ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_mul_of_le_of_one_le_of_nonneg
theorem le_mul_of_le_of_one_le_of_nonneg [PosMulMono α] (h : b ≤ c) (ha : 1 ≤ a) (hc : 0 ≤ c) :
    b ≤ c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : b ≤ c) (ha : 1 ≤ a) (hc : 0 ≤ c)
Goal: b ≤ c * a
-- lt_mul_of_le_of_one_lt_of_pos
theorem lt_mul_of_le_of_one_lt_of_pos [PosMulStrictMono α] (bc : b ≤ c) (ha : 1 < a) (c0 : 0 < c) :
    b < c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (bc : b ≤ c) (ha : 1 < a) (c0 : 0 < c)
Goal: b < c * a
-- lt_mul_of_lt_of_one_le_of_nonneg
theorem lt_mul_of_lt_of_one_le_of_nonneg [PosMulMono α] (h : b < c) (ha : 1 ≤ a) (hc : 0 ≤ c) :
    b < c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : b < c) (ha : 1 ≤ a) (hc : 0 ≤ c)
Goal: b < c * a
-- Left.one_le_mul_of_le_of_le
theorem Left.one_le_mul_of_le_of_le [PosMulMono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (a0 : 0 ≤ a) :
    1 ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (a0 : 0 ≤ a)
Goal: 1 ≤ a * b
-- Left.one_lt_mul_of_le_of_lt_of_pos
theorem Left.one_lt_mul_of_le_of_lt_of_pos [PosMulStrictMono α] (ha : 1 ≤ a) (hb : 1 < b)
    (a0 : 0 < a) : 1 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] (ha : 1 ≤ a) (hb : 1 < b) (a0 : 0 < a)
Goal: 1 < a * b
-- Left.lt_mul_of_lt_of_one_le_of_nonneg
theorem Left.lt_mul_of_lt_of_one_le_of_nonneg [PosMulMono α] (ha : 1 < a) (hb : 1 ≤ b)
    (a0 : 0 ≤ a) : 1 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (ha : 1 < a) (hb : 1 ≤ b) (a0 : 0 ≤ a)
Goal: 1 < a * b
-- le_mul_of_le_of_one_le'
theorem le_mul_of_le_of_one_le' [PosMulMono α] [MulPosMono α] (bc : b ≤ c) (ha : 1 ≤ a)
    (a0 : 0 ≤ a) (b0 : 0 ≤ b) : b ≤ c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosMono α] (bc : b ≤ c) (ha : 1 ≤ a) (a0 : 0 ≤ a) (b0 : 0 ≤ b)
Goal: b ≤ c * a
-- lt_mul_of_le_of_one_lt'
theorem lt_mul_of_le_of_one_lt' [PosMulStrictMono α] [MulPosMono α] (bc : b ≤ c) (ha : 1 < a)
    (a0 : 0 ≤ a) (b0 : 0 < b) : b < c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (bc : b ≤ c) (ha : 1 < a) (a0 : 0 ≤ a) (b0 : 0 < b)
Goal: b < c * a
-- lt_mul_of_lt_of_one_le'
theorem lt_mul_of_lt_of_one_le' [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : 1 ≤ a)
    (a0 : 0 < a) (b0 : 0 ≤ b) : b < c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (bc : b < c) (ha : 1 ≤ a) (a0 : 0 < a) (b0 : 0 ≤ b)
Goal: b < c * a
-- lt_mul_of_lt_of_one_lt_of_pos
theorem lt_mul_of_lt_of_one_lt_of_pos [PosMulStrictMono α] [MulPosStrictMono α] (bc : b < c)
    (ha : 1 < a) (a0 : 0 < a) (b0 : 0 < b) : b < c * a 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosStrictMono α] (bc : b < c) (ha : 1 < a) (a0 : 0 < a) (b0 : 0 < b)
Goal: b < c * a
-- mul_le_of_le_one_of_le_of_nonneg
theorem mul_le_of_le_one_of_le_of_nonneg [MulPosMono α] (ha : a ≤ 1) (h : b ≤ c) (hb : 0 ≤ b) :
    a * b ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : a ≤ 1) (h : b ≤ c) (hb : 0 ≤ b)
Goal: a * b ≤ c
-- mul_lt_of_lt_one_of_le_of_pos
theorem mul_lt_of_lt_one_of_le_of_pos [MulPosStrictMono α] (ha : a < 1) (h : b ≤ c) (hb : 0 < b) :
    a * b < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : a < 1) (h : b ≤ c) (hb : 0 < b)
Goal: a * b < c
-- mul_lt_of_le_one_of_lt_of_nonneg
theorem mul_lt_of_le_one_of_lt_of_nonneg [MulPosMono α] (ha : a ≤ 1) (h : b < c) (hb : 0 ≤ b) :
    a * b < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : a ≤ 1) (h : b < c) (hb : 0 ≤ b)
Goal: a * b < c
-- Right.mul_lt_one_of_lt_of_le_of_pos
theorem Right.mul_lt_one_of_lt_of_le_of_pos [MulPosStrictMono α] (ha : a < 1) (hb : b ≤ 1)
    (b0 : 0 < b) : a * b < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : a < 1) (hb : b ≤ 1) (b0 : 0 < b)
Goal: a * b < 1
-- Right.mul_lt_one_of_le_of_lt_of_nonneg
theorem Right.mul_lt_one_of_le_of_lt_of_nonneg [MulPosMono α] (ha : a ≤ 1) (hb : b < 1)
    (b0 : 0 ≤ b) : a * b < 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : a ≤ 1) (hb : b < 1) (b0 : 0 ≤ b)
Goal: a * b < 1
-- mul_lt_of_lt_one_of_lt_of_pos
theorem mul_lt_of_lt_one_of_lt_of_pos [PosMulStrictMono α] [MulPosStrictMono α] (ha : a < 1)
    (bc : b < c) (a0 : 0 < a) (c0 : 0 < c) : a * b < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosStrictMono α] (ha : a < 1) (bc : b < c) (a0 : 0 < a) (c0 : 0 < c)
Goal: a * b < c
-- Right.mul_le_one_of_le_of_le
theorem Right.mul_le_one_of_le_of_le [MulPosMono α] (ha : a ≤ 1) (hb : b ≤ 1) (b0 : 0 ≤ b) :
    a * b ≤ 1 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : a ≤ 1) (hb : b ≤ 1) (b0 : 0 ≤ b)
Goal: a * b ≤ 1
-- mul_le_of_le_one_of_le'
theorem mul_le_of_le_one_of_le' [PosMulMono α] [MulPosMono α] (ha : a ≤ 1) (bc : b ≤ c) (a0 : 0 ≤ a)
    (c0 : 0 ≤ c) : a * b ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosMono α] (ha : a ≤ 1) (bc : b ≤ c) (a0 : 0 ≤ a) (c0 : 0 ≤ c)
Goal: a * b ≤ c
-- mul_lt_of_lt_one_of_le'
theorem mul_lt_of_lt_one_of_le' [PosMulMono α] [MulPosStrictMono α] (ha : a < 1) (bc : b ≤ c)
    (a0 : 0 ≤ a) (c0 : 0 < c) : a * b < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] [MulPosStrictMono α] (ha : a < 1) (bc : b ≤ c) (a0 : 0 ≤ a) (c0 : 0 < c)
Goal: a * b < c
-- mul_lt_of_le_one_of_lt'
theorem mul_lt_of_le_one_of_lt' [PosMulStrictMono α] [MulPosMono α] (ha : a ≤ 1) (bc : b < c)
    (a0 : 0 < a) (c0 : 0 ≤ c) : a * b < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulStrictMono α] [MulPosMono α] (ha : a ≤ 1) (bc : b < c) (a0 : 0 < a) (c0 : 0 ≤ c)
Goal: a * b < c
-- lt_mul_of_one_lt_of_le_of_pos
theorem lt_mul_of_one_lt_of_le_of_pos [MulPosStrictMono α] (ha : 1 < a) (h : b ≤ c) (hc : 0 < c) :
    b < a * c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : 1 < a) (h : b ≤ c) (hc : 0 < c)
Goal: b < a * c
-- lt_mul_of_one_le_of_lt_of_nonneg
theorem lt_mul_of_one_le_of_lt_of_nonneg [MulPosMono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c) :
    b < a * c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c)
Goal: b < a * c
-- lt_mul_of_one_lt_of_lt_of_pos
theorem lt_mul_of_one_lt_of_lt_of_pos [MulPosStrictMono α] (ha : 1 < a) (h : b < c) (hc : 0 < c) :
    b < a * c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : 1 < a) (h : b < c) (hc : 0 < c)
Goal: b < a * c
-- Right.one_lt_mul_of_lt_of_le_of_pos
theorem Right.one_lt_mul_of_lt_of_le_of_pos [MulPosStrictMono α] (ha : 1 < a) (hb : 1 ≤ b)
    (b0 : 0 < b) : 1 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : 1 < a) (hb : 1 ≤ b) (b0 : 0 < b)
Goal: 1 < a * b
-- Right.one_lt_mul_of_le_of_lt_of_nonneg
theorem Right.one_lt_mul_of_le_of_lt_of_nonneg [MulPosMono α] (ha : 1 ≤ a) (hb : 1 < b)
    (b0 : 0 ≤ b) : 1 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : 1 ≤ a) (hb : 1 < b) (b0 : 0 ≤ b)
Goal: 1 < a * b
-- Right.one_lt_mul_of_lt_of_lt
theorem Right.one_lt_mul_of_lt_of_lt [MulPosStrictMono α] (ha : 1 < a) (hb : 1 < b) (b0 : 0 < b) :
    1 < a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosStrictMono α] (ha : 1 < a) (hb : 1 < b) (b0 : 0 < b)
Goal: 1 < a * b
-- lt_mul_of_one_lt_of_lt_of_nonneg
theorem lt_mul_of_one_lt_of_lt_of_nonneg [MulPosMono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c) :
    b < a * c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c)
Goal: b < a * c
-- lt_of_mul_lt_of_one_le_of_nonneg_left
theorem lt_of_mul_lt_of_one_le_of_nonneg_left [PosMulMono α] (h : a * b < c) (hle : 1 ≤ b)
    (ha : 0 ≤ a) : a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : a * b < c) (hle : 1 ≤ b) (ha : 0 ≤ a)
Goal: a < c
-- lt_of_lt_mul_of_le_one_of_nonneg_left
theorem lt_of_lt_mul_of_le_one_of_nonneg_left [PosMulMono α] (h : a < b * c) (hc : c ≤ 1)
    (hb : 0 ≤ b) : a < b 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : a < b * c) (hc : c ≤ 1) (hb : 0 ≤ b)
Goal: a < b
-- lt_of_lt_mul_of_le_one_of_nonneg_right
theorem lt_of_lt_mul_of_le_one_of_nonneg_right [MulPosMono α] (h : a < b * c) (hb : b ≤ 1)
    (hc : 0 ≤ c) : a < c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (h : a < b * c) (hb : b ≤ 1) (hc : 0 ≤ c)
Goal: a < c
-- le_mul_of_one_le_of_le_of_nonneg
theorem le_mul_of_one_le_of_le_of_nonneg [MulPosMono α] (ha : 1 ≤ a) (bc : b ≤ c) (c0 : 0 ≤ c) :
    b ≤ a * c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : 1 ≤ a) (bc : b ≤ c) (c0 : 0 ≤ c)
Goal: b ≤ a * c
-- Right.one_le_mul_of_le_of_le
theorem Right.one_le_mul_of_le_of_le [MulPosMono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (b0 : 0 ≤ b) :
    1 ≤ a * b 
Arguments: {α : Type u_1} {a b : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (b0 : 0 ≤ b)
Goal: 1 ≤ a * b
-- le_of_mul_le_of_one_le_of_nonneg_left
theorem le_of_mul_le_of_one_le_of_nonneg_left [PosMulMono α] (h : a * b ≤ c) (hb : 1 ≤ b)
    (ha : 0 ≤ a) : a ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : a * b ≤ c) (hb : 1 ≤ b) (ha : 0 ≤ a)
Goal: a ≤ c
-- le_of_le_mul_of_le_one_of_nonneg_left
theorem le_of_le_mul_of_le_one_of_nonneg_left [PosMulMono α] (h : a ≤ b * c) (hc : c ≤ 1)
    (hb : 0 ≤ b) : a ≤ b 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [PosMulMono α] (h : a ≤ b * c) (hc : c ≤ 1) (hb : 0 ≤ b)
Goal: a ≤ b
-- le_of_mul_le_of_one_le_nonneg_right
theorem le_of_mul_le_of_one_le_nonneg_right [MulPosMono α] (h : a * b ≤ c) (ha : 1 ≤ a)
    (hb : 0 ≤ b) : b ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (h : a * b ≤ c) (ha : 1 ≤ a) (hb : 0 ≤ b)
Goal: b ≤ c
-- le_of_le_mul_of_le_one_of_nonneg_right
theorem le_of_le_mul_of_le_one_of_nonneg_right [MulPosMono α] (h : a ≤ b * c) (hb : b ≤ 1)
    (hc : 0 ≤ c) : a ≤ c 
Arguments: {α : Type u_1} {a b c : α} [MulOneClass α] [Zero α] [Preorder α] [MulPosMono α] (h : a ≤ b * c) (hb : b ≤ 1) (hc : 0 ≤ c)
Goal: a ≤ c
-- exists_square_le'
theorem exists_square_le' [PosMulStrictMono α] (a0 : 0 < a) : ∃ b : α, b * b ≤ a 
Arguments: {α : Type u_1} {a : α} [MulOneClass α] [Zero α] [LinearOrder α] [PosMulStrictMono α] (a0 : 0 < a)
Goal: ∃ b, b * b ≤ a
-- PosMulMono.toPosMulStrictMono
theorem PosMulMono.toPosMulStrictMono [PosMulMono α] : PosMulStrictMono α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α] [PosMulMono α]
Goal: PosMulStrictMono α
-- posMulMono_iff_posMulStrictMono
theorem posMulMono_iff_posMulStrictMono : PosMulMono α ↔ PosMulStrictMono α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α]
Goal: PosMulMono α ↔ PosMulStrictMono α
-- MulPosMono.toMulPosStrictMono
theorem MulPosMono.toMulPosStrictMono [MulPosMono α] : MulPosStrictMono α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α] [MulPosMono α]
Goal: MulPosStrictMono α
-- mulPosMono_iff_mulPosStrictMono
theorem mulPosMono_iff_mulPosStrictMono : MulPosMono α ↔ MulPosStrictMono α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α]
Goal: MulPosMono α ↔ MulPosStrictMono α
-- PosMulReflectLT.toPosMulReflectLE
theorem PosMulReflectLT.toPosMulReflectLE [PosMulReflectLT α] : PosMulReflectLE α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α] [PosMulReflectLT α]
Goal: PosMulReflectLE α
-- posMulReflectLE_iff_posMulReflectLT
theorem posMulReflectLE_iff_posMulReflectLT : PosMulReflectLE α ↔ PosMulReflectLT α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α]
Goal: PosMulReflectLE α ↔ PosMulReflectLT α
-- MulPosReflectLT.toMulPosReflectLE
theorem MulPosReflectLT.toMulPosReflectLE [MulPosReflectLT α] : MulPosReflectLE α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α] [MulPosReflectLT α]
Goal: MulPosReflectLE α
-- mulPosReflectLE_iff_mulPosReflectLT
theorem mulPosReflectLE_iff_mulPosReflectLT : MulPosReflectLE α ↔ MulPosReflectLT α 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] [PartialOrder α]
Goal: MulPosReflectLE α ↔ MulPosReflectLT α
-- posMulStrictMono_iff_mulPosStrictMono
theorem posMulStrictMono_iff_mulPosStrictMono : PosMulStrictMono α ↔ MulPosStrictMono α 
Arguments: {α : Type u_1} [Mul α] [IsSymmOp α α fun x x_1 => x * x_1] [Zero α] [Preorder α]
Goal: PosMulStrictMono α ↔ MulPosStrictMono α
-- posMulReflectLT_iff_mulPosReflectLT
theorem posMulReflectLT_iff_mulPosReflectLT : PosMulReflectLT α ↔ MulPosReflectLT α 
Arguments: {α : Type u_1} [Mul α] [IsSymmOp α α fun x x_1 => x * x_1] [Zero α] [Preorder α]
Goal: PosMulReflectLT α ↔ MulPosReflectLT α
-- posMulMono_iff_mulPosMono
theorem posMulMono_iff_mulPosMono : PosMulMono α ↔ MulPosMono α 
Arguments: {α : Type u_1} [Mul α] [IsSymmOp α α fun x x_1 => x * x_1] [Zero α] [Preorder α]
Goal: PosMulMono α ↔ MulPosMono α
-- posMulReflectLE_iff_mulPosReflectLE
theorem posMulReflectLE_iff_mulPosReflectLE : PosMulReflectLE α ↔ MulPosReflectLE α 
Arguments: {α : Type u_1} [Mul α] [IsSymmOp α α fun x x_1 => x * x_1] [Zero α] [Preorder α]
Goal: PosMulReflectLE α ↔ MulPosReflectLE α
-- Nonneg.coe_smul
@[simp, norm_cast]
lemma coe_smul (a : 𝕜≥0) (x : 𝕜') : (a : 𝕜) • x = a • x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [OrderedSemiring 𝕜] [SMul 𝕜 𝕜'] (a : 𝕜≥0) (x : 𝕜')
Goal: ↑a • x = a • x
-- Nonneg.mk_smul
@[simp]
lemma mk_smul (a) (ha) (x : 𝕜') : (⟨a, ha⟩ : 𝕜≥0) • x = a • x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [OrderedSemiring 𝕜] [SMul 𝕜 𝕜'] (a : 𝕜) (ha : 0 ≤ a) (x : 𝕜')
Goal: ⟨a, ha⟩ • x = a • x
-- LinearOrderedCommMonoidWithZero.zero_le_one
class LinearOrderedCommMonoidWithZero (α : Type*) extends LinearOrderedCommMonoid α,
  CommMonoidWithZero α where
    zero_le_one : (0 : α) ≤ 1
Arguments: {α : Type u_2} [self : LinearOrderedCommMonoidWithZero α]
Goal: 0 ≤ 1
-- zero_le'
@[simp] lemma zero_le' : 0 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α}
Goal: 0 ≤ a
-- not_lt_zero'
@[simp]
theorem not_lt_zero' : ¬a < 0 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α}
Goal: ¬a < 0
-- le_zero_iff
@[simp]
theorem le_zero_iff : a ≤ 0 ↔ a = 0 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α}
Goal: a ≤ 0 ↔ a = 0
-- zero_lt_iff
theorem zero_lt_iff : 0 < a ↔ a ≠ 0 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α}
Goal: 0 < a ↔ a ≠ 0
-- ne_zero_of_lt
theorem ne_zero_of_lt (h : b < a) : a ≠ 0 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a b : α} (h : b < a)
Goal: a ≠ 0
-- pow_pos_iff
lemma pow_pos_iff (hn : n ≠ 0) : 0 < a ^ n ↔ 0 < a 
Arguments: {α : Type u_1} [LinearOrderedCommMonoidWithZero α] {a : α} {n : ℕ} [NoZeroDivisors α] (hn : n ≠ 0)
Goal: 0 < a ^ n ↔ 0 < a
-- mul_le_one₀
theorem mul_le_one₀ (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1)
Goal: a * b ≤ 1
-- one_le_mul₀
theorem one_le_mul₀ (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b)
Goal: 1 ≤ a * b
-- le_of_le_mul_right
theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : c ≠ 0) (hab : a * c ≤ b * c)
Goal: a ≤ b
-- le_mul_inv_of_mul_le
theorem le_mul_inv_of_mul_le (h : c ≠ 0) (hab : a * c ≤ b) : a ≤ b * c⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : c ≠ 0) (hab : a * c ≤ b)
Goal: a ≤ b * c⁻¹
-- mul_inv_le_of_le_mul
theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hab : a ≤ b * c)
Goal: a * c⁻¹ ≤ b
-- inv_le_one₀
theorem inv_le_one₀ (ha : a ≠ 0) : a⁻¹ ≤ 1 ↔ 1 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: a⁻¹ ≤ 1 ↔ 1 ≤ a
-- one_le_inv₀
theorem one_le_inv₀ (ha : a ≠ 0) : 1 ≤ a⁻¹ ↔ a ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: 1 ≤ a⁻¹ ↔ a ≤ 1
-- le_mul_inv_iff₀
theorem le_mul_inv_iff₀ (hc : c ≠ 0) : a ≤ b * c⁻¹ ↔ a * c ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a ≤ b * c⁻¹ ↔ a * c ≤ b
-- mul_inv_le_iff₀
theorem mul_inv_le_iff₀ (hc : c ≠ 0) : a * c⁻¹ ≤ b ↔ a ≤ b * c 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a * c⁻¹ ≤ b ↔ a ≤ b * c
-- div_le_div₀
theorem div_le_div₀ (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0) :
    a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0)
Goal: a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b
-- Units.zero_lt
@[simp]
theorem Units.zero_lt (u : αˣ) : (0 : α) < u 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] (u : αˣ)
Goal: 0 < ↑u
-- mul_lt_mul_of_lt_of_le₀
theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d)
Goal: a * c < b * d
-- mul_lt_mul₀
theorem mul_lt_mul₀ (hab : a < b) (hcd : c < d) : a * c < b * d 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (hab : a < b) (hcd : c < d)
Goal: a * c < b * d
-- mul_inv_lt_of_lt_mul₀
theorem mul_inv_lt_of_lt_mul₀ (h : a < b * c) : a * c⁻¹ < b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : a < b * c)
Goal: a * c⁻¹ < b
-- inv_mul_lt_of_lt_mul₀
theorem inv_mul_lt_of_lt_mul₀ (h : a < b * c) : b⁻¹ * a < c 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (h : a < b * c)
Goal: b⁻¹ * a < c
-- mul_lt_right₀
theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (c : α) (h : a < b) (hc : c ≠ 0)
Goal: a * c < b * c
-- inv_lt_one₀
theorem inv_lt_one₀ (ha : a ≠ 0) : a⁻¹ < 1 ↔ 1 < a 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: a⁻¹ < 1 ↔ 1 < a
-- one_lt_inv₀
theorem one_lt_inv₀ (ha : a ≠ 0) : 1 < a⁻¹ ↔ a < 1 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: 1 < a⁻¹ ↔ a < 1
-- inv_lt_inv₀
theorem inv_lt_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ < b⁻¹ ↔ b < a 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
Goal: a⁻¹ < b⁻¹ ↔ b < a
-- inv_le_inv₀
theorem inv_le_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
Goal: a⁻¹ ≤ b⁻¹ ↔ b ≤ a
-- lt_of_mul_lt_mul_of_le₀
theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c d : α} (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a)
Goal: b < d
-- mul_le_mul_right₀
theorem mul_le_mul_right₀ (hc : c ≠ 0) : a * c ≤ b * c ↔ a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a * c ≤ b * c ↔ a ≤ b
-- mul_le_mul_left₀
theorem mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (ha : a ≠ 0)
Goal: a * b ≤ a * c ↔ b ≤ c
-- div_le_div_right₀
theorem div_le_div_right₀ (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a / c ≤ b / c ↔ a ≤ b
-- div_le_div_left₀
theorem div_le_div_left₀ (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0)
Goal: a / b ≤ a / c ↔ c ≤ b
-- le_div_iff₀
theorem le_div_iff₀ (hc : c ≠ 0) : a ≤ b / c ↔ a * c ≤ b 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a ≤ b / c ↔ a * c ≤ b
-- div_le_iff₀
theorem div_le_iff₀ (hc : c ≠ 0) : a / c ≤ b ↔ a ≤ b * c 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a / c ≤ b ↔ a ≤ b * c
-- OrderIso.mulLeft₀'_toEquiv
@[simps! (config := { simpRhs := true }) apply toEquiv]
def OrderIso.mulLeft₀' {a : α} (ha : a ≠ 0) : α ≃o α 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: (OrderIso.mulLeft₀' ha).toEquiv = Equiv.mulLeft₀ a ha
-- OrderIso.mulLeft₀'_apply
@[simps! (config := { simpRhs := true }) apply toEquiv]
def OrderIso.mulLeft₀' {a : α} (ha : a ≠ 0) : α ≃o α 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) (x : α)
Goal: (OrderIso.mulLeft₀' ha) x = a * x
-- OrderIso.mulLeft₀'_symm
theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
    (OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' (inv_ne_zero ha) 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: (OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' ⋯
-- OrderIso.mulRight₀'_toEquiv
@[simps! (config := { simpRhs := true }) apply toEquiv]
def OrderIso.mulRight₀' {a : α} (ha : a ≠ 0) : α ≃o α 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: (OrderIso.mulRight₀' ha).toEquiv = Equiv.mulRight₀ a ha
-- OrderIso.mulRight₀'_apply
@[simps! (config := { simpRhs := true }) apply toEquiv]
def OrderIso.mulRight₀' {a : α} (ha : a ≠ 0) : α ≃o α 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0) (x : α)
Goal: (OrderIso.mulRight₀' ha) x = x * a
-- OrderIso.mulRight₀'_symm
theorem OrderIso.mulRight₀'_symm {a : α} (ha : a ≠ 0) :
    (OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' (inv_ne_zero ha) 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} (ha : a ≠ 0)
Goal: (OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' ⋯
-- pow_lt_pow_succ
lemma pow_lt_pow_succ (ha : 1 < a) : a ^ n < a ^ n.succ 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} {n : ℕ} (ha : 1 < a)
Goal: a ^ n < a ^ n.succ
-- pow_lt_pow_right₀
lemma pow_lt_pow_right₀ (ha : 1 < a) (hmn : m < n) : a ^ m < a ^ n 
Arguments: {α : Type u_1} [LinearOrderedCommGroupWithZero α] {a : α} {m n : ℕ} (ha : 1 < a) (hmn : m < n)
Goal: a ^ m < a ^ n
-- WithZero.zero_le
lemma zero_le (a : WithZero α) : 0 ≤ a 
Arguments: {α : Type u_1} [Preorder α] (a : WithZero α)
Goal: 0 ≤ a
-- WithZero.zero_lt_coe
lemma zero_lt_coe (a : α) : (0 : WithZero α) < a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: 0 < ↑a
-- WithZero.zero_eq_bot
lemma zero_eq_bot : (0 : WithZero α) = ⊥ 
Arguments: {α : Type u_1} [Preorder α]
Goal: 0 = ⊥
-- WithZero.coe_lt_coe
@[simp, norm_cast] lemma coe_lt_coe : (a : WithZero α) < b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: ↑a < ↑b ↔ a < b
-- WithZero.coe_le_coe
@[simp, norm_cast] lemma coe_le_coe : (a : WithZero α) ≤ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: ↑a ≤ ↑b ↔ a ≤ b
-- WithZero.one_lt_coe
@[simp, norm_cast] lemma one_lt_coe [One α] : 1 < (a : WithZero α) ↔ 1 < a 
Arguments: {α : Type u_1} [Preorder α] {a : α} [One α]
Goal: 1 < ↑a ↔ 1 < a
-- WithZero.one_le_coe
@[simp, norm_cast] lemma one_le_coe [One α] : 1 ≤ (a : WithZero α) ↔ 1 ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a : α} [One α]
Goal: 1 ≤ ↑a ↔ 1 ≤ a