonovaryOn.pow_left
code: @[to_additive] lemma MonovaryOn.pow_left (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
goal: MonovaryOn (f ^ n) g s

full_name: MonovaryOn.nsmul_left
code: @[to_additive] lemma MonovaryOn.pow_left (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
goal: MonovaryOn (n • f) g s

full_name: AntivaryOn.nsmul_left
code: @[to_additive] lemma AntivaryOn.pow_left (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
goal: AntivaryOn (n • f) g s

full_name: AntivaryOn.pow_left
code: @[to_additive] lemma AntivaryOn.pow_left (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
goal: AntivaryOn (f ^ n) g s

full_name: Monovary.mul_left
code: @[to_additive]
lemma Monovary.mul_left (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) : Monovary (f₁ * f₂) g 
goal: Monovary (f₁ * f₂) g

full_name: Monovary.add_left
code: @[to_additive]
lemma Monovary.mul_left (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) : Monovary (f₁ * f₂) g 
goal: Monovary (f₁ + f₂) g

full_name: Antivary.mul_left
code: @[to_additive]
lemma Antivary.mul_left (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) : Antivary (f₁ * f₂) g 
goal: Antivary (f₁ * f₂) g

full_name: Antivary.add_left
code: @[to_additive]
lemma Antivary.mul_left (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) : Antivary (f₁ * f₂) g 
goal: Antivary (f₁ + f₂) g

full_name: Monovary.div_left
code: @[to_additive]
lemma Monovary.div_left (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g) : Monovary (f₁ / f₂) g 
goal: Monovary (f₁ / f₂) g

full_name: Monovary.sub_left
code: @[to_additive]
lemma Monovary.div_left (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g) : Monovary (f₁ / f₂) g 
goal: Monovary (f₁ - f₂) g

full_name: Antivary.sub_left
code: @[to_additive]
lemma Antivary.div_left (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g) : Antivary (f₁ / f₂) g 
goal: Antivary (f₁ - f₂) g

full_name: Antivary.div_left
code: @[to_additive]
lemma Antivary.div_left (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g) : Antivary (f₁ / f₂) g 
goal: Antivary (f₁ / f₂) g

full_name: Monovary.nsmul_left
code: @[to_additive] lemma Monovary.pow_left (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
goal: Monovary (n • f) g

full_name: Monovary.pow_left
code: @[to_additive] lemma Monovary.pow_left (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
goal: Monovary (f ^ n) g

full_name: Antivary.pow_left
code: @[to_additive] lemma Antivary.pow_left (hfg : Antivary f g) (n : ℕ) : Antivary (f ^ n) g 
goal: Antivary (f ^ n) g

full_name: Antivary.nsmul_left
code: @[to_additive] lemma Antivary.pow_left (hfg : Antivary f g) (n : ℕ) : Antivary (f ^ n) g 
goal: Antivary (n • f) g

full_name: MonovaryOn.add_right
code: @[to_additive] lemma MonovaryOn.mul_right (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    MonovaryOn f (g₁ * g₂) s 
goal: MonovaryOn f (g₁ + g₂) s

full_name: MonovaryOn.mul_right
code: @[to_additive] lemma MonovaryOn.mul_right (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    MonovaryOn f (g₁ * g₂) s 
goal: MonovaryOn f (g₁ * g₂) s

full_name: AntivaryOn.mul_right
code: @[to_additive] lemma AntivaryOn.mul_right (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    AntivaryOn f (g₁ * g₂) s 
goal: AntivaryOn f (g₁ * g₂) s

full_name: AntivaryOn.add_right
code: @[to_additive] lemma AntivaryOn.mul_right (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    AntivaryOn f (g₁ * g₂) s 
goal: AntivaryOn f (g₁ + g₂) s

full_name: MonovaryOn.div_right
code: @[to_additive] lemma MonovaryOn.div_right (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    MonovaryOn f (g₁ / g₂) s 
goal: MonovaryOn f (g₁ / g₂) s

full_name: MonovaryOn.sub_right
code: @[to_additive] lemma MonovaryOn.div_right (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) :
    MonovaryOn f (g₁ / g₂) s 
goal: MonovaryOn f (g₁ - g₂) s

full_name: AntivaryOn.div_right
code: @[to_additive] lemma AntivaryOn.div_right (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    AntivaryOn f (g₁ / g₂) s 
goal: AntivaryOn f (g₁ / g₂) s

full_name: AntivaryOn.sub_right
code: @[to_additive] lemma AntivaryOn.div_right (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) :
    AntivaryOn f (g₁ / g₂) s 
goal: AntivaryOn f (g₁ - g₂) s

full_name: MonovaryOn.nsmul_right
code: @[to_additive] lemma MonovaryOn.pow_right (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn f (g ^ n) s 
goal: MonovaryOn f (n • g) s

full_name: MonovaryOn.pow_right
code: @[to_additive] lemma MonovaryOn.pow_right (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn f (g ^ n) s 
goal: MonovaryOn f (g ^ n) s

full_name: AntivaryOn.pow_right
code: @[to_additive] lemma AntivaryOn.pow_right (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn f (g ^ n) s 
goal: AntivaryOn f (g ^ n) s

full_name: AntivaryOn.nsmul_right
code: @[to_additive] lemma AntivaryOn.pow_right (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn f (g ^ n) s 
goal: AntivaryOn f (n • g) s

full_name: Monovary.mul_right
code: @[to_additive] lemma Monovary.mul_right (h₁ : Monovary f g₁) (h₂ : Monovary f g₂) :
    Monovary f (g₁ * g₂) 
goal: Monovary f (g₁ * g₂)

full_name: Monovary.add_right
code: @[to_additive] lemma Monovary.mul_right (h₁ : Monovary f g₁) (h₂ : Monovary f g₂) :
    Monovary f (g₁ * g₂) 
goal: Monovary f (g₁ + g₂)

full_name: Antivary.mul_right
code: @[to_additive] lemma Antivary.mul_right (h₁ : Antivary f g₁) (h₂ : Antivary f g₂) :
    Antivary f (g₁ * g₂) 
goal: Antivary f (g₁ * g₂)

full_name: Antivary.add_right
code: @[to_additive] lemma Antivary.mul_right (h₁ : Antivary f g₁) (h₂ : Antivary f g₂) :
    Antivary f (g₁ * g₂) 
goal: Antivary f (g₁ + g₂)

full_name: Monovary.sub_right
code: @[to_additive] lemma Monovary.div_right (h₁ : Monovary f g₁) (h₂ : Antivary f g₂) :
    Monovary f (g₁ / g₂) 
goal: Monovary f (g₁ - g₂)

full_name: Monovary.div_right
code: @[to_additive] lemma Monovary.div_right (h₁ : Monovary f g₁) (h₂ : Antivary f g₂) :
    Monovary f (g₁ / g₂) 
goal: Monovary f (g₁ / g₂)

full_name: Antivary.div_right
code: @[to_additive] lemma Antivary.div_right (h₁ : Antivary f g₁) (h₂ : Monovary f g₂) :
    Antivary f (g₁ / g₂) 
goal: Antivary f (g₁ / g₂)

full_name: Antivary.sub_right
code: @[to_additive] lemma Antivary.div_right (h₁ : Antivary f g₁) (h₂ : Monovary f g₂) :
    Antivary f (g₁ / g₂) 
goal: Antivary f (g₁ - g₂)

full_name: Monovary.nsmul_right
code: @[to_additive] lemma Monovary.pow_right (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
goal: Monovary f (n • g)

full_name: Monovary.pow_right
code: @[to_additive] lemma Monovary.pow_right (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
goal: Monovary f (g ^ n)

full_name: Antivary.pow_right
code: @[to_additive] lemma Antivary.pow_right (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
goal: Antivary f (g ^ n)

full_name: Antivary.nsmul_right
code: @[to_additive] lemma Antivary.pow_right (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
goal: Antivary f (n • g)

full_name: MonovaryOn.mul_left₀
code: 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 
goal: MonovaryOn (f₁ * f₂) g s

full_name: AntivaryOn.mul_left₀
code: 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 
goal: AntivaryOn (f₁ * f₂) g s

full_name: MonovaryOn.pow_left₀
code: lemma MonovaryOn.pow_left₀ (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn (f ^ n) g s 
goal: MonovaryOn (f ^ n) g s

full_name: AntivaryOn.pow_left₀
code: lemma AntivaryOn.pow_left₀ (hf : ∀ i ∈ s, 0 ≤ f i) (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn (f ^ n) g s 
goal: AntivaryOn (f ^ n) g s

full_name: Monovary.mul_left₀
code: lemma Monovary.mul_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Monovary f₁ g) (h₂ : Monovary f₂ g) :
    Monovary (f₁ * f₂) g 
goal: Monovary (f₁ * f₂) g

full_name: Antivary.mul_left₀
code: lemma Antivary.mul_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : 0 ≤ f₂) (h₁ : Antivary f₁ g) (h₂ : Antivary f₂ g) :
    Antivary (f₁ * f₂) g 
goal: Antivary (f₁ * f₂) g

full_name: Monovary.pow_left₀
code: lemma Monovary.pow_left₀ (hf : 0 ≤ f) (hfg : Monovary f g) (n : ℕ) : Monovary (f ^ n) g 
goal: Monovary (f ^ n) g

full_name: Antivary.pow_left₀
code: lemma Antivary.pow_left₀ (hf : 0 ≤ f) (hfg : Antivary f g) (n : ℕ) : Antivary (f ^ n) g 
goal: Antivary (f ^ n) g

full_name: MonovaryOn.mul_right₀
code: lemma MonovaryOn.mul_right₀ (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 ≤ g₂ i)
    (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) : MonovaryOn f (g₁ * g₂) s 
goal: MonovaryOn f (g₁ * g₂) s

full_name: AntivaryOn.mul_right₀
code: lemma AntivaryOn.mul_right₀ (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 ≤ g₂ i)
    (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) : AntivaryOn f (g₁ * g₂) s 
goal: AntivaryOn f (g₁ * g₂) s

full_name: MonovaryOn.pow_right₀
code: lemma MonovaryOn.pow_right₀ (hg : ∀ i ∈ s, 0 ≤ g i) (hfg : MonovaryOn f g s) (n : ℕ) :
    MonovaryOn f (g ^ n) s 
goal: MonovaryOn f (g ^ n) s

full_name: AntivaryOn.pow_right₀
code: lemma AntivaryOn.pow_right₀ (hg : ∀ i ∈ s, 0 ≤ g i) (hfg : AntivaryOn f g s) (n : ℕ) :
    AntivaryOn f (g ^ n) s 
goal: AntivaryOn f (g ^ n) s

full_name: Monovary.mul_right₀
code: lemma Monovary.mul_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (h₁ : Monovary f g₁) (h₂ : Monovary f g₂) :
    Monovary f (g₁ * g₂) 
goal: Monovary f (g₁ * g₂)

full_name: Antivary.mul_right₀
code: lemma Antivary.mul_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (h₁ : Antivary f g₁) (h₂ : Antivary f g₂) :
    Antivary f (g₁ * g₂) 
goal: Antivary f (g₁ * g₂)

full_name: Monovary.pow_right₀
code: lemma Monovary.pow_right₀ (hg : 0 ≤ g) (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
goal: Monovary f (g ^ n)

full_name: Antivary.pow_right₀
code: lemma Antivary.pow_right₀ (hg : 0 ≤ g) (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
goal: Antivary f (g ^ n)

full_name: monovaryOn_inv_left₀
code: @[simp]
lemma monovaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
goal: MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s

full_name: antivaryOn_inv_left₀
code: @[simp]
lemma antivaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
goal: AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s

full_name: monovaryOn_inv_right₀
code: @[simp]
lemma monovaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
goal: MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s

full_name: antivaryOn_inv_right₀
code: @[simp]
lemma antivaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
goal: AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s

full_name: monovaryOn_inv₀
code: lemma monovaryOn_inv₀ (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i) :
    MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s 
goal: MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s

full_name: antivaryOn_inv₀
code: lemma antivaryOn_inv₀ (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i) :
    AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s 
goal: AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s

full_name: monovary_inv_left₀
code: @[simp] lemma monovary_inv_left₀ (hf : StrongLT 0 f) : Monovary f⁻¹ g ↔ Antivary f g 
goal: Monovary f⁻¹ g ↔ Antivary f g

full_name: antivary_inv_left₀
code: @[simp] lemma antivary_inv_left₀ (hf : StrongLT 0 f) : Antivary f⁻¹ g ↔ Monovary f g 
goal: Antivary f⁻¹ g ↔ Monovary f g

full_name: monovary_inv_right₀
code: @[simp] lemma monovary_inv_right₀ (hg : StrongLT 0 g) : Monovary f g⁻¹ ↔ Antivary f g 
goal: Monovary f g⁻¹ ↔ Antivary f g

full_name: antivary_inv_right₀
code: @[simp] lemma antivary_inv_right₀ (hg : StrongLT 0 g) : Antivary f g⁻¹ ↔ Monovary f g 
goal: Antivary f g⁻¹ ↔ Monovary f g

full_name: monovary_inv₀
code: lemma monovary_inv₀ (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
goal: Monovary f⁻¹ g⁻¹ ↔ Monovary f g

full_name: antivary_inv₀
code: lemma antivary_inv₀ (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
goal: Antivary f⁻¹ g⁻¹ ↔ Antivary f g

full_name: MonovaryOn.div_left₀
code: lemma MonovaryOn.div_left₀ (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 < f₂ i)
    (h₁ : MonovaryOn f₁ g s) (h₂ : AntivaryOn f₂ g s) : MonovaryOn (f₁ / f₂) g s 
goal: MonovaryOn (f₁ / f₂) g s

full_name: AntivaryOn.div_left₀
code: lemma AntivaryOn.div_left₀ (hf₁ : ∀ i ∈ s, 0 ≤ f₁ i) (hf₂ : ∀ i ∈ s, 0 < f₂ i)
    (h₁ : AntivaryOn f₁ g s) (h₂ : MonovaryOn f₂ g s) : AntivaryOn (f₁ / f₂) g s 
goal: AntivaryOn (f₁ / f₂) g s

full_name: Monovary.div_left₀
code: lemma Monovary.div_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (h₁ : Monovary f₁ g)
    (h₂ : Antivary f₂ g) : Monovary (f₁ / f₂) g 
goal: Monovary (f₁ / f₂) g

full_name: Antivary.div_left₀
code: lemma Antivary.div_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (h₁ : Antivary f₁ g)
    (h₂ : Monovary f₂ g) : Antivary (f₁ / f₂) g 
goal: Antivary (f₁ / f₂) g

full_name: MonovaryOn.div_right₀
code: lemma MonovaryOn.div_right₀ (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 < g₂ i)
    (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s) : MonovaryOn f (g₁ / g₂) s 
goal: MonovaryOn f (g₁ / g₂) s

full_name: AntivaryOn.div_right₀
code: lemma AntivaryOn.div_right₀ (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 < g₂ i)
    (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s) : AntivaryOn f (g₁ / g₂) s 
goal: AntivaryOn f (g₁ / g₂) s

full_name: Monovary.div_right₀
code: lemma Monovary.div_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (h₁ : Monovary f g₁)
    (h₂ : Antivary f g₂) : Monovary f (g₁ / g₂) 
goal: Monovary f (g₁ / g₂)

full_name: Antivary.div_right₀
code: lemma Antivary.div_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (h₁ : Antivary f g₁)
    (h₂ : Monovary f g₂) : Antivary f (g₁ / g₂) 
goal: Antivary f (g₁ / g₂)

full_name: monovaryOn_iff_forall_smul_nonneg
code: lemma monovaryOn_iff_forall_smul_nonneg :
    MonovaryOn f g s ↔ ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i) 
goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)

full_name: antivaryOn_iff_forall_smul_nonpos
code: lemma antivaryOn_iff_forall_smul_nonpos :
    AntivaryOn f g s ↔ ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → (f j - f i) • (g j - g i) ≤ 0 
goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f j - f i) • (g j - g i) ≤ 0

full_name: monovary_iff_forall_smul_nonneg
code: lemma monovary_iff_forall_smul_nonneg : Monovary f g ↔ ∀ i j, 0 ≤ (f j - f i) • (g j - g i) 
goal: Monovary f g ↔ ∀ (i j : ι), 0 ≤ (f j - f i) • (g j - g i)

full_name: antivary_iff_forall_smul_nonpos
code: lemma antivary_iff_forall_smul_nonpos : Antivary f g ↔ ∀ i j, (f j - f i) • (g j - g i) ≤ 0 
goal: Antivary f g ↔ ∀ (i j : ι), (f j - f i) • (g j - g i) ≤ 0

full_name: monovaryOn_iff_smul_rearrangement
code: /-- Two functions monovary iff the rearrangement inequality holds. -/
lemma monovaryOn_iff_smul_rearrangement :
    MonovaryOn f g s ↔
      ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → f i • g j + f j • g i ≤ f i • g i + f j • g j 
goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → f i • g j + f j • g i ≤ f i • g i + f j • g j

full_name: antivaryOn_iff_smul_rearrangement
code: /-- Two functions antivary iff the rearrangement inequality holds. -/
lemma antivaryOn_iff_smul_rearrangement :
    AntivaryOn f g s ↔
      ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → f i • g i + f j • g j ≤ f i • g j + f j • g i 
goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → f i • g i + f j • g j ≤ f i • g j + f j • g i

full_name: monovary_iff_smul_rearrangement
code: /-- Two functions monovary iff the rearrangement inequality holds. -/
lemma monovary_iff_smul_rearrangement :
    Monovary f g ↔ ∀ i j, f i • g j + f j • g i ≤ f i • g i + f j • g j 
goal: Monovary f g ↔ ∀ (i j : ι), f i • g j + f j • g i ≤ f i • g i + f j • g j

full_name: antivary_iff_smul_rearrangement
code: /-- Two functions antivary iff the rearrangement inequality holds. -/
lemma antivary_iff_smul_rearrangement :
    Antivary f g ↔ ∀ i j, f i • g i + f j • g j ≤ f i • g j + f j • g i 
goal: Antivary f g ↔ ∀ (i j : ι), f i • g i + f j • g j ≤ f i • g j + f j • g i

full_name: monovaryOn_iff_forall_mul_nonneg
code: lemma monovaryOn_iff_forall_mul_nonneg :
    MonovaryOn f g s ↔ ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → 0 ≤ (f j - f i) * (g j - g i) 
goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) * (g j - g i)

full_name: antivaryOn_iff_forall_mul_nonpos
code: lemma antivaryOn_iff_forall_mul_nonpos :
    AntivaryOn f g s ↔ ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → (f j - f i) * (g j - g i) ≤ 0 
goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f j - f i) * (g j - g i) ≤ 0

full_name: monovary_iff_forall_mul_nonneg
code: lemma monovary_iff_forall_mul_nonneg : Monovary f g ↔ ∀ i j, 0 ≤ (f j - f i) * (g j - g i) 
goal: Monovary f g ↔ ∀ (i j : ι), 0 ≤ (f j - f i) * (g j - g i)

full_name: antivary_iff_forall_mul_nonpos
code: lemma antivary_iff_forall_mul_nonpos : Antivary f g ↔ ∀ i j, (f j - f i) * (g j - g i) ≤ 0 
goal: Antivary f g ↔ ∀ (i j : ι), (f j - f i) * (g j - g i) ≤ 0

full_name: monovaryOn_iff_mul_rearrangement
code: /-- Two functions monovary iff the rearrangement inequality holds. -/
lemma monovaryOn_iff_mul_rearrangement :
    MonovaryOn f g s ↔
      ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → f i * g j + f j * g i ≤ f i * g i + f j * g j 
goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → f i * g j + f j * g i ≤ f i * g i + f j * g j

full_name: antivaryOn_iff_mul_rearrangement
code: /-- Two functions antivary iff the rearrangement inequality holds. -/
lemma antivaryOn_iff_mul_rearrangement :
    AntivaryOn f g s ↔
      ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → f i * g i + f j * g j ≤ f i * g j + f j * g i 
goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → f i * g i + f j * g j ≤ f i * g j + f j * g i

full_name: monovary_iff_mul_rearrangement
code: /-- Two functions monovary iff the rearrangement inequality holds. -/
lemma monovary_iff_mul_rearrangement :
    Monovary f g ↔ ∀ i j, f i * g j + f j * g i ≤ f i * g i + f j * g j 
goal: Monovary f g ↔ ∀ (i j : ι), f i * g j + f j * g i ≤ f i * g i + f j * g j

full_name: antivary_iff_mul_rearrangement
code: /-- Two functions antivary iff the rearrangement inequality holds. -/
lemma antivary_iff_mul_rearrangement :
    Antivary f g ↔ ∀ i j, f i * g i + f j * g j ≤ f i * g j + f j * g i 
goal: Antivary f g ↔ ∀ (i j : ι), f i * g i + f j * g j ≤ f i * g j + f j * g i