# mathlibdocumentation

algebra.order_functions

# max and min#

This file proves basic properties about maxima and minima on a linear_order.

## Tags #

min, max

@[simp]
theorem le_min_iff {α : Type u} [linear_order α] {a b c : α} :
c min a b c a c b
@[simp]
theorem max_le_iff {α : Type u} [linear_order α] {a b c : α} :
max a b c a c b c
theorem max_le_max {α : Type u} [linear_order α] {a b c d : α} :
a cb dmax a b max c d
theorem min_le_min {α : Type u} [linear_order α] {a b c d : α} :
a cb dmin a b min c d
theorem le_max_of_le_left {α : Type u} [linear_order α] {a b c : α} :
a ba max b c
theorem le_max_of_le_right {α : Type u} [linear_order α] {a b c : α} :
a ca max b c
theorem lt_max_of_lt_left {α : Type u} [linear_order α] {a b c : α} (h : a < b) :
a < max b c
theorem lt_max_of_lt_right {α : Type u} [linear_order α] {a b c : α} (h : a < c) :
a < max b c
theorem min_le_of_left_le {α : Type u} [linear_order α] {a b c : α} :
a cmin a b c
theorem min_le_of_right_le {α : Type u} [linear_order α] {a b c : α} :
b cmin a b c
theorem min_lt_of_left_lt {α : Type u} [linear_order α] {a b c : α} (h : a < c) :
min a b < c
theorem min_lt_of_right_lt {α : Type u} [linear_order α] {a b c : α} (h : b < c) :
min a b < c
theorem max_min_distrib_left {α : Type u} [linear_order α] {a b c : α} :
max a (min b c) = min (max a b) (max a c)
theorem max_min_distrib_right {α : Type u} [linear_order α] {a b c : α} :
max (min a b) c = min (max a c) (max b c)
theorem min_max_distrib_left {α : Type u} [linear_order α] {a b c : α} :
min a (max b c) = max (min a b) (min a c)
theorem min_max_distrib_right {α : Type u} [linear_order α] {a b c : α} :
min (max a b) c = max (min a c) (min b c)
theorem min_le_max {α : Type u} [linear_order α] {a b : α} :
min a b max a b
@[simp]
theorem min_eq_left_iff {α : Type u} [linear_order α] {a b : α} :
min a b = a a b
@[simp]
theorem min_eq_right_iff {α : Type u} [linear_order α] {a b : α} :
min a b = b b a
@[simp]
theorem max_eq_left_iff {α : Type u} [linear_order α] {a b : α} :
max a b = a b a
@[simp]
theorem max_eq_right_iff {α : Type u} [linear_order α] {a b : α} :
max a b = b a b
@[instance]
def max_idem {α : Type u} [linear_order α] :

An instance asserting that max a a = a

@[instance]
def min_idem {α : Type u} [linear_order α] :

An instance asserting that min a a = a

@[simp]
theorem max_lt_iff {α : Type u} [linear_order α] {a b c : α} :
max a b < c a < c b < c
@[simp]
theorem lt_min_iff {α : Type u} [linear_order α] {a b c : α} :
a < min b c a < b a < c
@[simp]
theorem lt_max_iff {α : Type u} [linear_order α] {a b c : α} :
a < max b c a < b a < c
@[simp]
theorem min_lt_iff {α : Type u} [linear_order α] {a b c : α} :
min a b < c a < c b < c
@[simp]
theorem min_le_iff {α : Type u} [linear_order α] {a b c : α} :
min a b c a c b c
@[simp]
theorem le_max_iff {α : Type u} [linear_order α] {a b c : α} :
a max b c a b a c
theorem max_lt_max {α : Type u} [linear_order α] {a b c d : α} (h₁ : a < c) (h₂ : b < d) :
max a b < max c d
theorem min_lt_min {α : Type u} [linear_order α] {a b c d : α} (h₁ : a < c) (h₂ : b < d) :
min a b < min c d
theorem min_right_comm {α : Type u} [linear_order α] (a b c : α) :
min (min a b) c = min (min a c) b
theorem max.left_comm {α : Type u} [linear_order α] (a b c : α) :
max a (max b c) = max b (max a c)
theorem max.right_comm {α : Type u} [linear_order α] (a b c : α) :
max (max a b) c = max (max a c) b
theorem monotone.map_max {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : monotone f) :
f (max a b) = max (f a) (f b)
theorem monotone.map_min {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : monotone f) :
f (min a b) = min (f a) (f b)
theorem min_choice {α : Type u} [linear_order α] (a b : α) :
min a b = a min a b = b
theorem max_choice {α : Type u} [linear_order α] (a b : α) :
max a b = a max a b = b
theorem le_of_max_le_left {α : Type u} [linear_order α] {a b c : α} (h : max a b c) :
a c
theorem le_of_max_le_right {α : Type u} [linear_order α] {a b c : α} (h : max a b c) :
b c