ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- toDual_div
@[to_additive (attr := simp)]
theorem toDual_div [Div α] (a b : α) : toDual (a / b) = toDual a / toDual b 
Arguments: {α : Type u_1} [Div α] (a b : α)
Goal: OrderDual.toDual (a / b) = OrderDual.toDual a / OrderDual.toDual b
-- ofDual_div
@[to_additive (attr := simp)]
theorem ofDual_div [Div α] (a b : αᵒᵈ) : ofDual (a / b) = ofDual a / ofDual b 
Arguments: {α : Type u_1} [Div α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a / b) = OrderDual.ofDual a / OrderDual.ofDual b
-- ofDual_sub
@[to_additive (attr := simp)]
theorem ofDual_div [Div α] (a b : αᵒᵈ) : ofDual (a / b) = ofDual a / ofDual b 
Arguments: {α : Type u_1} [Sub α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a - b) = OrderDual.ofDual a - OrderDual.ofDual b
-- toDual_vadd
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul]
theorem toDual_pow [Pow α β] (a : α) (b : β) : toDual (a ^ b) = toDual a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α)
Goal: OrderDual.toDual (b +ᵥ a) = b +ᵥ OrderDual.toDual a
-- toDual_pow
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul]
theorem toDual_pow [Pow α β] (a : α) (b : β) : toDual (a ^ b) = toDual a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β)
Goal: OrderDual.toDual (a ^ b) = OrderDual.toDual a ^ b
-- toDual_smul
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul]
theorem toDual_pow [Pow α β] (a : α) (b : β) : toDual (a ^ b) = toDual a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α)
Goal: OrderDual.toDual (b • a) = b • OrderDual.toDual a
-- ofDual_vadd
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul]
theorem ofDual_pow [Pow α β] (a : αᵒᵈ) (b : β) : ofDual (a ^ b) = ofDual a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : αᵒᵈ)
Goal: OrderDual.ofDual (b +ᵥ a) = b +ᵥ OrderDual.ofDual a
-- ofDual_pow
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul]
theorem ofDual_pow [Pow α β] (a : αᵒᵈ) (b : β) : ofDual (a ^ b) = ofDual a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : αᵒᵈ) (b : β)
Goal: OrderDual.ofDual (a ^ b) = OrderDual.ofDual a ^ b
-- ofDual_smul
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul]
theorem ofDual_pow [Pow α β] (a : αᵒᵈ) (b : β) : ofDual (a ^ b) = ofDual a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : αᵒᵈ)
Goal: OrderDual.ofDual (b • a) = b • OrderDual.ofDual a
-- toDual_smul'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul']
theorem pow_toDual [Pow α β] (a : α) (b : β) : a ^ toDual b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α)
Goal: OrderDual.toDual b • a = b • a
-- toDual_vadd'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul']
theorem pow_toDual [Pow α β] (a : α) (b : β) : a ^ toDual b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α)
Goal: OrderDual.toDual b +ᵥ a = b +ᵥ a
-- pow_toDual
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toDual_smul']
theorem pow_toDual [Pow α β] (a : α) (b : β) : a ^ toDual b = a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β)
Goal: a ^ OrderDual.toDual b = a ^ b
-- ofDual_smul'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul']
theorem pow_ofDual [Pow α β] (a : α) (b : βᵒᵈ) : a ^ ofDual b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : βᵒᵈ) (a : α)
Goal: OrderDual.ofDual b • a = b • a
-- pow_ofDual
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul']
theorem pow_ofDual [Pow α β] (a : α) (b : βᵒᵈ) : a ^ ofDual b = a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : βᵒᵈ)
Goal: a ^ OrderDual.ofDual b = a ^ b
-- ofDual_vadd'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofDual_smul']
theorem pow_ofDual [Pow α β] (a : α) (b : βᵒᵈ) : a ^ ofDual b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : βᵒᵈ) (a : α)
Goal: OrderDual.ofDual b +ᵥ a = b +ᵥ a
-- toLex_one
@[to_additive (attr := simp)]
theorem toLex_one [One α] : toLex (1 : α) = 1 
Arguments: {α : Type u_1} [One α]
Goal: toLex 1 = 1
-- toLex_zero
@[to_additive (attr := simp)]
theorem toLex_one [One α] : toLex (1 : α) = 1 
Arguments: {α : Type u_1} [Zero α]
Goal: toLex 0 = 0
-- ofLex_one
@[to_additive (attr := simp)]
theorem ofLex_one [One α] : (ofLex 1 : α) = 1 
Arguments: {α : Type u_1} [One α]
Goal: ofLex 1 = 1
-- ofLex_zero
@[to_additive (attr := simp)]
theorem ofLex_one [One α] : (ofLex 1 : α) = 1 
Arguments: {α : Type u_1} [Zero α]
Goal: ofLex 0 = 0
-- toLex_add
@[to_additive (attr := simp)]
theorem toLex_mul [Mul α] (a b : α) : toLex (a * b) = toLex a * toLex b 
Arguments: {α : Type u_1} [Add α] (a b : α)
Goal: toLex (a + b) = toLex a + toLex b
-- toLex_mul
@[to_additive (attr := simp)]
theorem toLex_mul [Mul α] (a b : α) : toLex (a * b) = toLex a * toLex b 
Arguments: {α : Type u_1} [Mul α] (a b : α)
Goal: toLex (a * b) = toLex a * toLex b
-- ofLex_mul
@[to_additive (attr := simp)]
theorem ofLex_mul [Mul α] (a b : Lex α) : ofLex (a * b) = ofLex a * ofLex b 
Arguments: {α : Type u_1} [Mul α] (a b : Lex α)
Goal: ofLex (a * b) = ofLex a * ofLex b
-- ofLex_add
@[to_additive (attr := simp)]
theorem ofLex_mul [Mul α] (a b : Lex α) : ofLex (a * b) = ofLex a * ofLex b 
Arguments: {α : Type u_1} [Add α] (a b : Lex α)
Goal: ofLex (a + b) = ofLex a + ofLex b
-- toLex_inv
@[to_additive (attr := simp)]
theorem toLex_inv [Inv α] (a : α) : toLex a⁻¹ = (toLex a)⁻¹ 
Arguments: {α : Type u_1} [Inv α] (a : α)
Goal: toLex a⁻¹ = (toLex a)⁻¹
-- toLex_neg
@[to_additive (attr := simp)]
theorem toLex_inv [Inv α] (a : α) : toLex a⁻¹ = (toLex a)⁻¹ 
Arguments: {α : Type u_1} [Neg α] (a : α)
Goal: toLex (-a) = -toLex a
-- ofLex_neg
@[to_additive (attr := simp)]
theorem ofLex_inv [Inv α] (a : Lex α) : ofLex a⁻¹ = (ofLex a)⁻¹ 
Arguments: {α : Type u_1} [Neg α] (a : Lex α)
Goal: ofLex (-a) = -ofLex a
-- ofLex_inv
@[to_additive (attr := simp)]
theorem ofLex_inv [Inv α] (a : Lex α) : ofLex a⁻¹ = (ofLex a)⁻¹ 
Arguments: {α : Type u_1} [Inv α] (a : Lex α)
Goal: ofLex a⁻¹ = (ofLex a)⁻¹
-- toLex_div
@[to_additive (attr := simp)]
theorem toLex_div [Div α] (a b : α) : toLex (a / b) = toLex a / toLex b 
Arguments: {α : Type u_1} [Div α] (a b : α)
Goal: toLex (a / b) = toLex a / toLex b
-- toLex_sub
@[to_additive (attr := simp)]
theorem toLex_div [Div α] (a b : α) : toLex (a / b) = toLex a / toLex b 
Arguments: {α : Type u_1} [Sub α] (a b : α)
Goal: toLex (a - b) = toLex a - toLex b
-- ofLex_sub
@[to_additive (attr := simp)]
theorem ofLex_div [Div α] (a b : Lex α) : ofLex (a / b) = ofLex a / ofLex b 
Arguments: {α : Type u_1} [Sub α] (a b : Lex α)
Goal: ofLex (a - b) = ofLex a - ofLex b
-- ofLex_div
@[to_additive (attr := simp)]
theorem ofLex_div [Div α] (a b : Lex α) : ofLex (a / b) = ofLex a / ofLex b 
Arguments: {α : Type u_1} [Div α] (a b : Lex α)
Goal: ofLex (a / b) = ofLex a / ofLex b
-- toLex_pow
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul]
theorem toLex_pow [Pow α β] (a : α) (b : β) : toLex (a ^ b) = toLex a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β)
Goal: toLex (a ^ b) = toLex a ^ b
-- toLex_smul
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul]
theorem toLex_pow [Pow α β] (a : α) (b : β) : toLex (a ^ b) = toLex a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α)
Goal: toLex (b • a) = b • toLex a
-- toLex_vadd
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul]
theorem toLex_pow [Pow α β] (a : α) (b : β) : toLex (a ^ b) = toLex a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α)
Goal: toLex (b +ᵥ a) = b +ᵥ toLex a
-- ofLex_pow
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul]
theorem ofLex_pow [Pow α β] (a : Lex α) (b : β) : ofLex (a ^ b) = ofLex a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : Lex α) (b : β)
Goal: ofLex (a ^ b) = ofLex a ^ b
-- ofLex_smul
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul]
theorem ofLex_pow [Pow α β] (a : Lex α) (b : β) : ofLex (a ^ b) = ofLex a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : Lex α)
Goal: ofLex (b • a) = b • ofLex a
-- ofLex_vadd
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul]
theorem ofLex_pow [Pow α β] (a : Lex α) (b : β) : ofLex (a ^ b) = ofLex a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : Lex α)
Goal: ofLex (b +ᵥ a) = b +ᵥ ofLex a
-- toLex_smul'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul']
theorem pow_toLex [Pow α β] (a : α) (b : β) : a ^ toLex b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : β) (a : α)
Goal: toLex b • a = b • a
-- pow_toLex
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul']
theorem pow_toLex [Pow α β] (a : α) (b : β) : a ^ toLex b = a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : β)
Goal: a ^ toLex b = a ^ b
-- toLex_vadd'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) toLex_smul']
theorem pow_toLex [Pow α β] (a : α) (b : β) : a ^ toLex b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : β) (a : α)
Goal: toLex b +ᵥ a = b +ᵥ a
-- ofLex_vadd'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul']
theorem pow_ofLex [Pow α β] (a : α) (b : Lex β) : a ^ ofLex b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [VAdd β α] (b : Lex β) (a : α)
Goal: ofLex b +ᵥ a = b +ᵥ a
-- ofLex_smul'
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul']
theorem pow_ofLex [Pow α β] (a : α) (b : Lex β) : a ^ ofLex b = a ^ b 
Arguments: {β : Type u_2} {α : Type u_1} [SMul β α] (b : Lex β) (a : α)
Goal: ofLex b • a = b • a
-- pow_ofLex
@[to_additive (attr := simp, to_additive) (reorder := 1 2, 4 5) ofLex_smul']
theorem pow_ofLex [Pow α β] (a : α) (b : Lex β) : a ^ ofLex b = a ^ b 
Arguments: {α : Type u_1} {β : Type u_2} [Pow α β] (a : α) (b : Lex β)
Goal: a ^ ofLex b = a ^ b
-- Int.isCompl_even_odd
lemma isCompl_even_odd : IsCompl { n : ℤ | Even n } { n | Odd n } 
Goal: IsCompl {n | Even n} {n | Odd n}
-- Nat.cast_natAbs
lemma _root_.Nat.cast_natAbs {α : Type*} [AddGroupWithOne α] (n : ℤ) : (n.natAbs : α) = |n| 
Arguments: {α : Type u_1} [AddGroupWithOne α] (n : ℤ)
Goal: ↑n.natAbs = ↑|n|
-- Int.cast_mul_eq_zsmul_cast
lemma cast_mul_eq_zsmul_cast {α : Type*} [AddCommGroupWithOne α] :
    ∀ m n : ℤ, ↑(m * n) = m • (n : α) 
Arguments: {α : Type u_1} [AddCommGroupWithOne α] (m n : ℤ)
Goal: ↑(m * n) = m • ↑n
-- Int.two_le_iff_pos_of_even
lemma two_le_iff_pos_of_even {m : ℤ} (even : Even m) : 2 ≤ m ↔ 0 < m 
Arguments: {m : ℤ} (even : Even m)
Goal: 2 ≤ m ↔ 0 < m
-- Int.add_two_le_iff_lt_of_even_sub
lemma add_two_le_iff_lt_of_even_sub {m n : ℤ} (even : Even (n - m)) : m + 2 ≤ n ↔ m < n 
Arguments: {m n : ℤ} (even : Even (n - m))
Goal: m + 2 ≤ n ↔ m < n
-- PosSMulMono.elim
class PosSMulMono : Prop where
    protected elim ⦃a : α⦄ (ha : 0 ≤ a) ⦃b₁ b₂ : β⦄ (hb : b₁ ≤ b₂) : a • b₁ ≤ a • b₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero α] [self : PosSMulMono α β] ⦃a : α⦄ (ha : 0 ≤ a) ⦃b₁ b₂ : β⦄ (hb : b₁ ≤ b₂)
Goal: a • b₁ ≤ a • b₂
-- PosSMulStrictMono.elim
class PosSMulStrictMono : Prop where
    protected elim ⦃a : α⦄ (ha : 0 < a) ⦃b₁ b₂ : β⦄ (hb : b₁ < b₂) : a • b₁ < a • b₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero α] [self : PosSMulStrictMono α β] ⦃a : α⦄ (ha : 0 < a) ⦃b₁ b₂ : β⦄ (hb : b₁ < b₂)
Goal: a • b₁ < a • b₂
-- PosSMulReflectLT.elim
class PosSMulReflectLT : Prop where
    protected elim ⦃a : α⦄ (ha : 0 ≤ a) ⦃b₁ b₂ : β⦄ (hb : a • b₁ < a • b₂) : b₁ < b₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero α] [self : PosSMulReflectLT α β] ⦃a : α⦄ (ha : 0 ≤ a) ⦃b₁ b₂ : β⦄ (hb : a • b₁ < a • b₂)
Goal: b₁ < b₂
-- PosSMulReflectLE.elim
class PosSMulReflectLE : Prop where
    protected elim ⦃a : α⦄ (ha : 0 < a) ⦃b₁ b₂ : β⦄ (hb : a • b₁ ≤ a • b₂) : b₁ ≤ b₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero α] [self : PosSMulReflectLE α β] ⦃a : α⦄ (ha : 0 < a) ⦃b₁ b₂ : β⦄ (hb : a • b₁ ≤ a • b₂)
Goal: b₁ ≤ b₂
-- SMulPosMono.elim
class SMulPosMono : Prop where
    protected elim ⦃b : β⦄ (hb : 0 ≤ b) ⦃a₁ a₂ : α⦄ (ha : a₁ ≤ a₂) : a₁ • b ≤ a₂ • b
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero β] [self : SMulPosMono α β] ⦃b : β⦄ (hb : 0 ≤ b) ⦃a₁ a₂ : α⦄ (ha : a₁ ≤ a₂)
Goal: a₁ • b ≤ a₂ • b
-- SMulPosStrictMono.elim
class SMulPosStrictMono : Prop where
    protected elim ⦃b : β⦄ (hb : 0 < b) ⦃a₁ a₂ : α⦄ (ha : a₁ < a₂) : a₁ • b < a₂ • b
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero β] [self : SMulPosStrictMono α β] ⦃b : β⦄ (hb : 0 < b) ⦃a₁ a₂ : α⦄ (ha : a₁ < a₂)
Goal: a₁ • b < a₂ • b
-- SMulPosReflectLT.elim
class SMulPosReflectLT : Prop where
    protected elim ⦃b : β⦄ (hb : 0 ≤ b) ⦃a₁ a₂ : α⦄ (hb : a₁ • b < a₂ • b) : a₁ < a₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero β] [self : SMulPosReflectLT α β] ⦃b : β⦄ (hb : 0 ≤ b) ⦃a₁ a₂ : α⦄
Goal: a₁ • b < a₂ • b → a₁ < a₂
-- SMulPosReflectLE.elim
class SMulPosReflectLE : Prop where
    protected elim ⦃b : β⦄ (hb : 0 < b) ⦃a₁ a₂ : α⦄ (hb : a₁ • b ≤ a₂ • b) : a₁ ≤ a₂
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [Preorder β] [Zero β] [self : SMulPosReflectLE α β] ⦃b : β⦄ (hb : 0 < b) ⦃a₁ a₂ : α⦄
Goal: a₁ • b ≤ a₂ • b → a₁ ≤ a₂
-- monotone_smul_left_of_nonneg
lemma monotone_smul_left_of_nonneg [PosSMulMono α β] (ha : 0 ≤ a) : Monotone ((a • ·) : β → β) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulMono α β] (ha : 0 ≤ a)
Goal: Monotone fun x => a • x
-- strictMono_smul_left_of_pos
lemma strictMono_smul_left_of_pos [PosSMulStrictMono α β] (ha : 0 < a) :
    StrictMono ((a • ·) : β → β) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulStrictMono α β] (ha : 0 < a)
Goal: StrictMono fun x => a • x
-- smul_le_smul_of_nonneg_left
@[gcongr] lemma smul_le_smul_of_nonneg_left [PosSMulMono α β] (hb : b₁ ≤ b₂) (ha : 0 ≤ a) :
    a • b₁ ≤ a • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulMono α β] (hb : b₁ ≤ b₂) (ha : 0 ≤ a)
Goal: a • b₁ ≤ a • b₂
-- smul_lt_smul_of_pos_left
@[gcongr] lemma smul_lt_smul_of_pos_left [PosSMulStrictMono α β] (hb : b₁ < b₂) (ha : 0 < a) :
    a • b₁ < a • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulStrictMono α β] (hb : b₁ < b₂) (ha : 0 < a)
Goal: a • b₁ < a • b₂
-- lt_of_smul_lt_smul_left
lemma lt_of_smul_lt_smul_left [PosSMulReflectLT α β] (h : a • b₁ < a • b₂) (ha : 0 ≤ a) : b₁ < b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulReflectLT α β] (h : a • b₁ < a • b₂) (ha : 0 ≤ a)
Goal: b₁ < b₂
-- le_of_smul_le_smul_left
lemma le_of_smul_le_smul_left [PosSMulReflectLE α β] (h : a • b₁ ≤ a • b₂) (ha : 0 < a) : b₁ ≤ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulReflectLE α β] (h : a • b₁ ≤ a • b₂) (ha : 0 < a)
Goal: b₁ ≤ b₂
-- smul_le_smul_iff_of_pos_left
@[simp]
lemma smul_le_smul_iff_of_pos_left [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a) :
    a • b₁ ≤ a • b₂ ↔ b₁ ≤ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a)
Goal: a • b₁ ≤ a • b₂ ↔ b₁ ≤ b₂
-- smul_lt_smul_iff_of_pos_left
@[simp]
lemma smul_lt_smul_iff_of_pos_left [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a) :
    a • b₁ < a • b₂ ↔ b₁ < b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [PosSMulStrictMono α β] [PosSMulReflectLT α β] (ha : 0 < a)
Goal: a • b₁ < a • b₂ ↔ b₁ < b₂
-- monotone_smul_right_of_nonneg
lemma monotone_smul_right_of_nonneg [SMulPosMono α β] (hb : 0 ≤ b) : Monotone ((· • b) : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosMono α β] (hb : 0 ≤ b)
Goal: Monotone fun x => x • b
-- strictMono_smul_right_of_pos
lemma strictMono_smul_right_of_pos [SMulPosStrictMono α β] (hb : 0 < b) :
    StrictMono ((· • b) : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosStrictMono α β] (hb : 0 < b)
Goal: StrictMono fun x => x • b
-- smul_le_smul_of_nonneg_right
@[gcongr] lemma smul_le_smul_of_nonneg_right [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : 0 ≤ b) :
    a₁ • b ≤ a₂ • b 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : 0 ≤ b)
Goal: a₁ • b ≤ a₂ • b
-- smul_lt_smul_of_pos_right
@[gcongr] lemma smul_lt_smul_of_pos_right [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : 0 < b) :
    a₁ • b < a₂ • b 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : 0 < b)
Goal: a₁ • b < a₂ • b
-- lt_of_smul_lt_smul_right
lemma lt_of_smul_lt_smul_right [SMulPosReflectLT α β] (h : a₁ • b < a₂ • b) (hb : 0 ≤ b) :
    a₁ < a₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosReflectLT α β] (h : a₁ • b < a₂ • b) (hb : 0 ≤ b)
Goal: a₁ < a₂
-- le_of_smul_le_smul_right
lemma le_of_smul_le_smul_right [SMulPosReflectLE α β] (h : a₁ • b ≤ a₂ • b) (hb : 0 < b) :
    a₁ ≤ a₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosReflectLE α β] (h : a₁ • b ≤ a₂ • b) (hb : 0 < b)
Goal: a₁ ≤ a₂
-- smul_le_smul_iff_of_pos_right
@[simp]
lemma smul_le_smul_iff_of_pos_right [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b) :
    a₁ • b ≤ a₂ • b ↔ a₁ ≤ a₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosMono α β] [SMulPosReflectLE α β] (hb : 0 < b)
Goal: a₁ • b ≤ a₂ • b ↔ a₁ ≤ a₂
-- smul_lt_smul_iff_of_pos_right
@[simp]
lemma smul_lt_smul_iff_of_pos_right [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b) :
    a₁ • b < a₂ • b ↔ a₁ < a₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} [SMul α β] [Preorder α] [Preorder β] [Zero β] [SMulPosStrictMono α β] [SMulPosReflectLT α β] (hb : 0 < b)
Goal: a₁ • b < a₂ • b ↔ a₁ < a₂
-- smul_lt_smul_of_le_of_lt
lemma smul_lt_smul_of_le_of_lt [PosSMulStrictMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂)
    (hb : b₁ < b₂) (h₁ : 0 < a₁) (h₂ : 0 ≤ b₂) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulStrictMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ < b₂) (h₁ : 0 < a₁) (h₂ : 0 ≤ b₂)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_lt_smul_of_le_of_lt'
lemma smul_lt_smul_of_le_of_lt' [PosSMulStrictMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂)
    (hb : b₁ < b₂) (h₂ : 0 < a₂) (h₁ : 0 ≤ b₁) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulStrictMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ < b₂) (h₂ : 0 < a₂) (h₁ : 0 ≤ b₁)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_lt_smul_of_lt_of_le
lemma smul_lt_smul_of_lt_of_le [PosSMulMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂)
    (hb : b₁ ≤ b₂) (h₁ : 0 ≤ a₁) (h₂ : 0 < b₂) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ ≤ b₂) (h₁ : 0 ≤ a₁) (h₂ : 0 < b₂)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_lt_smul_of_lt_of_le'
lemma smul_lt_smul_of_lt_of_le' [PosSMulMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂)
    (hb : b₁ ≤ b₂) (h₂ : 0 ≤ a₂) (h₁ : 0 < b₁) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ ≤ b₂) (h₂ : 0 ≤ a₂) (h₁ : 0 < b₁)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_lt_smul
lemma smul_lt_smul [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ < b₂)
    (h₁ : 0 < a₁) (h₂ : 0 < b₂) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ < b₂) (h₁ : 0 < a₁) (h₂ : 0 < b₂)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_lt_smul'
lemma smul_lt_smul' [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ < b₂)
    (h₂ : 0 < a₂) (h₁ : 0 < b₁) : a₁ • b₁ < a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulStrictMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ < b₂) (h₂ : 0 < a₂) (h₁ : 0 < b₁)
Goal: a₁ • b₁ < a₂ • b₂
-- smul_le_smul
lemma smul_le_smul [PosSMulMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂)
    (h₁ : 0 ≤ a₁) (h₂ : 0 ≤ b₂) : a₁ • b₁ ≤ a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₁ : 0 ≤ a₁) (h₂ : 0 ≤ b₂)
Goal: a₁ • b₁ ≤ a₂ • b₂
-- smul_le_smul'
lemma smul_le_smul' [PosSMulMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₂ : 0 ≤ a₂)
    (h₁ : 0 ≤ b₁) : a₁ • b₁ ≤ a₂ • b₂ 
Arguments: {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} [SMul α β] [Preorder α] [Preorder β] [Zero α] [Zero β] [PosSMulMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) (h₂ : 0 ≤ a₂) (h₁ : 0 ≤ b₁)
Goal: a₁ • b₁ ≤ a₂ • b₂
-- PosSMulReflectLE.toPosSMulStrictMono
lemma PosSMulReflectLE.toPosSMulStrictMono [PosSMulReflectLE α β] : PosSMulStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero α] [PosSMulReflectLE α β]
Goal: PosSMulStrictMono α β
-- posSMulStrictMono_iff_PosSMulReflectLE
lemma posSMulStrictMono_iff_PosSMulReflectLE : PosSMulStrictMono α β ↔ PosSMulReflectLE α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero α]
Goal: PosSMulStrictMono α β ↔ PosSMulReflectLE α β
-- PosSMulReflectLT.toPosSMulMono
lemma PosSMulReflectLT.toPosSMulMono [PosSMulReflectLT α β] : PosSMulMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero α] [PosSMulReflectLT α β]
Goal: PosSMulMono α β
-- posSMulMono_iff_posSMulReflectLT
lemma posSMulMono_iff_posSMulReflectLT : PosSMulMono α β ↔ PosSMulReflectLT α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero α]
Goal: PosSMulMono α β ↔ PosSMulReflectLT α β
-- smul_max_of_nonneg
lemma smul_max_of_nonneg [PosSMulMono α β] (ha : 0 ≤ a) (b₁ b₂ : β) :
    a • max b₁ b₂ = max (a • b₁) (a • b₂) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [SMul α β] [Preorder α] [LinearOrder β] [Zero α] [PosSMulMono α β] (ha : 0 ≤ a) (b₁ b₂ : β)
Goal: a • max b₁ b₂ = max (a • b₁) (a • b₂)
-- smul_min_of_nonneg
lemma smul_min_of_nonneg [PosSMulMono α β] (ha : 0 ≤ a) (b₁ b₂ : β) :
    a • min b₁ b₂ = min (a • b₁) (a • b₂) 
Arguments: {α : Type u_1} {β : Type u_2} {a : α} [SMul α β] [Preorder α] [LinearOrder β] [Zero α] [PosSMulMono α β] (ha : 0 ≤ a) (b₁ b₂ : β)
Goal: a • min b₁ b₂ = min (a • b₁) (a • b₂)
-- SMulPosReflectLE.toSMulPosStrictMono
lemma SMulPosReflectLE.toSMulPosStrictMono [SMulPosReflectLE α β] : SMulPosStrictMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero β] [SMulPosReflectLE α β]
Goal: SMulPosStrictMono α β
-- SMulPosReflectLT.toSMulPosMono
lemma SMulPosReflectLT.toSMulPosMono [SMulPosReflectLT α β] : SMulPosMono α β 
Arguments: {α : Type u_1} {β : Type u_2} [SMul α β] [Preorder α] [LinearOrder β] [Zero β] [SMulPosReflectLT α β]
Goal: SMulPosMono α β