ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Left.one_lt_mul_of_le_of_lt
@[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} [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_pos_of_pos_of_nonneg
@[to_additive Left.add_pos_of_pos_of_nonneg
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_pos_of_pos_of_nonneg`."]
theorem Left.one_lt_mul_of_lt_of_le [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.one_lt_mul_of_lt_of_le
@[to_additive Left.add_pos_of_pos_of_nonneg
      "Assumes left covariance.
      The lemma assuming right covariance is `Right.add_pos_of_pos_of_nonneg`."]
theorem Left.one_lt_mul_of_lt_of_le [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_pos
@[to_additive Left.add_pos "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos`."]
theorem Left.one_lt_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.one_lt_mul
@[to_additive Left.add_pos "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos`."]
theorem Left.one_lt_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.one_lt_mul'
@[to_additive Left.add_pos' "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos'`."]
theorem Left.one_lt_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_pos'
@[to_additive Left.add_pos' "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos'`."]
theorem Left.one_lt_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
-- add_le_of_nonpos_of_le
@[to_additive]
theorem mul_le_of_le_one_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
    (hbc : b ≤ c) :
    a * b ≤ c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a ≤ 0) (hbc : b ≤ c)
Goal: a + b ≤ c
-- mul_le_of_le_one_of_le
@[to_additive]
theorem mul_le_of_le_one_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
    (hbc : b ≤ c) :
    a * b ≤ c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a ≤ 1) (hbc : b ≤ c)
Goal: a * b ≤ c
-- add_lt_of_neg_of_le
@[to_additive]
theorem mul_lt_of_lt_one_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
    (hbc : b ≤ c) :
    a * b < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (ha : a < 0) (hbc : b ≤ c)
Goal: a + b < c
-- mul_lt_of_lt_one_of_le
@[to_additive]
theorem mul_lt_of_lt_one_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
    (hbc : b ≤ c) :
    a * b < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (ha : a < 1) (hbc : b ≤ c)
Goal: a * b < c
-- add_lt_of_nonpos_of_lt
@[to_additive]
theorem mul_lt_of_le_one_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
    (hb : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a ≤ 0) (hb : b < c)
Goal: a + b < c
-- mul_lt_of_le_one_of_lt
@[to_additive]
theorem mul_lt_of_le_one_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
    (hb : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a ≤ 1) (hb : b < c)
Goal: a * b < c
-- add_lt_of_neg_of_lt
@[to_additive]
theorem mul_lt_of_lt_one_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
    (hb : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (ha : a < 0) (hb : b < c)
Goal: a + b < c
-- mul_lt_of_lt_one_of_lt
@[to_additive]
theorem mul_lt_of_lt_one_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
    (hb : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (ha : a < 1) (hb : b < c)
Goal: a * b < c
-- add_lt_of_neg_of_lt'
@[to_additive]
theorem mul_lt_of_lt_one_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a < 1)
    (hbc : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a < 0) (hbc : b < c)
Goal: a + b < c
-- mul_lt_of_lt_one_of_lt'
@[to_additive]
theorem mul_lt_of_lt_one_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a < 1)
    (hbc : b < c) :
    a * b < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : a < 1) (hbc : b < c)
Goal: a * b < c
-- Right.mul_le_one
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonpos`."]
theorem Right.mul_le_one [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a ≤ 1)
    (hb : b ≤ 1) :
    a * b ≤ 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_nonpos
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonpos`."]
theorem Right.mul_le_one [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a ≤ 1)
    (hb : b ≤ 1) :
    a * b ≤ 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.mul_lt_one_of_lt_of_le
@[to_additive Right.add_neg_of_neg_of_nonpos
      "Assumes right covariance.
      The lemma assuming left covariance is `Left.add_neg_of_neg_of_nonpos`."]
theorem Right.mul_lt_one_of_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (ha : a < 1) (hb : b ≤ 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_neg_of_neg_of_nonpos
@[to_additive Right.add_neg_of_neg_of_nonpos
      "Assumes right covariance.
      The lemma assuming left covariance is `Left.add_neg_of_neg_of_nonpos`."]
theorem Right.mul_lt_one_of_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (ha : a < 1) (hb : b ≤ 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.mul_lt_one_of_le_of_lt
@[to_additive Right.add_neg_of_nonpos_of_neg
      "Assumes right covariance.
      The lemma assuming left covariance is `Left.add_neg_of_nonpos_of_neg`."]
theorem Right.mul_lt_one_of_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : a ≤ 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_neg_of_nonpos_of_neg
@[to_additive Right.add_neg_of_nonpos_of_neg
      "Assumes right covariance.
      The lemma assuming left covariance is `Left.add_neg_of_nonpos_of_neg`."]
theorem Right.mul_lt_one_of_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : a ≤ 1) (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.mul_lt_one
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg`."]
theorem Right.mul_lt_one [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α} (ha : a < 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_neg
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg`."]
theorem Right.mul_lt_one [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α} (ha : a < 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_neg'
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg'`."]
theorem Right.mul_lt_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a < 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.mul_lt_one'
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg'`."]
theorem Right.mul_lt_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a < 1)
    (hb : b < 1) :
    a * b < 1 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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_mul_of_one_le_of_le
@[to_additive]
theorem le_mul_of_one_le_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
    (hbc : b ≤ c) :
    b ≤ a * c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 1 ≤ a) (hbc : b ≤ c)
Goal: b ≤ a * c
-- le_add_of_nonneg_of_le
@[to_additive]
theorem le_mul_of_one_le_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
    (hbc : b ≤ c) :
    b ≤ a * c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 0 ≤ a) (hbc : b ≤ c)
Goal: b ≤ a + c
-- lt_add_of_pos_of_le
@[to_additive]
theorem lt_mul_of_one_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
    (hbc : b ≤ c) :
    b < a * c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (ha : 0 < a) (hbc : b ≤ c)
Goal: b < a + c
-- lt_mul_of_one_lt_of_le
@[to_additive]
theorem lt_mul_of_one_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
    (hbc : b ≤ c) :
    b < a * c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (ha : 1 < a) (hbc : b ≤ c)
Goal: b < a * c
-- lt_mul_of_one_le_of_lt
@[to_additive]
theorem lt_mul_of_one_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 1 ≤ a) (hbc : b < c)
Goal: b < a * c
-- lt_add_of_nonneg_of_lt
@[to_additive]
theorem lt_mul_of_one_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 0 ≤ a) (hbc : b < c)
Goal: b < a + c
-- lt_mul_of_one_lt_of_lt
@[to_additive]
theorem lt_mul_of_one_lt_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α} (ha : 1 < a) (hbc : b < c)
Goal: b < a * c
-- lt_add_of_pos_of_lt
@[to_additive]
theorem lt_mul_of_one_lt_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α} (ha : 0 < a) (hbc : b < c)
Goal: b < a + c
-- lt_add_of_pos_of_lt'
@[to_additive]
theorem lt_mul_of_one_lt_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 < a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 0 < a) (hbc : b < c)
Goal: b < a + c
-- lt_mul_of_one_lt_of_lt'
@[to_additive]
theorem lt_mul_of_one_lt_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 < a)
    (hbc : b < c) :
    b < a * c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : 1 < a) (hbc : b < c)
Goal: b < a * c
-- Right.one_le_mul
@[to_additive Right.add_nonneg "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonneg`."]
theorem Right.one_le_mul [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a)
    (hb : 1 ≤ b) :
    1 ≤ a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_nonneg
@[to_additive Right.add_nonneg "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonneg`."]
theorem Right.one_le_mul [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a)
    (hb : 1 ≤ b) :
    1 ≤ a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.one_lt_mul_of_lt_of_le
@[to_additive Right.add_pos_of_pos_of_nonneg
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos_of_pos_of_nonneg`."]
theorem Right.one_lt_mul_of_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (ha : 1 < a) (hb : 1 ≤ b) :
    1 < a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_pos_of_pos_of_nonneg
@[to_additive Right.add_pos_of_pos_of_nonneg
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos_of_pos_of_nonneg`."]
theorem Right.one_lt_mul_of_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
    (ha : 1 < a) (hb : 1 ≤ b) :
    1 < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_pos_of_nonneg_of_pos
@[to_additive Right.add_pos_of_nonneg_of_pos
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos_of_nonneg_of_pos`."]
theorem Right.one_lt_mul_of_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.one_lt_mul_of_le_of_lt
@[to_additive Right.add_pos_of_nonneg_of_pos
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos_of_nonneg_of_pos`."]
theorem Right.one_lt_mul_of_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_pos
@[to_additive Right.add_pos "Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos`."]
theorem Right.one_lt_mul [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α} (ha : 1 < a)
    (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.one_lt_mul
@[to_additive Right.add_pos "Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos`."]
theorem Right.one_lt_mul [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α} (ha : 1 < a)
    (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.one_lt_mul'
@[to_additive Right.add_pos' "Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos'`."]
theorem Right.one_lt_mul' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 < a)
    (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- Right.add_pos'
@[to_additive Right.add_pos' "Assumes right covariance.
The lemma assuming left covariance is `Left.add_pos'`."]
theorem Right.one_lt_mul' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 < a)
    (hb : 1 < b) :
    1 < a * b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap 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
-- lt_of_add_lt_of_nonneg_left
@[to_additive]
theorem lt_of_mul_lt_of_one_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b < c)
    (hle : 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} (h : a + b < c) (hle : 0 ≤ b)
Goal: a < c
-- lt_of_mul_lt_of_one_le_left
@[to_additive]
theorem lt_of_mul_lt_of_one_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b < c)
    (hle : 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} (h : a * b < c) (hle : 1 ≤ b)
Goal: a < c
-- le_of_add_le_of_nonneg_left
@[to_additive]
theorem le_of_mul_le_of_one_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b ≤ c)
    (hle : 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} (h : a + b ≤ c) (hle : 0 ≤ b)
Goal: a ≤ c
-- le_of_mul_le_of_one_le_left
@[to_additive]
theorem le_of_mul_le_of_one_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b ≤ c)
    (hle : 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} (h : a * b ≤ c) (hle : 1 ≤ b)
Goal: a ≤ c
-- lt_of_lt_add_of_nonpos_left
@[to_additive]
theorem lt_of_lt_mul_of_le_one_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a < b * c)
    (hle : c ≤ 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 c : α} (h : a < b + c) (hle : c ≤ 0)
Goal: a < b
-- lt_of_lt_mul_of_le_one_left
@[to_additive]
theorem lt_of_lt_mul_of_le_one_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a < b * c)
    (hle : c ≤ 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 c : α} (h : a < b * c) (hle : c ≤ 1)
Goal: a < b
-- le_of_le_mul_of_le_one_left
@[to_additive]
theorem le_of_le_mul_of_le_one_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a ≤ b * c)
    (hle : c ≤ 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 c : α} (h : a ≤ b * c) (hle : c ≤ 1)
Goal: a ≤ b
-- le_of_le_add_of_nonpos_left
@[to_additive]
theorem le_of_le_mul_of_le_one_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a ≤ b * c)
    (hle : c ≤ 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 c : α} (h : a ≤ b + c) (hle : c ≤ 0)
Goal: a ≤ b
-- lt_of_mul_lt_of_one_le_right
@[to_additive]
theorem lt_of_mul_lt_of_one_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b < c) (hle : 1 ≤ a) :
    b < c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a * b < c) (hle : 1 ≤ a)
Goal: b < c
-- lt_of_add_lt_of_nonneg_right
@[to_additive]
theorem lt_of_mul_lt_of_one_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b < c) (hle : 1 ≤ a) :
    b < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a + b < c) (hle : 0 ≤ a)
Goal: b < c
-- le_of_mul_le_of_one_le_right
@[to_additive]
theorem le_of_mul_le_of_one_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b ≤ c) (hle : 1 ≤ a) :
    b ≤ c 
Arguments: {α : Type u_1} [MulOneClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a * b ≤ c) (hle : 1 ≤ a)
Goal: b ≤ c
-- le_of_add_le_of_nonneg_right
@[to_additive]
theorem le_of_mul_le_of_one_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a * b ≤ c) (hle : 1 ≤ a) :
    b ≤ c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a + b ≤ c) (hle : 0 ≤ a)
Goal: b ≤ c
-- lt_of_lt_add_of_nonpos_right
@[to_additive]
theorem lt_of_lt_mul_of_le_one_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
    (h : a < b * c) (hle : b ≤ 1) :
    a < c 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (h : a < b + c) (hle : b ≤ 0)
Goal: a < c