ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- add_neg_le_add_neg_iff
@[to_additive add_neg_le_add_neg_iff]
theorem mul_inv_le_mul_inv_iff' : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: a + -b ≤ c + -d ↔ a + d ≤ c + b
-- inv_mul_lt_iff_lt_mul'
@[to_additive]
theorem inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: c⁻¹ * a < b ↔ a < b * c
-- neg_add_lt_iff_lt_add'
@[to_additive]
theorem inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: -c + a < b ↔ a < b + c
-- add_neg_lt_iff_le_add'
@[to_additive]
theorem mul_inv_lt_iff_le_mul' : a * b⁻¹ < c ↔ a < b * c 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a + -b < c ↔ a < b + c
-- mul_inv_lt_iff_le_mul'
@[to_additive]
theorem mul_inv_lt_iff_le_mul' : a * b⁻¹ < c ↔ a < b * c 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a * b⁻¹ < c ↔ a < b * c
-- add_neg_lt_add_neg_iff
@[to_additive add_neg_lt_add_neg_iff]
theorem mul_inv_lt_mul_inv_iff' : a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α}
Goal: a + -b < c + -d ↔ a + d < c + b
-- mul_inv_lt_mul_inv_iff'
@[to_additive add_neg_lt_add_neg_iff]
theorem mul_inv_lt_mul_inv_iff' : a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α}
Goal: a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b
-- sub_le_sub_iff_right
@[to_additive]
theorem div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (c : α)
Goal: a - c ≤ b - c ↔ a ≤ b
-- div_le_div_iff_right
@[to_additive]
theorem div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (c : α)
Goal: a / c ≤ b / c ↔ a ≤ b
-- sub_le_sub_right
@[to_additive (attr := gcongr) sub_le_sub_right]
theorem div_le_div_right' (h : a ≤ b) (c : α) : a / c ≤ b / c 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a ≤ b) (c : α)
Goal: a - c ≤ b - c
-- div_le_div_right'
@[to_additive (attr := gcongr) sub_le_sub_right]
theorem div_le_div_right' (h : a ≤ b) (c : α) : a / c ≤ b / c 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : a ≤ b) (c : α)
Goal: a / c ≤ b / c
-- sub_nonneg
@[to_additive (attr := simp) sub_nonneg]
theorem one_le_div' : 1 ≤ a / b ↔ b ≤ a 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: 0 ≤ a - b ↔ b ≤ a
-- one_le_div'
@[to_additive (attr := simp) sub_nonneg]
theorem one_le_div' : 1 ≤ a / b ↔ b ≤ a 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: 1 ≤ a / b ↔ b ≤ a
-- div_le_one'
@[to_additive sub_nonpos]
theorem div_le_one' : a / b ≤ 1 ↔ a ≤ b 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a / b ≤ 1 ↔ a ≤ b
-- sub_nonpos
@[to_additive sub_nonpos]
theorem div_le_one' : a / b ≤ 1 ↔ a ≤ b 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a - b ≤ 0 ↔ a ≤ b
-- le_div_iff_mul_le
@[to_additive]
theorem le_div_iff_mul_le : a ≤ c / b ↔ a * b ≤ c 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a ≤ c / b ↔ a * b ≤ c
-- le_sub_iff_add_le
@[to_additive]
theorem le_div_iff_mul_le : a ≤ c / b ↔ a * b ≤ c 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a ≤ c - b ↔ a + b ≤ c
-- div_le_iff_le_mul
@[to_additive]
theorem div_le_iff_le_mul : a / c ≤ b ↔ a ≤ b * c 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a / c ≤ b ↔ a ≤ b * c
-- sub_le_iff_le_add
@[to_additive]
theorem div_le_iff_le_mul : a / c ≤ b ↔ a ≤ b * c 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a - c ≤ b ↔ a ≤ b + c
-- div_le_div_iff_left
@[to_additive]
theorem div_le_div_iff_left (a : α) : a / b ≤ a / c ↔ c ≤ b 
Arguments: {α : Type u} [Group α] [LE 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] {b c : α} (a : α)
Goal: a / b ≤ a / c ↔ c ≤ b
-- sub_le_sub_iff_left
@[to_additive]
theorem div_le_div_iff_left (a : α) : a / b ≤ a / c ↔ c ≤ b 
Arguments: {α : Type u} [AddGroup α] [LE 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] {b c : α} (a : α)
Goal: a - b ≤ a - c ↔ c ≤ b
-- sub_le_sub_left
@[to_additive (attr := gcongr) sub_le_sub_left]
theorem div_le_div_left' (h : a ≤ b) (c : α) : c / b ≤ c / a 
Arguments: {α : Type u} [AddGroup α] [LE 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 : α} (h : a ≤ b) (c : α)
Goal: c - b ≤ c - a
-- div_le_div_left'
@[to_additive (attr := gcongr) sub_le_sub_left]
theorem div_le_div_left' (h : a ≤ b) (c : α) : c / b ≤ c / a 
Arguments: {α : Type u} [Group α] [LE 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 : α} (h : a ≤ b) (c : α)
Goal: c / b ≤ c / a
-- sub_le_sub_iff
@[to_additive sub_le_sub_iff]
theorem div_le_div_iff' : a / b ≤ c / d ↔ a * d ≤ c * b 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: a - b ≤ c - d ↔ a + d ≤ c + b
-- div_le_div_iff'
@[to_additive sub_le_sub_iff]
theorem div_le_div_iff' : a / b ≤ c / d ↔ a * d ≤ c * b 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: a / b ≤ c / d ↔ a * d ≤ c * b
-- le_div_iff_mul_le'
@[to_additive]
theorem le_div_iff_mul_le' : b ≤ c / a ↔ a * b ≤ c 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b ≤ c / a ↔ a * b ≤ c
-- le_sub_iff_add_le'
@[to_additive]
theorem le_div_iff_mul_le' : b ≤ c / a ↔ a * b ≤ c 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b ≤ c - a ↔ a + b ≤ c
-- div_le_iff_le_mul'
@[to_additive]
theorem div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a / b ≤ c ↔ a ≤ b * c
-- sub_le_iff_le_add'
@[to_additive]
theorem div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a - b ≤ c ↔ a ≤ b + c
-- inv_le_div_iff_le_mul
@[to_additive (attr := simp)]
theorem inv_le_div_iff_le_mul : b⁻¹ ≤ a / c ↔ c ≤ a * b 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b⁻¹ ≤ a / c ↔ c ≤ a * b
-- neg_le_sub_iff_le_add
@[to_additive (attr := simp)]
theorem inv_le_div_iff_le_mul : b⁻¹ ≤ a / c ↔ c ≤ a * b 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: -b ≤ a - c ↔ c ≤ a + b
-- inv_le_div_iff_le_mul'
@[to_additive]
theorem inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a⁻¹ ≤ b / c ↔ c ≤ a * b
-- neg_le_sub_iff_le_add'
@[to_additive]
theorem inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: -a ≤ b - c ↔ c ≤ a + b
-- sub_le_comm
@[to_additive]
theorem div_le_comm : a / b ≤ c ↔ a / c ≤ b 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a - b ≤ c ↔ a - c ≤ b
-- div_le_comm
@[to_additive]
theorem div_le_comm : a / b ≤ c ↔ a / c ≤ b 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a / b ≤ c ↔ a / c ≤ b
-- le_sub_comm
@[to_additive]
theorem le_div_comm : a ≤ b / c ↔ c ≤ b / a 
Arguments: {α : Type u} [AddCommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a ≤ b - c ↔ c ≤ b - a
-- le_div_comm
@[to_additive]
theorem le_div_comm : a ≤ b / c ↔ c ≤ b / a 
Arguments: {α : Type u} [CommGroup α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a ≤ b / c ↔ c ≤ b / a
-- div_le_div''
@[to_additive (attr := gcongr) sub_le_sub]
theorem div_le_div'' (hab : a ≤ b) (hcd : c ≤ d) : a / d ≤ b / c 
Arguments: {α : Type u} [CommGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (hab : a ≤ b) (hcd : c ≤ d)
Goal: a / d ≤ b / c
-- sub_le_sub
@[to_additive (attr := gcongr) sub_le_sub]
theorem div_le_div'' (hab : a ≤ b) (hcd : c ≤ d) : a / d ≤ b / c 
Arguments: {α : Type u} [AddCommGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α} (hab : a ≤ b) (hcd : c ≤ d)
Goal: a - d ≤ b - c
-- sub_lt_sub_iff_right
@[to_additive (attr := simp)]
theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (c : α)
Goal: a - c < b - c ↔ a < b
-- div_lt_div_iff_right
@[to_additive (attr := simp)]
theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (c : α)
Goal: a / c < b / c ↔ a < b
-- div_lt_div_right'
@[to_additive (attr := gcongr) sub_lt_sub_right]
theorem div_lt_div_right' (h : a < b) (c : α) : a / c < b / c 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : a < b) (c : α)
Goal: a / c < b / c
-- sub_lt_sub_right
@[to_additive (attr := gcongr) sub_lt_sub_right]
theorem div_lt_div_right' (h : a < b) (c : α) : a / c < b / c 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : a < b) (c : α)
Goal: a - c < b - c
-- one_lt_div'
@[to_additive (attr := simp) sub_pos]
theorem one_lt_div' : 1 < a / b ↔ b < a 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α}
Goal: 1 < a / b ↔ b < a
-- sub_pos
@[to_additive (attr := simp) sub_pos]
theorem one_lt_div' : 1 < a / b ↔ b < a 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α}
Goal: 0 < a - b ↔ b < a
-- sub_neg
@[to_additive (attr := simp) sub_neg "For `a - -b = a + b`, see `sub_neg_eq_add`."]
theorem div_lt_one' : a / b < 1 ↔ a < b 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α}
Goal: a - b < 0 ↔ a < b
-- div_lt_one'
@[to_additive (attr := simp) sub_neg "For `a - -b = a + b`, see `sub_neg_eq_add`."]
theorem div_lt_one' : a / b < 1 ↔ a < b 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α}
Goal: a / b < 1 ↔ a < b
-- lt_sub_iff_add_lt
@[to_additive]
theorem lt_div_iff_mul_lt : a < c / b ↔ a * b < c 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a < c - b ↔ a + b < c
-- lt_div_iff_mul_lt
@[to_additive]
theorem lt_div_iff_mul_lt : a < c / b ↔ a * b < c 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a < c / b ↔ a * b < c
-- div_lt_iff_lt_mul
@[to_additive]
theorem div_lt_iff_lt_mul : a / c < b ↔ a < b * c 
Arguments: {α : Type u} [Group α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a / c < b ↔ a < b * c
-- sub_lt_iff_lt_add
@[to_additive]
theorem div_lt_iff_lt_mul : a / c < b ↔ a < b * c 
Arguments: {α : Type u} [AddGroup α] [LT α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a - c < b ↔ a < b + c
-- div_lt_div_iff_left
@[to_additive (attr := simp)]
theorem div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b 
Arguments: {α : Type u} [Group α] [LT 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] {b c : α} (a : α)
Goal: a / b < a / c ↔ c < b
-- sub_lt_sub_iff_left
@[to_additive (attr := simp)]
theorem div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b 
Arguments: {α : Type u} [AddGroup α] [LT 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] {b c : α} (a : α)
Goal: a - b < a - c ↔ c < b
-- inv_lt_div_iff_lt_mul
@[to_additive (attr := simp)]
theorem inv_lt_div_iff_lt_mul : a⁻¹ < b / c ↔ c < a * b 
Arguments: {α : Type u} [Group α] [LT 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 : α}
Goal: a⁻¹ < b / c ↔ c < a * b
-- neg_lt_sub_iff_lt_add
@[to_additive (attr := simp)]
theorem inv_lt_div_iff_lt_mul : a⁻¹ < b / c ↔ c < a * b 
Arguments: {α : Type u} [AddGroup α] [LT 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 : α}
Goal: -a < b - c ↔ c < a + b
-- sub_lt_sub_left
@[to_additive (attr := gcongr) sub_lt_sub_left]
theorem div_lt_div_left' (h : a < b) (c : α) : c / b < c / a 
Arguments: {α : Type u} [AddGroup α] [LT 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 : α} (h : a < b) (c : α)
Goal: c - b < c - a
-- div_lt_div_left'
@[to_additive (attr := gcongr) sub_lt_sub_left]
theorem div_lt_div_left' (h : a < b) (c : α) : c / b < c / a 
Arguments: {α : Type u} [Group α] [LT 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 : α} (h : a < b) (c : α)
Goal: c / b < c / a
-- sub_lt_sub_iff
@[to_additive sub_lt_sub_iff]
theorem div_lt_div_iff' : a / b < c / d ↔ a * d < c * b 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α}
Goal: a - b < c - d ↔ a + d < c + b
-- div_lt_div_iff'
@[to_additive sub_lt_sub_iff]
theorem div_lt_div_iff' : a / b < c / d ↔ a * d < c * b 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α}
Goal: a / b < c / d ↔ a * d < c * b
-- lt_div_iff_mul_lt'
@[to_additive]
theorem lt_div_iff_mul_lt' : b < c / a ↔ a * b < c 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: b < c / a ↔ a * b < c
-- lt_sub_iff_add_lt'
@[to_additive]
theorem lt_div_iff_mul_lt' : b < c / a ↔ a * b < c 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: b < c - a ↔ a + b < c
-- sub_lt_iff_lt_add'
@[to_additive]
theorem div_lt_iff_lt_mul' : a / b < c ↔ a < b * c 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a - b < c ↔ a < b + c
-- div_lt_iff_lt_mul'
@[to_additive]
theorem div_lt_iff_lt_mul' : a / b < c ↔ a < b * c 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a / b < c ↔ a < b * c
-- neg_lt_sub_iff_lt_add'
@[to_additive]
theorem inv_lt_div_iff_lt_mul' : b⁻¹ < a / c ↔ c < a * b 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: -b < a - c ↔ c < a + b
-- inv_lt_div_iff_lt_mul'
@[to_additive]
theorem inv_lt_div_iff_lt_mul' : b⁻¹ < a / c ↔ c < a * b 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: b⁻¹ < a / c ↔ c < a * b
-- sub_lt_comm
@[to_additive]
theorem div_lt_comm : a / b < c ↔ a / c < b 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a - b < c ↔ a - c < b
-- div_lt_comm
@[to_additive]
theorem div_lt_comm : a / b < c ↔ a / c < b 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a / b < c ↔ a / c < b
-- lt_div_comm
@[to_additive]
theorem lt_div_comm : a < b / c ↔ c < b / a 
Arguments: {α : Type u} [CommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a < b / c ↔ c < b / a
-- lt_sub_comm
@[to_additive]
theorem lt_div_comm : a < b / c ↔ c < b / a 
Arguments: {α : Type u} [AddCommGroup α] [LT α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}
Goal: a < b - c ↔ c < b - a
-- sub_lt_sub
@[to_additive (attr := gcongr) sub_lt_sub]
theorem div_lt_div'' (hab : a < b) (hcd : c < d) : a / d < b / c 
Arguments: {α : Type u} [AddCommGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c d : α} (hab : a < b) (hcd : c < d)
Goal: a - d < b - c
-- div_lt_div''
@[to_additive (attr := gcongr) sub_lt_sub]
theorem div_lt_div'' (hab : a < b) (hcd : c < d) : a / d < b / c 
Arguments: {α : Type u} [CommGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c d : α} (hab : a < b) (hcd : c < d)
Goal: a / d < b / c
-- lt_or_lt_of_div_lt_div
@[to_additive] lemma lt_or_lt_of_div_lt_div : a / d < b / c → a < b ∨ c < d 
Arguments: {α : Type u} [CommGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: a / d < b / c → a < b ∨ c < d
-- lt_or_lt_of_sub_lt_sub
@[to_additive] lemma lt_or_lt_of_div_lt_div : a / d < b / c → a < b ∨ c < d 
Arguments: {α : Type u} [AddCommGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c d : α}
Goal: a - d < b - c → a < b ∨ c < d
-- cmp_sub_zero
@[to_additive (attr := simp) cmp_sub_zero]
theorem cmp_div_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (a b : α) :
    cmp (a / b) 1 = cmp a b 
Arguments: {α : Type u} [AddGroup α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: cmp (a - b) 0 = cmp a b
-- cmp_div_one'
@[to_additive (attr := simp) cmp_sub_zero]
theorem cmp_div_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (a b : α) :
    cmp (a / b) 1 = cmp a b 
Arguments: {α : Type u} [Group α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: cmp (a / b) 1 = cmp a b
-- le_of_forall_pos_lt_add
@[to_additive]
theorem le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
Arguments: {α : Type u} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : ∀ (ε : α), 0 < ε → a < b + ε)
Goal: a ≤ b
-- le_of_forall_one_lt_lt_mul
@[to_additive]
theorem le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
Arguments: {α : Type u} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (h : ∀ (ε : α), 1 < ε → a < b * ε)
Goal: a ≤ b
-- le_iff_forall_pos_lt_add
@[to_additive]
theorem le_iff_forall_one_lt_lt_mul : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
Arguments: {α : Type u} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 0 < ε → a < b + ε
-- le_iff_forall_one_lt_lt_mul
@[to_additive]
theorem le_iff_forall_one_lt_lt_mul : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
Arguments: {α : Type u} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 1 < ε → a < b * ε
-- sub_le_neg_add_iff
@[to_additive]
theorem div_le_inv_mul_iff [CovariantClass α α (swap (· * ·)) (· ≤ ·)] :
    a / b ≤ a⁻¹ * b ↔ a ≤ b 
Arguments: {α : Type u} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a - b ≤ -a + b ↔ a ≤ b
-- div_le_inv_mul_iff
@[to_additive]
theorem div_le_inv_mul_iff [CovariantClass α α (swap (· * ·)) (· ≤ ·)] :
    a / b ≤ a⁻¹ * b ↔ a ≤ b 
Arguments: {α : Type u} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: a / b ≤ a⁻¹ * b ↔ a ≤ b
-- div_le_div_flip
@[to_additive]
theorem div_le_div_flip {α : Type*} [CommGroup α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} : a / b ≤ b / a ↔ a ≤ b 
Arguments: {α : Type u_1} [CommGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a / b ≤ b / a ↔ a ≤ b
-- sub_le_sub_flip
@[to_additive]
theorem div_le_div_flip {α : Type*} [CommGroup α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} : a / b ≤ b / a ↔ a ≤ b 
Arguments: {α : Type u_1} [AddCommGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a - b ≤ b - a ↔ a ≤ b
-- Monotone.inv
@[to_additive]
theorem Monotone.inv (hf : Monotone f) : Antitone fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [Group 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] [Preorder β] {f : β → α} (hf : Monotone f)
Goal: Antitone fun x => (f x)⁻¹
-- Monotone.neg
@[to_additive]
theorem Monotone.inv (hf : Monotone f) : Antitone fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [AddGroup 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] [Preorder β] {f : β → α} (hf : Monotone f)
Goal: Antitone fun x => -f x
-- Antitone.inv
@[to_additive]
theorem Antitone.inv (hf : Antitone f) : Monotone fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [Group 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] [Preorder β] {f : β → α} (hf : Antitone f)
Goal: Monotone fun x => (f x)⁻¹