-- 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 inl.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✝ : X
⊢ f (inl val✝) ∈ range Φ

Tactic:
apply mem_range_self

Result:
no goals

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
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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this✝¹ : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
I1 : ∀ (x : X), ⨅ y, Fb (inl x, inr y) ≤ r
y : Y
this✝ : f (inr y) ∈ ↑q
z : ↥(lp (fun n => ℝ) ⊤)
zq : z ∈ ↑p
hz : dist z (f (inr y)) < r
this : z ∈ range Φ
⊢ ⨅ x, Fb (inl x, inr y) ≤ r

Tactic:
rcases mem_range.1 this with ⟨x, hx⟩

Result:
case 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
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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this✝¹ : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
I1 : ∀ (x : X), ⨅ y, Fb (inl x, inr y) ≤ r
y : Y
this✝ : f (inr y) ∈ ↑q
z : ↥(lp (fun n => ℝ) ⊤)
zq : z ∈ ↑p
hz : dist z (f (inr y)) < r
this : z ∈ range Φ
x : X
hx : Φ x = z
⊢ ⨅ x, Fb (inl x, inr y) ≤ r

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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
I1 : ∀ (x : X), ⨅ y, Fb (inl x, inr y) ≤ r
⊢ ∀ (y : Y), ⨅ x, Fb (inl x, inr y) ≤ r

Tactic:
intro y

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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
I1 : ∀ (x : X), ⨅ y, Fb (inl x, inr y) ≤ r
y : Y
⊢ ⨅ x, Fb (inl x, inr y) ≤ r

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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
⊢ ∀ (x : X), ⨅ y, Fb (inl x, inr y) ≤ r

Tactic:
intro x

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)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
x : X
⊢ ⨅ y, Fb (inl x, inr y) ≤ r