# `min` and `max` in linearly ordered groups. #

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@[simp]
theorem max_one_div_max_inv_one_eq_self {α : Type u_1} [Group α] [LinearOrder α] [MulLeftMono α] (a : α) :
max a 1 / max a⁻¹ 1 = a
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@[simp]
theorem max_zero_sub_max_neg_zero_eq_self {α : Type u_1} [AddGroup α] [LinearOrder α] [AddLeftMono α] (a : α) :
max a 0 - max (-a) 0 = a
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theorem max_zero_sub_eq_self {α : Type u_1} [AddGroup α] [LinearOrder α] [AddLeftMono α] (a : α) :
max a 0 - max (-a) 0 = a

Alias of `max_zero_sub_max_neg_zero_eq_self`.

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theorem max_inv_one {α : Type u_1} [Group α] [LinearOrder α] [MulLeftMono α] (a : α) :
max a⁻¹ 1 = a⁻¹ * max a 1
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theorem max_neg_zero {α : Type u_1} [AddGroup α] [LinearOrder α] [AddLeftMono α] (a : α) :
max (-a) 0 = -a + max a 0
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theorem min_inv_inv' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b : α) :
min a⁻¹ b⁻¹ = (max a b)⁻¹
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theorem min_neg_neg {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b : α) :
min (-a) (-b) = -max a b
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theorem max_inv_inv' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b : α) :
max a⁻¹ b⁻¹ = (min a b)⁻¹
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theorem max_neg_neg {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b : α) :
max (-a) (-b) = -min a b
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theorem min_div_div_right' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b c : α) :
min (a / c) (b / c) = min a b / c
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theorem min_sub_sub_right {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c : α) :
min (a - c) (b - c) = min a b - c
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theorem max_div_div_right' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b c : α) :
max (a / c) (b / c) = max a b / c
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theorem max_sub_sub_right {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c : α) :
max (a - c) (b - c) = max a b - c
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theorem min_div_div_left' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b c : α) :
min (a / b) (a / c) = a / max b c
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theorem min_sub_sub_left {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c : α) :
min (a - b) (a - c) = a - max b c
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theorem max_div_div_left' {α : Type u_1} [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (a b c : α) :
max (a / b) (a / c) = a / min b c
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theorem max_sub_sub_left {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c : α) :
max (a - b) (a - c) = a - min b c
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theorem max_sub_max_le_max {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c d : α) :
max a b - max c d ≤ max (a - c) (b - d)
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theorem abs_max_sub_max_le_max {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c d : α) :
|max a b - max c d| ≤ max |a - c| |b - d|
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theorem abs_min_sub_min_le_max {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c d : α) :
|min a b - min c d| ≤ max |a - c| |b - d|
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theorem abs_max_sub_max_le_abs {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] (a b c : α) :
|max a c - max b c| ≤ |a - b|