ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- lt_of_lt_mul_of_le_one_right
@[to_additive]
theorem lt_of_lt_mul_of_le_one_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a < b * c) (hle : b ≤ 1) :
    a < c 
Arguments: {α : Type u_1} [MulOneClass 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 : α} (h : a < b * c) (hle : b ≤ 1)
Goal: a < c
-- le_of_le_mul_of_le_one_right
@[to_additive]
theorem le_of_le_mul_of_le_one_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a ≤ b * c) (hle : b ≤ 1) :
    a ≤ c 
Arguments: {α : Type u_1} [MulOneClass 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 : α} (h : a ≤ b * c) (hle : b ≤ 1)
Goal: a ≤ c
-- le_of_le_add_of_nonpos_right
@[to_additive]
theorem le_of_le_mul_of_le_one_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a ≤ b * c) (hle : b ≤ 1) :
    a ≤ c 
Arguments: {α : Type u_1} [AddZeroClass 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 : α} (h : a ≤ b + c) (hle : b ≤ 0)
Goal: a ≤ c
-- add_eq_zero_iff'
@[to_additive]
theorem mul_eq_one_iff' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    a * b = 1 ↔ a = 1 ∧ b = 1 
Arguments: {α : Type u_1} [AddZeroClass 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 : α} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: a + b = 0 ↔ a = 0 ∧ b = 0
-- mul_eq_one_iff'
@[to_additive]
theorem mul_eq_one_iff' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    a * b = 1 ↔ a = 1 ∧ b = 1 
Arguments: {α : Type u_1} [MulOneClass 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 : α} (ha : 1 ≤ a) (hb : 1 ≤ b)
Goal: a * b = 1 ↔ a = 1 ∧ b = 1
-- eq_zero_of_add_nonneg_left
@[to_additive eq_zero_of_add_nonneg_left]
theorem eq_one_of_one_le_mul_left (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : a = 1 
Arguments: {α : Type u_1} [AddZeroClass α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) (hab : 0 ≤ a + b)
Goal: a = 0
-- eq_one_of_one_le_mul_left
@[to_additive eq_zero_of_add_nonneg_left]
theorem eq_one_of_one_le_mul_left (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : a = 1 
Arguments: {α : Type u_1} [MulOneClass α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b)
Goal: a = 1
-- eq_one_of_mul_le_one_left
@[to_additive]
theorem eq_one_of_mul_le_one_left (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : a = 1 
Arguments: {α : Type u_1} [MulOneClass α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1)
Goal: a = 1
-- eq_zero_of_add_nonpos_left
@[to_additive]
theorem eq_one_of_mul_le_one_left (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : a = 1 
Arguments: {α : Type u_1} [AddZeroClass α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b ≤ 0)
Goal: a = 0
-- eq_one_of_one_le_mul_right
@[to_additive eq_zero_of_add_nonneg_right]
theorem eq_one_of_one_le_mul_right (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : b = 1 
Arguments: {α : Type u_1} [MulOneClass α] [PartialOrder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b)
Goal: b = 1
-- eq_zero_of_add_nonneg_right
@[to_additive eq_zero_of_add_nonneg_right]
theorem eq_one_of_one_le_mul_right (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : b = 1 
Arguments: {α : Type u_1} [AddZeroClass α] [PartialOrder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) (hab : 0 ≤ a + b)
Goal: b = 0
-- eq_one_of_mul_le_one_right
@[to_additive]
theorem eq_one_of_mul_le_one_right (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : b = 1 
Arguments: {α : Type u_1} [MulOneClass α] [PartialOrder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1)
Goal: b = 1
-- eq_zero_of_add_nonpos_right
@[to_additive]
theorem eq_one_of_mul_le_one_right (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : b = 1 
Arguments: {α : Type u_1} [AddZeroClass α] [PartialOrder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b ≤ 0)
Goal: b = 0
-- exists_square_le
theorem exists_square_le [CovariantClass α α (· * ·) (· < ·)] (a : α) : ∃ b : α, b * b ≤ a 
Arguments: {α : Type u_1} [MulOneClass α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α)
Goal: ∃ b, b * b ≤ a
-- Monotone.const_mul'
@[to_additive const_add]
theorem Monotone.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : Monotone f) (a : α) :
    Monotone fun x => a * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : Monotone f) (a : α)
Goal: Monotone fun x => a * f x
-- Monotone.const_add
@[to_additive const_add]
theorem Monotone.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : Monotone f) (a : α) :
    Monotone fun x => a * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : Monotone f) (a : α)
Goal: Monotone fun x => a + f x
-- MonotoneOn.const_add
@[to_additive const_add]
theorem MonotoneOn.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : MonotoneOn f s) (a : α) :
    MonotoneOn (fun x => a * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MonotoneOn f s) (a : α)
Goal: MonotoneOn (fun x => a + f x) s
-- MonotoneOn.const_mul'
@[to_additive const_add]
theorem MonotoneOn.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : MonotoneOn f s) (a : α) :
    MonotoneOn (fun x => a * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : MonotoneOn f s) (a : α)
Goal: MonotoneOn (fun x => a * f x) s
-- Antitone.const_mul'
@[to_additive const_add]
theorem Antitone.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : Antitone f) (a : α) :
    Antitone fun x => a * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : Antitone f) (a : α)
Goal: Antitone fun x => a * f x
-- Antitone.const_add
@[to_additive const_add]
theorem Antitone.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : Antitone f) (a : α) :
    Antitone fun x => a * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : Antitone f) (a : α)
Goal: Antitone fun x => a + f x
-- AntitoneOn.const_add
@[to_additive const_add]
theorem AntitoneOn.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : AntitoneOn f s) (a : α) :
    AntitoneOn (fun x => a * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : AntitoneOn f s) (a : α)
Goal: AntitoneOn (fun x => a + f x) s
-- AntitoneOn.const_mul'
@[to_additive const_add]
theorem AntitoneOn.const_mul' [CovariantClass α α (· * ·) (· ≤ ·)] (hf : AntitoneOn f s) (a : α) :
    AntitoneOn (fun x => a * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : AntitoneOn f s) (a : α)
Goal: AntitoneOn (fun x => a * f x) s
-- Monotone.mul_const'
@[to_additive add_const]
theorem Monotone.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Monotone f) (a : α) :
    Monotone fun x => f x * a 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : Monotone f) (a : α)
Goal: Monotone fun x => f x * a
-- Monotone.add_const
@[to_additive add_const]
theorem Monotone.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Monotone f) (a : α) :
    Monotone fun x => f x * a 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : Monotone f) (a : α)
Goal: Monotone fun x => f x + a
-- MonotoneOn.add_const
@[to_additive add_const]
theorem MonotoneOn.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : MonotoneOn f s)
    (a : α) :
    MonotoneOn (fun x => f x * a) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MonotoneOn f s) (a : α)
Goal: MonotoneOn (fun x => f x + a) s
-- MonotoneOn.mul_const'
@[to_additive add_const]
theorem MonotoneOn.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : MonotoneOn f s)
    (a : α) :
    MonotoneOn (fun x => f x * a) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : MonotoneOn f s) (a : α)
Goal: MonotoneOn (fun x => f x * a) s
-- Antitone.mul_const'
@[to_additive add_const]
theorem Antitone.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Antitone f) (a : α) :
    Antitone fun x => f x * a 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : Antitone f) (a : α)
Goal: Antitone fun x => f x * a
-- Antitone.add_const
@[to_additive add_const]
theorem Antitone.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Antitone f) (a : α) :
    Antitone fun x => f x * a 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : Antitone f) (a : α)
Goal: Antitone fun x => f x + a
-- AntitoneOn.add_const
@[to_additive add_const]
theorem AntitoneOn.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : AntitoneOn f s)
    (a : α) :
    AntitoneOn (fun x => f x * a) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : AntitoneOn f s) (a : α)
Goal: AntitoneOn (fun x => f x + a) s
-- AntitoneOn.mul_const'
@[to_additive add_const]
theorem AntitoneOn.mul_const' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : AntitoneOn f s)
    (a : α) :
    AntitoneOn (fun x => f x * a) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (hf : AntitoneOn f s) (a : α)
Goal: AntitoneOn (fun x => f x * a) s
-- Monotone.mul'
@[to_additive add "The sum of two monotone functions is monotone."]
theorem Monotone.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Monotone f) (hg : Monotone g) :
    Monotone fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : Monotone f) (hg : Monotone g)
Goal: Monotone fun x => f x * g x
-- Monotone.add
@[to_additive add "The sum of two monotone functions is monotone."]
theorem Monotone.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Monotone f) (hg : Monotone g) :
    Monotone fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : Monotone f) (hg : Monotone g)
Goal: Monotone fun x => f x + g x
-- MonotoneOn.add
@[to_additive add "The sum of two monotone functions is monotone."]
theorem MonotoneOn.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : MonotoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (fun x => f x + g x) s
-- MonotoneOn.mul'
@[to_additive add "The sum of two monotone functions is monotone."]
theorem MonotoneOn.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : MonotoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (fun x => f x * g x) s
-- Antitone.add
@[to_additive add "The sum of two antitone functions is antitone."]
theorem Antitone.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Antitone f) (hg : Antitone g) :
    Antitone fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : Antitone f) (hg : Antitone g)
Goal: Antitone fun x => f x + g x
-- Antitone.mul'
@[to_additive add "The sum of two antitone functions is antitone."]
theorem Antitone.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : Antitone f) (hg : Antitone g) :
    Antitone fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : Antitone f) (hg : Antitone g)
Goal: Antitone fun x => f x * g x
-- AntitoneOn.add
@[to_additive add "The sum of two antitone functions is antitone."]
theorem AntitoneOn.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : AntitoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (fun x => f x + g x) s
-- AntitoneOn.mul'
@[to_additive add "The sum of two antitone functions is antitone."]
theorem AntitoneOn.mul' [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (hf : AntitoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (fun x => f x * g x) s
-- StrictMono.const_add
@[to_additive const_add]
theorem StrictMono.const_mul' (hf : StrictMono f) (c : α) : StrictMono fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictMono f) (c : α)
Goal: StrictMono fun x => c + f x
-- StrictMono.const_mul'
@[to_additive const_add]
theorem StrictMono.const_mul' (hf : StrictMono f) (c : α) : StrictMono fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictMono f) (c : α)
Goal: StrictMono fun x => c * f x
-- StrictMonoOn.const_add
@[to_additive const_add]
theorem StrictMonoOn.const_mul' (hf : StrictMonoOn f s) (c : α) :
    StrictMonoOn (fun x => c * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictMonoOn f s) (c : α)
Goal: StrictMonoOn (fun x => c + f x) s
-- StrictMonoOn.const_mul'
@[to_additive const_add]
theorem StrictMonoOn.const_mul' (hf : StrictMonoOn f s) (c : α) :
    StrictMonoOn (fun x => c * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictMonoOn f s) (c : α)
Goal: StrictMonoOn (fun x => c * f x) s
-- StrictAnti.const_add
@[to_additive const_add]
theorem StrictAnti.const_mul' (hf : StrictAnti f) (c : α) : StrictAnti fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictAnti f) (c : α)
Goal: StrictAnti fun x => c + f x
-- StrictAnti.const_mul'
@[to_additive const_add]
theorem StrictAnti.const_mul' (hf : StrictAnti f) (c : α) : StrictAnti fun x => c * f x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictAnti f) (c : α)
Goal: StrictAnti fun x => c * f x
-- StrictAntiOn.const_add
@[to_additive const_add]
theorem StrictAntiOn.const_mul' (hf : StrictAntiOn f s) (c : α) :
    StrictAntiOn (fun x => c * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictAntiOn f s) (c : α)
Goal: StrictAntiOn (fun x => c + f x) s
-- StrictAntiOn.const_mul'
@[to_additive const_add]
theorem StrictAntiOn.const_mul' (hf : StrictAntiOn f s) (c : α) :
    StrictAntiOn (fun x => c * f x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictAntiOn f s) (c : α)
Goal: StrictAntiOn (fun x => c * f x) s
-- StrictMono.add_const
@[to_additive add_const]
theorem StrictMono.mul_const' (hf : StrictMono f) (c : α) : StrictMono fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictMono f) (c : α)
Goal: StrictMono fun x => f x + c
-- StrictMono.mul_const'
@[to_additive add_const]
theorem StrictMono.mul_const' (hf : StrictMono f) (c : α) : StrictMono fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictMono f) (c : α)
Goal: StrictMono fun x => f x * c
-- StrictMonoOn.add_const
@[to_additive add_const]
theorem StrictMonoOn.mul_const' (hf : StrictMonoOn f s) (c : α) :
    StrictMonoOn (fun x => f x * c) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictMonoOn f s) (c : α)
Goal: StrictMonoOn (fun x => f x + c) s
-- StrictMonoOn.mul_const'
@[to_additive add_const]
theorem StrictMonoOn.mul_const' (hf : StrictMonoOn f s) (c : α) :
    StrictMonoOn (fun x => f x * c) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictMonoOn f s) (c : α)
Goal: StrictMonoOn (fun x => f x * c) s
-- StrictAnti.mul_const'
@[to_additive add_const]
theorem StrictAnti.mul_const' (hf : StrictAnti f) (c : α) : StrictAnti fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictAnti f) (c : α)
Goal: StrictAnti fun x => f x * c
-- StrictAnti.add_const
@[to_additive add_const]
theorem StrictAnti.mul_const' (hf : StrictAnti f) (c : α) : StrictAnti fun x => f x * c 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictAnti f) (c : α)
Goal: StrictAnti fun x => f x + c
-- StrictAntiOn.mul_const'
@[to_additive add_const]
theorem StrictAntiOn.mul_const' (hf : StrictAntiOn f s) (c : α) :
    StrictAntiOn (fun x => f x * c) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (hf : StrictAntiOn f s) (c : α)
Goal: StrictAntiOn (fun x => f x * c) s
-- StrictAntiOn.add_const
@[to_additive add_const]
theorem StrictAntiOn.mul_const' (hf : StrictAntiOn f s) (c : α) :
    StrictAntiOn (fun x => f x * c) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f : β → α} {s : Set β} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hf : StrictAntiOn f s) (c : α)
Goal: StrictAntiOn (fun x => f x + c) s
-- StrictMono.mul'
@[to_additive add "The sum of two strictly monotone functions is strictly monotone."]
theorem StrictMono.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictMono f) (hg : StrictMono g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictMono f) (hg : StrictMono g)
Goal: StrictMono fun x => f x * g x
-- StrictMono.add
@[to_additive add "The sum of two strictly monotone functions is strictly monotone."]
theorem StrictMono.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictMono f) (hg : StrictMono g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictMono f) (hg : StrictMono g)
Goal: StrictMono fun x => f x + g x
-- StrictMonoOn.mul'
@[to_additive add "The sum of two strictly monotone functions is strictly monotone."]
theorem StrictMonoOn.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictMonoOn f s) (hg : StrictMonoOn g s) :
    StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictMonoOn f s) (hg : StrictMonoOn g s)
Goal: StrictMonoOn (fun x => f x * g x) s