ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SMulPosMono.toSMulPosReflectLT
lemma SMulPosMono.toSMulPosReflectLT [SMulPosMono α β] : SMulPosReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [LinearOrder α] [Preorder β] [Zero β] [SMulPosMono α β]
Goal: SMulPosReflectLT α β
-- smulPosStrictMono_iff_SMulPosReflectLE
lemma smulPosStrictMono_iff_SMulPosReflectLE : SMulPosStrictMono α β ↔ SMulPosReflectLE α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [LinearOrder α] [LinearOrder β] [Zero β]
Goal: SMulPosStrictMono α β ↔ SMulPosReflectLE α β
-- smulPosMono_iff_smulPosReflectLT
lemma smulPosMono_iff_smulPosReflectLT : SMulPosMono α β ↔ SMulPosReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [LinearOrder α] [LinearOrder β] [Zero β]
Goal: SMulPosMono α β ↔ SMulPosReflectLT α β
-- smul_pos
lemma smul_pos [PosSMulStrictMono α β] (ha : 0 < a) (hb : 0 < b) : 0 < a • b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulStrictMono α β] (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a • b
-- smul_neg_of_pos_of_neg
lemma smul_neg_of_pos_of_neg [PosSMulStrictMono α β] (ha : 0 < a) (hb : b < 0) : a • b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulStrictMono α β] (ha : 0 < a) (hb : b < 0)
Goal: a • b < 0
-- smul_pos_iff_of_pos_left
@[simp]
lemma smul_pos_iff_of_pos_left [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a) :
    0 < a • b ↔ 0 < b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a)
Goal: 0 < a • b ↔ 0 < b
-- smul_neg_iff_of_pos_left
lemma smul_neg_iff_of_pos_left [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a) :
    a • b < 0 ↔ b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a)
Goal: a • b < 0 ↔ b < 0
-- smul_nonneg
lemma smul_nonneg [PosSMulMono α β] (ha : 0 ≤ a) (hb : 0 ≤ b₁) : 0 ≤ a • b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulMono α β] (ha : 0 ≤ a) (hb : 0 ≤ b₁)
Goal: 0 ≤ a • b₁
-- smul_nonpos_of_nonneg_of_nonpos
lemma smul_nonpos_of_nonneg_of_nonpos [PosSMulMono α β] (ha : 0 ≤ a) (hb : b ≤ 0) : a • b ≤ 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulMono α β] (ha : 0 ≤ a) (hb : b ≤ 0)
Goal: a • b ≤ 0
-- pos_of_smul_pos_left
lemma pos_of_smul_pos_left [PosSMulReflectLT α β] (h : 0 < a • b) (ha : 0 ≤ a) : 0 < b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulReflectLT α β] (h : 0 < a • b) (ha : 0 ≤ a)
Goal: 0 < b
-- neg_of_smul_neg_left
lemma neg_of_smul_neg_left [PosSMulReflectLT α β] (h : a • b < 0) (ha : 0 ≤ a) : b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulZeroClass α β] [Preorder α] [Preorder β] [PosSMulReflectLT α β] (h : a • b < 0) (ha : 0 ≤ a)
Goal: b < 0
-- smul_pos'
lemma smul_pos' [SMulPosStrictMono α β] (ha : 0 < a) (hb : 0 < b) : 0 < a • b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a • b
-- smul_neg_of_neg_of_pos
lemma smul_neg_of_neg_of_pos [SMulPosStrictMono α β] (ha : a < 0) (hb : 0 < b) : a • b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] (ha : a < 0) (hb : 0 < b)
Goal: a • b < 0
-- smul_pos_iff_of_pos_right
@[simp]
lemma smul_pos_iff_of_pos_right [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b) :
    0 < a • b ↔ 0 < a 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b)
Goal: 0 < a • b ↔ 0 < a
-- smul_nonneg'
lemma smul_nonneg' [SMulPosMono α β] (ha : 0 ≤ a) (hb : 0 ≤ b₁) : 0 ≤ a • b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosMono α β] (ha : 0 ≤ a) (hb : 0 ≤ b₁)
Goal: 0 ≤ a • b₁
-- smul_nonpos_of_nonpos_of_nonneg
lemma smul_nonpos_of_nonpos_of_nonneg [SMulPosMono α β] (ha : a ≤ 0) (hb : 0 ≤ b) : a • b ≤ 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosMono α β] (ha : a ≤ 0) (hb : 0 ≤ b)
Goal: a • b ≤ 0
-- pos_of_smul_pos_right
lemma pos_of_smul_pos_right [SMulPosReflectLT α β] (h : 0 < a • b) (hb : 0 ≤ b) : 0 < a 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosReflectLT α β] (h : 0 < a • b) (hb : 0 ≤ b)
Goal: 0 < a
-- neg_of_smul_neg_right
lemma neg_of_smul_neg_right [SMulPosReflectLT α β] (h : a • b < 0) (hb : 0 ≤ b) : a < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [SMulPosReflectLT α β] (h : a • b < 0) (hb : 0 ≤ b)
Goal: a < 0
-- pos_iff_pos_of_smul_pos
lemma pos_iff_pos_of_smul_pos [PosSMulReflectLT α β] [SMulPosReflectLT α β] (hab : 0 < a • b) :
    0 < a ↔ 0 < b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [Preorder β] [PosSMulReflectLT α β] [SMulPosReflectLT α β] (hab : 0 < a • b)
Goal: 0 < a ↔ 0 < b
-- PosSMulMono.of_pos
lemma PosSMulMono.of_pos (h₀ : ∀ a : α, 0 < a → ∀ b₁ b₂ : β, b₁ ≤ b₂ → a • b₁ ≤ a • b₂) :
    PosSMulMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Zero α] [Zero β] [SMulWithZero α β] [PartialOrder α] [Preorder β] (h₀ : ∀ (a : α), 0 < a → ∀ (b₁ b₂ : β), b₁ ≤ b₂ → a • b₁ ≤ a • b₂)
Goal: PosSMulMono α β
-- PosSMulReflectLT.of_pos
lemma PosSMulReflectLT.of_pos (h₀ : ∀ a : α, 0 < a → ∀ b₁ b₂ : β, a • b₁ < a • b₂ → b₁ < b₂) :
    PosSMulReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [Zero α] [Zero β] [SMulWithZero α β] [PartialOrder α] [Preorder β] (h₀ : ∀ (a : α), 0 < a → ∀ (b₁ b₂ : β), a • b₁ < a • b₂ → b₁ < b₂)
Goal: PosSMulReflectLT α β
-- SMulPosMono.of_pos
lemma SMulPosMono.of_pos (h₀ : ∀ b : β, 0 < b → ∀ a₁ a₂ : α, a₁ ≤ a₂ → a₁ • b ≤ a₂ • b) :
    SMulPosMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [PartialOrder β] (h₀ : ∀ (b : β), 0 < b → ∀ (a₁ a₂ : α), a₁ ≤ a₂ → a₁ • b ≤ a₂ • b)
Goal: SMulPosMono α β
-- SMulPosReflectLT.of_pos
lemma SMulPosReflectLT.of_pos (h₀ : ∀ b : β, 0 < b → ∀ a₁ a₂ : α, a₁ • b < a₂ • b → a₁ < a₂) :
    SMulPosReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [Zero α] [Zero β] [SMulWithZero α β] [Preorder α] [PartialOrder β] (h₀ : ∀ (b : β), 0 < b → ∀ (a₁ a₂ : α), a₁ • b < a₂ • b → a₁ < a₂)
Goal: SMulPosReflectLT α β
-- smul_eq_smul_iff_eq_and_eq_of_pos
lemma smul_eq_smul_iff_eq_and_eq_of_pos [PosSMulStrictMono α β] [SMulPosStrictMono α β]
    (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₁ : 0 < a₁) (h₂ : 0 < b₂) :
    a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [Zero α] [Zero β] [SMulWithZero α β] [PartialOrder α] [PartialOrder β] [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₁ : 0 < a₁) (h₂ : 0 < b₂)
Goal: a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂
-- smul_eq_smul_iff_eq_and_eq_of_pos'
lemma smul_eq_smul_iff_eq_and_eq_of_pos' [PosSMulStrictMono α β] [SMulPosStrictMono α β]
    (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₂ : 0 < a₂) (h₁ : 0 < b₁) :
    a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [Zero α] [Zero β] [SMulWithZero α β] [PartialOrder α] [PartialOrder β] [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₂ : 0 < a₂) (h₁ : 0 < b₁)
Goal: a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂
-- pos_and_pos_or_neg_and_neg_of_smul_pos
lemma pos_and_pos_or_neg_and_neg_of_smul_pos [PosSMulMono α β] [SMulPosMono α β] (hab : 0 < a • b) :
    0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [PosSMulMono α β] [SMulPosMono α β] (hab : 0 < a • b)
Goal: 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0
-- neg_of_smul_pos_right
lemma neg_of_smul_pos_right [PosSMulMono α β] [SMulPosMono α β] (h : 0 < a • b) (ha : a ≤ 0) :
    b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [PosSMulMono α β] [SMulPosMono α β] (h : 0 < a • b) (ha : a ≤ 0)
Goal: b < 0
-- neg_of_smul_pos_left
lemma neg_of_smul_pos_left [PosSMulMono α β] [SMulPosMono α β] (h : 0 < a • b) (ha : b ≤ 0) :
    a < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [PosSMulMono α β] [SMulPosMono α β] (h : 0 < a • b) (ha : b ≤ 0)
Goal: a < 0
-- neg_iff_neg_of_smul_pos
lemma neg_iff_neg_of_smul_pos [PosSMulMono α β] [SMulPosMono α β] (hab : 0 < a • b) :
    a < 0 ↔ b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [PosSMulMono α β] [SMulPosMono α β] (hab : 0 < a • b)
Goal: a < 0 ↔ b < 0
-- neg_of_smul_neg_left'
lemma neg_of_smul_neg_left' [SMulPosMono α β] (h : a • b < 0) (ha : 0 ≤ a) : b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [SMulPosMono α β] (h : a • b < 0) (ha : 0 ≤ a)
Goal: b < 0
-- neg_of_smul_neg_right'
lemma neg_of_smul_neg_right' [PosSMulMono α β] (h : a • b < 0) (hb : 0 ≤ b) : a < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Zero α] [Zero β] [SMulWithZero α β] [LinearOrder α] [LinearOrder β] [PosSMulMono α β] (h : a • b < 0) (hb : 0 ≤ b)
Goal: a < 0
-- le_smul_iff_one_le_left
@[simp]
lemma le_smul_iff_one_le_left [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b) :
    b ≤ a • b ↔ 1 ≤ a 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b)
Goal: b ≤ a • b ↔ 1 ≤ a
-- lt_smul_iff_one_lt_left
@[simp]
lemma lt_smul_iff_one_lt_left [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b) :
    b < a • b ↔ 1 < a 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b)
Goal: b < a • b ↔ 1 < a
-- smul_le_iff_le_one_left
@[simp]
lemma smul_le_iff_le_one_left [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b) :
    a • b ≤ b ↔ a ≤ 1 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b)
Goal: a • b ≤ b ↔ a ≤ 1
-- smul_lt_iff_lt_one_left
@[simp]
lemma smul_lt_iff_lt_one_left [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b) :
    a • b < b ↔ a < 1 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b)
Goal: a • b < b ↔ a < 1
-- smul_le_of_le_one_left
lemma smul_le_of_le_one_left [SMulPosMono α β] (hb : 0 ≤ b) (h : a ≤ 1) : a • b ≤ b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosMono α β] (hb : 0 ≤ b) (h : a ≤ 1)
Goal: a • b ≤ b
-- le_smul_of_one_le_left
lemma le_smul_of_one_le_left [SMulPosMono α β] (hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a • b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosMono α β] (hb : 0 ≤ b) (h : 1 ≤ a)
Goal: b ≤ a • b
-- smul_lt_of_lt_one_left
lemma smul_lt_of_lt_one_left [SMulPosStrictMono α β] (hb : 0 < b) (h : a < 1) : a • b < b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] (hb : 0 < b) (h : a < 1)
Goal: a • b < b
-- lt_smul_of_one_lt_left
lemma lt_smul_of_one_lt_left [SMulPosStrictMono α β] (hb : 0 < b) (h : 1 < a) : b < a • b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [Monoid α] [Zero β] [MulAction α β] [Preorder α] [Preorder β] [SMulPosStrictMono α β] (hb : 0 < b) (h : 1 < a)
Goal: b < a • b
-- PosSMulMono.toPosSMulStrictMono
lemma PosSMulMono.toPosSMulStrictMono [PosSMulMono α β] : PosSMulStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Semiring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [Preorder α] [PartialOrder β] [PosSMulMono α β]
Goal: PosSMulStrictMono α β
-- posSMulMono_iff_posSMulStrictMono
lemma posSMulMono_iff_posSMulStrictMono : PosSMulMono α β ↔ PosSMulStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Semiring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β]
Goal: PosSMulMono α β ↔ PosSMulStrictMono α β
-- PosSMulReflectLE_iff_posSMulReflectLT
lemma PosSMulReflectLE_iff_posSMulReflectLT : PosSMulReflectLE α β ↔ PosSMulReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [Semiring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β]
Goal: PosSMulReflectLE α β ↔ PosSMulReflectLT α β
-- SMulPosMono.toSMulPosStrictMono
lemma SMulPosMono.toSMulPosStrictMono [SMulPosMono α β] : SMulPosStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Ring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β] [SMulPosMono α β]
Goal: SMulPosStrictMono α β
-- smulPosMono_iff_smulPosStrictMono
lemma smulPosMono_iff_smulPosStrictMono : SMulPosMono α β ↔ SMulPosStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [Ring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β]
Goal: SMulPosMono α β ↔ SMulPosStrictMono α β
-- SMulPosReflectLT.toSMulPosReflectLE
lemma SMulPosReflectLT.toSMulPosReflectLE [SMulPosReflectLT α β] : SMulPosReflectLE α β 
Arguments: {α : Type u_1} {β : Type u_2} [Ring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β] [SMulPosReflectLT α β]
Goal: SMulPosReflectLE α β
-- SMulPosReflectLE_iff_smulPosReflectLT
lemma SMulPosReflectLE_iff_smulPosReflectLT : SMulPosReflectLE α β ↔ SMulPosReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [Ring α] [AddCommGroup β] [Module α β] [NoZeroSMulDivisors α β] [PartialOrder α] [PartialOrder β]
Goal: SMulPosReflectLE α β ↔ SMulPosReflectLT α β
-- inv_smul_le_iff_of_pos
lemma inv_smul_le_iff_of_pos [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a) :
    a⁻¹ • b₁ ≤ b₂ ↔ b₁ ≤ a • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a)
Goal: a⁻¹ • b₁ ≤ b₂ ↔ b₁ ≤ a • b₂
-- le_inv_smul_iff_of_pos
lemma le_inv_smul_iff_of_pos [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a) :
    b₁ ≤ a⁻¹ • b₂ ↔ a • b₁ ≤ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a)
Goal: b₁ ≤ a⁻¹ • b₂ ↔ a • b₁ ≤ b₂
-- inv_smul_lt_iff_of_pos
lemma inv_smul_lt_iff_of_pos [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a) :
    a⁻¹ • b₁ < b₂ ↔ b₁ < a • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a)
Goal: a⁻¹ • b₁ < b₂ ↔ b₁ < a • b₂
-- lt_inv_smul_iff_of_pos
lemma lt_inv_smul_iff_of_pos [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a) :
    b₁ < a⁻¹ • b₂ ↔ a • b₁ < b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a)
Goal: b₁ < a⁻¹ • b₂ ↔ a • b₁ < b₂
-- OrderIso.smulRight_symm_apply
@[simps!]
def OrderIso.smulRight [PosSMulMono α β] [PosSMulReflectLE α β] {a : α} (ha : 0 < a) : β ≃o β 
Arguments: {α : Type u_1} {β : Type u_2} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulMono α β] [PosSMulReflectLE α β] {a : α} (ha : 0 < a) (b : β)
Goal: (RelIso.symm (OrderIso.smulRight ha)) b = a⁻¹ • b
-- OrderIso.smulRight_apply
@[simps!]
def OrderIso.smulRight [PosSMulMono α β] [PosSMulReflectLE α β] {a : α} (ha : 0 < a) : β ≃o β 
Arguments: {α : Type u_1} {β : Type u_2} [GroupWithZero α] [Preorder α] [Preorder β] [MulAction α β] [PosSMulMono α β] [PosSMulReflectLE α β] {a : α} (ha : 0 < a) (b : β)
Goal: (OrderIso.smulRight ha) b = a • b
-- smul_le_smul_of_nonpos_left
lemma smul_le_smul_of_nonpos_left (h : b₁ ≤ b₂) (ha : a ≤ 0) : a • b₂ ≤ a • b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulMono α β] (h : b₁ ≤ b₂) (ha : a ≤ 0)
Goal: a • b₂ ≤ a • b₁
-- antitone_smul_left
lemma antitone_smul_left (ha : a ≤ 0) : Antitone ((a • ·) : β → β) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulMono α β] (ha : a ≤ 0)
Goal: Antitone fun x => a • x
-- smul_lt_smul_of_neg_left
lemma smul_lt_smul_of_neg_left (hb : b₁ < b₂) (ha : a < 0) : a • b₂ < a • b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] (hb : b₁ < b₂) (ha : a < 0)
Goal: a • b₂ < a • b₁
-- strictAnti_smul_left
lemma strictAnti_smul_left (ha : a < 0) : StrictAnti ((a • ·) : β → β) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] (ha : a < 0)
Goal: StrictAnti fun x => a • x
-- le_of_smul_le_smul_of_neg
lemma le_of_smul_le_smul_of_neg [PosSMulReflectLE α β] (h : a • b₁ ≤ a • b₂) (ha : a < 0) :
    b₂ ≤ b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulReflectLE α β] (h : a • b₁ ≤ a • b₂) (ha : a < 0)
Goal: b₂ ≤ b₁
-- lt_of_smul_lt_smul_of_nonpos
lemma lt_of_smul_lt_smul_of_nonpos [PosSMulReflectLT α β] (h : a • b₁ < a • b₂) (ha : a ≤ 0) :
    b₂ < b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulReflectLT α β] (h : a • b₁ < a • b₂) (ha : a ≤ 0)
Goal: b₂ < b₁
-- smul_nonneg_of_nonpos_of_nonpos
lemma smul_nonneg_of_nonpos_of_nonpos [SMulPosMono α β] (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a • b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [SMulPosMono α β] (ha : a ≤ 0) (hb : b ≤ 0)
Goal: 0 ≤ a • b
-- smul_le_smul_iff_of_neg_left
lemma smul_le_smul_iff_of_neg_left [PosSMulMono α β] [PosSMulReflectLE α β] (ha : a < 0) :
    a • b₁ ≤ a • b₂ ↔ b₂ ≤ b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulMono α β] [PosSMulReflectLE α β] (ha : a < 0)
Goal: a • b₁ ≤ a • b₂ ↔ b₂ ≤ b₁
-- smul_lt_smul_iff_of_neg_left
lemma smul_lt_smul_iff_of_neg_left (ha : a < 0) : a • b₁ < a • b₂ ↔ b₂ < b₁ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : a < 0)
Goal: a • b₁ < a • b₂ ↔ b₂ < b₁
-- smul_pos_iff_of_neg_left
lemma smul_pos_iff_of_neg_left (ha : a < 0) : 0 < a • b ↔ b < 0 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : a < 0)
Goal: 0 < a • b ↔ b < 0
-- smul_neg_iff_of_neg_left
lemma smul_neg_iff_of_neg_left (ha : a < 0) : a • b < 0 ↔ 0 < b 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b : β} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : a < 0)
Goal: a • b < 0 ↔ 0 < b
-- smul_add_smul_le_smul_add_smul
lemma smul_add_smul_le_smul_add_smul [PosSMulMono α β]
    {b₁ b₂ : α} {a d : β} (hab : b₁ ≤ b₂) (hcd : a ≤ d) : b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulMono α β] {b₁ b₂ : α} {a d : β} (hab : b₁ ≤ b₂) (hcd : a ≤ d)
Goal: b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d
-- smul_add_smul_le_smul_add_smul'
lemma smul_add_smul_le_smul_add_smul' [PosSMulMono α β]
    {b₁ b₂ : α} {a d : β} (hba : b₂ ≤ b₁) (hdc : d ≤ a) : b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulMono α β] {b₁ b₂ : α} {a d : β} (hba : b₂ ≤ b₁) (hdc : d ≤ a)
Goal: b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d
-- smul_add_smul_lt_smul_add_smul
lemma smul_add_smul_lt_smul_add_smul [PosSMulStrictMono α β]
    {b₁ b₂ : α} {a d : β} (hab : b₁ < b₂) (hcd : a < d) :
    b₁ • d + b₂ • a < b₁ • a + b₂ • d 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {b₁ b₂ : α} {a d : β} (hab : b₁ < b₂) (hcd : a < d)
Goal: b₁ • d + b₂ • a < b₁ • a + b₂ • d
-- smul_add_smul_lt_smul_add_smul'
lemma smul_add_smul_lt_smul_add_smul' [PosSMulStrictMono α β]
    {b₁ b₂ : α} {a d : β} (hba : b₂ < b₁) (hdc : d < a) :
    b₁ • d + b₂ • a < b₁ • a + b₂ • d 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [OrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {b₁ b₂ : α} {a d : β} (hba : b₂ < b₁) (hdc : d < a)
Goal: b₁ • d + b₂ • a < b₁ • a + b₂ • d
-- smul_max_of_nonpos
lemma smul_max_of_nonpos (ha : a ≤ 0) (b₁ b₂ : β) : a • max b₁ b₂ = min (a • b₁) (a • b₂) 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulMono α β] {a : α} (ha : a ≤ 0) (b₁ b₂ : β)
Goal: a • max b₁ b₂ = min (a • b₁) (a • b₂)
-- smul_min_of_nonpos
lemma smul_min_of_nonpos (ha : a ≤ 0) (b₁ b₂ : β) : a • min b₁ b₂ = max (a • b₁) (a • b₂) 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulMono α β] {a : α} (ha : a ≤ 0) (b₁ b₂ : β)
Goal: a • min b₁ b₂ = max (a • b₁) (a • b₂)
-- nonneg_and_nonneg_or_nonpos_and_nonpos_of_smul_nonneg
lemma nonneg_and_nonneg_or_nonpos_and_nonpos_of_smul_nonneg (hab : 0 ≤ a • b) :
    0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β} (hab : 0 ≤ a • b)
Goal: 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- smul_nonneg_iff
lemma smul_nonneg_iff : 0 ≤ a • b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β}
Goal: 0 ≤ a • b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- smul_nonpos_iff
lemma smul_nonpos_iff : a • b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β}
Goal: a • b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b
-- smul_nonneg_iff_pos_imp_nonneg
lemma smul_nonneg_iff_pos_imp_nonneg : 0 ≤ a • b ↔ (0 < a → 0 ≤ b) ∧ (0 < b → 0 ≤ a) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β}
Goal: 0 ≤ a • b ↔ (0 < a → 0 ≤ b) ∧ (0 < b → 0 ≤ a)
-- smul_nonneg_iff_neg_imp_nonpos
lemma smul_nonneg_iff_neg_imp_nonpos : 0 ≤ a • b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β}
Goal: 0 ≤ a • b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0)
-- smul_nonpos_iff_pos_imp_nonpos
lemma smul_nonpos_iff_pos_imp_nonpos : a • b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [PosSMulStrictMono α β] {a : α} {b : β}
Goal: a • b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)