# Documentation

Mathlib.Init.Algebra.Functions

theorem min_def {α : Type u} [inst : ] (a : α) (b : α) :
min a b = if a b then a else b
theorem min_le_left {α : Type u} [inst : ] (a : α) (b : α) :
min a b a
theorem min_le_right {α : Type u} [inst : ] (a : α) (b : α) :
min a b b
theorem le_min {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : c a) (h₂ : c b) :
c min a b
theorem max_def {α : Type u} [inst : ] (a : α) (b : α) :
max a b = if a b then b else a
theorem le_max_left {α : Type u} [inst : ] (a : α) (b : α) :
a max a b
theorem le_max_right {α : Type u} [inst : ] (a : α) (b : α) :
b max a b
theorem max_le {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : a c) (h₂ : b c) :
max a b c
theorem eq_min {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : c a) (h₂ : c b) (h₃ : ∀ {d : α}, d ad bd c) :
c = min a b
theorem min_comm {α : Type u} [inst : ] (a : α) (b : α) :
min a b = min b a
theorem min_assoc {α : Type u} [inst : ] (a : α) (b : α) (c : α) :
min (min a b) c = min a (min b c)
theorem min_left_comm {α : Type u} [inst : ] :
@[simp]
theorem min_self {α : Type u} [inst : ] (a : α) :
min a a = a
theorem min_eq_left {α : Type u} [inst : ] {a : α} {b : α} (h : a b) :
min a b = a
theorem min_eq_right {α : Type u} [inst : ] {a : α} {b : α} (h : b a) :
min a b = b
theorem eq_max {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : a c) (h₂ : b c) (h₃ : ∀ {d : α}, a db dc d) :
c = max a b
theorem max_comm {α : Type u} [inst : ] (a : α) (b : α) :
max a b = max b a
theorem max_assoc {α : Type u} [inst : ] (a : α) (b : α) (c : α) :
max (max a b) c = max a (max b c)
theorem max_left_comm {α : Type u} [inst : ] (a : α) (b : α) (c : α) :
max a (max b c) = max b (max a c)
@[simp]
theorem max_self {α : Type u} [inst : ] (a : α) :
max a a = a
theorem max_eq_left {α : Type u} [inst : ] {a : α} {b : α} (h : b a) :
max a b = a
theorem max_eq_right {α : Type u} [inst : ] {a : α} {b : α} (h : a b) :
max a b = b
theorem min_eq_left_of_lt {α : Type u} [inst : ] {a : α} {b : α} (h : a < b) :
min a b = a
theorem min_eq_right_of_lt {α : Type u} [inst : ] {a : α} {b : α} (h : b < a) :
min a b = b
theorem max_eq_left_of_lt {α : Type u} [inst : ] {a : α} {b : α} (h : b < a) :
max a b = a
theorem max_eq_right_of_lt {α : Type u} [inst : ] {a : α} {b : α} (h : a < b) :
max a b = b
theorem lt_min {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : a < b) (h₂ : a < c) :
a < min b c
theorem max_lt {α : Type u} [inst : ] {a : α} {b : α} {c : α} (h₁ : a < c) (h₂ : b < c) :
max a b < c