Theorem: GromovHausdorff.hausdorffDist_optimal
File: Mathlib/Topology/MetricSpace/GromovHausdorff.lean

Proof steps:
state_before: 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
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y
state_after: 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
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y
tactic: inhabit X
state_before: 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
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y
state_after: 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
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y
tactic: inhabit Y
state_before: 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
state_after: 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

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: refine le_antisymm ?_ ?_