ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- div_lt_comm₀
lemma div_lt_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b < c ↔ a / c < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 < b) (hc : 0 < c)
Goal: a / b < c ↔ a / c < b
-- inv_mul_le_iff
theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: b⁻¹ * a ≤ c ↔ a ≤ b * c
-- inv_mul_le_iff'
theorem inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: b⁻¹ * a ≤ c ↔ a ≤ c * b
-- mul_inv_le_iff
theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: a * b⁻¹ ≤ c ↔ a ≤ b * c
-- mul_inv_le_iff'
theorem mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: a * b⁻¹ ≤ c ↔ a ≤ c * b
-- div_self_le_one
theorem div_self_le_one (a : α) : a / a ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α)
Goal: a / a ≤ 1
-- inv_mul_lt_iff
theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: b⁻¹ * a < c ↔ a < b * c
-- inv_mul_lt_iff'
theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: b⁻¹ * a < c ↔ a < c * b
-- mul_inv_lt_iff
theorem mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: a * b⁻¹ < c ↔ a < b * c
-- mul_inv_lt_iff'
theorem mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : 0 < b)
Goal: a * b⁻¹ < c ↔ a < c * b
-- inv_pos_le_iff_one_le_mul
theorem inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a)
Goal: a⁻¹ ≤ b ↔ 1 ≤ b * a
-- inv_pos_le_iff_one_le_mul'
theorem inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a)
Goal: a⁻¹ ≤ b ↔ 1 ≤ a * b
-- inv_pos_lt_iff_one_lt_mul
theorem inv_pos_lt_iff_one_lt_mul (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a)
Goal: a⁻¹ < b ↔ 1 < b * a
-- inv_pos_lt_iff_one_lt_mul'
theorem inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a)
Goal: a⁻¹ < b ↔ 1 < a * b
-- div_le_of_nonneg_of_le_mul
theorem div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b)
Goal: a / b ≤ c
-- mul_le_of_nonneg_of_le_div
lemma mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c)
Goal: a * c ≤ b
-- div_le_one_of_le
theorem div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (h : a ≤ b) (hb : 0 ≤ b)
Goal: a / b ≤ 1
-- mul_inv_le_one_of_le
lemma mul_inv_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (h : a ≤ b) (hb : 0 ≤ b)
Goal: a * b⁻¹ ≤ 1
-- inv_mul_le_one_of_le
lemma inv_mul_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (h : a ≤ b) (hb : 0 ≤ b)
Goal: b⁻¹ * a ≤ 1
-- inv_le_inv_of_le
@[gcongr]
theorem inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : a ≤ b)
Goal: b⁻¹ ≤ a⁻¹
-- inv_le_inv
theorem inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a⁻¹ ≤ b⁻¹ ↔ b ≤ a
-- inv_le
theorem inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a⁻¹ ≤ b ↔ b⁻¹ ≤ a
-- inv_le_of_inv_le
theorem inv_le_of_inv_le (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : a⁻¹ ≤ b)
Goal: b⁻¹ ≤ a
-- le_inv
theorem le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a ≤ b⁻¹ ↔ b ≤ a⁻¹
-- inv_lt_inv
theorem inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a⁻¹ < b⁻¹ ↔ b < a
-- inv_lt_inv_of_lt
@[gcongr]
theorem inv_lt_inv_of_lt (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b) (h : b < a)
Goal: a⁻¹ < b⁻¹
-- inv_lt
theorem inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a⁻¹ < b ↔ b⁻¹ < a
-- inv_lt_of_inv_lt
theorem inv_lt_of_inv_lt (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : a⁻¹ < b)
Goal: b⁻¹ < a
-- lt_inv
theorem lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a < b⁻¹ ↔ b < a⁻¹
-- inv_lt_one
theorem inv_lt_one (ha : 1 < a) : a⁻¹ < 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 < a)
Goal: a⁻¹ < 1
-- one_lt_inv
theorem one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a < 1)
Goal: 1 < a⁻¹
-- inv_le_one
theorem inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 ≤ a)
Goal: a⁻¹ ≤ 1
-- one_le_inv
theorem one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a ≤ 1)
Goal: 1 ≤ a⁻¹
-- inv_lt_one_iff_of_pos
theorem inv_lt_one_iff_of_pos (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a)
Goal: a⁻¹ < 1 ↔ 1 < a
-- inv_lt_one_iff
theorem inv_lt_one_iff : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a
-- one_lt_inv_iff
theorem one_lt_inv_iff : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: 1 < a⁻¹ ↔ 0 < a ∧ a < 1
-- inv_le_one_iff
theorem inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a
-- one_le_inv_iff
theorem one_le_inv_iff : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1
-- div_le_div_of_nonneg_right
@[mono, gcongr]
lemma div_le_div_of_nonneg_right (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hab : a ≤ b) (hc : 0 ≤ c)
Goal: a / c ≤ b / c
-- div_lt_div_of_pos_right
@[gcongr]
lemma div_lt_div_of_pos_right (h : a < b) (hc : 0 < c) : a / c < b / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (h : a < b) (hc : 0 < c)
Goal: a / c < b / c
-- div_le_div_of_nonneg_left
@[gcongr]
lemma div_le_div_of_nonneg_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b)
Goal: a / b ≤ a / c
-- div_lt_div_of_pos_left
@[gcongr]
lemma div_lt_div_of_pos_left (ha : 0 < a) (hc : 0 < c) (h : c < b) : a / b < a / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (ha : 0 < a) (hc : 0 < c) (h : c < b)
Goal: a / b < a / c
-- div_le_div_of_le
@[deprecated div_le_div_of_nonneg_right (since := "2024-02-16")]
lemma div_le_div_of_le (hc : 0 ≤ c) (hab : a ≤ b) : a / c ≤ b / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 ≤ c) (hab : a ≤ b)
Goal: a / c ≤ b / c
-- div_le_div_right
theorem div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a / c ≤ b / c ↔ a ≤ b
-- div_lt_div_right
theorem div_lt_div_right (hc : 0 < c) : a / c < b / c ↔ a < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a / c < b / c ↔ a < b
-- div_lt_div_left
theorem div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c)
Goal: a / b < a / c ↔ c < b
-- div_le_div_left
theorem div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c)
Goal: a / b ≤ a / c ↔ c ≤ b
-- div_lt_div_iff
theorem div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b < c / d ↔ a * d < c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (b0 : 0 < b) (d0 : 0 < d)
Goal: a / b < c / d ↔ a * d < c * b
-- div_le_div_iff
theorem div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (b0 : 0 < b) (d0 : 0 < d)
Goal: a / b ≤ c / d ↔ a * d ≤ c * b
-- div_le_div
@[mono, gcongr]
theorem div_le_div (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b)
Goal: a / b ≤ c / d
-- div_lt_div
@[gcongr]
theorem div_lt_div (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) : a / b < c / d 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d)
Goal: a / b < c / d
-- div_lt_div'
theorem div_lt_div' (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) : a / b < c / d 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d)
Goal: a / b < c / d
-- div_le_self
theorem div_le_self (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 ≤ a) (hb : 1 ≤ b)
Goal: a / b ≤ a
-- div_lt_self
theorem div_lt_self (ha : 0 < a) (hb : 1 < b) : a / b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 1 < b)
Goal: a / b < a
-- le_div_self
theorem le_div_self (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) : a ≤ a / b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1)
Goal: a ≤ a / b
-- one_le_div
theorem one_le_div (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b)
Goal: 1 ≤ a / b ↔ b ≤ a
-- div_le_one
theorem div_le_one (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b)
Goal: a / b ≤ 1 ↔ a ≤ b
-- one_lt_div
theorem one_lt_div (hb : 0 < b) : 1 < a / b ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b)
Goal: 1 < a / b ↔ b < a
-- div_lt_one
theorem div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b)
Goal: a / b < 1 ↔ a < b
-- one_div_le
theorem one_div_le (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: 1 / a ≤ b ↔ 1 / b ≤ a
-- one_div_lt
theorem one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: 1 / a < b ↔ 1 / b < a
-- le_one_div
theorem le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a ≤ 1 / b ↔ b ≤ 1 / a
-- lt_one_div
theorem lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: a < 1 / b ↔ b < 1 / a
-- one_div_le_one_div_of_le
theorem one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : a ≤ b)
Goal: 1 / b ≤ 1 / a
-- one_div_lt_one_div_of_lt
theorem one_div_lt_one_div_of_lt (ha : 0 < a) (h : a < b) : 1 / b < 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : a < b)
Goal: 1 / b < 1 / a
-- le_of_one_div_le_one_div
theorem le_of_one_div_le_one_div (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : 1 / a ≤ 1 / b)
Goal: b ≤ a
-- lt_of_one_div_lt_one_div
theorem lt_of_one_div_lt_one_div (ha : 0 < a) (h : 1 / a < 1 / b) : b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (h : 1 / a < 1 / b)
Goal: b < a
-- one_div_le_one_div
theorem one_div_le_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ 1 / b ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: 1 / a ≤ 1 / b ↔ b ≤ a
-- one_div_lt_one_div
theorem one_div_lt_one_div (ha : 0 < a) (hb : 0 < b) : 1 / a < 1 / b ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 < b)
Goal: 1 / a < 1 / b ↔ b < a
-- one_lt_one_div
theorem one_lt_one_div (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a < 1)
Goal: 1 < 1 / a
-- one_le_one_div
theorem one_le_one_div (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a ≤ 1)
Goal: 1 ≤ 1 / a
-- half_pos
theorem half_pos (h : 0 < a) : 0 < a / 2 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a)
Goal: 0 < a / 2
-- one_half_pos
theorem one_half_pos : (0 : α) < 1 / 2 
Arguments: {α : Type u_2} [LinearOrderedSemifield α]
Goal: 0 < 1 / 2
-- half_le_self_iff
@[simp]
theorem half_le_self_iff : a / 2 ≤ a ↔ 0 ≤ a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: a / 2 ≤ a ↔ 0 ≤ a
-- half_lt_self_iff
@[simp]
theorem half_lt_self_iff : a / 2 < a ↔ 0 < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α}
Goal: a / 2 < a ↔ 0 < a
-- one_half_lt_one
theorem one_half_lt_one : (1 / 2 : α) < 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α]
Goal: 1 / 2 < 1
-- two_inv_lt_one
theorem two_inv_lt_one : (2⁻¹ : α) < 1 
Arguments: {α : Type u_2} [LinearOrderedSemifield α]
Goal: 2⁻¹ < 1
-- left_lt_add_div_two
theorem left_lt_add_div_two : a < (a + b) / 2 ↔ a < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α}
Goal: a < (a + b) / 2 ↔ a < b
-- add_div_two_lt_right
theorem add_div_two_lt_right : (a + b) / 2 < b ↔ a < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α}
Goal: (a + b) / 2 < b ↔ a < b
-- add_thirds
theorem add_thirds (a : α) : a / 3 + a / 3 + a / 3 = a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α)
Goal: a / 3 + a / 3 + a / 3 = a
-- div_pos_iff_of_pos_left
@[simp] lemma div_pos_iff_of_pos_left (ha : 0 < a) : 0 < a / b ↔ 0 < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a)
Goal: 0 < a / b ↔ 0 < b
-- div_pos_iff_of_pos_right
@[simp] lemma div_pos_iff_of_pos_right (hb : 0 < b) : 0 < a / b ↔ 0 < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} (hb : 0 < b)
Goal: 0 < a / b ↔ 0 < a
-- mul_le_mul_of_mul_div_le
theorem mul_le_mul_of_mul_div_le (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d : α} (h : a * (b / c) ≤ d) (hc : 0 < c)
Goal: b * a ≤ d * c
-- div_mul_le_div_mul_of_div_le_div
theorem div_mul_le_div_mul_of_div_le_div (h : a / b ≤ c / d) (he : 0 ≤ e) :
    a / (b * e) ≤ c / (d * e) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c d e : α} (h : a / b ≤ c / d) (he : 0 ≤ e)
Goal: a / (b * e) ≤ c / (d * e)
-- exists_pos_mul_lt
theorem exists_pos_mul_lt {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b * c < a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α)
Goal: ∃ c, 0 < c ∧ b * c < a
-- exists_pos_lt_mul
theorem exists_pos_lt_mul {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b < c * a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α)
Goal: ∃ c, 0 < c ∧ b < c * a
-- monotone_div_right_of_nonneg
lemma monotone_div_right_of_nonneg (ha : 0 ≤ a) : Monotone (· / a) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 ≤ a)
Goal: Monotone fun x => x / a
-- strictMono_div_right_of_pos
lemma strictMono_div_right_of_pos (ha : 0 < a) : StrictMono (· / a) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 < a)
Goal: StrictMono fun x => x / a
-- Monotone.div_const
theorem Monotone.div_const {β : Type*} [Preorder β] {f : β → α} (hf : Monotone f) {c : α}
    (hc : 0 ≤ c) : Monotone fun x => f x / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : Monotone f) {c : α} (hc : 0 ≤ c)
Goal: Monotone fun x => f x / c
-- StrictMono.div_const
theorem StrictMono.div_const {β : Type*} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α}
    (hc : 0 < c) : StrictMono fun x => f x / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α} (hc : 0 < c)
Goal: StrictMono fun x => f x / c
-- min_div_div_right
theorem min_div_div_right {c : α} (hc : 0 ≤ c) (a b : α) : min (a / c) (b / c) = min a b / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a b : α)
Goal: min (a / c) (b / c) = min a b / c
-- max_div_div_right
theorem max_div_div_right {c : α} (hc : 0 ≤ c) (a b : α) : max (a / c) (b / c) = max a b / c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a b : α)
Goal: max (a / c) (b / c) = max a b / c
-- one_div_strictAntiOn
theorem one_div_strictAntiOn : StrictAntiOn (fun x : α => 1 / x) (Set.Ioi 0) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α]
Goal: StrictAntiOn (fun x => 1 / x) (Set.Ioi 0)
-- one_div_pow_le_one_div_pow_of_le
theorem one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
    1 / a ^ n ≤ 1 / a ^ m 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n)
Goal: 1 / a ^ n ≤ 1 / a ^ m
-- one_div_pow_lt_one_div_pow_of_lt
theorem one_div_pow_lt_one_div_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) :
    1 / a ^ n < 1 / a ^ m 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m n : ℕ} (mn : m < n)
Goal: 1 / a ^ n < 1 / a ^ m
-- one_div_pow_anti
theorem one_div_pow_anti (a1 : 1 ≤ a) : Antitone fun n : ℕ => 1 / a ^ n 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a)
Goal: Antitone fun n => 1 / a ^ n
-- one_div_pow_strictAnti
theorem one_div_pow_strictAnti (a1 : 1 < a) : StrictAnti fun n : ℕ => 1 / a ^ n 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a)
Goal: StrictAnti fun n => 1 / a ^ n
-- inv_strictAntiOn
theorem inv_strictAntiOn : StrictAntiOn (fun x : α => x⁻¹) (Set.Ioi 0) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α]
Goal: StrictAntiOn (fun x => x⁻¹) (Set.Ioi 0)
-- inv_pow_le_inv_pow_of_le
theorem inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n)
Goal: (a ^ n)⁻¹ ≤ (a ^ m)⁻¹
-- inv_pow_lt_inv_pow_of_lt
theorem inv_pow_lt_inv_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) : (a ^ n)⁻¹ < (a ^ m)⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m n : ℕ} (mn : m < n)
Goal: (a ^ n)⁻¹ < (a ^ m)⁻¹
-- inv_pow_anti
theorem inv_pow_anti (a1 : 1 ≤ a) : Antitone fun n : ℕ => (a ^ n)⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a)
Goal: Antitone fun n => (a ^ n)⁻¹
-- inv_pow_strictAnti
theorem inv_pow_strictAnti (a1 : 1 < a) : StrictAnti fun n : ℕ => (a ^ n)⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a)
Goal: StrictAnti fun n => (a ^ n)⁻¹
-- IsGLB.mul_left
theorem IsGLB.mul_left {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :
    IsGLB ((fun b => a * b) '' s) (a * b) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b)
Goal: IsGLB ((fun b => a * b) '' s) (a * b)
-- IsGLB.mul_right
theorem IsGLB.mul_right {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :
    IsGLB ((fun b => b * a) '' s) (b * a) 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b)
Goal: IsGLB ((fun b => b * a) '' s) (b * a)
-- div_pos_iff
theorem div_pos_iff : 0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: 0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0
-- div_neg_iff
theorem div_neg_iff : a / b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: a / b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b
-- div_nonneg_iff
theorem div_nonneg_iff : 0 ≤ a / b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: 0 ≤ a / b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- div_nonpos_iff
theorem div_nonpos_iff : a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b
-- div_nonneg_of_nonpos
theorem div_nonneg_of_nonpos (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a / b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0)
Goal: 0 ≤ a / b
-- div_pos_of_neg_of_neg
theorem div_pos_of_neg_of_neg (ha : a < 0) (hb : b < 0) : 0 < a / b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: 0 < a / b
-- div_neg_of_neg_of_pos
theorem div_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a / b < 0 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : 0 < b)
Goal: a / b < 0
-- div_neg_of_pos_of_neg
theorem div_neg_of_pos_of_neg (ha : 0 < a) (hb : b < 0) : a / b < 0 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : 0 < a) (hb : b < 0)
Goal: a / b < 0
-- div_le_iff_of_neg
theorem div_le_iff_of_neg (hc : c < 0) : b / c ≤ a ↔ a * c ≤ b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: b / c ≤ a ↔ a * c ≤ b
-- div_le_iff_of_neg'
theorem div_le_iff_of_neg' (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: b / c ≤ a ↔ c * a ≤ b
-- le_div_iff_of_neg
theorem le_div_iff_of_neg (hc : c < 0) : a ≤ b / c ↔ b ≤ a * c 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a ≤ b / c ↔ b ≤ a * c
-- le_div_iff_of_neg'
theorem le_div_iff_of_neg' (hc : c < 0) : a ≤ b / c ↔ b ≤ c * a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a ≤ b / c ↔ b ≤ c * a
-- div_lt_iff_of_neg
theorem div_lt_iff_of_neg (hc : c < 0) : b / c < a ↔ a * c < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: b / c < a ↔ a * c < b
-- div_lt_iff_of_neg'
theorem div_lt_iff_of_neg' (hc : c < 0) : b / c < a ↔ c * a < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: b / c < a ↔ c * a < b
-- lt_div_iff_of_neg
theorem lt_div_iff_of_neg (hc : c < 0) : a < b / c ↔ b < a * c 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a < b / c ↔ b < a * c
-- lt_div_iff_of_neg'
theorem lt_div_iff_of_neg' (hc : c < 0) : a < b / c ↔ b < c * a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a < b / c ↔ b < c * a
-- div_le_one_of_ge
theorem div_le_one_of_ge (h : b ≤ a) (hb : b ≤ 0) : a / b ≤ 1 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (h : b ≤ a) (hb : b ≤ 0)
Goal: a / b ≤ 1