order.bounds

# Upper / lower bounds

In this file we define:

• upper_bounds, lower_bounds : the set of upper bounds (resp., lower bounds) of a set;
• bdd_above s, bdd_below s : the set s is bounded above (resp., below), i.e., the set of upper (resp., lower) bounds of s is nonempty;
• is_least s a, is_greatest s a : a is a least (resp., greatest) element of s; for a partial order, it is unique if exists;
• is_lub s a, is_glb s a : a is a least upper bound (resp., a greatest lower bound) of s; for a partial order, it is unique if exists.

We also prove various lemmas about monotonicity, behaviour under ∪, ∩, insert, and provide formulas for ∅, univ, and intervals.

### Definitions

def upper_bounds {α : Type u} [preorder α] (s : set α) :
set α

The set of upper bounds of a set.

Equations
• = {x : α | ∀ ⦃a : α⦄, a sa x}
def lower_bounds {α : Type u} [preorder α] (s : set α) :
set α

The set of lower bounds of a set.

Equations
• = {x : α | ∀ ⦃a : α⦄, a sx a}
def bdd_above {α : Type u} [preorder α] (s : set α) :
Prop

A set is bounded above if there exists an upper bound.

Equations
def bdd_below {α : Type u} [preorder α] (s : set α) :
Prop

A set is bounded below if there exists a lower bound.

Equations
def is_least {α : Type u} [preorder α] (s : set α) (a : α) :
Prop

a is a least element of a set s; for a partial order, it is unique if exists.

Equations
def is_greatest {α : Type u} [preorder α] (s : set α) (a : α) :
Prop

a is a greatest element of a set s; for a partial order, it is unique if exists

Equations
def is_lub {α : Type u} [preorder α] (s : set α) :
α → Prop

a is a least upper bound of a set s; for a partial order, it is unique if exists.

Equations
def is_glb {α : Type u} [preorder α] (s : set α) :
α → Prop

a is a greatest lower bound of a set s; for a partial order, it is unique if exists.

Equations
theorem mem_upper_bounds {α : Type u} [preorder α] {s : set α} {a : α} :
∀ (x : α), x sx a

theorem mem_lower_bounds {α : Type u} [preorder α] {s : set α} {a : α} :
∀ (x : α), x sa x

theorem not_bdd_above_iff' {α : Type u} [preorder α] {s : set α} :
∀ (x : α), ∃ (y : α) (H : y s), ¬y x

A set s is not bounded above if and only if for each x there exists y ∈ s such that x is not greater than or equal to y. This version only assumes preorder structure and uses ¬(y ≤ x). A version for linear orders is called not_bdd_above_iff.

theorem not_bdd_below_iff' {α : Type u} [preorder α] {s : set α} :
∀ (x : α), ∃ (y : α) (H : y s), ¬x y

A set s is not bounded below if and only if for each x there exists y ∈ s such that x is not less than or equal to y. This version only assumes preorder structure and uses ¬(x ≤ y). A version for linear orders is called not_bdd_below_iff.

theorem not_bdd_above_iff {α : Type u_1} [linear_order α] {s : set α} :
∀ (x : α), ∃ (y : α) (H : y s), x < y

A set s is not bounded above if and only if for each x there exists y ∈ s that is greater than x. A version for preorders is called not_bdd_above_iff'.

theorem not_bdd_below_iff {α : Type u_1} [linear_order α] {s : set α} :
∀ (x : α), ∃ (y : α) (H : y s), y < x

A set s is not bounded below if and only if for each x there exists y ∈ s that is less than x. A version for preorders is called not_bdd_below_iff'.

### Monotonicity

theorem upper_bounds_mono_set {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s t) :

theorem lower_bounds_mono_set {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s t) :

theorem upper_bounds_mono_mem {α : Type u} [preorder α] {s : set α} ⦃a b : α⦄ (hab : a b) :

theorem lower_bounds_mono_mem {α : Type u} [preorder α] {s : set α} ⦃a b : α⦄ (hab : a b) :

theorem upper_bounds_mono {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s t) ⦃a b : α⦄ (hab : a b) :

theorem lower_bounds_mono {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s t) ⦃a b : α⦄ (hab : a b) :

theorem bdd_above.mono {α : Type u} [preorder α] ⦃s t : set α⦄ (h : s t) :

If s ⊆ t and t is bounded above, then so is s.

theorem bdd_below.mono {α : Type u} [preorder α] ⦃s t : set α⦄ (h : s t) :

If s ⊆ t and t is bounded below, then so is s.

theorem is_lub.of_subset_of_superset {α : Type u} [preorder α] {a : α} {s t p : set α} (hs : a) (hp : a) (hst : s t) (htp : t p) :
a

If a is a least upper bound for sets s and p, then it is a least upper bound for any set t, s ⊆ t ⊆ p.

theorem is_glb.of_subset_of_superset {α : Type u} [preorder α] {a : α} {s t p : set α} (hs : a) (hp : a) (hst : s t) (htp : t p) :
a

If a is a greatest lower bound for sets s and p, then it is a greater lower bound for any set t, s ⊆ t ⊆ p.

theorem is_least.mono {α : Type u} [preorder α] {s t : set α} {a b : α} (ha : a) (hb : b) (hst : s t) :
b a

theorem is_greatest.mono {α : Type u} [preorder α] {s t : set α} {a b : α} (ha : a) (hb : b) (hst : s t) :
a b

theorem is_lub.mono {α : Type u} [preorder α] {s t : set α} {a b : α} (ha : a) (hb : b) (hst : s t) :
a b

theorem is_glb.mono {α : Type u} [preorder α] {s t : set α} {a b : α} (ha : a) (hb : b) (hst : s t) :
b a

### Conversions

theorem is_least.is_glb {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :
a

theorem is_greatest.is_lub {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :
a

theorem is_lub.upper_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

theorem is_glb.lower_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

theorem is_least.lower_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

theorem is_greatest.upper_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

theorem is_lub_le_iff {α : Type u} [preorder α] {s : set α} {a b : α} (h : a) :
a b

theorem le_is_glb_iff {α : Type u} [preorder α] {s : set α} {a b : α} (h : a) :
b a

theorem is_lub.bdd_above {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

If s has a least upper bound, then it is bounded above.

theorem is_glb.bdd_below {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

If s has a greatest lower bound, then it is bounded below.

theorem is_greatest.bdd_above {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

If s has a greatest element, then it is bounded above.

theorem is_least.bdd_below {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

If s has a least element, then it is bounded below.

theorem is_least.nonempty {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

theorem is_greatest.nonempty {α : Type u} [preorder α] {s : set α} {a : α} (h : a) :

### Union and intersection

@[simp]
theorem upper_bounds_union {α : Type u} [preorder α] {s t : set α} :

@[simp]
theorem lower_bounds_union {α : Type u} [preorder α] {s t : set α} :

theorem union_upper_bounds_subset_upper_bounds_inter {α : Type u} [preorder α] {s t : set α} :

theorem union_lower_bounds_subset_lower_bounds_inter {α : Type u} [preorder α] {s t : set α} :

theorem is_least_union_iff {α : Type u} [preorder α] {a : α} {s t : set α} :
is_least (s t) a a a

theorem is_greatest_union_iff {α : Type u} [preorder α] {s t : set α} {a : α} :

theorem bdd_above.inter_of_left {α : Type u} [preorder α] {s t : set α} (h : bdd_above s) :

If s is bounded, then so is s ∩ t

theorem bdd_above.inter_of_right {α : Type u} [preorder α] {s t : set α} (h : bdd_above t) :

If t is bounded, then so is s ∩ t

theorem bdd_below.inter_of_left {α : Type u} [preorder α] {s t : set α} (h : bdd_below s) :

If s is bounded, then so is s ∩ t

theorem bdd_below.inter_of_right {α : Type u} [preorder α] {s t : set α} (h : bdd_below t) :

If t is bounded, then so is s ∩ t

theorem bdd_above.union {γ : Type w} {s t : set γ} :
bdd_above (s t)

If s and t are bounded above sets in a semilattice_sup, then so is s ∪ t.

theorem bdd_above_union {γ : Type w} {s t : set γ} :

The union of two sets is bounded above if and only if each of the sets is.

theorem bdd_below.union {γ : Type w} {s t : set γ} :
bdd_below (s t)

theorem bdd_below_union {γ : Type w} {s t : set γ} :

The union of two sets is bounded above if and only if each of the sets is.

theorem is_lub.union {γ : Type w} {a b : γ} {s t : set γ} (hs : a) (ht : b) :
is_lub (s t) (a b)

If a is the least upper bound of s and b is the least upper bound of t, then a ⊔ b is the least upper bound of s ∪ t.

theorem is_glb.union {γ : Type w} {a₁ a₂ : γ} {s t : set γ} (hs : a₁) (ht : a₂) :
is_glb (s t) (a₁ a₂)

If a is the greatest lower bound of s and b is the greatest lower bound of t, then a ⊓ b is the greatest lower bound of s ∪ t.

theorem is_least.union {γ : Type w} [linear_order γ] {a b : γ} {s t : set γ} (ha : a) (hb : b) :
is_least (s t) (min a b)

If a is the least element of s and b is the least element of t, then min a b is the least element of s ∪ t.

theorem is_greatest.union {γ : Type w} [linear_order γ] {a b : γ} {s t : set γ} (ha : a) (hb : b) :
is_greatest (s t) (max a b)

If a is the greatest element of s and b is the greatest element of t, then max a b is the greatest element of s ∪ t.

### Specific sets

#### Unbounded intervals

theorem is_least_Ici {α : Type u} [preorder α] {a : α} :

theorem is_greatest_Iic {α : Type u} [preorder α] {a : α} :
a

theorem is_lub_Iic {α : Type u} [preorder α] {a : α} :

theorem is_glb_Ici {α : Type u} [preorder α] {a : α} :

theorem upper_bounds_Iic {α : Type u} [preorder α] {a : α} :
=

theorem lower_bounds_Ici {α : Type u} [preorder α] {a : α} :
=

theorem bdd_above_Iic {α : Type u} [preorder α] {a : α} :

theorem bdd_below_Ici {α : Type u} [preorder α] {a : α} :

theorem bdd_above_Iio {α : Type u} [preorder α] {a : α} :

theorem bdd_below_Ioi {α : Type u} [preorder α] {a : α} :

theorem is_lub_Iio {γ : Type w} [linear_order γ] {a : γ} :

theorem is_glb_Ioi {γ : Type w} [linear_order γ] {a : γ} :

theorem upper_bounds_Iio {γ : Type w} [linear_order γ] {a : γ} :
=

theorem lower_bounds_Ioi {γ : Type w} [linear_order γ] {a : γ} :
=

#### Singleton

theorem is_greatest_singleton {α : Type u} [preorder α] {a : α} :

theorem is_least_singleton {α : Type u} [preorder α] {a : α} :
is_least {a} a

theorem is_lub_singleton {α : Type u} [preorder α] {a : α} :
is_lub {a} a

theorem is_glb_singleton {α : Type u} [preorder α] {a : α} :
is_glb {a} a

theorem bdd_above_singleton {α : Type u} [preorder α] {a : α} :

theorem bdd_below_singleton {α : Type u} [preorder α] {a : α} :

@[simp]
theorem upper_bounds_singleton {α : Type u} [preorder α] {a : α} :

@[simp]
theorem lower_bounds_singleton {α : Type u} [preorder α] {a : α} :

#### Bounded intervals

theorem bdd_above_Icc {α : Type u} [preorder α] {a b : α} :

theorem bdd_below_Icc {α : Type u} [preorder α] {a b : α} :

theorem bdd_above_Ico {α : Type u} [preorder α] {a b : α} :

theorem bdd_below_Ico {α : Type u} [preorder α] {a b : α} :

theorem bdd_above_Ioc {α : Type u} [preorder α] {a b : α} :

theorem bdd_below_Ioc {α : Type u} [preorder α] {a b : α} :

theorem bdd_above_Ioo {α : Type u} [preorder α] {a b : α} :

theorem bdd_below_Ioo {α : Type u} [preorder α] {a b : α} :

theorem is_greatest_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :

theorem is_lub_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :
is_lub (set.Icc a b) b

theorem upper_bounds_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :

theorem is_least_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :

theorem is_glb_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :
is_glb (set.Icc a b) a

theorem lower_bounds_Icc {α : Type u} [preorder α] {a b : α} (h : a b) :

theorem is_greatest_Ioc {α : Type u} [preorder α] {a b : α} (h : a < b) :

theorem is_lub_Ioc {α : Type u} [preorder α] {a b : α} (h : a < b) :
is_lub (set.Ioc a b) b

theorem upper_bounds_Ioc {α : Type u} [preorder α] {a b : α} (h : a < b) :

theorem is_least_Ico {α : Type u} [preorder α] {a b : α} (h : a < b) :

theorem is_glb_Ico {α : Type u} [preorder α] {a b : α} (h : a < b) :
is_glb (set.Ico a b) a

theorem lower_bounds_Ico {α : Type u} [preorder α] {a b : α} (h : a < b) :

theorem is_glb_Ioo {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :
is_glb (set.Ioo a b) a

theorem lower_bounds_Ioo {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :

theorem is_glb_Ioc {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :
is_glb (set.Ioc a b) a

theorem lower_bound_Ioc {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :

theorem is_lub_Ioo {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :
is_lub (set.Ioo a b) b

theorem upper_bounds_Ioo {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :

theorem is_lub_Ico {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :
is_lub (set.Ico a b) b

theorem upper_bounds_Ico {γ : Type w} [linear_order γ] {a b : γ} (hab : a < b) :

theorem bdd_below_iff_subset_Ici {α : Type u} [preorder α] {s : set α} :
∃ (a : α), s

theorem bdd_above_iff_subset_Iic {α : Type u} [preorder α] {s : set α} :
∃ (a : α), s

theorem bdd_below_bdd_above_iff_subset_Icc {α : Type u} [preorder α] {s : set α} :
∃ (a b : α), s b

### Univ

theorem order_top.upper_bounds_univ {γ : Type w} [order_top γ] :

theorem is_greatest_univ {γ : Type w} [order_top γ] :

theorem is_lub_univ {γ : Type w} [order_top γ] :

theorem order_bot.lower_bounds_univ {γ : Type w} [order_bot γ] :

theorem is_least_univ {γ : Type w} [order_bot γ] :

theorem is_glb_univ {γ : Type w} [order_bot γ] :

theorem no_top_order.upper_bounds_univ {α : Type u} [preorder α] [no_top_order α] :

theorem no_bot_order.lower_bounds_univ {α : Type u} [preorder α] [no_bot_order α] :

### Empty set

@[simp]
theorem upper_bounds_empty {α : Type u} [preorder α] :

@[simp]
theorem lower_bounds_empty {α : Type u} [preorder α] :

@[simp]
theorem bdd_above_empty {α : Type u} [preorder α] [nonempty α] :

@[simp]
theorem bdd_below_empty {α : Type u} [preorder α] [nonempty α] :

theorem is_glb_empty {γ : Type w} [order_top γ] :

theorem is_lub_empty {γ : Type w} [order_bot γ] :

theorem is_lub.nonempty {α : Type u} [preorder α] {s : set α} {a : α} [no_bot_order α] (hs : a) :

theorem is_glb.nonempty {α : Type u} [preorder α] {s : set α} {a : α} [no_top_order α] (hs : a) :

theorem nonempty_of_not_bdd_above {α : Type u} [preorder α] {s : set α} [ha : nonempty α] (h : ¬) :

theorem nonempty_of_not_bdd_below {α : Type u} [preorder α] {s : set α} [ha : nonempty α] (h : ¬) :

### insert

@[simp]
theorem bdd_above_insert {γ : Type w} (a : γ) {s : set γ} :

Adding a point to a set preserves its boundedness above.

theorem bdd_above.insert {γ : Type w} (a : γ) {s : set γ} (hs : bdd_above s) :

@[simp]
theorem bdd_below_insert {γ : Type w} (a : γ) {s : set γ} :

Adding a point to a set preserves its boundedness below.

theorem bdd_below.insert {γ : Type w} (a : γ) {s : set γ} (hs : bdd_below s) :

theorem is_lub.insert {γ : Type w} (a : γ) {b : γ} {s : set γ} (hs : b) :
is_lub (insert a s) (a b)

theorem is_glb.insert {γ : Type w} (a : γ) {b : γ} {s : set γ} (hs : b) :
is_glb (insert a s) (a b)

theorem is_greatest.insert {γ : Type w} [linear_order γ] (a : γ) {b : γ} {s : set γ} (hs : b) :
is_greatest (insert a s) (max a b)

theorem is_least.insert {γ : Type w} [linear_order γ] (a : γ) {b : γ} {s : set γ} (hs : b) :
is_least (insert a s) (min a b)

@[simp]
theorem upper_bounds_insert {α : Type u} [preorder α] (a : α) (s : set α) :

@[simp]
theorem lower_bounds_insert {α : Type u} [preorder α] (a : α) (s : set α) :

@[simp]
theorem order_top.bdd_above {γ : Type w} [order_top γ] (s : set γ) :

When there is a global maximum, every set is bounded above.

@[simp]
theorem order_bot.bdd_below {γ : Type w} [order_bot γ] (s : set γ) :

When there is a global minimum, every set is bounded below.

### Pair

theorem is_lub_pair {γ : Type w} {a b : γ} :
is_lub {a, b} (a b)

theorem is_glb_pair {γ : Type w} {a b : γ} :
is_glb {a, b} (a b)

theorem is_least_pair {γ : Type w} [linear_order γ] {a b : γ} :
is_least {a, b} (min a b)

theorem is_greatest_pair {γ : Type w} [linear_order γ] {a b : γ} :
is_greatest {a, b} (max a b)

### (In)equalities with the least upper bound and the greatest lower bound

theorem lower_bounds_le_upper_bounds {α : Type u} [preorder α] {s : set α} {a b : α} (ha : a ) (hb : b ) :
s.nonemptya b

theorem is_glb_le_is_lub {α : Type u} [preorder α] {s : set α} {a b : α} (ha : a) (hb : b) (hs : s.nonempty) :
a b

theorem is_lub_lt_iff {α : Type u} [preorder α] {s : set α} {a b : α} (ha : a) :
a < b ∃ (c : α) (H : c , c < b

theorem lt_is_glb_iff {α : Type u} [preorder α] {s : set α} {a b : α} (ha : a) :
b < a ∃ (c : α) (H : c , b < c

theorem is_least.unique {α : Type u} {s : set α} {a b : α} (Ha : a) (Hb : b) :
a = b

theorem is_least.is_least_iff_eq {α : Type u} {s : set α} {a b : α} (Ha : a) :
b a = b

theorem is_greatest.unique {α : Type u} {s : set α} {a b : α} (Ha : a) (Hb : b) :
a = b

theorem is_greatest.is_greatest_iff_eq {α : Type u} {s : set α} {a b : α} (Ha : a) :
b a = b

theorem is_lub.unique {α : Type u} {s : set α} {a b : α} (Ha : a) (Hb : b) :
a = b

theorem is_glb.unique {α : Type u} {s : set α} {a b : α} (Ha : a) (Hb : b) :
a = b

theorem lt_is_lub_iff {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) :
b < a ∃ (c : α) (H : c s), b < c

theorem is_glb_lt_iff {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) :
a < b ∃ (c : α) (H : c s), c < b

theorem is_lub.exists_between {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) (hb : b < a) :
∃ (c : α) (H : c s), b < c c a

theorem is_lub.exists_between' {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) (h' : a s) (hb : b < a) :
∃ (c : α) (H : c s), b < c c < a

theorem is_glb.exists_between {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) (hb : a < b) :
∃ (c : α) (H : c s), a c c < b

theorem is_glb.exists_between' {α : Type u} [linear_order α] {s : set α} {a b : α} (h : a) (h' : a s) (hb : a < b) :
∃ (c : α) (H : c s), a < c c < b

### Least upper bound and the greatest lower bound in linear ordered additive commutative groups

theorem is_glb.exists_between_self_add {α : Type u} {s : set α} {a ε : α} (h : a) (hε : 0 < ε) :
∃ (b : α) (H : b s), a b b < a + ε

theorem is_glb.exists_between_self_add' {α : Type u} {s : set α} {a ε : α} (h : a) (h₂ : a s) (hε : 0 < ε) :
∃ (b : α) (H : b s), a < b b < a + ε

theorem is_lub.exists_between_sub_self {α : Type u} {s : set α} {a ε : α} (h : a) (hε : 0 < ε) :
∃ (b : α) (H : b s), a - ε < b b a

theorem is_lub.exists_between_sub_self' {α : Type u} {s : set α} {a ε : α} (h : a) (h₂ : a s) (hε : 0 < ε) :
∃ (b : α) (H : b s), a - ε < b b < a

### Images of upper/lower bounds under monotone functions

theorem monotone.mem_upper_bounds_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a ) :
f a upper_bounds (f '' s)

theorem monotone.mem_lower_bounds_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a ) :
f a lower_bounds (f '' s)

theorem monotone.map_bdd_above {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : monotone f) :
bdd_above (f '' s)

The image under a monotone function of a set which is bounded above is bounded above.

theorem monotone.map_bdd_below {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : monotone f) :
bdd_below (f '' s)

The image under a monotone function of a set which is bounded below is bounded below.

theorem monotone.map_is_least {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a) :
is_least (f '' s) (f a)

A monotone map sends a least element of a set to a least element of its image.

theorem monotone.map_is_greatest {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a) :
is_greatest (f '' s) (f a)

A monotone map sends a greatest element of a set to a greatest element of its image.

theorem monotone.is_lub_image_le {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a) {b : β} (Hb : is_lub (f '' s) b) :
b f a

theorem monotone.le_is_glb_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : a) {b : β} (Hb : is_glb (f '' s) b) :
f a b

theorem is_glb.of_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : ∀ {x y : α}, f x f y x y) {s : set α} {x : α} (hx : is_glb (f '' s) (f x)) :
x

theorem is_lub.of_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : ∀ {x y : α}, f x f y x y) {s : set α} {x : α} (hx : is_lub (f '' s) (f x)) :
x