ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Right.mul_lt_mul
@[to_additive "Only assumes right strict covariance"]
theorem Right.mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
    (h₁ : a < b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a * c < b * d
-- Right.add_lt_add
@[to_additive "Only assumes right strict covariance"]
theorem Right.mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
    (h₁ : a < b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a + c < b + d
-- mul_le_mul'
@[to_additive (attr := gcongr) add_le_add]
theorem mul_le_mul' [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) :
    a * c ≤ b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d)
Goal: a * c ≤ b * d
-- add_le_add
@[to_additive (attr := gcongr) add_le_add]
theorem mul_le_mul' [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) :
    a * c ≤ b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d)
Goal: a + c ≤ b + d
-- mul_le_mul_three
@[to_additive]
theorem mul_le_mul_three [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e)
    (h₃ : c ≤ f) :
    a * b * c ≤ d * e * f 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e) (h₃ : c ≤ f)
Goal: a * b * c ≤ d * e * f
-- add_le_add_three
@[to_additive]
theorem mul_le_mul_three [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e)
    (h₃ : c ≤ f) :
    a * b * c ≤ d * e * f 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e) (h₃ : c ≤ f)
Goal: a + b + c ≤ d + e + f
-- add_lt_of_add_lt_left
@[to_additive]
theorem mul_lt_of_mul_lt_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b < c)
    (hle : d ≤ b) :
    a * d < c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a + b < c) (hle : d ≤ b)
Goal: a + d < c
-- mul_lt_of_mul_lt_left
@[to_additive]
theorem mul_lt_of_mul_lt_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b < c)
    (hle : d ≤ b) :
    a * d < c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a * b < c) (hle : d ≤ b)
Goal: a * d < c
-- mul_le_of_mul_le_left
@[to_additive]
theorem mul_le_of_mul_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b ≤ c)
    (hle : d ≤ b) :
    a * d ≤ c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a * b ≤ c) (hle : d ≤ b)
Goal: a * d ≤ c
-- add_le_of_add_le_left
@[to_additive]
theorem mul_le_of_mul_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b ≤ c)
    (hle : d ≤ b) :
    a * d ≤ c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a + b ≤ c) (hle : d ≤ b)
Goal: a + d ≤ c
-- add_lt_of_add_lt_right
@[to_additive]
theorem mul_lt_of_mul_lt_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a * b < c) (hle : d ≤ a) :
    d * b < c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a + b < c) (hle : d ≤ a)
Goal: d + b < c
-- mul_lt_of_mul_lt_right
@[to_additive]
theorem mul_lt_of_mul_lt_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a * b < c) (hle : d ≤ a) :
    d * b < c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a * b < c) (hle : d ≤ a)
Goal: d * b < c
-- mul_le_of_mul_le_right
@[to_additive]
theorem mul_le_of_mul_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a * b ≤ c) (hle : d ≤ a) :
    d * b ≤ c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a * b ≤ c) (hle : d ≤ a)
Goal: d * b ≤ c
-- add_le_of_add_le_right
@[to_additive]
theorem mul_le_of_mul_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a * b ≤ c) (hle : d ≤ a) :
    d * b ≤ c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a + b ≤ c) (hle : d ≤ a)
Goal: d + b ≤ c
-- lt_add_of_lt_add_left
@[to_additive]
theorem lt_mul_of_lt_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a < b * c)
    (hle : c ≤ d) :
    a < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a < b + c) (hle : c ≤ d)
Goal: a < b + d
-- lt_mul_of_lt_mul_left
@[to_additive]
theorem lt_mul_of_lt_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a < b * c)
    (hle : c ≤ d) :
    a < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a < b * c) (hle : c ≤ d)
Goal: a < b * d
-- le_mul_of_le_mul_left
@[to_additive]
theorem le_mul_of_le_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a ≤ b * c)
    (hle : c ≤ d) :
    a ≤ b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a ≤ b * c) (hle : c ≤ d)
Goal: a ≤ b * d
-- le_add_of_le_add_left
@[to_additive]
theorem le_mul_of_le_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a ≤ b * c)
    (hle : c ≤ d) :
    a ≤ b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a ≤ b + c) (hle : c ≤ d)
Goal: a ≤ b + d
-- lt_add_of_lt_add_right
@[to_additive]
theorem lt_mul_of_lt_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a < b * c) (hle : b ≤ d) :
    a < d * c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a < b + c) (hle : b ≤ d)
Goal: a < d + c
-- lt_mul_of_lt_mul_right
@[to_additive]
theorem lt_mul_of_lt_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a < b * c) (hle : b ≤ d) :
    a < d * c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a < b * c) (hle : b ≤ d)
Goal: a < d * c
-- le_mul_of_le_mul_right
@[to_additive]
theorem le_mul_of_le_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a ≤ b * c) (hle : b ≤ d) :
    a ≤ d * c 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a ≤ b * c) (hle : b ≤ d)
Goal: a ≤ d * c
-- le_add_of_le_add_right
@[to_additive]
theorem le_mul_of_le_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
    (h : a ≤ b * c) (hle : b ≤ d) :
    a ≤ d * c 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h : a ≤ b + c) (hle : b ≤ d)
Goal: a ≤ d + c
-- add_left_cancel''
@[to_additive]
theorem mul_left_cancel'' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b = a * c) :
    b = c 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a + b = a + c)
Goal: b = c
-- mul_left_cancel''
@[to_additive]
theorem mul_left_cancel'' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b = a * c) :
    b = c 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a * b = a * c)
Goal: b = c
-- add_right_cancel''
@[to_additive]
theorem mul_right_cancel'' [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b = c * b) :
    a = c 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a + b = c + b)
Goal: a = c
-- mul_right_cancel''
@[to_additive]
theorem mul_right_cancel'' [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b = c * b) :
    a = c 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a * b = c * b)
Goal: a = c
-- mul_le_mul_iff_of_ge
@[to_additive] lemma mul_le_mul_iff_of_ge [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
    a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂ 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂)
Goal: a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂
-- add_le_add_iff_of_ge
@[to_additive] lemma mul_le_mul_iff_of_ge [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
    a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂ 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂)
Goal: a₂ + b₂ ≤ a₁ + b₁ ↔ a₁ = a₂ ∧ b₁ = b₂
-- mul_eq_mul_iff_eq_and_eq
@[to_additive] theorem mul_eq_mul_iff_eq_and_eq [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) :
    a * b = c * d ↔ a = c ∧ b = d 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d)
Goal: a * b = c * d ↔ a = c ∧ b = d
-- add_eq_add_iff_eq_and_eq
@[to_additive] theorem mul_eq_mul_iff_eq_and_eq [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) :
    a * b = c * d ↔ a = c ∧ b = d 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d)
Goal: a + b = c + d ↔ a = c ∧ b = d
-- min_lt_max_of_add_lt_add
@[to_additive] lemma min_lt_max_of_mul_lt_mul
    [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    (h : a * b < c * d) : min a b < max c d 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a + b < c + d)
Goal: min a b < max c d
-- min_lt_max_of_mul_lt_mul
@[to_additive] lemma min_lt_max_of_mul_lt_mul
    [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    (h : a * b < c * d) : min a b < max c d 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : a * b < c * d)
Goal: min a b < max c d
-- Left.min_le_max_of_mul_le_mul
@[to_additive] lemma Left.min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : a * b ≤ c * d)
Goal: min a b ≤ max c d
-- Left.min_le_max_of_add_le_add
@[to_additive] lemma Left.min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a + b ≤ c + d)
Goal: min a b ≤ max c d
-- Right.min_le_max_of_mul_le_mul
@[to_additive] lemma Right.min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (h : a * b ≤ c * d)
Goal: min a b ≤ max c d
-- Right.min_le_max_of_add_le_add
@[to_additive] lemma Right.min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (h : a + b ≤ c + d)
Goal: min a b ≤ max c d
-- min_le_max_of_mul_le_mul
@[to_additive] lemma min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (h : a * b ≤ c * d)
Goal: min a b ≤ max c d
-- min_le_max_of_add_le_add
@[to_additive] lemma min_le_max_of_mul_le_mul
    [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
    (h : a * b ≤ c * d) : min a b ≤ max c d 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (h : a + b ≤ c + d)
Goal: min a b ≤ max c d
-- max_mul_mul_le_max_mul_max'
@[to_additive max_add_add_le_max_add_max]
theorem max_mul_mul_le_max_mul_max' : max (a * b) (c * d) ≤ max a c * max b d 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: max (a * b) (c * d) ≤ max a c * max b d
-- max_add_add_le_max_add_max
@[to_additive max_add_add_le_max_add_max]
theorem max_mul_mul_le_max_mul_max' : max (a * b) (c * d) ≤ max a c * max b d 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: max (a + b) (c + d) ≤ max a c + max b d
-- min_add_min_le_min_add_add
@[to_additive min_add_min_le_min_add_add]
theorem min_mul_min_le_min_mul_mul' : min a c * min b d ≤ min (a * b) (c * d) 
Arguments: {α : Type u_1} [Add α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: min a c + min b d ≤ min (a + b) (c + d)
-- min_mul_min_le_min_mul_mul'
@[to_additive min_add_min_le_min_add_add]
theorem min_mul_min_le_min_mul_mul' : min a c * min b d ≤ min (a * b) (c * d) 
Arguments: {α : Type u_1} [Mul α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: min a c * min b d ≤ min (a * b) (c * d)
-- le_mul_of_one_le_right'
@[to_additive le_add_of_nonneg_right]
theorem le_mul_of_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
    a ≤ a * b 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : 1 ≤ b)
Goal: a ≤ a * b
-- le_add_of_nonneg_right
@[to_additive le_add_of_nonneg_right]
theorem le_mul_of_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
    a ≤ a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : 0 ≤ b)
Goal: a ≤ a + b
-- mul_le_of_le_one_right'
@[to_additive add_le_of_nonpos_right]
theorem mul_le_of_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
    a * b ≤ a 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ 1)
Goal: a * b ≤ a
-- add_le_of_nonpos_right
@[to_additive add_le_of_nonpos_right]
theorem mul_le_of_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
    a * b ≤ a 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ 0)
Goal: a + b ≤ a
-- le_add_of_nonneg_left
@[to_additive le_add_of_nonneg_left]
theorem le_mul_of_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
    a ≤ b * a 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : 0 ≤ b)
Goal: a ≤ b + a
-- le_mul_of_one_le_left'
@[to_additive le_add_of_nonneg_left]
theorem le_mul_of_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
    a ≤ b * a 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : 1 ≤ b)
Goal: a ≤ b * a
-- mul_le_of_le_one_left'
@[to_additive add_le_of_nonpos_left]
theorem mul_le_of_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
    b * a ≤ a 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ 1)
Goal: b * a ≤ a
-- add_le_of_nonpos_left
@[to_additive add_le_of_nonpos_left]
theorem mul_le_of_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
    b * a ≤ a 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ 0)
Goal: b + a ≤ a
-- one_le_of_le_mul_right
@[to_additive]
theorem one_le_of_le_mul_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a ≤ a * b) :
    1 ≤ b 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a ≤ a * b)
Goal: 1 ≤ b
-- nonneg_of_le_add_right
@[to_additive]
theorem one_le_of_le_mul_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a ≤ a * b) :
    1 ≤ b 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a ≤ a + b)
Goal: 0 ≤ b
-- nonpos_of_add_le_right
@[to_additive]
theorem le_one_of_mul_le_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a * b ≤ a) :
    b ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a + b ≤ a)
Goal: b ≤ 0
-- le_one_of_mul_le_right
@[to_additive]
theorem le_one_of_mul_le_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a * b ≤ a) :
    b ≤ 1 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b ≤ a)
Goal: b ≤ 1
-- nonneg_of_le_add_left
@[to_additive]
theorem one_le_of_le_mul_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (h : b ≤ a * b) :
    1 ≤ a 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ a + b)
Goal: 0 ≤ a
-- one_le_of_le_mul_left
@[to_additive]
theorem one_le_of_le_mul_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (h : b ≤ a * b) :
    1 ≤ a 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : b ≤ a * b)
Goal: 1 ≤ a
-- le_one_of_mul_le_left
@[to_additive]
theorem le_one_of_mul_le_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (h : a * b ≤ b) :
    a ≤ 1 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a * b ≤ b)
Goal: a ≤ 1
-- nonpos_of_add_le_left
@[to_additive]
theorem le_one_of_mul_le_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (h : a * b ≤ b) :
    a ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a + b ≤ b)
Goal: a ≤ 0
-- le_add_iff_nonneg_right
@[to_additive (attr := simp) le_add_iff_nonneg_right]
theorem le_mul_iff_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
    a ≤ a * b ↔ 1 ≤ b 
Arguments: {α : Type u_1} [AddZeroClass α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) {b : α}
Goal: a ≤ a + b ↔ 0 ≤ b
-- le_mul_iff_one_le_right'
@[to_additive (attr := simp) le_add_iff_nonneg_right]
theorem le_mul_iff_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
    a ≤ a * b ↔ 1 ≤ b 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α) {b : α}
Goal: a ≤ a * b ↔ 1 ≤ b
-- le_mul_iff_one_le_left'
@[to_additive (attr := simp) le_add_iff_nonneg_left]
theorem le_mul_iff_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b : α} :
    a ≤ b * a ↔ 1 ≤ b 
Arguments: {α : Type u_1} [MulOneClass α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α) {b : α}
Goal: a ≤ b * a ↔ 1 ≤ b