/-- If there are subsets which are `ε₁`-dense and `ε₃`-dense in two spaces, and
isometric up to `ε₂`, then the Gromov-Hausdorff distance between the spaces is bounded by
`ε₁ + ε₂/2 + ε₃`. -/
theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
    (hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
    (H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ := by
  refine' le_of_forall_pos_le_add fun δ δ0 => _
  rcases exists_mem_of_nonempty X with ⟨xX, _⟩
  rcases hs xX with ⟨xs, hxs, Dxs⟩
  have sne : s.Nonempty := ⟨xs, hxs⟩
  letI : Nonempty s := sne.to_subtype
  have : 0 ≤ ε₂ := le_trans (abs_nonneg _) (H ⟨xs, hxs⟩ ⟨xs, hxs⟩)
  have : ∀ p q : s, |dist p q - dist (Φ p) (Φ q)| ≤ 2 * (ε₂ / 2 + δ) := fun p q =>
    calc
      |dist p q - dist (Φ p) (Φ q)| ≤ ε₂ := H p q
      _ ≤ 2 * (ε₂ / 2 + δ) := by linarith
  -- glue `X` and `Y` along the almost matching subsets
  letI : MetricSpace (Sum X Y) :=
    glueMetricApprox (fun x : s => (x : X)) (fun x => Φ x) (ε₂ / 2 + δ) (by linarith) this
  let Fl := @Sum.inl X Y
  let Fr := @Sum.inr X Y
  have Il : Isometry Fl := Isometry.of_dist_eq fun x y => rfl
  have Ir : Isometry Fr := Isometry.of_dist_eq fun x y => rfl
  /- The proof goes as follows : the `GH_dist` is bounded by the Hausdorff distance of the images
    in the coupling, which is bounded (using the triangular inequality) by the sum of the Hausdorff
    distances of `X` and `s` (in the coupling or, equivalently in the original space), of `s` and
    `Φ s`, and of `Φ s` and `Y` (in the coupling or, equivalently, in the original space).
    The first term is bounded by `ε₁`, by `ε₁`-density. The third one is bounded by `ε₃`.
    And the middle one is bounded by `ε₂/2` as in the coupling the points `x` and `Φ x` are
    at distance `ε₂/2` by construction of the coupling (in fact `ε₂/2 + δ` where `δ` is an
    arbitrarily small positive constant where positivity is used to ensure that the coupling
    is really a metric space and not a premetric space on `X ⊕ Y`). -/
  have : ghDist X Y ≤ hausdorffDist (range Fl) (range Fr) := ghDist_le_hausdorffDist Il Ir
  have :
    hausdorffDist (range Fl) (range Fr) ≤
      hausdorffDist (range Fl) (Fl '' s) + hausdorffDist (Fl '' s) (range Fr) :=
    have B : IsBounded (range Fl) := (isCompact_range Il.continuous).isBounded
    hausdorffDist_triangle
      (hausdorffEdist_ne_top_of_nonempty_of_bounded (range_nonempty _) (sne.image _) B
        (B.subset (image_subset_range _ _)))
  have :
    hausdorffDist (Fl '' s) (range Fr) ≤
      hausdorffDist (Fl '' s) (Fr '' range Φ) + hausdorffDist (Fr '' range Φ) (range Fr) :=
    have B : IsBounded (range Fr) := (isCompact_range Ir.continuous).isBounded
    hausdorffDist_triangle'
      (hausdorffEdist_ne_top_of_nonempty_of_bounded ((range_nonempty _).image _) (range_nonempty _)
        (B.subset (image_subset_range _ _)) B)
  have : hausdorffDist (range Fl) (Fl '' s) ≤ ε₁ := by
    rw [← image_univ, hausdorffDist_image Il]
    have : 0 ≤ ε₁ := le_trans dist_nonneg Dxs
    refine' hausdorffDist_le_of_mem_dist this (fun x _ => hs x) fun x _ =>
      ⟨x, mem_univ _, by simpa only [dist_self]⟩
  have : hausdorffDist (Fl '' s) (Fr '' range Φ) ≤ ε₂ / 2 + δ := by
    refine' hausdorffDist_le_of_mem_dist (by linarith) _ _
    · intro x' hx'
      rcases (Set.mem_image _ _ _).1 hx' with ⟨x, ⟨x_in_s, xx'⟩⟩
      rw [← xx']
      use Fr (Φ ⟨x, x_in_s⟩), mem_image_of_mem Fr (mem_range_self _)
      exact le_of_eq (glueDist_glued_points (fun x : s => (x : X)) Φ (ε₂ / 2 + δ) ⟨x, x_in_s⟩)
    · intro x' hx'
      rcases (Set.mem_image _ _ _).1 hx' with ⟨y, ⟨y_in_s', yx'⟩⟩
      rcases mem_range.1 y_in_s' with ⟨x, xy⟩
      use Fl x, mem_image_of_mem _ x.2
      rw [← yx', ← xy, dist_comm]
      exact le_of_eq (glueDist_glued_points (Z := s) (@Subtype.val X s) Φ (ε₂ / 2 + δ) x)
  have : hausdorffDist (Fr '' range Φ) (range Fr) ≤ ε₃ := by
    rw [← @image_univ _ _ Fr, hausdorffDist_image Ir]
    rcases exists_mem_of_nonempty Y with ⟨xY, _⟩
    rcases hs' xY with ⟨xs', Dxs'⟩
    have : 0 ≤ ε₃ := le_trans dist_nonneg Dxs'
    refine hausdorffDist_le_of_mem_dist this
      (fun x _ => ⟨x, mem_univ _, by simpa only [dist_self]⟩)
      fun x _ => ?_
    

Proof state:
case intro.intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝¹ : xX ∈ univ
xs : X
hxs : xs ∈ s
Dxs : dist xX xs ≤ ε₁
sne : Set.Nonempty s
this✝⁸ : Nonempty ↑s := Nonempty.to_subtype sne
this✝⁷ : 0 ≤ ε₂
this✝⁶ : ∀ (p q : ↑s), |dist p q - dist (Φ p) (Φ q)| ≤ 2 * (ε₂ / 2 + δ)
this✝⁵ : MetricSpace (X ⊕ Y) := glueMetricApprox (fun x => ↑x) (fun x => Φ x) (ε₂ / 2 + δ) (_ : 0 < ε₂ / 2 + δ) this✝⁶
Fl : X → X ⊕ Y := inl
Fr : Y → X ⊕ Y := inr
Il : Isometry Fl
Ir : Isometry Fr
this✝⁴ : ghDist X Y ≤ hausdorffDist (range Fl) (range Fr)
this✝³ : hausdorffDist (range Fl) (range Fr) ≤ hausdorffDist (range Fl) (Fl '' s) + hausdorffDist (Fl '' s) (range Fr)
this✝² :
  hausdorffDist (Fl '' s) (range Fr) ≤
    hausdorffDist (Fl '' s) (Fr '' range Φ) + hausdorffDist (Fr '' range Φ) (range Fr)
this✝¹ : hausdorffDist (range Fl) (Fl '' s) ≤ ε₁
this✝ : hausdorffDist (Fl '' s) (Fr '' range Φ) ≤ ε₂ / 2 + δ
xY : Y
h✝ : xY ∈ univ
xs' : ↑s
Dxs' : dist xY (Φ xs') ≤ ε₃
this : 0 ≤ ε₃
x : Y
x✝ : x ∈ univ
⊢ ∃ y ∈ range Φ, dist x y ≤ ε₃

Next tactic:
rcases hs' x with ⟨y, Dy⟩