ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_add_iff_nonneg_left
@[to_additive (attr := simp) le_add_iff_nonneg_left]
theorem le_mul_iff_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b : α} :
    a ≤ b * a ↔ 1 ≤ b 
Arguments: {α : Type u_1} [AddZeroClass 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 : α}
Goal: a ≤ b + a ↔ 0 ≤ b
-- mul_le_iff_le_one_right'
@[to_additive (attr := simp) add_le_iff_nonpos_right]
theorem mul_le_iff_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
    a * b ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [MulOneClass 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 : α}
Goal: a * b ≤ a ↔ b ≤ 1
-- add_le_iff_nonpos_right
@[to_additive (attr := simp) add_le_iff_nonpos_right]
theorem mul_le_iff_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)]
    [ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
    a * b ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass 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 : α}
Goal: a + b ≤ a ↔ b ≤ 0
-- add_le_iff_nonpos_left
@[to_additive (attr := simp) add_le_iff_nonpos_left]
theorem mul_le_iff_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} :
    a * b ≤ b ↔ a ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass 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 b : α}
Goal: a + b ≤ b ↔ a ≤ 0
-- mul_le_iff_le_one_left'
@[to_additive (attr := simp) add_le_iff_nonpos_left]
theorem mul_le_iff_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
    [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} :
    a * b ≤ b ↔ a ≤ 1 
Arguments: {α : Type u_1} [MulOneClass 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 b : α}
Goal: a * b ≤ b ↔ a ≤ 1
-- lt_mul_of_one_lt_right'
@[to_additive lt_add_of_pos_right]
theorem lt_mul_of_one_lt_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : 1 < b) :
    a < a * b 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : 1 < b)
Goal: a < a * b
-- lt_add_of_pos_right
@[to_additive lt_add_of_pos_right]
theorem lt_mul_of_one_lt_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : 1 < b) :
    a < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : 0 < b)
Goal: a < a + b
-- mul_lt_of_lt_one_right'
@[to_additive add_lt_of_neg_right]
theorem mul_lt_of_lt_one_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : b < 1) :
    a * b < a 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : b < 1)
Goal: a * b < a
-- add_lt_of_neg_right
@[to_additive add_lt_of_neg_right]
theorem mul_lt_of_lt_one_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : b < 1) :
    a * b < a 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : b < 0)
Goal: a + b < a
-- lt_add_of_pos_left
@[to_additive lt_add_of_pos_left]
theorem lt_mul_of_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
    (h : 1 < b) :
    a < b * a 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : 0 < b)
Goal: a < b + a
-- lt_mul_of_one_lt_left'
@[to_additive lt_add_of_pos_left]
theorem lt_mul_of_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
    (h : 1 < b) :
    a < b * a 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : 1 < b)
Goal: a < b * a
-- add_lt_of_neg_left
@[to_additive add_lt_of_neg_left]
theorem mul_lt_of_lt_one_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
    (h : b < 1) :
    b * a < a 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : b < 0)
Goal: b + a < a
-- mul_lt_of_lt_one_left'
@[to_additive add_lt_of_neg_left]
theorem mul_lt_of_lt_one_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
    (h : b < 1) :
    b * a < a 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a : α) {b : α} (h : b < 1)
Goal: b * a < a
-- pos_of_lt_add_right
@[to_additive]
theorem one_lt_of_lt_mul_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a < a * b) :
    1 < b 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : a < a + b)
Goal: 0 < b
-- one_lt_of_lt_mul_right
@[to_additive]
theorem one_lt_of_lt_mul_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a < a * b) :
    1 < b 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : a < a * b)
Goal: 1 < b
-- neg_of_add_lt_right
@[to_additive]
theorem lt_one_of_mul_lt_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a * b < a) :
    b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : a + b < a)
Goal: b < 0
-- lt_one_of_mul_lt_right
@[to_additive]
theorem lt_one_of_mul_lt_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a * b < a) :
    b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : a * b < a)
Goal: b < 1
-- pos_of_lt_add_left
@[to_additive]
theorem one_lt_of_lt_mul_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (h : b < a * b) :
    1 < a 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : b < a + b)
Goal: 0 < a
-- one_lt_of_lt_mul_left
@[to_additive]
theorem one_lt_of_lt_mul_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (h : b < a * b) :
    1 < a 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : b < a * b)
Goal: 1 < a
-- lt_one_of_mul_lt_left
@[to_additive]
theorem lt_one_of_mul_lt_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (h : a * b < b) :
    a < 1 
Arguments: {α : Type u_1} [MulOneClass α] [LT α] [ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : a * b < b)
Goal: a < 1
-- neg_of_add_lt_left
@[to_additive]
theorem lt_one_of_mul_lt_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (h : a * b < b) :
    a < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [LT α] [ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : a + b < b)
Goal: a < 0
-- lt_mul_iff_one_lt_right'
@[to_additive (attr := simp) lt_add_iff_pos_right]
theorem lt_mul_iff_one_lt_right' [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] (a : α) {b : α} :
    a < a * b ↔ 1 < b 
Arguments: {α : Type u_1} [MulOneClass 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 : α}
Goal: a < a * b ↔ 1 < b
-- lt_add_iff_pos_right
@[to_additive (attr := simp) lt_add_iff_pos_right]
theorem lt_mul_iff_one_lt_right' [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] (a : α) {b : α} :
    a < a * b ↔ 1 < b 
Arguments: {α : Type u_1} [AddZeroClass 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 : α}
Goal: a < a + b ↔ 0 < b
-- lt_add_iff_pos_left
@[to_additive (attr := simp) lt_add_iff_pos_left]
theorem lt_mul_iff_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α} : a < b * a ↔ 1 < b 
Arguments: {α : Type u_1} [AddZeroClass 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 : α}
Goal: a < b + a ↔ 0 < b
-- lt_mul_iff_one_lt_left'
@[to_additive (attr := simp) lt_add_iff_pos_left]
theorem lt_mul_iff_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α} : a < b * a ↔ 1 < b 
Arguments: {α : Type u_1} [MulOneClass 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 : α}
Goal: a < b * a ↔ 1 < b
-- add_lt_iff_neg_left
@[to_additive (attr := simp) add_lt_iff_neg_left]
theorem mul_lt_iff_lt_one_left' [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] {a b : α} :
    a * b < a ↔ b < 1 
Arguments: {α : Type u_1} [AddZeroClass 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 b : α}
Goal: a + b < a ↔ b < 0
-- mul_lt_iff_lt_one_left'
@[to_additive (attr := simp) add_lt_iff_neg_left]
theorem mul_lt_iff_lt_one_left' [CovariantClass α α (· * ·) (· < ·)]
    [ContravariantClass α α (· * ·) (· < ·)] {a b : α} :
    a * b < a ↔ b < 1 
Arguments: {α : Type u_1} [MulOneClass 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 b : α}
Goal: a * b < a ↔ b < 1
-- mul_lt_iff_lt_one_right'
@[to_additive (attr := simp) add_lt_iff_neg_right]
theorem mul_lt_iff_lt_one_right' [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] {a : α} (b : α) : a * b < b ↔ a < 1 
Arguments: {α : Type u_1} [MulOneClass 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 : α)
Goal: a * b < b ↔ a < 1
-- add_lt_iff_neg_right
@[to_additive (attr := simp) add_lt_iff_neg_right]
theorem mul_lt_iff_lt_one_right' [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] {a : α} (b : α) : a * b < b ↔ a < 1 
Arguments: {α : Type u_1} [AddZeroClass 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 : α)
Goal: a + b < b ↔ a < 0
-- add_le_of_le_of_nonpos
@[to_additive]
theorem mul_le_of_le_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
    (ha : a ≤ 1) :
    b * a ≤ c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b ≤ c) (ha : a ≤ 0)
Goal: b + a ≤ c
-- mul_le_of_le_of_le_one
@[to_additive]
theorem mul_le_of_le_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
    (ha : a ≤ 1) :
    b * a ≤ c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b ≤ c) (ha : a ≤ 1)
Goal: b * a ≤ c
-- add_lt_of_le_of_neg
@[to_additive]
theorem mul_lt_of_le_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b ≤ c) (ha : a < 0)
Goal: b + a < c
-- mul_lt_of_le_of_lt_one
@[to_additive]
theorem mul_lt_of_le_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b ≤ c) (ha : a < 1)
Goal: b * a < c
-- mul_lt_of_lt_of_le_one
@[to_additive]
theorem mul_lt_of_lt_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : a ≤ 1) :
    b * a < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : a ≤ 1)
Goal: b * a < c
-- add_lt_of_lt_of_nonpos
@[to_additive]
theorem mul_lt_of_lt_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : a ≤ 1) :
    b * a < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : a ≤ 0)
Goal: b + a < c
-- add_lt_of_lt_of_neg
@[to_additive]
theorem mul_lt_of_lt_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b < c) (ha : a < 0)
Goal: b + a < c
-- mul_lt_of_lt_of_lt_one
@[to_additive]
theorem mul_lt_of_lt_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b < c) (ha : a < 1)
Goal: b * a < c
-- add_lt_of_lt_of_neg'
@[to_additive]
theorem mul_lt_of_lt_of_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : a < 0)
Goal: b + a < c
-- mul_lt_of_lt_of_lt_one'
@[to_additive]
theorem mul_lt_of_lt_of_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : a < 1) :
    b * a < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : a < 1)
Goal: b * a < c
-- Left.add_nonpos
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonpos`."]
theorem Left.mul_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1) :
    a * b ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0)
Goal: a + b ≤ 0
-- Left.mul_le_one
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonpos`."]
theorem Left.mul_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1) :
    a * b ≤ 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1)
Goal: a * b ≤ 1
-- Left.mul_lt_one_of_le_of_lt
@[to_additive Left.add_neg_of_nonpos_of_neg
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_neg_of_nonpos_of_neg`."]
theorem Left.mul_lt_one_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a ≤ 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (ha : a ≤ 1) (hb : b < 1)
Goal: a * b < 1
-- Left.add_neg_of_nonpos_of_neg
@[to_additive Left.add_neg_of_nonpos_of_neg
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_neg_of_nonpos_of_neg`."]
theorem Left.mul_lt_one_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a ≤ 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (ha : a ≤ 0) (hb : b < 0)
Goal: a + b < 0
-- Left.add_neg_of_neg_of_nonpos
@[to_additive Left.add_neg_of_neg_of_nonpos
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_neg_of_neg_of_nonpos`."]
theorem Left.mul_lt_one_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1)
    (hb : b ≤ 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a < 0) (hb : b ≤ 0)
Goal: a + b < 0
-- Left.mul_lt_one_of_lt_of_le
@[to_additive Left.add_neg_of_neg_of_nonpos
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_neg_of_neg_of_nonpos`."]
theorem Left.mul_lt_one_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1)
    (hb : b ≤ 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a < 1) (hb : b ≤ 1)
Goal: a * b < 1
-- Left.mul_lt_one
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg`."]
theorem Left.mul_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (ha : a < 1) (hb : b < 1)
Goal: a * b < 1
-- Left.add_neg
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg`."]
theorem Left.mul_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a + b < 0
-- Left.add_neg'
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg'`."]
theorem Left.mul_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a + b < 0
-- Left.mul_lt_one'
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg'`."]
theorem Left.mul_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : a < 1) (hb : b < 1)
Goal: a * b < 1
-- le_add_of_le_of_nonneg
@[to_additive]
theorem le_mul_of_le_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
    (ha : 1 ≤ a) :
    b ≤ c * a 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b ≤ c) (ha : 0 ≤ a)
Goal: b ≤ c + a
-- le_mul_of_le_of_one_le
@[to_additive]
theorem le_mul_of_le_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
    (ha : 1 ≤ a) :
    b ≤ c * a 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b ≤ c) (ha : 1 ≤ a)
Goal: b ≤ c * a
-- lt_mul_of_le_of_one_lt
@[to_additive]
theorem lt_mul_of_le_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b ≤ c) (ha : 1 < a)
Goal: b < c * a
-- lt_add_of_le_of_pos
@[to_additive]
theorem lt_mul_of_le_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b ≤ c) (ha : 0 < a)
Goal: b < c + a
-- lt_mul_of_lt_of_one_le
@[to_additive]
theorem lt_mul_of_lt_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : 1 ≤ a) :
    b < c * a 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : 1 ≤ a)
Goal: b < c * a
-- lt_add_of_lt_of_nonneg
@[to_additive]
theorem lt_mul_of_lt_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : 1 ≤ a) :
    b < c * a 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : 0 ≤ a)
Goal: b < c + a
-- lt_add_of_lt_of_pos
@[to_additive]
theorem lt_mul_of_lt_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b < c) (ha : 0 < a)
Goal: b < c + a
-- lt_mul_of_lt_of_one_lt
@[to_additive]
theorem lt_mul_of_lt_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (hbc : b < c) (ha : 1 < a)
Goal: b < c * a
-- lt_mul_of_lt_of_one_lt'
@[to_additive]
theorem lt_mul_of_lt_of_one_lt' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : 1 < a)
Goal: b < c * a
-- lt_add_of_lt_of_pos'
@[to_additive]
theorem lt_mul_of_lt_of_one_lt' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
    (ha : 1 < a) :
    b < c * a 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (hbc : b < c) (ha : 0 < a)
Goal: b < c + a
-- Left.one_le_mul
@[to_additive Left.add_nonneg "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonneg`."]
theorem Left.one_le_mul [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    1 ≤ a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b)
Goal: 1 ≤ a * b
-- Left.add_nonneg
@[to_additive Left.add_nonneg "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonneg`."]
theorem Left.one_le_mul [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    1 ≤ a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: 0 ≤ a + b
-- Left.add_pos_of_nonneg_of_pos
@[to_additive Left.add_pos_of_nonneg_of_pos
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_pos_of_nonneg_of_pos`."]
theorem Left.one_lt_mul_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : 1 ≤ a)
    (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (ha : 0 ≤ a) (hb : 0 < b)
Goal: 0 < a + b