-- Mathlib/Order/Monotone/Extension.lean
theorem MonotoneOn.exists_monotone_extension (h : MonotoneOn f s) (hl : BddBelow (f '' s))
    (hu : BddAbove (f '' s)) : ∃ g : α → β, Monotone g ∧ EqOn f g s

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : (Iic x ∩ s).Nonempty
hy : (Iic y ∩ s).Nonempty
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))

Tactic:
refine csSup_le_csSup (hu' _) (hx.image _) (image_subset _ ?_)

Result:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : (Iic x ∩ s).Nonempty
hy : (Iic y ∩ s).Nonempty
⊢ Iic x ∩ s ⊆ Iic y ∩ s

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
x : α
hx : x ∈ s
this : IsGreatest (Iic x ∩ s) x
⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))

Tactic:
rw [if_neg this.nonempty.not_disjoint,
  ((h.mono inter_subset_right).map_isGreatest this).csSup_eq]

Result:
no goals

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ a

Tactic:
exact (hx <| hy.mono_left <| Iic_subset_Iic.2 hxy).elim

Result:
no goals

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
⊢ g x ≤ g y

Tactic:
by_cases hx : Disjoint (Iic x) s <;> by_cases hy : Disjoint (Iic y) s <;>
  simp only [g, if_pos, if_neg, not_false_iff, *, refl]

Result:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ a ≤ sSup (f '' (Iic y ∩ s))

case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ a

case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : ¬Disjoint (Iic x) s
hy : ¬Disjoint (Iic y) s
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
hx : (Iic x ∩ s).Nonempty
hy : (Iic y ∩ s).Nonempty
⊢ Iic x ∩ s ⊆ Iic y ∩ s

Tactic:
exact inter_subset_inter_left _ (Iic_subset_Iic.2 hxy)

Result:
no goals

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
⊢ ∃ g, Monotone g ∧ EqOn f g s

Tactic:
refine ⟨g, fun x y hxy => ?_, hgs⟩

Result:
case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
hgs : EqOn f g s
x y : α
hxy : x ≤ y
⊢ g x ≤ g y

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
x : α
hx : x ∈ s
⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))

Tactic:
have : IsGreatest (Iic x ∩ s) x := ⟨⟨right_mem_Iic, hx⟩, fun y hy => hy.1⟩

Result:
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
hu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))
g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
x : α
hx : x ∈ s
this : IsGreatest (Iic x ∩ s) x
⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))