min and max in linearly ordered groups.

@[simp]

theorem max_zero_sub_max_neg_zero_eq_self {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) : max a 0 - max (-a) 0 = a

@[simp]

theorem max_one_div_max_inv_one_eq_self {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) : max a 1 / max a⁻¹ 1 = a

theorem max_zero_sub_eq_self {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) : max a 0 - max (-a) 0 = a

Alias of max_zero_sub_max_neg_zero_eq_self.

theorem max_neg_zero {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) : max (-a) 0 = -a + max a 0

theorem max_inv_one {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) : max a⁻¹ 1 = a⁻¹ * max a 1

theorem min_neg_neg {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : min (-a) (-b) = -max a b

theorem min_inv_inv' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) : min a⁻¹ b⁻¹ = (max a b)⁻¹

theorem max_neg_neg {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : max (-a) (-b) = -min a b

theorem max_inv_inv' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) : max a⁻¹ b⁻¹ = (min a b)⁻¹

theorem min_sub_sub_right {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : min (a - c) (b - c) = min a b - c

theorem min_div_div_right' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) (c : α) : min (a / c) (b / c) = min a b / c

theorem max_sub_sub_right {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : max (a - c) (b - c) = max a b - c

theorem max_div_div_right' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) (c : α) : max (a / c) (b / c) = max a b / c

theorem min_sub_sub_left {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : min (a - b) (a - c) = a - max b c

theorem min_div_div_left' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) (c : α) : min (a / b) (a / c) = a / max b c

theorem max_sub_sub_left {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : max (a - b) (a - c) = a - min b c

theorem max_div_div_left' {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) (c : α) : max (a / b) (a / c) = a / min b c

theorem max_sub_max_le_max {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) (d : α) : max a b - max c d ≤ max (a - c) (b - d)

theorem abs_max_sub_max_le_max {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) (d : α) : |max a b - max c d| ≤ max |a - c| |b - d|

theorem abs_min_sub_min_le_max {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) (d : α) : |min a b - min c d| ≤ max |a - c| |b - d|

theorem abs_max_sub_max_le_abs {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : |max a c - max b c| ≤ |a - b|