Mathlib.Init.Algebra.Functions

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theorem min_def {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

min a b = if a ≤ b then a else b

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theorem min_le_left {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

min a b ≤ a

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theorem min_le_right {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

min a b ≤ b

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theorem le_min {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) :

c ≤ min a b

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theorem max_def {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

max a b = if a ≤ b then b else a

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theorem le_max_left {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

a ≤ max a b

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theorem le_max_right {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

b ≤ max a b

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theorem max_le {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) :

max a b ≤ c

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theorem eq_min {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d : α}, d ≤ a → d ≤ b → d ≤ c) :

c = min a b

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theorem min_comm {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

min a b = min b a

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theorem min_assoc {α : Type u} [inst : LinearOrder α] (a : α) (b : α) (c : α) :

min (min a b) c = min a (min b c)

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theorem min_left_comm {α : Type u} [inst : LinearOrder α] :

LeftCommutative min

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@[simp]

theorem min_self {α : Type u} [inst : LinearOrder α] (a : α) :

min a a = a

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theorem min_eq_left {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

min a b = a

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theorem min_eq_right {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : b ≤ a) :

min a b = b

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theorem eq_max {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) (h₃ : ∀ {d : α}, a ≤ d → b ≤ d → c ≤ d) :

c = max a b

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theorem max_comm {α : Type u} [inst : LinearOrder α] (a : α) (b : α) :

max a b = max b a

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theorem max_assoc {α : Type u} [inst : LinearOrder α] (a : α) (b : α) (c : α) :

max (max a b) c = max a (max b c)

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theorem max_left_comm {α : Type u} [inst : LinearOrder α] (a : α) (b : α) (c : α) :

max a (max b c) = max b (max a c)

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@[simp]

theorem max_self {α : Type u} [inst : LinearOrder α] (a : α) :

max a a = a

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theorem max_eq_left {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : b ≤ a) :

max a b = a

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theorem max_eq_right {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

max a b = b

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theorem min_eq_left_of_lt {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : a < b) :

min a b = a

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theorem min_eq_right_of_lt {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : b < a) :

min a b = b

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theorem max_eq_left_of_lt {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : b < a) :

max a b = a

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theorem max_eq_right_of_lt {α : Type u} [inst : LinearOrder α] {a : α} {b : α} (h : a < b) :

max a b = b

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theorem lt_min {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a < b) (h₂ : a < c) :

a < min b c

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theorem max_lt {α : Type u} [inst : LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a < c) (h₂ : b < c) :

max a b < c