ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Antivary.pow_left₀
lemma Antivary.pow_left₀ (hf : 0 ≤ f) (hfg : Antivary f g) (n : ℕ) : Antivary (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 : Antivary f g) (n : ℕ)
Goal: Antivary (f ^ n) g
-- MonovaryOn.mul_right₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 ≤ g₂ i) (h₁ : MonovaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: MonovaryOn f (g₁ * g₂) s
-- AntivaryOn.mul_right₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 ≤ g₂ i) (h₁ : AntivaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: AntivaryOn f (g₁ * g₂) s
-- MonovaryOn.pow_right₀
lemma MonovaryOn.pow_right₀ (hg : ∀ i ∈ s, 0 ≤ g i) (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} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {s : Set ι} {f : ι → α} {g : ι → β} (hg : ∀ i ∈ s, 0 ≤ g i) (hfg : MonovaryOn f g s) (n : ℕ)
Goal: MonovaryOn f (g ^ n) s
-- AntivaryOn.pow_right₀
lemma AntivaryOn.pow_right₀ (hg : ∀ i ∈ s, 0 ≤ g i) (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} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {s : Set ι} {f : ι → α} {g : ι → β} (hg : ∀ i ∈ s, 0 ≤ g i) (hfg : AntivaryOn f g s) (n : ℕ)
Goal: AntivaryOn f (g ^ n) s
-- Monovary.mul_right₀
lemma Monovary.mul_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (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} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (h₁ : Monovary f g₁) (h₂ : Monovary f g₂)
Goal: Monovary f (g₁ * g₂)
-- Antivary.mul_right₀
lemma Antivary.mul_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (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} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : 0 ≤ g₁) (hg₂ : 0 ≤ g₂) (h₁ : Antivary f g₁) (h₂ : Antivary f g₂)
Goal: Antivary f (g₁ * g₂)
-- Monovary.pow_right₀
lemma Monovary.pow_right₀ (hg : 0 ≤ g) (hfg : Monovary f g) (n : ℕ) : Monovary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {f : ι → α} {g : ι → β} (hg : 0 ≤ g) (hfg : Monovary f g) (n : ℕ)
Goal: Monovary f (g ^ n)
-- Antivary.pow_right₀
lemma Antivary.pow_right₀ (hg : 0 ≤ g) (hfg : Antivary f g) (n : ℕ) : Antivary f (g ^ n) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] {f : ι → α} {g : ι → β} (hg : 0 ≤ g) (hfg : Antivary f g) (n : ℕ)
Goal: Antivary f (g ^ n)
-- monovaryOn_inv_left₀
@[simp]
lemma monovaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 < f i)
Goal: MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s
-- antivaryOn_inv_left₀
@[simp]
lemma antivaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 < f i)
Goal: AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s
-- monovaryOn_inv_right₀
@[simp]
lemma monovaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hg : ∀ i ∈ s, 0 < g i)
Goal: MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s
-- antivaryOn_inv_right₀
@[simp]
lemma antivaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hg : ∀ i ∈ s, 0 < g i)
Goal: AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s
-- monovaryOn_inv₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i)
Goal: MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s
-- antivaryOn_inv₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g : ι → β} (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i)
Goal: AntivaryOn f⁻¹ g⁻¹ s ↔ AntivaryOn f g s
-- monovary_inv_left₀
@[simp] lemma monovary_inv_left₀ (hf : StrongLT 0 f) : Monovary f⁻¹ g ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hf : StrongLT 0 f)
Goal: Monovary f⁻¹ g ↔ Antivary f g
-- antivary_inv_left₀
@[simp] lemma antivary_inv_left₀ (hf : StrongLT 0 f) : Antivary f⁻¹ g ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hf : StrongLT 0 f)
Goal: Antivary f⁻¹ g ↔ Monovary f g
-- monovary_inv_right₀
@[simp] lemma monovary_inv_right₀ (hg : StrongLT 0 g) : Monovary f g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hg : StrongLT 0 g)
Goal: Monovary f g⁻¹ ↔ Antivary f g
-- antivary_inv_right₀
@[simp] lemma antivary_inv_right₀ (hg : StrongLT 0 g) : Antivary f g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hg : StrongLT 0 g)
Goal: Antivary f g⁻¹ ↔ Monovary f g
-- monovary_inv₀
lemma monovary_inv₀ (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Monovary f⁻¹ g⁻¹ ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hf : StrongLT 0 f) (hg : StrongLT 0 g)
Goal: Monovary f⁻¹ g⁻¹ ↔ Monovary f g
-- antivary_inv₀
lemma antivary_inv₀ (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Antivary f⁻¹ g⁻¹ ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g : ι → β} (hf : StrongLT 0 f) (hg : StrongLT 0 g)
Goal: Antivary f⁻¹ g⁻¹ ↔ Antivary f g
-- MonovaryOn.div_left₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield 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 : AntivaryOn f₂ g s)
Goal: MonovaryOn (f₁ / f₂) g s
-- AntivaryOn.div_left₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield 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 : MonovaryOn f₂ g s)
Goal: AntivaryOn (f₁ / f₂) g s
-- Monovary.div_left₀
lemma Monovary.div_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (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} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (h₁ : Monovary f₁ g) (h₂ : Antivary f₂ g)
Goal: Monovary (f₁ / f₂) g
-- Antivary.div_left₀
lemma Antivary.div_left₀ (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (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} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f₁ f₂ : ι → α} {g : ι → β} (hf₁ : 0 ≤ f₁) (hf₂ : StrongLT 0 f₂) (h₁ : Antivary f₁ g) (h₂ : Monovary f₂ g)
Goal: Antivary (f₁ / f₂) g
-- MonovaryOn.div_right₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 < g₂ i) (h₁ : MonovaryOn f g₁ s) (h₂ : AntivaryOn f g₂ s)
Goal: MonovaryOn f (g₁ / g₂) s
-- AntivaryOn.div_right₀
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : ∀ i ∈ s, 0 ≤ g₁ i) (hg₂ : ∀ i ∈ s, 0 < g₂ i) (h₁ : AntivaryOn f g₁ s) (h₂ : MonovaryOn f g₂ s)
Goal: AntivaryOn f (g₁ / g₂) s
-- Monovary.div_right₀
lemma Monovary.div_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (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} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (h₁ : Monovary f g₁) (h₂ : Antivary f g₂)
Goal: Monovary f (g₁ / g₂)
-- Antivary.div_right₀
lemma Antivary.div_right₀ (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (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} [LinearOrderedSemifield α] [LinearOrderedSemifield β] {f : ι → α} {g₁ g₂ : ι → β} (hg₁ : 0 ≤ g₁) (hg₂ : StrongLT 0 g₂) (h₁ : Antivary f g₁) (h₂ : Monovary f g₂)
Goal: Antivary f (g₁ / g₂)
-- monovaryOn_iff_forall_smul_nonneg
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) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)
-- antivaryOn_iff_forall_smul_nonpos
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f j - f i) • (g j - g i) ≤ 0
-- monovary_iff_forall_smul_nonneg
lemma monovary_iff_forall_smul_nonneg : Monovary f g ↔ ∀ i j, 0 ≤ (f j - f i) • (g j - g i) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β}
Goal: Monovary f g ↔ ∀ (i j : ι), 0 ≤ (f j - f i) • (g j - g i)
-- antivary_iff_forall_smul_nonpos
lemma antivary_iff_forall_smul_nonpos : Antivary f g ↔ ∀ i j, (f j - f i) • (g j - g i) ≤ 0 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β}
Goal: Antivary f g ↔ ∀ (i j : ι), (f j - f i) • (g j - g i) ≤ 0
-- monovaryOn_iff_smul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} {s : Set ι}
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
-- antivaryOn_iff_smul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β} {s : Set ι}
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
-- monovary_iff_smul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β}
Goal: Monovary f g ↔ ∀ (i j : ι), f i • g j + f j • g i ≤ f i • g i + f j • g j
-- antivary_iff_smul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β] {f : ι → α} {g : ι → β}
Goal: Antivary f g ↔ ∀ (i j : ι), f i • g i + f j • g j ≤ f i • g j + f j • g i
-- monovaryOn_iff_forall_mul_nonneg
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) 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} {s : Set ι}
Goal: MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) * (g j - g i)
-- antivaryOn_iff_forall_mul_nonpos
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 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} {s : Set ι}
Goal: AntivaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f j - f i) * (g j - g i) ≤ 0
-- monovary_iff_forall_mul_nonneg
lemma monovary_iff_forall_mul_nonneg : Monovary f g ↔ ∀ i j, 0 ≤ (f j - f i) * (g j - g i) 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α}
Goal: Monovary f g ↔ ∀ (i j : ι), 0 ≤ (f j - f i) * (g j - g i)
-- antivary_iff_forall_mul_nonpos
lemma antivary_iff_forall_mul_nonpos : Antivary f g ↔ ∀ i j, (f j - f i) * (g j - g i) ≤ 0 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α}
Goal: Antivary f g ↔ ∀ (i j : ι), (f j - f i) * (g j - g i) ≤ 0
-- monovaryOn_iff_mul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} {s : Set ι}
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
-- antivaryOn_iff_mul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α} {s : Set ι}
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
-- monovary_iff_mul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α}
Goal: Monovary f g ↔ ∀ (i j : ι), f i * g j + f j * g i ≤ f i * g i + f j * g j
-- antivary_iff_mul_rearrangement
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 
Arguments: {ι : Type u_1} {α : Type u_2} [LinearOrderedRing α] {f g : ι → α}
Goal: Antivary f g ↔ ∀ (i j : ι), f i * g i + f j * g j ≤ f i * g j + f j * g i
-- Finset.mem_finsuppAntidiag
@[simp] lemma mem_finsuppAntidiag : f ∈ finsuppAntidiag s n ↔ s.sum f = n ∧ f.support ⊆ s 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {s : Finset ι} {n : μ} {f : ι →₀ μ}
Goal: f ∈ s.finsuppAntidiag n ↔ s.sum ⇑f = n ∧ f.support ⊆ s
-- Finset.mem_finsuppAntidiag'
lemma mem_finsuppAntidiag' :
    f ∈ finsuppAntidiag s n ↔ f.sum (fun _ x ↦ x) = n ∧ f.support ⊆ s 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {s : Finset ι} {n : μ} {f : ι →₀ μ}
Goal: f ∈ s.finsuppAntidiag n ↔ (f.sum fun x x => x) = n ∧ f.support ⊆ s
-- Finset.finsuppAntidiag_empty_zero
@[simp] lemma finsuppAntidiag_empty_zero : finsuppAntidiag (∅ : Finset ι) (0 : μ) = {0} 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ]
Goal: ∅.finsuppAntidiag 0 = {0}
-- Finset.finsuppAntidiag_empty_of_ne_zero
@[simp] lemma finsuppAntidiag_empty_of_ne_zero (hn : n ≠ 0) :
    finsuppAntidiag (∅ : Finset ι) n = ∅ 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {n : μ} (hn : n ≠ 0)
Goal: ∅.finsuppAntidiag n = ∅
-- Finset.finsuppAntidiag_empty
lemma finsuppAntidiag_empty (n : μ) :
    finsuppAntidiag (∅ : Finset ι) n = if n = 0 then {0} else ∅ 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] (n : μ)
Goal: ∅.finsuppAntidiag n = if n = 0 then {0} else ∅
-- Finset.mem_finsuppAntidiag_insert
theorem mem_finsuppAntidiag_insert {a : ι} {s : Finset ι}
    (h : a ∉ s) (n : μ) {f : ι →₀ μ} :
    f ∈ finsuppAntidiag (insert a s) n ↔
      ∃ m ∈ antidiagonal n, ∃ (g : ι →₀ μ),
        f = Finsupp.update g a m.1 ∧ g ∈ finsuppAntidiag s m.2 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {a : ι} {s : Finset ι} (h : a ∉ s) (n : μ) {f : ι →₀ μ}
Goal: f ∈ (insert a s).finsuppAntidiag n ↔ ∃ m ∈ Finset.antidiagonal n, ∃ g, f = g.update a m.1 ∧ g ∈ s.finsuppAntidiag m.2
-- Finset.finsuppAntidiag_insert
theorem finsuppAntidiag_insert [DecidableEq μ] {a : ι} {s : Finset ι}
    (h : a ∉ s) (n : μ) :
    finsuppAntidiag (insert a s) n = (antidiagonal n).biUnion
      (fun p : μ × μ =>
        (finsuppAntidiag s p.snd).attach.map
        ⟨fun f => Finsupp.update f.val a p.fst,
        (fun ⟨f, hf⟩ ⟨g, hg⟩ hfg => Subtype.ext <| by
          simp only [mem_val, mem_finsuppAntidiag] at hf hg
          simp only [DFunLike.ext_iff] at hfg ⊢
          intro x
          obtain rfl | hx := eq_or_ne x a
          · replace hf := mt (hf.2 ·) h
            replace hg := mt (hg.2 ·) h
            rw [not_mem_support_iff.mp hf, not_mem_support_iff.mp hg]
          · simpa only [coe_update, Function.update, dif_neg hx] using hfg x)⟩) 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {a : ι} {s : Finset ι} (h : a ∉ s) (n : μ)
Goal: (insert a s).finsuppAntidiag n =
  (Finset.antidiagonal n).biUnion fun p =>
    Finset.map { toFun := fun f => (↑f).update a p.1, inj' := ⋯ } (s.finsuppAntidiag p.2).attach
-- Finset.mapRange_finsuppAntidiag_subset
lemma mapRange_finsuppAntidiag_subset {e : μ ≃+ μ'} {s : Finset ι} {n : μ} :
    (finsuppAntidiag s n).map (mapRange.addEquiv e).toEmbedding ⊆ finsuppAntidiag s (e n) 
Arguments: {ι : Type u_1} {μ : Type u_2} {μ' : Type u_3} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] [AddCommMonoid μ'] [Finset.HasAntidiagonal μ'] [DecidableEq μ'] {e : μ ≃+ μ'} {s : Finset ι} {n : μ}
Goal: Finset.map (Finsupp.mapRange.addEquiv e).toEmbedding (s.finsuppAntidiag n) ⊆ s.finsuppAntidiag (e n)
-- Finset.mapRange_finsuppAntidiag_eq
lemma mapRange_finsuppAntidiag_eq {e : μ ≃+ μ'} {s : Finset ι} {n : μ} :
    (finsuppAntidiag s n).map (mapRange.addEquiv e).toEmbedding = finsuppAntidiag s (e n) 
Arguments: {ι : Type u_1} {μ : Type u_2} {μ' : Type u_3} [DecidableEq ι] [AddCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] [AddCommMonoid μ'] [Finset.HasAntidiagonal μ'] [DecidableEq μ'] {e : μ ≃+ μ'} {s : Finset ι} {n : μ}
Goal: Finset.map (Finsupp.mapRange.addEquiv e).toEmbedding (s.finsuppAntidiag n) = s.finsuppAntidiag (e n)
-- Finset.finsuppAntidiag_zero
@[simp] lemma finsuppAntidiag_zero (s : Finset ι) : finsuppAntidiag s (0 : μ) = {0} 
Arguments: {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [DecidableEq μ] [CanonicallyOrderedAddCommMonoid μ] [Finset.HasAntidiagonal μ] (s : Finset ι)
Goal: s.finsuppAntidiag 0 = {0}
-- OrderedAddCommGroup.add_le_add_left
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
    protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
Arguments: {α : Type u} [self : OrderedAddCommGroup α] (a b : α)
Goal: a ≤ b → ∀ (c : α), c + a ≤ c + b
-- OrderedCommGroup.mul_le_mul_left
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
    protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
Arguments: {α : Type u} [self : OrderedCommGroup α] (a b : α)
Goal: a ≤ b → ∀ (c : α), c * a ≤ c * b
-- OrderedAddCommGroup.to_contravariantClass_left_le
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
    ContravariantClass α α (· * ·) (· ≤ ·) 
Arguments: (α : Type u) [OrderedAddCommGroup α]
Goal: ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
-- OrderedCommGroup.to_contravariantClass_left_le
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
    ContravariantClass α α (· * ·) (· ≤ ·) 
Arguments: (α : Type u) [OrderedCommGroup α]
Goal: ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
-- OrderedAddCommGroup.to_contravariantClass_right_le
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
    ContravariantClass α α (swap (· * ·)) (· ≤ ·) 
Arguments: (α : Type u) [OrderedAddCommGroup α]
Goal: ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
-- OrderedCommGroup.to_contravariantClass_right_le
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
    ContravariantClass α α (swap (· * ·)) (· ≤ ·) 
Arguments: (α : Type u) [OrderedCommGroup α]
Goal: ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
-- LinearOrderedCommGroup.mul_lt_mul_left'
@[to_additive LinearOrderedAddCommGroup.add_lt_add_left]
theorem LinearOrderedCommGroup.mul_lt_mul_left' (a b : α) (h : a < b) (c : α) : c * a < c * b 
Arguments: {α : Type u} [LinearOrderedCommGroup α] (a b : α) (h : a < b) (c : α)
Goal: c * a < c * b
-- LinearOrderedAddCommGroup.add_lt_add_left
@[to_additive LinearOrderedAddCommGroup.add_lt_add_left]
theorem LinearOrderedCommGroup.mul_lt_mul_left' (a b : α) (h : a < b) (c : α) : c * a < c * b 
Arguments: {α : Type u} [LinearOrderedAddCommGroup α] (a b : α) (h : a < b) (c : α)
Goal: c + a < c + b
-- eq_zero_of_neg_eq
@[to_additive eq_zero_of_neg_eq]
theorem eq_one_of_inv_eq' (h : a⁻¹ = a) : a = 1 
Arguments: {α : Type u} [LinearOrderedAddCommGroup α] {a : α} (h : -a = a)
Goal: a = 0
-- eq_one_of_inv_eq'
@[to_additive eq_zero_of_neg_eq]
theorem eq_one_of_inv_eq' (h : a⁻¹ = a) : a = 1 
Arguments: {α : Type u} [LinearOrderedCommGroup α] {a : α} (h : a⁻¹ = a)
Goal: a = 1