Theorem: MonotoneOn.exists_monotone_extension
File: Mathlib/Order/Monotone/Extension.lean

tactic: rcases hl with ⟨a, ha⟩
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a b : α
h : MonotoneOn f s
hl : BddBelow (f '' s)
hu : BddAbove (f '' s)
⊢ ∃ g, Monotone g ∧ EqOn f g s

tactic: have hu' : ∀ x, BddAbove (f '' (Iic x ∩ s)) := fun x =>
  hu.mono (image_subset _ (inter_subset_right _ _))
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : α
h : MonotoneOn f s
hu : BddAbove (f '' s)
a : β
ha : a ∈ lowerBounds (f '' s)
⊢ ∃ g, Monotone g ∧ EqOn f g s

tactic: let g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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, Monotone g ∧ EqOn f g s

tactic: have hgs : EqOn f g s := by
  intro x hx
  simp only []
  have : IsGreatest (Iic x ∩ s) x := ⟨⟨right_mem_Iic, hx⟩, fun y hy => hy.1⟩
  rw [if_neg this.nonempty.not_disjoint,
    ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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))
⊢ ∃ g, Monotone g ∧ EqOn f g s

tactic: refine' ⟨g, fun x y hxy => _, hgs⟩
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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: by_cases hx : Disjoint (Iic x) s <;> by_cases hy : Disjoint (Iic y) s <;>
  simp only [if_pos, if_neg, not_false_iff, *, refl]
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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: intro x hx
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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))
⊢ EqOn f g s

tactic: simp only []
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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 = g x

tactic: have : IsGreatest (Iic x ∩ s) x := ⟨⟨right_mem_Iic, hx⟩, fun y hy => hy.1⟩
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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: rw [if_neg this.nonempty.not_disjoint,
  ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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: rcases not_disjoint_iff_nonempty_inter.1 hy with ⟨z, hz⟩
state_before: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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))

tactic: exact le_csSup_of_le (hu' _) (mem_image_of_mem _ hz) (ha <| mem_image_of_mem _ hz.2)
state_before: case neg.intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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
z : α
hz : z ∈ Iic y ∩ s
⊢ a ≤ sSup (f '' (Iic y ∩ s))

tactic: exact (hx <| hy.mono_left <| Iic_subset_Iic.2 hxy).elim
state_before: case pos
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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: rw [not_disjoint_iff_nonempty_inter] at hx hy
state_before: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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))

tactic: refine' csSup_le_csSup (hu' _) (hx.image _) (image_subset _ _)
state_before: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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 : Set.Nonempty (Iic x ∩ s)
hy : Set.Nonempty (Iic y ∩ s)
⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))

tactic: exact inter_subset_inter_left _ (Iic_subset_Iic.2 hxy)
state_before: case neg
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : ConditionallyCompleteLinearOrder β
f : α → β
s : Set α
a✝ b : 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 : Set.Nonempty (Iic x ∩ s)
hy : Set.Nonempty (Iic y ∩ s)
⊢ Iic x ∩ s ⊆ Iic y ∩ s