ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_max_of_two_nsmul_le_add
@[to_additive le_max_of_two_nsmul_le_add]
theorem le_max_of_sq_le_mul {a b c : M} (h : a ^ 2 ≤ b * c) : a ≤ max b c 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : M} (h : 2 • a ≤ b + c)
Goal: a ≤ max b c
-- le_max_of_sq_le_mul
@[to_additive le_max_of_two_nsmul_le_add]
theorem le_max_of_sq_le_mul {a b c : M} (h : a ^ 2 ≤ b * c) : a ≤ max b c 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : M} (h : a ^ 2 ≤ b * c)
Goal: a ≤ max b c
-- Left.pow_lt_one_iff'
@[to_additive Left.nsmul_neg_iff]
theorem Left.pow_lt_one_iff' [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n) :
    x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n)
Goal: x ^ n < 1 ↔ x < 1
-- Left.nsmul_neg_iff
@[to_additive Left.nsmul_neg_iff]
theorem Left.pow_lt_one_iff' [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n) :
    x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n)
Goal: n • x < 0 ↔ x < 0
-- Left.pow_lt_one_iff
theorem Left.pow_lt_one_iff [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n) :
    x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n)
Goal: x ^ n < 1 ↔ x < 1
-- Right.pow_lt_one_iff
@[to_additive]
theorem Right.pow_lt_one_iff [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
    (hn : 0 < n) : x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n)
Goal: x ^ n < 1 ↔ x < 1
-- Right.nsmul_neg_iff
@[to_additive]
theorem Right.pow_lt_one_iff [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
    (hn : 0 < n) : x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n)
Goal: n • x < 0 ↔ x < 0
-- zsmul_nonneg
@[to_additive zsmul_nonneg]
theorem one_le_zpow {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) : 1 ≤ x ^ n 
Arguments: {G : Type u_2} [SubNegMonoid G] [Preorder G] [CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : G} (H : 0 ≤ x) {n : ℤ} (hn : 0 ≤ n)
Goal: 0 ≤ n • x
-- one_le_zpow
@[to_additive zsmul_nonneg]
theorem one_le_zpow {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) : 1 ≤ x ^ n 
Arguments: {G : Type u_2} [DivInvMonoid G] [Preorder G] [CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n)
Goal: 1 ≤ x ^ n
-- Additive.ofMul_le
@[simp]
theorem ofMul_le {a b : α} : ofMul a ≤ ofMul b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Additive.ofMul a ≤ Additive.ofMul b ↔ a ≤ b
-- Additive.ofMul_lt
@[simp]
theorem ofMul_lt {a b : α} : ofMul a < ofMul b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Additive.ofMul a < Additive.ofMul b ↔ a < b
-- Additive.toMul_le
@[simp]
theorem toMul_le {a b : Additive α} : toMul a ≤ toMul b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Additive α}
Goal: Additive.toMul a ≤ Additive.toMul b ↔ a ≤ b
-- Additive.toMul_lt
@[simp]
theorem toMul_lt {a b : Additive α} : toMul a < toMul b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : Additive α}
Goal: Additive.toMul a < Additive.toMul b ↔ a < b
-- Multiplicative.ofAdd_le
@[simp]
theorem ofAdd_le {a b : α} : ofAdd a ≤ ofAdd b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Multiplicative.ofAdd a ≤ Multiplicative.ofAdd b ↔ a ≤ b
-- Multiplicative.ofAdd_lt
@[simp]
theorem ofAdd_lt {a b : α} : ofAdd a < ofAdd b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Multiplicative.ofAdd a < Multiplicative.ofAdd b ↔ a < b
-- Multiplicative.toAdd_le
@[simp]
theorem toAdd_le {a b : Multiplicative α} : toAdd a ≤ toAdd b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Multiplicative α}
Goal: Multiplicative.toAdd a ≤ Multiplicative.toAdd b ↔ a ≤ b
-- Multiplicative.toAdd_lt
@[simp]
theorem toAdd_lt {a b : Multiplicative α} : toAdd a < toAdd b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : Multiplicative α}
Goal: Multiplicative.toAdd a < Multiplicative.toAdd b ↔ a < b
-- algebraMap_monotone
theorem algebraMap_monotone : Monotone (algebraMap R A) 
Arguments: {R : Type u_1} {A : Type u_2} [OrderedCommRing R] [OrderedRing A] [Algebra R A] [OrderedSMul R A]
Goal: Monotone ⇑(algebraMap R A)
-- Function.one_le_const_of_one_le
@[to_additive const_nonneg_of_nonneg]
theorem one_le_const_of_one_le (ha : 1 ≤ a) : 1 ≤ const β a 
Arguments: {α : Type u_3} (β : Type u_4) [One α] [Preorder α] {a : α} (ha : 1 ≤ a)
Goal: 1 ≤ Function.const β a
-- Function.const_nonneg_of_nonneg
@[to_additive const_nonneg_of_nonneg]
theorem one_le_const_of_one_le (ha : 1 ≤ a) : 1 ≤ const β a 
Arguments: {α : Type u_3} (β : Type u_4) [Zero α] [Preorder α] {a : α} (ha : 0 ≤ a)
Goal: 0 ≤ Function.const β a
-- Function.const_le_one_of_le_one
@[to_additive]
theorem const_le_one_of_le_one (ha : a ≤ 1) : const β a ≤ 1 
Arguments: {α : Type u_3} (β : Type u_4) [One α] [Preorder α] {a : α} (ha : a ≤ 1)
Goal: Function.const β a ≤ 1
-- Function.const_nonpos_of_nonpos
@[to_additive]
theorem const_le_one_of_le_one (ha : a ≤ 1) : const β a ≤ 1 
Arguments: {α : Type u_3} (β : Type u_4) [Zero α] [Preorder α] {a : α} (ha : a ≤ 0)
Goal: Function.const β a ≤ 0
-- Function.one_le_const
@[to_additive (attr := simp) const_nonneg]
theorem one_le_const : 1 ≤ const β a ↔ 1 ≤ a 
Arguments: {α : Type u_3} {β : Type u_4} [One α] [Preorder α] {a : α} [Nonempty β]
Goal: 1 ≤ Function.const β a ↔ 1 ≤ a
-- Function.const_nonneg
@[to_additive (attr := simp) const_nonneg]
theorem one_le_const : 1 ≤ const β a ↔ 1 ≤ a 
Arguments: {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] {a : α} [Nonempty β]
Goal: 0 ≤ Function.const β a ↔ 0 ≤ a
-- Function.const_pos
@[to_additive (attr := simp) const_pos]
theorem one_lt_const : 1 < const β a ↔ 1 < a 
Arguments: {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] {a : α} [Nonempty β]
Goal: 0 < Function.const β a ↔ 0 < a
-- Function.one_lt_const
@[to_additive (attr := simp) const_pos]
theorem one_lt_const : 1 < const β a ↔ 1 < a 
Arguments: {α : Type u_3} {β : Type u_4} [One α] [Preorder α] {a : α} [Nonempty β]
Goal: 1 < Function.const β a ↔ 1 < a
-- Function.const_nonpos
@[to_additive (attr := simp)]
theorem const_le_one : const β a ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] {a : α} [Nonempty β]
Goal: Function.const β a ≤ 0 ↔ a ≤ 0
-- Function.const_le_one
@[to_additive (attr := simp)]
theorem const_le_one : const β a ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_3} {β : Type u_4} [One α] [Preorder α] {a : α} [Nonempty β]
Goal: Function.const β a ≤ 1 ↔ a ≤ 1
-- Function.const_lt_one
@[to_additive (attr := simp) const_neg']
theorem const_lt_one : const β a < 1 ↔ a < 1 
Arguments: {α : Type u_3} {β : Type u_4} [One α] [Preorder α] {a : α} [Nonempty β]
Goal: Function.const β a < 1 ↔ a < 1
-- Function.const_neg'
@[to_additive (attr := simp) const_neg']
theorem const_lt_one : const β a < 1 ↔ a < 1 
Arguments: {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] {a : α} [Nonempty β]
Goal: Function.const β a < 0 ↔ a < 0
-- Function.one_le_extend
@[to_additive extend_nonneg] lemma one_le_extend (hg : 1 ≤ g) (he : 1 ≤ e) : 1 ≤ extend f g e 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [One γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : 1 ≤ g) (he : 1 ≤ e)
Goal: 1 ≤ Function.extend f g e
-- Function.extend_nonneg
@[to_additive extend_nonneg] lemma one_le_extend (hg : 1 ≤ g) (he : 1 ≤ e) : 1 ≤ extend f g e 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Zero γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : 0 ≤ g) (he : 0 ≤ e)
Goal: 0 ≤ Function.extend f g e
-- Function.extend_le_one
@[to_additive] lemma extend_le_one (hg : g ≤ 1) (he : e ≤ 1) : extend f g e ≤ 1 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [One γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : g ≤ 1) (he : e ≤ 1)
Goal: Function.extend f g e ≤ 1
-- Function.extend_nonpos
@[to_additive] lemma extend_le_one (hg : g ≤ 1) (he : e ≤ 1) : extend f g e ≤ 1 
Arguments: {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Zero γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ} (hg : g ≤ 0) (he : e ≤ 0)
Goal: Function.extend f g e ≤ 0
-- NNRat.coe_inv
@[simp, norm_cast] lemma coe_inv (q : ℚ≥0) : ((q⁻¹ : ℚ≥0) : ℚ) = (q : ℚ)⁻¹ 
Arguments: (q : ℚ≥0)
Goal: ↑q⁻¹ = (↑q)⁻¹
-- NNRat.coe_div
@[simp, norm_cast] lemma coe_div (p q : ℚ≥0) : ((p / q : ℚ≥0) : ℚ) = p / q 
Arguments: (p q : ℚ≥0)
Goal: ↑(p / q) = ↑p / ↑q
-- NNRat.inv_def
lemma inv_def (q : ℚ≥0) : q⁻¹ = divNat q.den q.num 
Arguments: (q : ℚ≥0)
Goal: q⁻¹ = NNRat.divNat q.den q.num
-- NNRat.div_def
lemma div_def (p q : ℚ≥0) : p / q = divNat (p.num * q.den) (p.den * q.num) 
Arguments: (p q : ℚ≥0)
Goal: p / q = NNRat.divNat (p.num * q.den) (p.den * q.num)
-- NNRat.num_inv_of_ne_zero
lemma num_inv_of_ne_zero {q : ℚ≥0} (hq : q ≠ 0) : q⁻¹.num = q.den 
Arguments: {q : ℚ≥0} (hq : q ≠ 0)
Goal: q⁻¹.num = q.den
-- NNRat.den_inv_of_ne_zero
lemma den_inv_of_ne_zero {q : ℚ≥0} (hq : q ≠ 0) : q⁻¹.den = q.num 
Arguments: {q : ℚ≥0} (hq : q ≠ 0)
Goal: q⁻¹.den = q.num
-- invOf_pos
@[simp]
theorem invOf_pos [Invertible a] : 0 < ⅟ a ↔ 0 < a 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] {a : α} [Invertible a]
Goal: 0 < ⅟a ↔ 0 < a
-- invOf_nonpos
@[simp]
theorem invOf_nonpos [Invertible a] : ⅟ a ≤ 0 ↔ a ≤ 0 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] {a : α} [Invertible a]
Goal: ⅟a ≤ 0 ↔ a ≤ 0
-- invOf_nonneg
@[simp]
theorem invOf_nonneg [Invertible a] : 0 ≤ ⅟ a ↔ 0 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] {a : α} [Invertible a]
Goal: 0 ≤ ⅟a ↔ 0 ≤ a
-- invOf_lt_zero
@[simp]
theorem invOf_lt_zero [Invertible a] : ⅟ a < 0 ↔ a < 0 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] {a : α} [Invertible a]
Goal: ⅟a < 0 ↔ a < 0
-- invOf_le_one
@[simp]
theorem invOf_le_one [Invertible a] (h : 1 ≤ a) : ⅟ a ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] {a : α} [Invertible a] (h : 1 ≤ a)
Goal: ⅟a ≤ 1
-- pos_invOf_of_invertible_cast
theorem pos_invOf_of_invertible_cast [Nontrivial α] (n : ℕ)
    [Invertible (n : α)] : 0 < ⅟(n : α) 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] [Nontrivial α] (n : ℕ) [Invertible ↑n]
Goal: 0 < ⅟↑n
-- add_le_add_left
@[to_additive (attr := gcongr) add_le_add_left]
theorem mul_le_mul_left' [CovariantClass α α (· * ·) (· ≤ ·)] {b c : α} (bc : b ≤ c) (a : α) :
    a * b ≤ a * c 
Arguments: {α : Type u_1} [Add α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {b c : α} (bc : b ≤ c) (a : α)
Goal: a + b ≤ a + c
-- mul_le_mul_left'
@[to_additive (attr := gcongr) add_le_add_left]
theorem mul_le_mul_left' [CovariantClass α α (· * ·) (· ≤ ·)] {b c : α} (bc : b ≤ c) (a : α) :
    a * b ≤ a * c 
Arguments: {α : Type u_1} [Mul α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {b c : α} (bc : b ≤ c) (a : α)
Goal: a * b ≤ a * c
-- le_of_mul_le_mul_left'
@[to_additive le_of_add_le_add_left]
theorem le_of_mul_le_mul_left' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
    (bc : a * b ≤ a * c) :
    b ≤ c 
Arguments: {α : Type u_1} [Mul α] [LE α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (bc : a * b ≤ a * c)
Goal: b ≤ c
-- le_of_add_le_add_left
@[to_additive le_of_add_le_add_left]
theorem le_of_mul_le_mul_left' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
    (bc : a * b ≤ a * c) :
    b ≤ c 
Arguments: {α : Type u_1} [Add α] [LE α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (bc : a + b ≤ a + c)
Goal: b ≤ c
-- add_le_add_right
@[to_additive (attr := gcongr) add_le_add_right]
theorem mul_le_mul_right' [i : CovariantClass α α (swap (· * ·)) (· ≤ ·)] {b c : α} (bc : b ≤ c)
    (a : α) :
    b * a ≤ c * a 
Arguments: {α : Type u_1} [Add α] [LE α] [i : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {b c : α} (bc : b ≤ c) (a : α)
Goal: b + a ≤ c + a
-- mul_le_mul_right'
@[to_additive (attr := gcongr) add_le_add_right]
theorem mul_le_mul_right' [i : CovariantClass α α (swap (· * ·)) (· ≤ ·)] {b c : α} (bc : b ≤ c)
    (a : α) :
    b * a ≤ c * a 
Arguments: {α : Type u_1} [Mul α] [LE α] [i : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {b c : α} (bc : b ≤ c) (a : α)
Goal: b * a ≤ c * a
-- le_of_add_le_add_right
@[to_additive le_of_add_le_add_right]
theorem le_of_mul_le_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (bc : b * a ≤ c * a) :
    b ≤ c 
Arguments: {α : Type u_1} [Add α] [LE α] [i : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (bc : b + a ≤ c + a)
Goal: b ≤ c
-- le_of_mul_le_mul_right'
@[to_additive le_of_add_le_add_right]
theorem le_of_mul_le_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (bc : b * a ≤ c * a) :
    b ≤ c 
Arguments: {α : Type u_1} [Mul α] [LE α] [i : ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (bc : b * a ≤ c * a)
Goal: b ≤ c
-- add_le_add_iff_left
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_left [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b c : α} :
    a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} [Add α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) {b c : α}
Goal: a + b ≤ a + c ↔ b ≤ c
-- mul_le_mul_iff_left
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_left [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b c : α} :
    a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} [Mul α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α) {b c : α}
Goal: a * b ≤ a * c ↔ b ≤ c
-- mul_le_mul_iff_right
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b c : α} :
    b * a ≤ c * a ↔ b ≤ c 
Arguments: {α : Type u_1} [Mul α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α) {b c : α}
Goal: b * a ≤ c * a ↔ b ≤ c
-- add_le_add_iff_right
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b c : α} :
    b * a ≤ c * a ↔ b ≤ c 
Arguments: {α : Type u_1} [Add α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) {b c : α}
Goal: b + a ≤ c + a ↔ b ≤ c
-- add_lt_add_iff_left
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_left [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] (a : α) {b c : α} :
    a * b < a * c ↔ b < c 
Arguments: {α : Type u_1} [Add α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b c : α}
Goal: a + b < a + c ↔ b < c
-- mul_lt_mul_iff_left
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_left [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] (a : α) {b c : α} :
    a * b < a * c ↔ b < c 
Arguments: {α : Type u_1} [Mul α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b c : α}
Goal: a * b < a * c ↔ b < c
-- add_lt_add_iff_right
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_right [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b c : α} :
    b * a < c * a ↔ b < c 
Arguments: {α : Type u_1} [Add α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b c : α}
Goal: b + a < c + a ↔ b < c
-- mul_lt_mul_iff_right
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_right [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b c : α} :
    b * a < c * a ↔ b < c 
Arguments: {α : Type u_1} [Mul α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b c : α}
Goal: b * a < c * a ↔ b < c
-- add_lt_add_left
@[to_additive (attr := gcongr) add_lt_add_left]
theorem mul_lt_mul_left' [CovariantClass α α (· * ·) (· < ·)] {b c : α} (bc : b < c) (a : α) :
    a * b < a * c 
Arguments: {α : Type u_1} [Add α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {b c : α} (bc : b < c) (a : α)
Goal: a + b < a + c
-- mul_lt_mul_left'
@[to_additive (attr := gcongr) add_lt_add_left]
theorem mul_lt_mul_left' [CovariantClass α α (· * ·) (· < ·)] {b c : α} (bc : b < c) (a : α) :
    a * b < a * c 
Arguments: {α : Type u_1} [Mul α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {b c : α} (bc : b < c) (a : α)
Goal: a * b < a * c
-- lt_of_mul_lt_mul_left'
@[to_additive lt_of_add_lt_add_left]
theorem lt_of_mul_lt_mul_left' [ContravariantClass α α (· * ·) (· < ·)] {a b c : α}
    (bc : a * b < a * c) :
    b < c 
Arguments: {α : Type u_1} [Mul α] [LT α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (bc : a * b < a * c)
Goal: b < c
-- lt_of_add_lt_add_left
@[to_additive lt_of_add_lt_add_left]
theorem lt_of_mul_lt_mul_left' [ContravariantClass α α (· * ·) (· < ·)] {a b c : α}
    (bc : a * b < a * c) :
    b < c 
Arguments: {α : Type u_1} [Add α] [LT α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (bc : a + b < a + c)
Goal: b < c
-- mul_lt_mul_right'
@[to_additive (attr := gcongr) add_lt_add_right]
theorem mul_lt_mul_right' [i : CovariantClass α α (swap (· * ·)) (· < ·)] {b c : α} (bc : b < c)
    (a : α) :
    b * a < c * a 
Arguments: {α : Type u_1} [Mul α] [LT α] [i : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {b c : α} (bc : b < c) (a : α)
Goal: b * a < c * a
-- add_lt_add_right
@[to_additive (attr := gcongr) add_lt_add_right]
theorem mul_lt_mul_right' [i : CovariantClass α α (swap (· * ·)) (· < ·)] {b c : α} (bc : b < c)
    (a : α) :
    b * a < c * a 
Arguments: {α : Type u_1} [Add α] [LT α] [i : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {b c : α} (bc : b < c) (a : α)
Goal: b + a < c + a
-- lt_of_mul_lt_mul_right'
@[to_additive lt_of_add_lt_add_right]
theorem lt_of_mul_lt_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
    (bc : b * a < c * a) :
    b < c 
Arguments: {α : Type u_1} [Mul α] [LT α] [i : ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (bc : b * a < c * a)
Goal: b < c
-- lt_of_add_lt_add_right
@[to_additive lt_of_add_lt_add_right]
theorem lt_of_mul_lt_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
    (bc : b * a < c * a) :
    b < c 
Arguments: {α : Type u_1} [Add α] [LT α] [i : ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (bc : b + a < c + a)
Goal: b < c
-- add_lt_add_of_lt_of_lt
@[to_additive (attr := gcongr)]
theorem mul_lt_mul_of_lt_of_lt [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)]
    {a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a + c < b + d
-- mul_lt_mul_of_lt_of_lt
@[to_additive (attr := gcongr)]
theorem mul_lt_mul_of_lt_of_lt [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)]
    {a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a * c < b * d
-- mul_lt_mul_of_le_of_lt
@[to_additive]
theorem mul_lt_mul_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a ≤ b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a ≤ b) (h₂ : c < d)
Goal: a * c < b * d
-- add_lt_add_of_le_of_lt
@[to_additive]
theorem mul_lt_mul_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a ≤ b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a ≤ b) (h₂ : c < d)
Goal: a + c < b + d
-- add_lt_add_of_lt_of_le
@[to_additive]
theorem mul_lt_mul_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (h₁ : a < b) (h₂ : c ≤ d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c ≤ d)
Goal: a + c < b + d
-- mul_lt_mul_of_lt_of_le
@[to_additive]
theorem mul_lt_mul_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (h₁ : a < b) (h₂ : c ≤ d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α} (h₁ : a < b) (h₂ : c ≤ d)
Goal: a * c < b * d
-- Left.mul_lt_mul
@[to_additive "Only assumes left strict covariance"]
theorem Left.mul_lt_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a < b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Mul α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a * c < b * d
-- Left.add_lt_add
@[to_additive "Only assumes left strict covariance"]
theorem Left.mul_lt_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a < b) (h₂ : c < d) :
    a * c < b * d 
Arguments: {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (h₁ : a < b) (h₂ : c < d)
Goal: a + c < b + d