-- 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:
case inr.a
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 ⊕ Y
val✝ : Y
⊢ f (inr val✝) ∈ range Ψ

Tactic:
apply mem_range_self

Result:
no goals

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
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
DΨ : diam (range Ψ) = diam univ
⊢ diam univ + (diam univ + 1 + diam univ) + diam univ = 2 * diam univ + 1 + 2 * diam univ

Tactic:
ring

Result:
no goals

Proof state:
case left.left.left.left.right
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 : Y), dist (f (inr x)) (f (inr y)) = dist x y

Tactic:
exact fun x y =>
  calc
    F (inr x, inr y) = dist (Ψ x) (Ψ y) := rfl
    _ = dist x y := Ψisom.dist_eq x y

Result:
no goals

Proof state:
case inr
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 ⊕ Y
val✝ : Y
⊢ f (inr val✝) ∈ range Φ ∪ range Ψ

Tactic:
apply mem_union_right

Result:
case inr.a
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 ⊕ Y
val✝ : Y
⊢ f (inr val✝) ∈ range Ψ

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
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
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q

Tactic:
by_cases h :
  hausdorffDist (p : Set ℓ_infty_ℝ) q < diam (univ : Set X) + 1 + diam (univ : Set Y)

Result:
case pos
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
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
h : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q

case neg
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
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
h : ¬hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ hausdorffDist ↑p ↑q

Proof state:
case refine_2
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
⊢ ghDist X Y ≤ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y))

Tactic:
exact ghDist_le_hausdorffDist (isometry_optimalGHInjl X Y) (isometry_optimalGHInjr X Y)

Result:
no goals