ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Int.abs_ediv_le_abs
theorem abs_ediv_le_abs : ∀ a b : ℤ, |a / b| ≤ |a| 
Arguments: (a b : ℤ)
Goal: |a / b| ≤ |a|
-- Int.abs_sign_of_nonzero
theorem abs_sign_of_nonzero {z : ℤ} (hz : z ≠ 0) : |z.sign| = 1 
Arguments: {z : ℤ} (hz : z ≠ 0)
Goal: |z.sign| = 1
-- Int.sign_eq_ediv_abs
protected theorem sign_eq_ediv_abs (a : ℤ) : sign a = a / |a| 
Arguments: (a : ℤ)
Goal: a.sign = a / |a|
-- abs_zsmul_eq_zero
@[to_additive (attr := simp) abs_zsmul_eq_zero]
lemma zpow_abs_eq_one (a : G) (n : ℤ) : a ^ |n| = 1 ↔ a ^ n = 1 
Arguments: {G : Type u_1} [AddGroup G] (a : G) (n : ℤ)
Goal: |n| • a = 0 ↔ n • a = 0
-- zpow_abs_eq_one
@[to_additive (attr := simp) abs_zsmul_eq_zero]
lemma zpow_abs_eq_one (a : G) (n : ℤ) : a ^ |n| = 1 ↔ a ^ n = 1 
Arguments: {G : Type u_1} [Group G] (a : G) (n : ℤ)
Goal: a ^ |n| = 1 ↔ a ^ n = 1
-- max_one_div_max_inv_one_eq_self
@[to_additive (attr := simp)]
theorem max_one_div_max_inv_one_eq_self (a : α) : max a 1 / max a⁻¹ 1 = a 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: max a 1 / max a⁻¹ 1 = a
-- max_zero_sub_max_neg_zero_eq_self
@[to_additive (attr := simp)]
theorem max_one_div_max_inv_one_eq_self (a : α) : max a 1 / max a⁻¹ 1 = a 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: max a 0 - max (-a) 0 = a
-- max_neg_zero
@[to_additive]
lemma max_inv_one (a : α) : max a⁻¹ 1 = a⁻¹ * max a 1 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: max (-a) 0 = -a + max a 0
-- max_inv_one
@[to_additive]
lemma max_inv_one (a : α) : max a⁻¹ 1 = a⁻¹ * max a 1 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: max a⁻¹ 1 = a⁻¹ * max a 1
-- min_inv_inv'
@[to_additive min_neg_neg]
theorem min_inv_inv' (a b : α) : min a⁻¹ b⁻¹ = (max a b)⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b : α)
Goal: min a⁻¹ b⁻¹ = (max a b)⁻¹
-- min_neg_neg
@[to_additive min_neg_neg]
theorem min_inv_inv' (a b : α) : min a⁻¹ b⁻¹ = (max a b)⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: min (-a) (-b) = -max a b
-- max_inv_inv'
@[to_additive max_neg_neg]
theorem max_inv_inv' (a b : α) : max a⁻¹ b⁻¹ = (min a b)⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b : α)
Goal: max a⁻¹ b⁻¹ = (min a b)⁻¹
-- max_neg_neg
@[to_additive max_neg_neg]
theorem max_inv_inv' (a b : α) : max a⁻¹ b⁻¹ = (min a b)⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: max (-a) (-b) = -min a b
-- min_sub_sub_right
@[to_additive min_sub_sub_right]
theorem min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: min (a - c) (b - c) = min a b - c
-- min_div_div_right'
@[to_additive min_sub_sub_right]
theorem min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b c : α)
Goal: min (a / c) (b / c) = min a b / c
-- max_div_div_right'
@[to_additive max_sub_sub_right]
theorem max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b c : α)
Goal: max (a / c) (b / c) = max a b / c
-- max_sub_sub_right
@[to_additive max_sub_sub_right]
theorem max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: max (a - c) (b - c) = max a b - c
-- min_sub_sub_left
@[to_additive min_sub_sub_left]
theorem min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: min (a - b) (a - c) = a - max b c
-- min_div_div_left'
@[to_additive min_sub_sub_left]
theorem min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b c : α)
Goal: min (a / b) (a / c) = a / max b c
-- max_sub_sub_left
@[to_additive max_sub_sub_left]
theorem max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: max (a - b) (a - c) = a - min b c
-- max_div_div_left'
@[to_additive max_sub_sub_left]
theorem max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b c : α)
Goal: max (a / b) (a / c) = a / min b c
-- max_sub_max_le_max
theorem max_sub_max_le_max (a b c d : α) : max a b - max c d ≤ max (a - c) (b - d) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c d : α)
Goal: max a b - max c d ≤ max (a - c) (b - d)
-- abs_max_sub_max_le_max
theorem abs_max_sub_max_le_max (a b c d : α) : |max a b - max c d| ≤ max |a - c| |b - d| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c d : α)
Goal: |max a b - max c d| ≤ max |a - c| |b - d|
-- abs_min_sub_min_le_max
theorem abs_min_sub_min_le_max (a b c d : α) : |min a b - min c d| ≤ max |a - c| |b - d| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c d : α)
Goal: |min a b - min c d| ≤ max |a - c| |b - d|
-- abs_max_sub_max_le_abs
theorem abs_max_sub_max_le_abs (a b c : α) : |max a c - max b c| ≤ |a - b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: |max a c - max b c| ≤ |a - b|
-- zpow_le_of_le
@[gcongr]
theorem zpow_le_of_le (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m n : ℤ} (ha : 1 ≤ a) (h : m ≤ n)
Goal: a ^ m ≤ a ^ n
-- zpow_le_one_of_nonpos
theorem zpow_le_one_of_nonpos (ha : 1 ≤ a) (hn : n ≤ 0) : a ^ n ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : n ≤ 0)
Goal: a ^ n ≤ 1
-- one_le_zpow_of_nonneg
theorem one_le_zpow_of_nonneg (ha : 1 ≤ a) (hn : 0 ≤ n) : 1 ≤ a ^ n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : 0 ≤ n)
Goal: 1 ≤ a ^ n
-- Nat.zpow_pos_of_pos
protected theorem Nat.zpow_pos_of_pos {a : ℕ} (h : 0 < a) (n : ℤ) : 0 < (a : α) ^ n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ)
Goal: 0 < ↑a ^ n
-- Nat.zpow_ne_zero_of_pos
theorem Nat.zpow_ne_zero_of_pos {a : ℕ} (h : 0 < a) (n : ℤ) : (a : α) ^ n ≠ 0 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ)
Goal: ↑a ^ n ≠ 0
-- one_lt_zpow
theorem one_lt_zpow (ha : 1 < a) : ∀ n : ℤ, 0 < n → 1 < a ^ n
  
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} (ha : 1 < a) (n : ℤ)
Goal: 0 < n → 1 < a ^ n
-- zpow_strictMono
theorem zpow_strictMono (hx : 1 < a) : StrictMono (a ^ · : ℤ → α) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} (hx : 1 < a)
Goal: StrictMono fun x => a ^ x
-- zpow_strictAnti
theorem zpow_strictAnti (h₀ : 0 < a) (h₁ : a < 1) : StrictAnti (a ^ · : ℤ → α) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a < 1)
Goal: StrictAnti fun x => a ^ x
-- zpow_lt_iff_lt
@[simp]
theorem zpow_lt_iff_lt (hx : 1 < a) : a ^ m < a ^ n ↔ m < n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m n : ℤ} (hx : 1 < a)
Goal: a ^ m < a ^ n ↔ m < n
-- zpow_le_iff_le
@[simp]
theorem zpow_le_iff_le (hx : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m n : ℤ} (hx : 1 < a)
Goal: a ^ m ≤ a ^ n ↔ m ≤ n
-- div_pow_le
@[simp]
theorem div_pow_le (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ) : a / b ^ k ≤ a 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a b : α} (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ)
Goal: a / b ^ k ≤ a
-- zpow_injective
theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a ≠ 1)
Goal: Function.Injective fun x => a ^ x
-- zpow_inj
@[simp]
theorem zpow_inj (h₀ : 0 < a) (h₁ : a ≠ 1) : a ^ m = a ^ n ↔ m = n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m n : ℤ} (h₀ : 0 < a) (h₁ : a ≠ 1)
Goal: a ^ m = a ^ n ↔ m = n
-- zpow_le_max_of_min_le
theorem zpow_le_max_of_min_le {x : α} (hx : 1 ≤ x) {a b c : ℤ} (h : min a b ≤ c) :
    x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 ≤ x) {a b c : ℤ} (h : min a b ≤ c)
Goal: x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b))
-- zpow_le_max_iff_min_le
theorem zpow_le_max_iff_min_le {x : α} (hx : 1 < x) {a b c : ℤ} :
    x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 < x) {a b c : ℤ}
Goal: x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c
-- Even.zpow_nonneg
protected theorem Even.zpow_nonneg (hn : Even n) (a : α) : 0 ≤ a ^ n 
Arguments: {α : Type u_1} [LinearOrderedField α] {n : ℤ} (hn : Even n) (a : α)
Goal: 0 ≤ a ^ n
-- zpow_two_nonneg
lemma zpow_two_nonneg (a : α) : 0 ≤ a ^ (2 : ℤ) 
Arguments: {α : Type u_1} [LinearOrderedField α] (a : α)
Goal: 0 ≤ a ^ 2
-- zpow_neg_two_nonneg
lemma zpow_neg_two_nonneg (a : α) : 0 ≤ a ^ (-2 : ℤ) 
Arguments: {α : Type u_1} [LinearOrderedField α] (a : α)
Goal: 0 ≤ a ^ (-2)
-- Even.zpow_pos
protected lemma Even.zpow_pos (hn : Even n) (ha : a ≠ 0) : 0 < a ^ n 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (ha : a ≠ 0)
Goal: 0 < a ^ n
-- zpow_two_pos_of_ne_zero
lemma zpow_two_pos_of_ne_zero (ha : a ≠ 0) : 0 < a ^ (2 : ℤ) 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} (ha : a ≠ 0)
Goal: 0 < a ^ 2
-- Even.zpow_pos_iff
theorem Even.zpow_pos_iff (hn : Even n) (h : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (h : n ≠ 0)
Goal: 0 < a ^ n ↔ a ≠ 0
-- Odd.zpow_neg_iff
theorem Odd.zpow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n)
Goal: a ^ n < 0 ↔ a < 0
-- Odd.zpow_nonneg_iff
protected lemma Odd.zpow_nonneg_iff (hn : Odd n) : 0 ≤ a ^ n ↔ 0 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n)
Goal: 0 ≤ a ^ n ↔ 0 ≤ a
-- Odd.zpow_nonpos_iff
theorem Odd.zpow_nonpos_iff (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n)
Goal: a ^ n ≤ 0 ↔ a ≤ 0
-- Odd.zpow_pos_iff
lemma Odd.zpow_pos_iff (hn : Odd n) : 0 < a ^ n ↔ 0 < a 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n)
Goal: 0 < a ^ n ↔ 0 < a
-- Even.zpow_abs
theorem Even.zpow_abs {p : ℤ} (hp : Even p) (a : α) : |a| ^ p = a ^ p 
Arguments: {α : Type u_1} [LinearOrderedField α] {p : ℤ} (hp : Even p) (a : α)
Goal: |a| ^ p = a ^ p
-- Nat.cast_le_pow_sub_div_sub
theorem Nat.cast_le_pow_sub_div_sub (H : 1 < a) (n : ℕ) : (n : α) ≤ (a ^ n - 1) / (a - 1) 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} (H : 1 < a) (n : ℕ)
Goal: ↑n ≤ (a ^ n - 1) / (a - 1)
-- Nat.cast_le_pow_div_sub
theorem Nat.cast_le_pow_div_sub (H : 1 < a) (n : ℕ) : (n : α) ≤ a ^ n / (a - 1) 
Arguments: {α : Type u_1} [LinearOrderedField α] {a : α} (H : 1 < a) (n : ℕ)
Goal: ↑n ≤ a ^ n / (a - 1)
-- le_of_forall_neg_add_le
@[to_additive]
theorem le_of_forall_lt_one_mul_le (h : ∀ ε < 1, a * ε ≤ b) : a ≤ b 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α} (h : ∀ (ε : α), ε < 0 → a + ε ≤ b)
Goal: a ≤ b
-- le_of_forall_lt_one_mul_le
@[to_additive]
theorem le_of_forall_lt_one_mul_le (h : ∀ ε < 1, a * ε ≤ b) : a ≤ b 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α} (h : ∀ (ε : α), ε < 1 → a * ε ≤ b)
Goal: a ≤ b
-- le_of_forall_pos_sub_le
@[to_additive]
theorem le_of_forall_one_lt_div_le (h : ∀ ε : α, 1 < ε → a / ε ≤ b) : a ≤ b 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α} (h : ∀ (ε : α), 0 < ε → a - ε ≤ b)
Goal: a ≤ b
-- le_of_forall_one_lt_div_le
@[to_additive]
theorem le_of_forall_one_lt_div_le (h : ∀ ε : α, 1 < ε → a / ε ≤ b) : a ≤ b 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α} (h : ∀ (ε : α), 1 < ε → a / ε ≤ b)
Goal: a ≤ b
-- le_iff_forall_one_lt_le_mul
@[to_additive]
theorem le_iff_forall_one_lt_le_mul : a ≤ b ↔ ∀ ε, 1 < ε → a ≤ b * ε 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 1 < ε → a ≤ b * ε
-- le_iff_forall_pos_le_add
@[to_additive]
theorem le_iff_forall_one_lt_le_mul : a ≤ b ↔ ∀ ε, 1 < ε → a ≤ b * ε 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 0 < ε → a ≤ b + ε
-- le_iff_forall_lt_one_mul_le
@[to_additive]
theorem le_iff_forall_lt_one_mul_le : a ≤ b ↔ ∀ ε < 1, a * ε ≤ b 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), ε < 1 → a * ε ≤ b
-- le_iff_forall_neg_add_le
@[to_additive]
theorem le_iff_forall_lt_one_mul_le : a ≤ b ↔ ∀ ε < 1, a * ε ≤ b 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [DenselyOrdered α] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), ε < 0 → a + ε ≤ b
-- AddCommGroup.PositiveCone.pos_iff
-- Porting note(#5171): @[nolint has_nonempty_instance]
structure PositiveCone (α : Type*) [AddCommGroup α] where
    nonneg : α → Prop
    pos : α → Prop := fun a => nonneg a ∧ ¬nonneg (-a)
  pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬nonneg (-a) := by intros; rfl
  zero_nonneg : nonneg 0
  add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b)
  nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0
Arguments: {α : Type u_1} [AddCommGroup α] (self : AddCommGroup.PositiveCone α) (a : α)
Goal: self.pos a ↔ self.nonneg a ∧ ¬self.nonneg (-a)
-- AddCommGroup.PositiveCone.add_nonneg
-- Porting note(#5171): @[nolint has_nonempty_instance]
structure PositiveCone (α : Type*) [AddCommGroup α] where
    nonneg : α → Prop
    pos : α → Prop := fun a => nonneg a ∧ ¬nonneg (-a)
  pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬nonneg (-a) := by intros; rfl
  zero_nonneg : nonneg 0
  add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b)
  nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0
Arguments: {α : Type u_1} [AddCommGroup α] (self : AddCommGroup.PositiveCone α) {a b : α}
Goal: self.nonneg a → self.nonneg b → self.nonneg (a + b)
-- AddCommGroup.PositiveCone.nonneg_antisymm
-- Porting note(#5171): @[nolint has_nonempty_instance]
structure PositiveCone (α : Type*) [AddCommGroup α] where
    nonneg : α → Prop
    pos : α → Prop := fun a => nonneg a ∧ ¬nonneg (-a)
  pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬nonneg (-a) := by intros; rfl
  zero_nonneg : nonneg 0
  add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b)
  nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0
Arguments: {α : Type u_1} [AddCommGroup α] (self : AddCommGroup.PositiveCone α) {a : α}
Goal: self.nonneg a → self.nonneg (-a) → a = 0
-- AddCommGroup.PositiveCone.zero_nonneg
-- Porting note(#5171): @[nolint has_nonempty_instance]
structure PositiveCone (α : Type*) [AddCommGroup α] where
    nonneg : α → Prop
    pos : α → Prop := fun a => nonneg a ∧ ¬nonneg (-a)
  pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬nonneg (-a) := by intros; rfl
  zero_nonneg : nonneg 0
  add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b)
  nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0
Arguments: {α : Type u_1} [AddCommGroup α] (self : AddCommGroup.PositiveCone α)
Goal: self.nonneg 0
-- AddCommGroup.TotalPositiveCone.nonneg_total
-- Porting note(#5171): @[nolint has_nonempty_instance]
structure TotalPositiveCone (α : Type*) [AddCommGroup α] extends PositiveCone α where
    nonnegDecidable : DecidablePred nonneg
    nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a)
Arguments: {α : Type u_1} [AddCommGroup α] (self : AddCommGroup.TotalPositiveCone α) (a : α)
Goal: self.nonneg a ∨ self.nonneg (-a)
-- mul_sup
@[to_additive]
lemma mul_sup (a b c : α) : c * (a ⊔ b) = c * a ⊔ c * b 
Arguments: {α : Type u_1} [Lattice α] [Group 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) = c * a ⊔ c * b
-- add_sup
@[to_additive]
lemma mul_sup (a b c : α) : c * (a ⊔ b) = c * a ⊔ c * b 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 = (c + a) ⊔ (c + b)
-- sup_mul
@[to_additive]
lemma sup_mul (a b c : α) : (a ⊔ b) * c = a * c ⊔ b * c 
Arguments: {α : Type u_1} [Lattice α] [Group 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 * c
-- sup_add
@[to_additive]
lemma sup_mul (a b c : α) : (a ⊔ b) * c = a * c ⊔ b * c 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 + c)
-- mul_inf
@[to_additive]
lemma mul_inf (a b c : α) : c * (a ⊓ b) = c * a ⊓ c * b 
Arguments: {α : Type u_1} [Lattice α] [Group 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) = c * a ⊓ c * b
-- add_inf
@[to_additive]
lemma mul_inf (a b c : α) : c * (a ⊓ b) = c * a ⊓ c * b 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 = (c + a) ⊓ (c + b)
-- inf_add
@[to_additive]
lemma inf_mul (a b c : α) : (a ⊓ b) * c = a * c ⊓ b * c 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 + c)
-- inf_mul
@[to_additive]
lemma inf_mul (a b c : α) : (a ⊓ b) * c = a * c ⊓ b * c 
Arguments: {α : Type u_1} [Lattice α] [Group 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 * c
-- inv_sup
@[to_additive] lemma inv_sup (a b : α) : (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group 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 = a⁻¹ ⊓ b⁻¹
-- neg_sup
@[to_additive] lemma inv_sup (a b : α) : (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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) = -a ⊓ -b
-- inv_inf
@[to_additive] lemma inv_inf (a b : α) : (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group 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 = a⁻¹ ⊔ b⁻¹
-- neg_inf
@[to_additive] lemma inv_inf (a b : α) : (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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) = -a ⊔ -b
-- sub_sup
@[to_additive]
lemma div_sup (a b c : α) : c / (a ⊔ b) = c / a ⊓ c / b 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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: c - a ⊔ b = (c - a) ⊓ (c - b)
-- div_sup
@[to_additive]
lemma div_sup (a b c : α) : c / (a ⊔ b) = c / a ⊓ c / b 
Arguments: {α : Type u_1} [Lattice α] [Group 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: c / (a ⊔ b) = c / a ⊓ c / b
-- sup_sub
@[to_additive]
lemma sup_div (a b c : α) : (a ⊔ b) / c = a / c ⊔ b / c 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 - c)
-- sup_div
@[to_additive]
lemma sup_div (a b c : α) : (a ⊔ b) / c = a / c ⊔ b / c 
Arguments: {α : Type u_1} [Lattice α] [Group 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 / c
-- div_inf
@[to_additive]
lemma div_inf (a b c : α) : c / (a ⊓ b) = c / a ⊔ c / b 
Arguments: {α : Type u_1} [Lattice α] [Group 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: c / (a ⊓ b) = c / a ⊔ c / b
-- sub_inf
@[to_additive]
lemma div_inf (a b c : α) : c / (a ⊓ b) = c / a ⊔ c / b 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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: c - a ⊓ b = (c - a) ⊔ (c - b)
-- inf_div
@[to_additive]
lemma inf_div (a b c : α) : (a ⊓ b) / c = a / c ⊓ b / c 
Arguments: {α : Type u_1} [Lattice α] [Group 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 / c
-- inf_sub
@[to_additive]
lemma inf_div (a b c : α) : (a ⊓ b) / c = a / c ⊓ b / c 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 - c)
-- pow_two_semiclosed
@[to_additive]
lemma pow_two_semiclosed
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a : α} (ha : 1 ≤ a ^ 2) : 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [Group 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 : α} (ha : 1 ≤ a ^ 2)
Goal: 1 ≤ a
-- nsmul_two_semiclosed
@[to_additive]
lemma pow_two_semiclosed
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a : α} (ha : 1 ≤ a ^ 2) : 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup 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 : α} (ha : 0 ≤ 2 • a)
Goal: 0 ≤ a
-- inf_add_sup
@[to_additive]
lemma inf_mul_sup [CovariantClass α α (· * ·) (· ≤ ·)] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊓ b + a ⊔ b = a + b
-- inf_mul_sup
@[to_additive]
lemma inf_mul_sup [CovariantClass α α (· * ·) (· ≤ ·)] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: (a ⊓ b) * (a ⊔ b) = a * b
-- monovaryOn_inv_left
@[to_additive (attr := simp)]
lemma monovaryOn_inv_left : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s
-- monovaryOn_neg_left
@[to_additive (attr := simp)]
lemma monovaryOn_inv_left : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: MonovaryOn (-f) g s ↔ AntivaryOn f g s
-- antivaryOn_inv_left
@[to_additive (attr := simp)]
lemma antivaryOn_inv_left : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedCommGroup α] [OrderedCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s
-- antivaryOn_neg_left
@[to_additive (attr := simp)]
lemma antivaryOn_inv_left : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} [OrderedAddCommGroup α] [OrderedAddCommGroup β] {s : Set ι} {f : ι → α} {g : ι → β}
Goal: AntivaryOn (-f) g s ↔ MonovaryOn f g s