# mathlib3documentation

data.list.min_max

# Minimum and maximum of lists #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

## Main definitions #

The main definitions are argmax, argmin, minimum and maximum for lists.

argmax f l returns some a, where a of l that maximises f a. If there are a b such that f a = f b, it returns whichever of a or b comes first in the list. argmax f [] = none

minimum l returns an with_top α, the smallest element of l for nonempty lists, and ⊤ for []

def list.arg_aux {α : Type u_1} (r : α α Prop) (a : option α) (b : α) :

Auxiliary definition for argmax and argmin.

Equations
@[simp]
theorem list.foldl_arg_aux_eq_none {α : Type u_1} (r : α α Prop) {l : list α} {o : option α} :
@[simp]
theorem list.arg_aux_self {α : Type u_1} (r : α α Prop) (hr₀ : irreflexive r) (a : α) :
(option.some a) a = a
theorem list.not_of_mem_foldl_arg_aux {α : Type u_1} (r : α α Prop) {l : list α} (hr₀ : irreflexive r) (hr₁ : transitive r) {a m : α} {o : option α} :
a l m o l ¬r a m
def list.argmax {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) (l : list α) :

argmax f l returns some a, where f a is maximal among the elements of l, in the sense that there is no b ∈ l with f a < f b. If a, b are such that f a = f b, it returns whichever of a or b comes first in the list. argmax f [] = none.

Equations
def list.argmin {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) (l : list α) :

argmin f l returns some a, where f a is minimal among the elements of l, in the sense that there is no b ∈ l with f b < f a. If a, b are such that f a = f b, it returns whichever of a or b comes first in the list. argmin f [] = none.

Equations
@[simp]
theorem list.argmax_nil {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) :
@[simp]
theorem list.argmin_nil {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) :
@[simp]
theorem list.argmax_singleton {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {a : α} :
[a] = a
@[simp]
theorem list.argmin_singleton {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {a : α} :
[a] = a
theorem list.not_lt_of_mem_argmax {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} {a m : α} :
a l m l ¬f m < f a
theorem list.not_lt_of_mem_argmin {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} {a m : α} :
a l m l ¬f a < f m
theorem list.argmax_concat {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) (a : α) (l : list α) :
(l ++ [a]) = l).cases_on (option.some a) (λ (c : α), ite (f c < f a) (option.some a) (option.some c))
theorem list.argmin_concat {α : Type u_1} {β : Type u_2} [preorder β] (f : α β) (a : α) (l : list α) :
(l ++ [a]) = l).cases_on (option.some a) (λ (c : α), ite (f a < f c) (option.some a) (option.some c))
theorem list.argmax_mem {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} {m : α} :
m l m l
theorem list.argmin_mem {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} {m : α} :
m l m l
@[simp]
theorem list.argmax_eq_none {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} :
@[simp]
theorem list.argmin_eq_none {α : Type u_1} {β : Type u_2} [preorder β] {f : α β} {l : list α} :
theorem list.le_of_mem_argmax {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {a m : α} :
a l m l f a f m
theorem list.le_of_mem_argmin {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {a m : α} :
a l m l f m f a
theorem list.argmax_cons {α : Type u_1} {β : Type u_2} [linear_order β] (f : α β) (a : α) (l : list α) :
(a :: l) = l).cases_on (option.some a) (λ (c : α), ite (f a < f c) (option.some c) (option.some a))
theorem list.argmin_cons {α : Type u_1} {β : Type u_2} [linear_order β] (f : α β) (a : α) (l : list α) :
(a :: l) = l).cases_on (option.some a) (λ (c : α), ite (f c < f a) (option.some c) (option.some a))
theorem list.index_of_argmax {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} [decidable_eq α] {l : list α} {m : α} :
m l {a : α}, a l f m f a
theorem list.index_of_argmin {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} [decidable_eq α] {l : list α} {m : α} :
m l {a : α}, a l f a f m
theorem list.mem_argmax_iff {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {m : α} [decidable_eq α] :
m l m l ( (a : α), a l f a f m) (a : α), a l f m f a
theorem list.argmax_eq_some_iff {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {m : α} [decidable_eq α] :
l = m l ( (a : α), a l f a f m) (a : α), a l f m f a
theorem list.mem_argmin_iff {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {m : α} [decidable_eq α] :
m l m l ( (a : α), a l f m f a) (a : α), a l f a f m
theorem list.argmin_eq_some_iff {α : Type u_1} {β : Type u_2} [linear_order β] {f : α β} {l : list α} {m : α} [decidable_eq α] :
l = m l ( (a : α), a l f m f a) (a : α), a l f a f m
def list.maximum {α : Type u_1} [preorder α] (l : list α) :

maximum l returns an with_bot α, the largest element of l for nonempty lists, and ⊥ for []

Equations
def list.minimum {α : Type u_1} [preorder α] (l : list α) :

minimum l returns an with_top α, the smallest element of l for nonempty lists, and ⊤ for []`

Equations
@[simp]
theorem list.maximum_nil {α : Type u_1} [preorder α]  :
@[simp]
theorem list.minimum_nil {α : Type u_1} [preorder α]  :
@[simp]
theorem list.maximum_singleton {α : Type u_1} [preorder α] (a : α) :
@[simp]
theorem list.minimum_singleton {α : Type u_1} [preorder α] (a : α) :
theorem list.maximum_mem {α : Type u_1} [preorder α] {l : list α} {m : α} :
theorem list.minimum_mem {α : Type u_1} [preorder α] {l : list α} {m : α} :
@[simp]
theorem list.maximum_eq_none {α : Type u_1} [preorder α] {l : list α} :
@[simp]
theorem list.minimum_eq_none {α : Type u_1} [preorder α] {l : list α} :
theorem list.not_lt_maximum_of_mem {α : Type u_1} [preorder α] {l : list α} {a m : α} :
a l l.maximum = m ¬m < a
theorem list.minimum_not_lt_of_mem {α : Type u_1} [preorder α] {l : list α} {a m : α} :
a l l.minimum = m ¬a < m
theorem list.not_lt_maximum_of_mem' {α : Type u_1} [preorder α] {l : list α} {a : α} (ha : a l) :
theorem list.not_lt_minimum_of_mem' {α : Type u_1} [preorder α] {l : list α} {a : α} (ha : a l) :
theorem list.maximum_concat {α : Type u_1} [linear_order α] (a : α) (l : list α) :
(l ++ [a]).maximum =
theorem list.le_maximum_of_mem {α : Type u_1} [linear_order α] {l : list α} {a m : α} :
a l l.maximum = m a m
theorem list.minimum_le_of_mem {α : Type u_1} [linear_order α] {l : list α} {a m : α} :
a l l.minimum = m m a
theorem list.le_maximum_of_mem' {α : Type u_1} [linear_order α] {l : list α} {a : α} (ha : a l) :
theorem list.le_minimum_of_mem' {α : Type u_1} [linear_order α] {l : list α} {a : α} (ha : a l) :
theorem list.minimum_concat {α : Type u_1} [linear_order α] (a : α) (l : list α) :
(l ++ [a]).minimum =
theorem list.maximum_cons {α : Type u_1} [linear_order α] (a : α) (l : list α) :
(a :: l).maximum =
theorem list.minimum_cons {α : Type u_1} [linear_order α] (a : α) (l : list α) :
(a :: l).minimum =
theorem list.maximum_eq_coe_iff {α : Type u_1} [linear_order α] {l : list α} {m : α} :
l.maximum = m m l (a : α), a l a m
theorem list.minimum_eq_coe_iff {α : Type u_1} [linear_order α] {l : list α} {m : α} :
l.minimum = m m l (a : α), a l m a
@[simp]
theorem list.foldr_max_of_ne_nil {α : Type u_1} [linear_order α] [order_bot α] {l : list α} (h : l list.nil) :
theorem list.max_le_of_forall_le {α : Type u_1} [linear_order α] [order_bot α] (l : list α) (a : α) (h : (x : α), x l x a) :
theorem list.le_max_of_le {α : Type u_1} [linear_order α] [order_bot α] {l : list α} {a x : α} (hx : x l) (h : a x) :
@[simp]
theorem list.foldr_min_of_ne_nil {α : Type u_1} [linear_order α] [order_top α] {l : list α} (h : l list.nil) :
theorem list.le_min_of_forall_le {α : Type u_1} [linear_order α] [order_top α] (l : list α) (a : α) (h : (x : α), x l a x) :
theorem list.min_le_of_le {α : Type u_1} [linear_order α] [order_top α] (l : list α) (a : α) {x : α} (hx : x l) (h : x a) :