/-- **Tietze extension theorem** for real-valued bounded continuous maps, a version for a closed
embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Then there
exists a bounded continuous function `g : Y →ᵇ ℝ` such that `g ∘ e = f` and each value `g y` belongs
to a closed interval `[f x₁, f x₂]` for some `x₁` and `x₂`.  -/
theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
    {e : X → Y} (he : ClosedEmbedding e) :
    ∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f := by
  inhabit X
  -- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
  obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a
  exact ⟨_, isGLB_ciInf (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).1⟩
  obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b
  exact ⟨_, isLUB_ciSup (Real.isBounded_iff_bddBelow_bddAbove.1 f.isBounded_range).2⟩
  -- Then `f x ∈ [a, b]` for all `x`
  have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.1 ⟨x, rfl⟩, hb.1 ⟨x, rfl⟩⟩
  -- Rule out the trivial case `a = b`
  have hle : a ≤ b := (hmem default).1.trans (hmem default).2
  rcases hle.eq_or_lt with (rfl | hlt)
  · have : ∀ x, f x = a := by simpa using hmem
    use const Y a
    simp [this, Function.funext_iff]
  -- Put `c = (a + b) / 2`. Then `a < c < b` and `c - a = b - c`.
  set c := (a + b) / 2
  have hac : a < c := left_lt_add_div_two.2 hlt
  have hcb : c < b := add_div_two_lt_right.2 hlt
  have hsub : c - a = b - c := by
    field_simp
    ring
  /- Due to `exists_extension_forall_mem_Icc_of_closedEmbedding`, there exists an extension `g`
    such that `g y ∈ [a, b]` for all `y`. However, if `a` and/or `b` do not belong to the range of
    `f`, then we need to ensure that these points do not belong to the range of `g`. This is done
    in two almost identical steps. First we deal with the case `∀ x, f x ≠ a`. -/
  obtain ⟨g, hg_mem, hgf⟩ : ∃ g : Y →ᵇ ℝ, (∀ y, ∃ x, g y ∈ Icc (f x) b) ∧ g ∘ e = f := by
    rcases exists_extension_forall_mem_Icc_of_closedEmbedding f hmem hle he with ⟨g, hg_mem, hgf⟩
    -- If `a ∈ range f`, then we are done.
    rcases em (∃ x, f x = a) with (⟨x, rfl⟩ | ha')
    · exact ⟨g, fun y => ⟨x, hg_mem _⟩, hgf⟩
    /- Otherwise, `g ⁻¹' {a}` is disjoint with `range e ∪ g ⁻¹' (Ici c)`, hence there exists a
        function `dg : Y → ℝ` such that `dg ∘ e = 0`, `dg y = 0` whenever `c ≤ g y`, `dg y = c - a`
        whenever `g y = a`, and `0 ≤ dg y ≤ c - a` for all `y`.  -/
    have hd : Disjoint (range e ∪ g ⁻¹' Ici c) (g ⁻¹' {a}) := by
      refine' disjoint_union_left.2 ⟨_, Disjoint.preimage _ _⟩
      · rw [Set.disjoint_left]
        rintro _ ⟨x, rfl⟩ (rfl : g (e x) = a)
        exact ha' ⟨x, (congr_fun hgf x).symm⟩
      · exact Set.disjoint_singleton_right.2 hac.not_le
    rcases exists_bounded_mem_Icc_of_closed_of_le
        (he.closed_range.union <| isClosed_Ici.preimage g.continuous)
        (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with
      ⟨dg, dg0, dga, dgmem⟩
    replace hgf : ∀ x, (g + dg) (e x) = f x
    · intro x
      simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
    refine' ⟨g + dg, fun y => _, funext hgf⟩
    · have hay : a < (g + dg) y := by
        rcases (hg_mem y).1.eq_or_lt with (rfl | hlt)
        · refine' (lt_add_iff_pos_right _).2 _
          calc
            0 < c - g y := sub_pos.2 hac
            _ = dg y := (dga rfl).symm
        · exact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)
      rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, _, hxy⟩
      refine' ⟨x, hxy.le, _⟩
      rcases le_total c (g y) with hc | hc
      · simp [dg0 (Or.inr hc), (hg_mem y).2]
      · calc
          g y + dg y ≤ c + (c - a) := add_le_add hc (dgmem _).2
          _ = b := by rw [hsub, add_sub_cancel'_right]
  /- Now we deal with the case `∀ x, f x ≠ b`. The proof is the same as in the first case, with
    minor modifications that make it hard to deduplicate code. -/
  choose xl hxl hgb using hg_mem
  rcases em (∃ x, f x = b) with (⟨x, rfl⟩ | hb')
  · exact ⟨g, fun y => ⟨xl y, x, hxl y, hgb y⟩, hgf⟩
  have hd : Disjoint (range e ∪ g ⁻¹' Iic c) (g ⁻¹' {b}) := by
    refine' disjoint_union_left.2 ⟨_, Disjoint.preimage _ _⟩
    · rw [Set.disjoint_left]
      rintro _ ⟨x, rfl⟩ (rfl : g (e x) = b)
      exact hb' ⟨x, (congr_fun hgf x).symm⟩
    · exact Set.disjoint_singleton_right.2 hcb.not_le
  rcases exists_bounded_mem_Icc_of_closed_of_le
      (he.closed_range.union <| isClosed_Iic.preimage g.continuous)
      (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with
    ⟨dg, dg0, dgb, dgmem⟩
  replace hgf : ∀ x, (g - dg) (e x) = f x
  · intro x
    simp [dg0 (Or.inl <| mem_range_self _), ← hgf]
  refine' ⟨g - dg, fun y => _, funext hgf⟩
  · have hyb : (g - dg) y < b := by
      rcases (hgb y).eq_or_lt with (rfl | hlt)
      · refine' (sub_lt_self_iff _).2 _
        calc
          0 < g y - c := sub_pos.2 hcb
          _ = dg y := (dgb rfl).symm
      · exact ((sub_le_self_iff _).2 (dgmem _).1).trans_lt hlt
    rcases hb.exists_between hyb with ⟨_, ⟨xu, rfl⟩, hyxu, _⟩
    cases' lt_or_le c (g y) with hc hc
    · rcases em (a ∈ range f) with (⟨x, rfl⟩ | _)
      · refine' ⟨x, xu, _, hyxu.le⟩
        calc
          f x = c - (b - c) := by rw [← hsub, sub_sub_cancel]
          _ ≤ g y - dg y := sub_le_sub hc.le (dgmem _).2
      · have hay : a < (g - dg) y := by
          calc
            a = c - (b - c) := by rw [← hsub, sub_sub_cancel]
            _ < g y - (b - c) := (sub_lt_sub_right hc _)
            _ ≤ g y - dg y := sub_le_sub_left (dgmem _).2 _
        rcases ha.exists_between hay with ⟨_, ⟨x, rfl⟩, _, hxy⟩
        exact ⟨x, xu, hxy.le, hyxu.le⟩
    · refine' ⟨xl y, xu, _, hyxu.le⟩
      

Proof state:
case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : ClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
xl : Y → X
hxl : ∀ (y : Y), f (xl y) ≤ g y
hgb : ∀ (y : Y), g y ≤ b
hb' : ¬∃ x, f x = b
hd : Disjoint (range e ∪ ⇑g ⁻¹' Iic c) (⇑g ⁻¹' {b})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Iic c)
dgb : EqOn (⇑dg) (Function.const Y (b - c)) (⇑g ⁻¹' {b})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (b - c)
hgf : ∀ (x : X), (g - dg) (e x) = f x
y : Y
hyb : (g - dg) y < b
xu : X
hyxu : (g - dg) y < f xu
right✝ : f xu ≤ b
hc : g y ≤ c
⊢ f (xl y) ≤ (g - dg) y

Next tactic:
simp [dg0 (Or.inr hc), hxl]