-- Mathlib/Topology/MetricSpace/GromovHausdorff.lean
theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
    {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y

Proof state:
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q

Tactic:
rcases eq_toGHSpace_iff.1 hp with ⟨Φ, ⟨Φisom, Φrange⟩⟩

Result:
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q

Proof state:
case intro.intro.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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
xX : X
h✝ : xX ∈ univ
y : ↥(lp (fun n => ℝ) ⊤)
hy : y ∈ range Ψ
z : Y
dy : dist (Φ xX) (Ψ z) < diam univ + 1 + diam univ
hzy : Ψ z = y
⊢ diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ

Tactic:
have DΦ : diam (range Φ) = diam (univ : Set X) := Φisom.diam_range

Result:
case intro.intro.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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
xX : X
h✝ : xX ∈ univ
y : ↥(lp (fun n => ℝ) ⊤)
hy : y ∈ range Ψ
z : Y
dy : dist (Φ xX) (Ψ z) < diam univ + 1 + diam univ
hzy : Ψ z = y
DΦ : diam (range Φ) = diam univ
⊢ diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ

Proof state:
case refine_1
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ ghDist X Y

Tactic:
apply le_csInf

Result:
case refine_1.h₁
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y}).Nonempty

case refine_1.h₂
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y →
        hausdorffDist ↑p ↑q < diam univ + 1 + diam univ →
          hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
B :
  ∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
    ⟦p⟧ = toGHSpace X →
      ⟦q⟧ = toGHSpace Y → hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q
⊢ ∀ b ∈ (fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y},
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ b

Proof state:
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n => ℝ) ⊤) :=
  fun x =>
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
⊢ F ∈ candidates X Y

Tactic:
simp only [F, candidates, forall_const, add_comm, eq_self_iff_true,
  dist_eq_zero, and_self_iff, Set.mem_setOf_eq]

Result:
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
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n => ℝ) ⊤)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n => ℝ) ⊤)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n => ℝ) ⊤) :=
  fun x =>
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
⊢ (((((∀ (x y : X), dist (f (inl x)) (f (inl y)) = dist x y) ∧ ∀ (x y : Y), dist (f (inr x)) (f (inr y)) = dist x y) ∧
          ∀ (x y : X ⊕ Y), dist (f x) (f y) = dist (f y) (f x)) ∧
        ∀ (x y z : X ⊕ Y), dist (f x) (f z) ≤ dist (f x) (f y) + dist (f y) (f z)) ∧
      True) ∧
    ∀ (x y : X ⊕ Y), dist (f x) (f y) ≤ ↑(GromovHausdorff.maxVar✝ X Y)