ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- neg_add_nonpos_iff
@[to_additive]
theorem inv_mul_le_one_iff : a⁻¹ * b ≤ 1 ↔ b ≤ 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] {a b : α}
Goal: -a + b ≤ 0 ↔ b ≤ a
-- Left.one_lt_inv_iff
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 
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] {a : α}
Goal: 1 < a⁻¹ ↔ a < 1
-- Left.neg_pos_iff
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 
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] {a : α}
Goal: 0 < -a ↔ a < 0
-- Left.inv_lt_one_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < 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] {a : α}
Goal: a⁻¹ < 1 ↔ 1 < a
-- Left.neg_neg_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < 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] {a : α}
Goal: -a < 0 ↔ 0 < a
-- lt_neg_add_iff_add_lt
@[to_additive (attr := simp)]
theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c 
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] {a b c : α}
Goal: b < -a + c ↔ a + b < c
-- lt_inv_mul_iff_mul_lt
@[to_additive (attr := simp)]
theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c 
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] {a b c : α}
Goal: b < a⁻¹ * c ↔ a * b < c
-- inv_mul_lt_iff_lt_mul
@[to_additive (attr := simp)]
theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c 
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] {a b c : α}
Goal: b⁻¹ * a < c ↔ a < b * c
-- neg_add_lt_iff_lt_add
@[to_additive (attr := simp)]
theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c 
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] {a b c : α}
Goal: -b + a < c ↔ a < b + c
-- neg_lt_iff_pos_add'
@[to_additive]
theorem inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < 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] {a b : α}
Goal: -a < b ↔ 0 < a + b
-- inv_lt_iff_one_lt_mul'
@[to_additive]
theorem inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < 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] {a b : α}
Goal: a⁻¹ < b ↔ 1 < a * b
-- lt_inv_iff_mul_lt_one'
@[to_additive]
theorem lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 
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] {a b : α}
Goal: a < b⁻¹ ↔ b * a < 1
-- lt_neg_iff_add_neg'
@[to_additive]
theorem lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 
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] {a b : α}
Goal: a < -b ↔ b + a < 0
-- lt_neg_add_iff_lt
@[to_additive]
theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < 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] {a b : α}
Goal: 0 < -b + a ↔ b < a
-- lt_inv_mul_iff_lt
@[to_additive]
theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < 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] {a b : α}
Goal: 1 < b⁻¹ * a ↔ b < a
-- inv_mul_lt_one_iff
@[to_additive]
theorem inv_mul_lt_one_iff : a⁻¹ * b < 1 ↔ b < 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] {a b : α}
Goal: a⁻¹ * b < 1 ↔ b < a
-- neg_add_neg_iff
@[to_additive]
theorem inv_mul_lt_one_iff : a⁻¹ * b < 1 ↔ b < 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] {a b : α}
Goal: -a + b < 0 ↔ b < a
-- Right.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ 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 : α}
Goal: -a ≤ 0 ↔ 0 ≤ a
-- Right.inv_le_one_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ 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 : α}
Goal: a⁻¹ ≤ 1 ↔ 1 ≤ a
-- Right.one_le_inv_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 
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 : α}
Goal: 1 ≤ a⁻¹ ↔ a ≤ 1
-- Right.nonneg_neg_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 
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 : α}
Goal: 0 ≤ -a ↔ a ≤ 0
-- inv_le_iff_one_le_mul
@[to_additive neg_le_iff_add_nonneg]
theorem inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ 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: a⁻¹ ≤ b ↔ 1 ≤ b * a
-- neg_le_iff_add_nonneg
@[to_additive neg_le_iff_add_nonneg]
theorem inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ 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: -a ≤ b ↔ 0 ≤ b + a
-- le_neg_iff_add_nonpos_right
@[to_additive]
theorem le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 
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 ↔ a + b ≤ 0
-- le_inv_iff_mul_le_one_right
@[to_additive]
theorem le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 
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 ≤ 1
-- add_neg_le_iff_le_add
@[to_additive (attr := simp)]
theorem mul_inv_le_iff_le_mul : a * b⁻¹ ≤ c ↔ a ≤ c * 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 c : α}
Goal: a + -b ≤ c ↔ a ≤ c + b
-- mul_inv_le_iff_le_mul
@[to_additive (attr := simp)]
theorem mul_inv_le_iff_le_mul : a * b⁻¹ ≤ c ↔ a ≤ c * 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 c : α}
Goal: a * b⁻¹ ≤ c ↔ a ≤ c * b
-- le_add_neg_iff_add_le
@[to_additive (attr := simp)]
theorem le_mul_inv_iff_mul_le : c ≤ a * b⁻¹ ↔ c * 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 c : α}
Goal: c ≤ a + -b ↔ c + b ≤ a
-- le_mul_inv_iff_mul_le
@[to_additive (attr := simp)]
theorem le_mul_inv_iff_mul_le : c ≤ a * b⁻¹ ↔ c * 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 c : α}
Goal: c ≤ a * b⁻¹ ↔ c * b ≤ a
-- add_neg_nonpos_iff_le
@[to_additive]
theorem mul_inv_le_one_iff_le : a * b⁻¹ ≤ 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
-- mul_inv_le_one_iff_le
@[to_additive]
theorem mul_inv_le_one_iff_le : a * b⁻¹ ≤ 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 ≤ 1 ↔ a ≤ b
-- le_add_neg_iff_le
@[to_additive]
theorem le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ 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
-- le_mul_inv_iff_le
@[to_additive]
theorem le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ 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⁻¹ ↔ b ≤ a
-- mul_inv_le_one_iff
@[to_additive]
theorem mul_inv_le_one_iff : b * a⁻¹ ≤ 1 ↔ 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: b * a⁻¹ ≤ 1 ↔ b ≤ a
-- add_neg_nonpos_iff
@[to_additive]
theorem mul_inv_le_one_iff : b * a⁻¹ ≤ 1 ↔ 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: b + -a ≤ 0 ↔ b ≤ a
-- Right.neg_neg_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < 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 : α}
Goal: -a < 0 ↔ 0 < a
-- Right.inv_lt_one_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < 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 : α}
Goal: a⁻¹ < 1 ↔ 1 < a
-- Right.neg_pos_iff
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 
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 : α}
Goal: 0 < -a ↔ a < 0
-- Right.one_lt_inv_iff
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 
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 : α}
Goal: 1 < a⁻¹ ↔ a < 1
-- neg_lt_iff_pos_add
@[to_additive]
theorem inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < 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: -a < b ↔ 0 < b + a
-- inv_lt_iff_one_lt_mul
@[to_additive]
theorem inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < 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: a⁻¹ < b ↔ 1 < b * a
-- lt_inv_iff_mul_lt_one
@[to_additive]
theorem lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 
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 < 1
-- lt_neg_iff_add_neg
@[to_additive]
theorem lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 
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 ↔ a + b < 0
-- add_neg_lt_iff_lt_add
@[to_additive (attr := simp)]
theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * 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 c : α}
Goal: a + -b < c ↔ a < c + b
-- mul_inv_lt_iff_lt_mul
@[to_additive (attr := simp)]
theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * 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 c : α}
Goal: a * b⁻¹ < c ↔ a < c * b
-- lt_add_neg_iff_add_lt
@[to_additive (attr := simp)]
theorem lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * 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 c : α}
Goal: c < a + -b ↔ c + b < a
-- lt_mul_inv_iff_mul_lt
@[to_additive (attr := simp)]
theorem lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * 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 c : α}
Goal: c < a * b⁻¹ ↔ c * b < a
-- neg_add_neg_iff_lt
@[to_additive]
theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 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
-- inv_mul_lt_one_iff_lt
@[to_additive]
theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 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 < 1 ↔ a < b
-- lt_add_neg_iff_lt
@[to_additive]
theorem lt_mul_inv_iff_lt : 1 < a * b⁻¹ ↔ 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
-- lt_mul_inv_iff_lt
@[to_additive]
theorem lt_mul_inv_iff_lt : 1 < a * b⁻¹ ↔ 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⁻¹ ↔ b < a
-- mul_inv_lt_one_iff
@[to_additive]
theorem mul_inv_lt_one_iff : b * a⁻¹ < 1 ↔ 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: b * a⁻¹ < 1 ↔ b < a
-- add_neg_neg_iff
@[to_additive]
theorem mul_inv_lt_one_iff : b * a⁻¹ < 1 ↔ 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: b + -a < 0 ↔ b < a
-- inv_le_inv_iff
@[to_additive (attr := simp)]
theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ 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}
Goal: a⁻¹ ≤ b⁻¹ ↔ b ≤ a
-- neg_le_neg_iff
@[to_additive (attr := simp)]
theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ 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}
Goal: -a ≤ -b ↔ b ≤ a
-- mul_inv_le_inv_mul_iff
@[to_additive]
theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ 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] {a b c d : α}
Goal: a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b
-- add_neg_le_neg_add_iff
@[to_additive]
theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ 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] {a b c d : α}
Goal: a + -b ≤ -d + c ↔ d + a ≤ c + b
-- div_le_self_iff
@[to_additive (attr := simp)]
theorem div_le_self_iff (a : α) {b : α} : a / b ≤ a ↔ 1 ≤ 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] (a : α) {b : α}
Goal: a / b ≤ a ↔ 1 ≤ b
-- sub_le_self_iff
@[to_additive (attr := simp)]
theorem div_le_self_iff (a : α) {b : α} : a / b ≤ a ↔ 1 ≤ 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] (a : α) {b : α}
Goal: a - b ≤ a ↔ 0 ≤ b
-- le_div_self_iff
@[to_additive (attr := simp)]
theorem le_div_self_iff (a : α) {b : α} : a ≤ a / b ↔ b ≤ 1 
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] (a : α) {b : α}
Goal: a ≤ a / b ↔ b ≤ 1
-- le_sub_self_iff
@[to_additive (attr := simp)]
theorem le_div_self_iff (a : α) {b : α} : a ≤ a / b ↔ b ≤ 1 
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] (a : α) {b : α}
Goal: a ≤ a - b ↔ b ≤ 0
-- neg_lt_neg_iff
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < 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}
Goal: -a < -b ↔ b < a
-- inv_lt_inv_iff
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < 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}
Goal: a⁻¹ < b⁻¹ ↔ b < a
-- inv_lt'
@[to_additive neg_lt]
theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < 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}
Goal: a⁻¹ < b ↔ b⁻¹ < a
-- neg_lt
@[to_additive neg_lt]
theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < 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}
Goal: -a < b ↔ -b < a
-- lt_neg
@[to_additive lt_neg]
theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ 
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}
Goal: a < -b ↔ b < -a
-- lt_inv'
@[to_additive lt_neg]
theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ 
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}
Goal: a < b⁻¹ ↔ b < a⁻¹
-- mul_inv_lt_inv_mul_iff
@[to_additive]
theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < 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] {a b c d : α}
Goal: a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b
-- add_neg_lt_neg_add_iff
@[to_additive]
theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < 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] {a b c d : α}
Goal: a + -b < -d + c ↔ d + a < c + b
-- sub_lt_self_iff
@[to_additive (attr := simp)]
theorem div_lt_self_iff (a : α) {b : α} : a / b < a ↔ 1 < 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] (a : α) {b : α}
Goal: a - b < a ↔ 0 < b
-- div_lt_self_iff
@[to_additive (attr := simp)]
theorem div_lt_self_iff (a : α) {b : α} : a / b < a ↔ 1 < 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] (a : α) {b : α}
Goal: a / b < a ↔ 1 < b
-- Left.neg_le_self
@[to_additive]
theorem Left.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 0 ≤ a)
Goal: -a ≤ a
-- Left.inv_le_self
@[to_additive]
theorem Left.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 1 ≤ a)
Goal: a⁻¹ ≤ a
-- Left.self_le_inv
@[to_additive]
theorem Left.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a ≤ 1)
Goal: a ≤ a⁻¹
-- Left.self_le_neg
@[to_additive]
theorem Left.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a ≤ 0)
Goal: a ≤ -a
-- Left.neg_lt_self
@[to_additive]
theorem Left.inv_lt_self (h : 1 < a) : a⁻¹ < a 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (h : 0 < a)
Goal: -a < a
-- Left.inv_lt_self
@[to_additive]
theorem Left.inv_lt_self (h : 1 < a) : a⁻¹ < a 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : α} (h : 1 < a)
Goal: a⁻¹ < a
-- Left.self_lt_neg
@[to_additive]
theorem Left.self_lt_inv (h : a < 1) : a < a⁻¹ 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (h : a < 0)
Goal: a < -a
-- Left.self_lt_inv
@[to_additive]
theorem Left.self_lt_inv (h : a < 1) : a < a⁻¹ 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : α} (h : a < 1)
Goal: a < a⁻¹
-- Right.neg_le_self
@[to_additive]
theorem Right.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 0 ≤ a)
Goal: -a ≤ a
-- Right.inv_le_self
@[to_additive]
theorem Right.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 1 ≤ a)
Goal: a⁻¹ ≤ a
-- Right.self_le_inv
@[to_additive]
theorem Right.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a ≤ 1)
Goal: a ≤ a⁻¹
-- Right.self_le_neg
@[to_additive]
theorem Right.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a ≤ 0)
Goal: a ≤ -a
-- Right.inv_lt_self
@[to_additive]
theorem Right.inv_lt_self (h : 1 < a) : a⁻¹ < a 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : α} (h : 1 < a)
Goal: a⁻¹ < a
-- Right.neg_lt_self
@[to_additive]
theorem Right.inv_lt_self (h : 1 < a) : a⁻¹ < a 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (h : 0 < a)
Goal: -a < a
-- Right.self_lt_inv
@[to_additive]
theorem Right.self_lt_inv (h : a < 1) : a < a⁻¹ 
Arguments: {α : Type u} [Group α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : α} (h : a < 1)
Goal: a < a⁻¹
-- Right.self_lt_neg
@[to_additive]
theorem Right.self_lt_inv (h : a < 1) : a < a⁻¹ 
Arguments: {α : Type u} [AddGroup α] [Preorder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : α} (h : a < 0)
Goal: a < -a
-- inv_mul_le_iff_le_mul'
@[to_additive]
theorem inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ 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: c⁻¹ * a ≤ b ↔ a ≤ b * c
-- neg_add_le_iff_le_add'
@[to_additive]
theorem inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ 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: -c + a ≤ b ↔ a ≤ b + c
-- mul_inv_le_iff_le_mul'
@[to_additive]
theorem mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ 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⁻¹ ≤ c ↔ a ≤ b * c
-- add_neg_le_iff_le_add'
@[to_additive]
theorem mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ 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
-- mul_inv_le_mul_inv_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} [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⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b