ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- antitone_mul_left
theorem antitone_mul_left [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a : α} (ha : a ≤ 0) : Antitone (a * ·) 
Arguments: {α : Type u} [Semiring α] [Preorder α] [ExistsAddOfLE α] [PosMulMono 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} (ha : a ≤ 0)
Goal: Antitone fun x => a * x
-- antitone_mul_right
theorem antitone_mul_right [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    {a : α} (ha : a ≤ 0) : Antitone fun x => x * a 
Arguments: {α : Type u} [Semiring α] [Preorder α] [ExistsAddOfLE α] [MulPosMono 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} (ha : a ≤ 0)
Goal: Antitone fun x => x * a
-- Monotone.const_mul_of_nonpos
theorem Monotone.const_mul_of_nonpos [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Monotone f) (ha : a ≤ 0) : Antitone fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [PosMulMono 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] (hf : Monotone f) (ha : a ≤ 0)
Goal: Antitone fun x => a * f x
-- Monotone.mul_const_of_nonpos
theorem Monotone.mul_const_of_nonpos [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Monotone f) (ha : a ≤ 0) : Antitone fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [MulPosMono 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] (hf : Monotone f) (ha : a ≤ 0)
Goal: Antitone fun x => f x * a
-- Antitone.const_mul_of_nonpos
theorem Antitone.const_mul_of_nonpos [ExistsAddOfLE α] [PosMulMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Antitone f) (ha : a ≤ 0) : Monotone fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [PosMulMono 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] (hf : Antitone f) (ha : a ≤ 0)
Goal: Monotone fun x => a * f x
-- Antitone.mul_const_of_nonpos
theorem Antitone.mul_const_of_nonpos [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Antitone f) (ha : a ≤ 0) : Monotone fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [MulPosMono 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] (hf : Antitone f) (ha : a ≤ 0)
Goal: Monotone fun x => f x * a
-- Antitone.mul_monotone
theorem Antitone.mul_monotone [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Antitone f) (hg : Monotone g) (hf₀ : ∀ x, f x ≤ 0)
    (hg₀ : ∀ x, 0 ≤ g x) : Antitone (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] [Preorder β] {f g : β → α} [ExistsAddOfLE α] [PosMulMono α] [MulPosMono 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] (hf : Antitone f) (hg : Monotone g) (hf₀ : ∀ (x : β), f x ≤ 0) (hg₀ : ∀ (x : β), 0 ≤ g x)
Goal: Antitone (f * g)
-- Monotone.mul_antitone
theorem Monotone.mul_antitone [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Monotone f) (hg : Antitone g) (hf₀ : ∀ x, 0 ≤ f x)
    (hg₀ : ∀ x, g x ≤ 0) : Antitone (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] [Preorder β] {f g : β → α} [ExistsAddOfLE α] [PosMulMono α] [MulPosMono 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] (hf : Monotone f) (hg : Antitone g) (hf₀ : ∀ (x : β), 0 ≤ f x) (hg₀ : ∀ (x : β), g x ≤ 0)
Goal: Antitone (f * g)
-- Antitone.mul
theorem Antitone.mul [ExistsAddOfLE α] [PosMulMono α] [MulPosMono α]
    [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
    (hf : Antitone f) (hg : Antitone g) (hf₀ : ∀ x, f x ≤ 0) (hg₀ : ∀ x, g x ≤ 0) :
    Monotone (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Preorder α] [Preorder β] {f g : β → α} [ExistsAddOfLE α] [PosMulMono α] [MulPosMono 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] (hf : Antitone f) (hg : Antitone g) (hf₀ : ∀ (x : β), f x ≤ 0) (hg₀ : ∀ (x : β), g x ≤ 0)
Goal: Monotone (f * g)
-- le_iff_exists_nonneg_add
lemma le_iff_exists_nonneg_add
    [ExistsAddOfLE α] [ContravariantClass α α (· + ·) (· ≤ ·)] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a b : α) : a ≤ b ↔ ∃ c ≥ 0, b = a + c 
Arguments: {α : Type u} [Semiring α] [Preorder α] [ExistsAddOfLE α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ≤ b ↔ ∃ c, c ≥ 0 ∧ b = a + c
-- pow_pos
@[simp]
theorem pow_pos [ZeroLEOneClass α] [PosMulStrictMono α]
    (H : 0 < a) : ∀ n : ℕ, 0 < a ^ n
  
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a : α} [ZeroLEOneClass α] [PosMulStrictMono α] (H : 0 < a) (n : ℕ)
Goal: 0 < a ^ n
-- mul_self_lt_mul_self
theorem mul_self_lt_mul_self [PosMulStrictMono α] [MulPosMono α]
    (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [PosMulStrictMono α] [MulPosMono α] (h1 : 0 ≤ a) (h2 : a < b)
Goal: a * a < b * b
-- strictMonoOn_mul_self
theorem strictMonoOn_mul_self [PosMulStrictMono α] [MulPosMono α] :
    StrictMonoOn (fun x : α => x * x) { x | 0 ≤ x } 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] [PosMulStrictMono α] [MulPosMono α]
Goal: StrictMonoOn (fun x => x * x) {x | 0 ≤ x}
-- Decidable.mul_lt_mul''
protected theorem Decidable.mul_lt_mul'' [PosMulMono α] [PosMulStrictMono α] [MulPosStrictMono α]
    [@DecidableRel α (· ≤ ·)] (h1 : a < c) (h2 : b < d)
    (h3 : 0 ≤ a) (h4 : 0 ≤ b) : a * b < c * d 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c d : α} [PosMulMono α] [PosMulStrictMono α] [MulPosStrictMono α] [DecidableRel fun x x_1 => x ≤ x_1] (h1 : a < c) (h2 : b < d) (h3 : 0 ≤ a) (h4 : 0 ≤ b)
Goal: a * b < c * d
-- lt_mul_left
theorem lt_mul_left [MulPosStrictMono α]
    (hn : 0 < a) (hm : 1 < b) : a < b * a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [MulPosStrictMono α] (hn : 0 < a) (hm : 1 < b)
Goal: a < b * a
-- lt_mul_right
theorem lt_mul_right [PosMulStrictMono α]
    (hn : 0 < a) (hm : 1 < b) : a < a * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [PosMulStrictMono α] (hn : 0 < a) (hm : 1 < b)
Goal: a < a * b
-- lt_mul_self
theorem lt_mul_self [ZeroLEOneClass α] [MulPosStrictMono α]
    (hn : 1 < a) : a < a * a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a : α} [ZeroLEOneClass α] [MulPosStrictMono α] (hn : 1 < a)
Goal: a < a * a
-- strictMono_mul_left_of_pos
theorem strictMono_mul_left_of_pos [PosMulStrictMono α]
    (ha : 0 < a) : StrictMono fun x => a * x 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a : α} [PosMulStrictMono α] (ha : 0 < a)
Goal: StrictMono fun x => a * x
-- strictMono_mul_right_of_pos
theorem strictMono_mul_right_of_pos [MulPosStrictMono α]
    (ha : 0 < a) : StrictMono fun x => x * a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a : α} [MulPosStrictMono α] (ha : 0 < a)
Goal: StrictMono fun x => x * a
-- StrictMono.mul_const
theorem StrictMono.mul_const [MulPosStrictMono α]
    (hf : StrictMono f) (ha : 0 < a) : StrictMono fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [MulPosStrictMono α] (hf : StrictMono f) (ha : 0 < a)
Goal: StrictMono fun x => f x * a
-- StrictMono.const_mul
theorem StrictMono.const_mul [PosMulStrictMono α]
    (hf : StrictMono f) (ha : 0 < a) : StrictMono fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [PosMulStrictMono α] (hf : StrictMono f) (ha : 0 < a)
Goal: StrictMono fun x => a * f x
-- StrictAnti.mul_const
theorem StrictAnti.mul_const [MulPosStrictMono α]
    (hf : StrictAnti f) (ha : 0 < a) : StrictAnti fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [MulPosStrictMono α] (hf : StrictAnti f) (ha : 0 < a)
Goal: StrictAnti fun x => f x * a
-- StrictAnti.const_mul
theorem StrictAnti.const_mul [PosMulStrictMono α]
    (hf : StrictAnti f) (ha : 0 < a) : StrictAnti fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [PosMulStrictMono α] (hf : StrictAnti f) (ha : 0 < a)
Goal: StrictAnti fun x => a * f x
-- StrictMono.mul_monotone
theorem StrictMono.mul_monotone [PosMulMono α] [MulPosStrictMono α]
    (hf : StrictMono f) (hg : Monotone g) (hf₀ : ∀ x, 0 ≤ f x)
    (hg₀ : ∀ x, 0 < g x) : StrictMono (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] [Preorder β] {f g : β → α} [PosMulMono α] [MulPosStrictMono α] (hf : StrictMono f) (hg : Monotone g) (hf₀ : ∀ (x : β), 0 ≤ f x) (hg₀ : ∀ (x : β), 0 < g x)
Goal: StrictMono (f * g)
-- Monotone.mul_strictMono
theorem Monotone.mul_strictMono [PosMulStrictMono α] [MulPosMono α]
    (hf : Monotone f) (hg : StrictMono g) (hf₀ : ∀ x, 0 < f x)
    (hg₀ : ∀ x, 0 ≤ g x) : StrictMono (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] [Preorder β] {f g : β → α} [PosMulStrictMono α] [MulPosMono α] (hf : Monotone f) (hg : StrictMono g) (hf₀ : ∀ (x : β), 0 < f x) (hg₀ : ∀ (x : β), 0 ≤ g x)
Goal: StrictMono (f * g)
-- StrictMono.mul
theorem StrictMono.mul [PosMulStrictMono α] [MulPosStrictMono α]
    (hf : StrictMono f) (hg : StrictMono g) (hf₀ : ∀ x, 0 ≤ f x)
    (hg₀ : ∀ x, 0 ≤ g x) : StrictMono (f * g) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] [Preorder β] {f g : β → α} [PosMulStrictMono α] [MulPosStrictMono α] (hf : StrictMono f) (hg : StrictMono g) (hf₀ : ∀ (x : β), 0 ≤ f x) (hg₀ : ∀ (x : β), 0 ≤ g x)
Goal: StrictMono (f * g)
-- lt_two_mul_self
theorem lt_two_mul_self [ZeroLEOneClass α] [MulPosStrictMono α] [NeZero (R := α) 1]
    [CovariantClass α α (· + ·) (· < ·)] (ha : 0 < a) : a < 2 * a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a : α} [ZeroLEOneClass α] [MulPosStrictMono α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (ha : 0 < a)
Goal: a < 2 * a
-- mul_lt_mul_of_neg_left
theorem mul_lt_mul_of_neg_left [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (h : b < a) (hc : c < 0) : c * a < c * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c : α} [ExistsAddOfLE α] [PosMulStrictMono 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] (h : b < a) (hc : c < 0)
Goal: c * a < c * b
-- mul_lt_mul_of_neg_right
theorem mul_lt_mul_of_neg_right [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (h : b < a) (hc : c < 0) : a * c < b * c 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c : α} [ExistsAddOfLE α] [MulPosStrictMono 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] (h : b < a) (hc : c < 0)
Goal: a * c < b * c
-- mul_pos_of_neg_of_neg
theorem mul_pos_of_neg_of_neg [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    {a b : α} (ha : a < 0) (hb : b < 0) : 0 < a * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] [ExistsAddOfLE α] [MulPosStrictMono 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 b : α} (ha : a < 0) (hb : b < 0)
Goal: 0 < a * b
-- lt_mul_of_lt_one_left
theorem lt_mul_of_lt_one_left [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hb : b < 0) (h : a < 1) : b < a * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hb : b < 0) (h : a < 1)
Goal: b < a * b
-- mul_lt_of_one_lt_left
theorem mul_lt_of_one_lt_left [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hb : b < 0) (h : 1 < a) : a * b < b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hb : b < 0) (h : 1 < a)
Goal: a * b < b
-- lt_mul_of_lt_one_right
theorem lt_mul_of_lt_one_right [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (ha : a < 0) (h : b < 1) : a < a * b 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [ExistsAddOfLE α] [PosMulStrictMono 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] (ha : a < 0) (h : b < 1)
Goal: a < a * b
-- mul_lt_of_one_lt_right
theorem mul_lt_of_one_lt_right [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (ha : a < 0) (h : 1 < b) : a * b < a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b : α} [ExistsAddOfLE α] [PosMulStrictMono 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] (ha : a < 0) (h : 1 < b)
Goal: a * b < a
-- strictAnti_mul_left
theorem strictAnti_mul_left [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    {a : α} (ha : a < 0) : StrictAnti (a * ·) 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] [ExistsAddOfLE α] [PosMulStrictMono 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} (ha : a < 0)
Goal: StrictAnti fun x => a * x
-- strictAnti_mul_right
theorem strictAnti_mul_right [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    {a : α} (ha : a < 0) : StrictAnti fun x => x * a 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] [ExistsAddOfLE α] [MulPosStrictMono 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} (ha : a < 0)
Goal: StrictAnti fun x => x * a
-- StrictMono.const_mul_of_neg
theorem StrictMono.const_mul_of_neg [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hf : StrictMono f) (ha : a < 0) : StrictAnti fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [PosMulStrictMono 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] (hf : StrictMono f) (ha : a < 0)
Goal: StrictAnti fun x => a * f x
-- StrictMono.mul_const_of_neg
theorem StrictMono.mul_const_of_neg [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hf : StrictMono f) (ha : a < 0) : StrictAnti fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hf : StrictMono f) (ha : a < 0)
Goal: StrictAnti fun x => f x * a
-- StrictAnti.const_mul_of_neg
theorem StrictAnti.const_mul_of_neg [ExistsAddOfLE α] [PosMulStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hf : StrictAnti f) (ha : a < 0) : StrictMono fun x => a * f x 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [PosMulStrictMono 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] (hf : StrictAnti f) (ha : a < 0)
Goal: StrictMono fun x => a * f x
-- StrictAnti.mul_const_of_neg
theorem StrictAnti.mul_const_of_neg [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (swap (· + ·)) (· < ·)] [ContravariantClass α α (swap (· + ·)) (· < ·)]
    (hf : StrictAnti f) (ha : a < 0) : StrictMono fun x => f x * a 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [PartialOrder α] {a : α} [Preorder β] {f : β → α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hf : StrictAnti f) (ha : a < 0)
Goal: StrictMono fun x => f x * a
-- mul_add_mul_le_mul_add_mul
lemma mul_add_mul_le_mul_add_mul [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)]
    (hab : a ≤ b) (hcd : c ≤ d) : a * d + b * c ≤ a * c + b * d 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c d : α} [ExistsAddOfLE α] [MulPosMono 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] (hab : a ≤ b) (hcd : c ≤ d)
Goal: a * d + b * c ≤ a * c + b * d
-- mul_add_mul_le_mul_add_mul'
lemma mul_add_mul_le_mul_add_mul' [ExistsAddOfLE α] [MulPosMono α]
    [CovariantClass α α (· + ·) (· ≤ ·)] [ContravariantClass α α (· + ·) (· ≤ ·)]
    (hba : b ≤ a) (hdc : d ≤ c) : a * d + b * c ≤ a * c + b * d 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c d : α} [ExistsAddOfLE α] [MulPosMono 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] (hba : b ≤ a) (hdc : d ≤ c)
Goal: a * d + b * c ≤ a * c + b * d
-- mul_add_mul_lt_mul_add_mul
lemma mul_add_mul_lt_mul_add_mul [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· < ·)] [ContravariantClass α α (· + ·) (· < ·)]
    (hab : a < b) (hcd : c < d) : a * d + b * c < a * c + b * d 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c d : α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hab : a < b) (hcd : c < d)
Goal: a * d + b * c < a * c + b * d
-- mul_add_mul_lt_mul_add_mul'
lemma mul_add_mul_lt_mul_add_mul' [ExistsAddOfLE α] [MulPosStrictMono α]
    [CovariantClass α α (· + ·) (· < ·)] [ContravariantClass α α (· + ·) (· < ·)]
    (hba : b < a) (hdc : d < c) : a * d + b * c < a * c + b * d 
Arguments: {α : Type u} [Semiring α] [PartialOrder α] {a b c d : α} [ExistsAddOfLE α] [MulPosStrictMono 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] (hba : b < a) (hdc : d < c)
Goal: a * d + b * c < a * c + b * d
-- nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonneg
theorem nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonneg
    [MulPosStrictMono α] [PosMulStrictMono α]
    (hab : 0 ≤ a * b) : 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [MulPosStrictMono α] [PosMulStrictMono α] (hab : 0 ≤ a * b)
Goal: 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- nonneg_of_mul_nonneg_left
theorem nonneg_of_mul_nonneg_left [MulPosStrictMono α]
    (h : 0 ≤ a * b) (hb : 0 < b) : 0 ≤ a 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [MulPosStrictMono α] (h : 0 ≤ a * b) (hb : 0 < b)
Goal: 0 ≤ a
-- nonneg_of_mul_nonneg_right
theorem nonneg_of_mul_nonneg_right [PosMulStrictMono α]
    (h : 0 ≤ a * b) (ha : 0 < a) : 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (h : 0 ≤ a * b) (ha : 0 < a)
Goal: 0 ≤ b
-- nonpos_of_mul_nonpos_left
theorem nonpos_of_mul_nonpos_left [PosMulStrictMono α]
    (h : a * b ≤ 0) (hb : 0 < b) : a ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (h : a * b ≤ 0) (hb : 0 < b)
Goal: a ≤ 0
-- nonpos_of_mul_nonpos_right
theorem nonpos_of_mul_nonpos_right [PosMulStrictMono α]
    (h : a * b ≤ 0) (ha : 0 < a) : b ≤ 0 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [PosMulStrictMono α] (h : a * b ≤ 0) (ha : 0 < a)
Goal: b ≤ 0
-- mul_nonneg_iff_of_pos_left
@[simp]
theorem mul_nonneg_iff_of_pos_left [PosMulStrictMono α]
    (h : 0 < c) : 0 ≤ c * b ↔ 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {b c : α} [PosMulStrictMono α] (h : 0 < c)
Goal: 0 ≤ c * b ↔ 0 ≤ b
-- mul_nonneg_iff_of_pos_right
@[simp]
theorem mul_nonneg_iff_of_pos_right [MulPosStrictMono α]
    (h : 0 < c) : 0 ≤ b * c ↔ 0 ≤ b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {b c : α} [MulPosStrictMono α] (h : 0 < c)
Goal: 0 ≤ b * c ↔ 0 ≤ b
-- add_le_mul_of_left_le_right
theorem add_le_mul_of_left_le_right [ZeroLEOneClass α] [NeZero (R := α) 1]
    [MulPosStrictMono α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a2 : 2 ≤ a) (ab : a ≤ b) : a + b ≤ a * b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ZeroLEOneClass α] [NeZero 1] [MulPosStrictMono α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a2 : 2 ≤ a) (ab : a ≤ b)
Goal: a + b ≤ a * b
-- add_le_mul_of_right_le_left
theorem add_le_mul_of_right_le_left [ZeroLEOneClass α] [NeZero (R := α) 1]
    [CovariantClass α α (· + ·) (· ≤ ·)] [PosMulStrictMono α]
    (b2 : 2 ≤ b) (ba : b ≤ a) : a + b ≤ a * b 
Arguments: {α : Type u} [Semiring α] [LinearOrder α] {a b : α} [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [PosMulStrictMono α] (b2 : 2 ≤ b) (ba : b ≤ a)
Goal: a + b ≤ a * b