ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mul_le_mul_left
@[simp]
theorem mul_le_mul_left [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} {a b c : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a)
Goal: a * b ≤ a * c ↔ b ≤ c
-- mul_le_mul_right
@[simp]
theorem mul_le_mul_right [MulPosMono α] [MulPosReflectLE α] (a0 : 0 < a) : b * a ≤ c * a ↔ b ≤ c 
Arguments: {α : Type u_1} {a b c : α} [Mul α] [Zero α] [Preorder α] [MulPosMono α] [MulPosReflectLE α] (a0 : 0 < a)
Goal: b * a ≤ c * a ↔ b ≤ c
-- mul_le_mul_of_nonneg
theorem mul_le_mul_of_nonneg [PosMulMono α] [MulPosMono α]
    (h₁ : a ≤ b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 ≤ d) : a * c ≤ b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] [MulPosMono α] (h₁ : a ≤ b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 ≤ d)
Goal: a * c ≤ b * d
-- mul_le_mul_of_nonneg'
theorem mul_le_mul_of_nonneg' [PosMulMono α] [MulPosMono α]
    (h₁ : a ≤ b) (h₂ : c ≤ d) (c0 : 0 ≤ c) (b0 : 0 ≤ b) : a * c ≤ b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] [MulPosMono α] (h₁ : a ≤ b) (h₂ : c ≤ d) (c0 : 0 ≤ c) (b0 : 0 ≤ b)
Goal: a * c ≤ b * d
-- mul_lt_mul_of_le_of_lt_of_pos_of_nonneg
theorem mul_lt_mul_of_le_of_lt_of_pos_of_nonneg [PosMulStrictMono α] [MulPosMono α]
    (h₁ : a ≤ b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 ≤ d) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (h₁ : a ≤ b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 ≤ d)
Goal: a * c < b * d
-- mul_lt_mul_of_le_of_lt_of_nonneg_of_pos
theorem mul_lt_mul_of_le_of_lt_of_nonneg_of_pos [PosMulStrictMono α] [MulPosMono α]
    (h₁ : a ≤ b) (h₂ : c < d) (c0 : 0 ≤ c) (b0 : 0 < b) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (h₁ : a ≤ b) (h₂ : c < d) (c0 : 0 ≤ c) (b0 : 0 < b)
Goal: a * c < b * d
-- mul_lt_mul_of_lt_of_le_of_nonneg_of_pos
theorem mul_lt_mul_of_lt_of_le_of_nonneg_of_pos [PosMulMono α] [MulPosStrictMono α]
    (h₁ : a < b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 < d) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (h₁ : a < b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 < d)
Goal: a * c < b * d
-- mul_lt_mul_of_lt_of_le_of_pos_of_nonneg
theorem mul_lt_mul_of_lt_of_le_of_pos_of_nonneg [PosMulMono α] [MulPosStrictMono α]
    (h₁ : a < b) (h₂ : c ≤ d) (c0 : 0 < c) (b0 : 0 ≤ b) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (h₁ : a < b) (h₂ : c ≤ d) (c0 : 0 < c) (b0 : 0 ≤ b)
Goal: a * c < b * d
-- mul_lt_mul_of_pos
theorem mul_lt_mul_of_pos [PosMulStrictMono α] [MulPosStrictMono α]
    (h₁ : a < b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 < d) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosStrictMono α] (h₁ : a < b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 < d)
Goal: a * c < b * d
-- mul_lt_mul_of_pos'
theorem mul_lt_mul_of_pos' [PosMulStrictMono α] [MulPosStrictMono α]
    (h₁ : a < b) (h₂ : c < d) (c0 : 0 < c) (b0 : 0 < b) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosStrictMono α] (h₁ : a < b) (h₂ : c < d) (c0 : 0 < c) (b0 : 0 < b)
Goal: a * c < b * d
-- mul_le_of_mul_le_of_nonneg_left
theorem mul_le_of_mul_le_of_nonneg_left [PosMulMono α] (h : a * b ≤ c) (hle : d ≤ b) (a0 : 0 ≤ a) :
    a * d ≤ c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] (h : a * b ≤ c) (hle : d ≤ b) (a0 : 0 ≤ a)
Goal: a * d ≤ c
-- mul_lt_of_mul_lt_of_nonneg_left
theorem mul_lt_of_mul_lt_of_nonneg_left [PosMulMono α] (h : a * b < c) (hle : d ≤ b) (a0 : 0 ≤ a) :
    a * d < c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] (h : a * b < c) (hle : d ≤ b) (a0 : 0 ≤ a)
Goal: a * d < c
-- le_mul_of_le_mul_of_nonneg_left
theorem le_mul_of_le_mul_of_nonneg_left [PosMulMono α] (h : a ≤ b * c) (hle : c ≤ d) (b0 : 0 ≤ b) :
    a ≤ b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] (h : a ≤ b * c) (hle : c ≤ d) (b0 : 0 ≤ b)
Goal: a ≤ b * d
-- lt_mul_of_lt_mul_of_nonneg_left
theorem lt_mul_of_lt_mul_of_nonneg_left [PosMulMono α] (h : a < b * c) (hle : c ≤ d) (b0 : 0 ≤ b) :
    a < b * d 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [PosMulMono α] (h : a < b * c) (hle : c ≤ d) (b0 : 0 ≤ b)
Goal: a < b * d
-- mul_le_of_mul_le_of_nonneg_right
theorem mul_le_of_mul_le_of_nonneg_right [MulPosMono α] (h : a * b ≤ c) (hle : d ≤ a) (b0 : 0 ≤ b) :
    d * b ≤ c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [MulPosMono α] (h : a * b ≤ c) (hle : d ≤ a) (b0 : 0 ≤ b)
Goal: d * b ≤ c
-- mul_lt_of_mul_lt_of_nonneg_right
theorem mul_lt_of_mul_lt_of_nonneg_right [MulPosMono α] (h : a * b < c) (hle : d ≤ a) (b0 : 0 ≤ b) :
    d * b < c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [MulPosMono α] (h : a * b < c) (hle : d ≤ a) (b0 : 0 ≤ b)
Goal: d * b < c
-- le_mul_of_le_mul_of_nonneg_right
theorem le_mul_of_le_mul_of_nonneg_right [MulPosMono α] (h : a ≤ b * c) (hle : b ≤ d) (c0 : 0 ≤ c) :
    a ≤ d * c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [MulPosMono α] (h : a ≤ b * c) (hle : b ≤ d) (c0 : 0 ≤ c)
Goal: a ≤ d * c
-- lt_mul_of_lt_mul_of_nonneg_right
theorem lt_mul_of_lt_mul_of_nonneg_right [MulPosMono α] (h : a < b * c) (hle : b ≤ d) (c0 : 0 ≤ c) :
    a < d * c 
Arguments: {α : Type u_1} {a b c d : α} [Mul α] [Zero α] [Preorder α] [MulPosMono α] (h : a < b * c) (hle : b ≤ d) (c0 : 0 ≤ c)
Goal: a < d * c
-- PosMulReflectLE.toPosMulStrictMono
theorem PosMulReflectLE.toPosMulStrictMono [PosMulReflectLE α] : PosMulStrictMono α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [PosMulReflectLE α]
Goal: PosMulStrictMono α
-- MulPosReflectLE.toMulPosStrictMono
theorem MulPosReflectLE.toMulPosStrictMono [MulPosReflectLE α] : MulPosStrictMono α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [MulPosReflectLE α]
Goal: MulPosStrictMono α
-- posMulStrictMono_iff_posMulReflectLE
theorem posMulStrictMono_iff_posMulReflectLE : PosMulStrictMono α ↔ PosMulReflectLE α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α]
Goal: PosMulStrictMono α ↔ PosMulReflectLE α
-- mulPosStrictMono_iff_mulPosReflectLE
theorem mulPosStrictMono_iff_mulPosReflectLE : MulPosStrictMono α ↔ MulPosReflectLE α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α]
Goal: MulPosStrictMono α ↔ MulPosReflectLE α
-- PosMulReflectLT.toPosMulMono
theorem PosMulReflectLT.toPosMulMono [PosMulReflectLT α] : PosMulMono α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [PosMulReflectLT α]
Goal: PosMulMono α
-- MulPosReflectLT.toMulPosMono
theorem MulPosReflectLT.toMulPosMono [MulPosReflectLT α] : MulPosMono α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [MulPosReflectLT α]
Goal: MulPosMono α
-- PosMulMono.toPosMulReflectLT
theorem PosMulMono.toPosMulReflectLT [PosMulMono α] : PosMulReflectLT α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [PosMulMono α]
Goal: PosMulReflectLT α
-- MulPosMono.toMulPosReflectLT
theorem MulPosMono.toMulPosReflectLT [MulPosMono α] : MulPosReflectLT α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α] [MulPosMono α]
Goal: MulPosReflectLT α
-- posMulMono_iff_posMulReflectLT
theorem posMulMono_iff_posMulReflectLT : PosMulMono α ↔ PosMulReflectLT α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α]
Goal: PosMulMono α ↔ PosMulReflectLT α
-- mulPosMono_iff_mulPosReflectLT
theorem mulPosMono_iff_mulPosReflectLT : MulPosMono α ↔ MulPosReflectLT α 
Arguments: {α : Type u_1} [Mul α] [Zero α] [LinearOrder α]
Goal: MulPosMono α ↔ MulPosReflectLT α
-- Left.mul_pos
theorem Left.mul_pos [PosMulStrictMono α] (ha : 0 < a) (hb : 0 < b) : 0 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulStrictMono α] (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a * b
-- mul_neg_of_pos_of_neg
theorem mul_neg_of_pos_of_neg [PosMulStrictMono α] (ha : 0 < a) (hb : b < 0) : a * b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulStrictMono α] (ha : 0 < a) (hb : b < 0)
Goal: a * b < 0
-- mul_pos_iff_of_pos_left
@[simp]
theorem mul_pos_iff_of_pos_left [PosMulStrictMono α] [PosMulReflectLT α] (h : 0 < a) :
    0 < a * b ↔ 0 < b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulStrictMono α] [PosMulReflectLT α] (h : 0 < a)
Goal: 0 < a * b ↔ 0 < b
-- Right.mul_pos
theorem Right.mul_pos [MulPosStrictMono α] (ha : 0 < a) (hb : 0 < b) : 0 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosStrictMono α] (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a * b
-- mul_neg_of_neg_of_pos
theorem mul_neg_of_neg_of_pos [MulPosStrictMono α] (ha : a < 0) (hb : 0 < b) : a * b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosStrictMono α] (ha : a < 0) (hb : 0 < b)
Goal: a * b < 0
-- mul_pos_iff_of_pos_right
@[simp]
theorem mul_pos_iff_of_pos_right [MulPosStrictMono α] [MulPosReflectLT α] (h : 0 < b) :
    0 < a * b ↔ 0 < a 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosStrictMono α] [MulPosReflectLT α] (h : 0 < b)
Goal: 0 < a * b ↔ 0 < a
-- Left.mul_nonneg
theorem Left.mul_nonneg [PosMulMono α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulMono α] (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: 0 ≤ a * b
-- mul_nonpos_of_nonneg_of_nonpos
theorem mul_nonpos_of_nonneg_of_nonpos [PosMulMono α] (ha : 0 ≤ a) (hb : b ≤ 0) : a * b ≤ 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulMono α] (ha : 0 ≤ a) (hb : b ≤ 0)
Goal: a * b ≤ 0
-- Right.mul_nonneg
theorem Right.mul_nonneg [MulPosMono α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosMono α] (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: 0 ≤ a * b
-- mul_nonpos_of_nonpos_of_nonneg
theorem mul_nonpos_of_nonpos_of_nonneg [MulPosMono α] (ha : a ≤ 0) (hb : 0 ≤ b) : a * b ≤ 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosMono α] (ha : a ≤ 0) (hb : 0 ≤ b)
Goal: a * b ≤ 0
-- pos_of_mul_pos_right
theorem pos_of_mul_pos_right [PosMulReflectLT α] (h : 0 < a * b) (ha : 0 ≤ a) : 0 < b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulReflectLT α] (h : 0 < a * b) (ha : 0 ≤ a)
Goal: 0 < b
-- pos_of_mul_pos_left
theorem pos_of_mul_pos_left [MulPosReflectLT α] (h : 0 < a * b) (hb : 0 ≤ b) : 0 < a 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [MulPosReflectLT α] (h : 0 < a * b) (hb : 0 ≤ b)
Goal: 0 < a
-- pos_iff_pos_of_mul_pos
theorem pos_iff_pos_of_mul_pos [PosMulReflectLT α] [MulPosReflectLT α] (hab : 0 < a * b) :
    0 < a ↔ 0 < b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulReflectLT α] [MulPosReflectLT α] (hab : 0 < a * b)
Goal: 0 < a ↔ 0 < b
-- Left.mul_lt_mul_of_nonneg
theorem Left.mul_lt_mul_of_nonneg [PosMulStrictMono α] [MulPosMono α]
    (h₁ : a < b) (h₂ : c < d) (a0 : 0 ≤ a) (c0 : 0 ≤ c) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [MulZeroClass α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (h₁ : a < b) (h₂ : c < d) (a0 : 0 ≤ a) (c0 : 0 ≤ c)
Goal: a * c < b * d
-- Right.mul_lt_mul_of_nonneg
theorem Right.mul_lt_mul_of_nonneg [PosMulMono α] [MulPosStrictMono α]
    (h₁ : a < b) (h₂ : c < d) (a0 : 0 ≤ a) (c0 : 0 ≤ c) : a * c < b * d 
Arguments: {α : Type u_1} {a b c d : α} [MulZeroClass α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (h₁ : a < b) (h₂ : c < d) (a0 : 0 ≤ a) (c0 : 0 ≤ c)
Goal: a * c < b * d
-- mul_self_le_mul_self
theorem mul_self_le_mul_self [PosMulMono α] [MulPosMono α] (ha : 0 ≤ a) (hab : a ≤ b) :
    a * a ≤ b * b 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [Preorder α] [PosMulMono α] [MulPosMono α] (ha : 0 ≤ a) (hab : a ≤ b)
Goal: a * a ≤ b * b
-- posMulMono_iff_covariant_pos
theorem posMulMono_iff_covariant_pos :
    PosMulMono α ↔ CovariantClass α>0 α (fun x y => x * y) (· ≤ ·) 
Arguments: {α : Type u_1} [MulZeroClass α] [PartialOrder α]
Goal: PosMulMono α ↔ CovariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x ≤ x_1
-- mulPosMono_iff_covariant_pos
theorem mulPosMono_iff_covariant_pos :
    MulPosMono α ↔ CovariantClass α>0 α (fun x y => y * x) (· ≤ ·) 
Arguments: {α : Type u_1} [MulZeroClass α] [PartialOrder α]
Goal: MulPosMono α ↔ CovariantClass { x // 0 < x } α (fun x y => y * ↑x) fun x x_1 => x ≤ x_1
-- posMulReflectLT_iff_contravariant_pos
theorem posMulReflectLT_iff_contravariant_pos :
    PosMulReflectLT α ↔ ContravariantClass α>0 α (fun x y => x * y) (· < ·) 
Arguments: {α : Type u_1} [MulZeroClass α] [PartialOrder α]
Goal: PosMulReflectLT α ↔ ContravariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x < x_1
-- mulPosReflectLT_iff_contravariant_pos
theorem mulPosReflectLT_iff_contravariant_pos :
    MulPosReflectLT α ↔ ContravariantClass α>0 α (fun x y => y * x) (· < ·) 
Arguments: {α : Type u_1} [MulZeroClass α] [PartialOrder α]
Goal: MulPosReflectLT α ↔ ContravariantClass { x // 0 < x } α (fun x y => y * ↑x) fun x x_1 => x < x_1
-- mul_left_cancel_iff_of_pos
theorem mul_left_cancel_iff_of_pos [PosMulReflectLE α] (a0 : 0 < a) : a * b = a * c ↔ b = c 
Arguments: {α : Type u_1} {a b c : α} [MulZeroClass α] [PartialOrder α] [PosMulReflectLE α] (a0 : 0 < a)
Goal: a * b = a * c ↔ b = c
-- mul_right_cancel_iff_of_pos
theorem mul_right_cancel_iff_of_pos [MulPosReflectLE α] (b0 : 0 < b) : a * b = c * b ↔ a = c 
Arguments: {α : Type u_1} {a b c : α} [MulZeroClass α] [PartialOrder α] [MulPosReflectLE α] (b0 : 0 < b)
Goal: a * b = c * b ↔ a = c
-- mul_eq_mul_iff_eq_and_eq_of_pos
theorem mul_eq_mul_iff_eq_and_eq_of_pos [PosMulStrictMono α] [MulPosStrictMono α]
    (hab : a ≤ b) (hcd : c ≤ d) (a0 : 0 < a) (d0 : 0 < d) :
    a * c = b * d ↔ a = b ∧ c = d 
Arguments: {α : Type u_1} {a b c d : α} [MulZeroClass α] [PartialOrder α] [PosMulStrictMono α] [MulPosStrictMono α] (hab : a ≤ b) (hcd : c ≤ d) (a0 : 0 < a) (d0 : 0 < d)
Goal: a * c = b * d ↔ a = b ∧ c = d
-- mul_eq_mul_iff_eq_and_eq_of_pos'
theorem mul_eq_mul_iff_eq_and_eq_of_pos' [PosMulStrictMono α] [MulPosStrictMono α]
    (hab : a ≤ b) (hcd : c ≤ d) (b0 : 0 < b) (c0 : 0 < c) :
    a * c = b * d ↔ a = b ∧ c = d 
Arguments: {α : Type u_1} {a b c d : α} [MulZeroClass α] [PartialOrder α] [PosMulStrictMono α] [MulPosStrictMono α] (hab : a ≤ b) (hcd : c ≤ d) (b0 : 0 < b) (c0 : 0 < c)
Goal: a * c = b * d ↔ a = b ∧ c = d
-- pos_and_pos_or_neg_and_neg_of_mul_pos
theorem pos_and_pos_or_neg_and_neg_of_mul_pos [PosMulMono α] [MulPosMono α] (hab : 0 < a * b) :
    0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] [MulPosMono α] (hab : 0 < a * b)
Goal: 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0
-- neg_of_mul_pos_right
theorem neg_of_mul_pos_right [PosMulMono α] [MulPosMono α] (h : 0 < a * b) (ha : a ≤ 0) : b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] [MulPosMono α] (h : 0 < a * b) (ha : a ≤ 0)
Goal: b < 0
-- neg_of_mul_pos_left
theorem neg_of_mul_pos_left [PosMulMono α] [MulPosMono α] (h : 0 < a * b) (ha : b ≤ 0) : a < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] [MulPosMono α] (h : 0 < a * b) (ha : b ≤ 0)
Goal: a < 0
-- neg_iff_neg_of_mul_pos
theorem neg_iff_neg_of_mul_pos [PosMulMono α] [MulPosMono α] (hab : 0 < a * b) : a < 0 ↔ b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] [MulPosMono α] (hab : 0 < a * b)
Goal: a < 0 ↔ b < 0
-- Left.neg_of_mul_neg_right
theorem Left.neg_of_mul_neg_right [PosMulMono α] (h : a * b < 0) (a0 : 0 ≤ a) : b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] (h : a * b < 0) (a0 : 0 ≤ a)
Goal: b < 0
-- Right.neg_of_mul_neg_right
theorem Right.neg_of_mul_neg_right [MulPosMono α] (h : a * b < 0) (a0 : 0 ≤ a) : b < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [MulPosMono α] (h : a * b < 0) (a0 : 0 ≤ a)
Goal: b < 0
-- Left.neg_of_mul_neg_left
theorem Left.neg_of_mul_neg_left [PosMulMono α] (h : a * b < 0) (b0 : 0 ≤ b) : a < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [PosMulMono α] (h : a * b < 0) (b0 : 0 ≤ b)
Goal: a < 0
-- Right.neg_of_mul_neg_left
theorem Right.neg_of_mul_neg_left [MulPosMono α] (h : a * b < 0) (b0 : 0 ≤ b) : a < 0 
Arguments: {α : Type u_1} {a b : α} [MulZeroClass α] [LinearOrder α] [MulPosMono α] (h : a * b < 0) (b0 : 0 ≤ b)
Goal: a < 0
-- le_mul_iff_one_le_right
@[simp]
lemma le_mul_iff_one_le_right [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a) : a ≤ a * b ↔ 1 ≤ b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a)
Goal: a ≤ a * b ↔ 1 ≤ b
-- lt_mul_iff_one_lt_right
@[simp]
theorem lt_mul_iff_one_lt_right [PosMulStrictMono α] [PosMulReflectLT α] (a0 : 0 < a) :
    a < a * b ↔ 1 < b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [PosMulReflectLT α] (a0 : 0 < a)
Goal: a < a * b ↔ 1 < b
-- mul_le_iff_le_one_right
@[simp]
lemma mul_le_iff_le_one_right [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a) : a * b ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [PosMulReflectLE α] (a0 : 0 < a)
Goal: a * b ≤ a ↔ b ≤ 1
-- mul_lt_iff_lt_one_right
@[simp]
theorem mul_lt_iff_lt_one_right [PosMulStrictMono α] [PosMulReflectLT α] (a0 : 0 < a) :
    a * b < a ↔ b < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [PosMulReflectLT α] (a0 : 0 < a)
Goal: a * b < a ↔ b < 1
-- le_mul_iff_one_le_left
@[simp]
lemma le_mul_iff_one_le_left [MulPosMono α] [MulPosReflectLE α] (a0 : 0 < a) : a ≤ b * a ↔ 1 ≤ b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] [MulPosReflectLE α] (a0 : 0 < a)
Goal: a ≤ b * a ↔ 1 ≤ b
-- lt_mul_iff_one_lt_left
@[simp]
theorem lt_mul_iff_one_lt_left [MulPosStrictMono α] [MulPosReflectLT α] (a0 : 0 < a) :
    a < b * a ↔ 1 < b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] [MulPosReflectLT α] (a0 : 0 < a)
Goal: a < b * a ↔ 1 < b
-- mul_le_iff_le_one_left
@[simp]
lemma mul_le_iff_le_one_left [MulPosMono α] [MulPosReflectLE α] (b0 : 0 < b) : a * b ≤ b ↔ a ≤ 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] [MulPosReflectLE α] (b0 : 0 < b)
Goal: a * b ≤ b ↔ a ≤ 1
-- mul_lt_iff_lt_one_left
@[simp]
theorem mul_lt_iff_lt_one_left [MulPosStrictMono α] [MulPosReflectLT α] (b0 : 0 < b) :
    a * b < b ↔ a < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] [MulPosReflectLT α] (b0 : 0 < b)
Goal: a * b < b ↔ a < 1
-- mul_le_of_le_one_left
theorem mul_le_of_le_one_left [MulPosMono α] (hb : 0 ≤ b) (h : a ≤ 1) : a * b ≤ b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (hb : 0 ≤ b) (h : a ≤ 1)
Goal: a * b ≤ b
-- le_mul_of_one_le_left
theorem le_mul_of_one_le_left [MulPosMono α] (hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (hb : 0 ≤ b) (h : 1 ≤ a)
Goal: b ≤ a * b
-- mul_le_of_le_one_right
theorem mul_le_of_le_one_right [PosMulMono α] (ha : 0 ≤ a) (h : b ≤ 1) : a * b ≤ a 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : 0 ≤ a) (h : b ≤ 1)
Goal: a * b ≤ a
-- le_mul_of_one_le_right
theorem le_mul_of_one_le_right [PosMulMono α] (ha : 0 ≤ a) (h : 1 ≤ b) : a ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : 0 ≤ a) (h : 1 ≤ b)
Goal: a ≤ a * b
-- mul_lt_of_lt_one_left
theorem mul_lt_of_lt_one_left [MulPosStrictMono α] (hb : 0 < b) (h : a < 1) : a * b < b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (hb : 0 < b) (h : a < 1)
Goal: a * b < b
-- lt_mul_of_one_lt_left
theorem lt_mul_of_one_lt_left [MulPosStrictMono α] (hb : 0 < b) (h : 1 < a) : b < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (hb : 0 < b) (h : 1 < a)
Goal: b < a * b
-- mul_lt_of_lt_one_right
theorem mul_lt_of_lt_one_right [PosMulStrictMono α] (ha : 0 < a) (h : b < 1) : a * b < a 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (ha : 0 < a) (h : b < 1)
Goal: a * b < a
-- lt_mul_of_one_lt_right
theorem lt_mul_of_one_lt_right [PosMulStrictMono α] (ha : 0 < a) (h : 1 < b) : a < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (ha : 0 < a) (h : 1 < b)
Goal: a < a * b
-- mul_le_of_le_of_le_one_of_nonneg
theorem mul_le_of_le_of_le_one_of_nonneg [PosMulMono α] (h : b ≤ c) (ha : a ≤ 1) (hb : 0 ≤ b) :
    b * a ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : b ≤ c) (ha : a ≤ 1) (hb : 0 ≤ b)
Goal: b * a ≤ c
-- mul_lt_of_le_of_lt_one_of_pos
theorem mul_lt_of_le_of_lt_one_of_pos [PosMulStrictMono α] (bc : b ≤ c) (ha : a < 1) (b0 : 0 < b) :
    b * a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (bc : b ≤ c) (ha : a < 1) (b0 : 0 < b)
Goal: b * a < c
-- mul_lt_of_lt_of_le_one_of_nonneg
theorem mul_lt_of_lt_of_le_one_of_nonneg [PosMulMono α] (h : b < c) (ha : a ≤ 1) (hb : 0 ≤ b) :
    b * a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : b < c) (ha : a ≤ 1) (hb : 0 ≤ b)
Goal: b * a < c
-- Left.mul_le_one_of_le_of_le
theorem Left.mul_le_one_of_le_of_le [PosMulMono α] (ha : a ≤ 1) (hb : b ≤ 1) (a0 : 0 ≤ a) :
    a * b ≤ 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : a ≤ 1) (hb : b ≤ 1) (a0 : 0 ≤ a)
Goal: a * b ≤ 1
-- Left.mul_lt_of_le_of_lt_one_of_pos
theorem Left.mul_lt_of_le_of_lt_one_of_pos [PosMulStrictMono α] (ha : a ≤ 1) (hb : b < 1)
    (a0 : 0 < a) : a * b < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (ha : a ≤ 1) (hb : b < 1) (a0 : 0 < a)
Goal: a * b < 1
-- Left.mul_lt_of_lt_of_le_one_of_nonneg
theorem Left.mul_lt_of_lt_of_le_one_of_nonneg [PosMulMono α] (ha : a < 1) (hb : b ≤ 1)
    (a0 : 0 ≤ a) : a * b < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : a < 1) (hb : b ≤ 1) (a0 : 0 ≤ a)
Goal: a * b < 1
-- mul_le_of_le_of_le_one'
theorem mul_le_of_le_of_le_one' [PosMulMono α] [MulPosMono α] (bc : b ≤ c) (ha : a ≤ 1) (a0 : 0 ≤ a)
    (c0 : 0 ≤ c) : b * a ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosMono α] (bc : b ≤ c) (ha : a ≤ 1) (a0 : 0 ≤ a) (c0 : 0 ≤ c)
Goal: b * a ≤ c
-- mul_lt_of_lt_of_le_one'
theorem mul_lt_of_lt_of_le_one' [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : a ≤ 1)
    (a0 : 0 < a) (c0 : 0 ≤ c) : b * a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : a ≤ 1) (a0 : 0 < a) (c0 : 0 ≤ c)
Goal: b * a < c
-- mul_lt_of_le_of_lt_one'
theorem mul_lt_of_le_of_lt_one' [PosMulStrictMono α] [MulPosMono α] (bc : b ≤ c) (ha : a < 1)
    (a0 : 0 ≤ a) (c0 : 0 < c) : b * a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (bc : b ≤ c) (ha : a < 1) (a0 : 0 ≤ a) (c0 : 0 < c)
Goal: b * a < c
-- mul_lt_of_lt_of_lt_one_of_pos
theorem mul_lt_of_lt_of_lt_one_of_pos [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : a ≤ 1)
    (a0 : 0 < a) (c0 : 0 ≤ c) : b * a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : a ≤ 1) (a0 : 0 < a) (c0 : 0 ≤ c)
Goal: b * a < c