awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
⊢ hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
      (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) ≤
    hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
        (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
      hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
        (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z))
have refine_1 : Bornology.IsBounded (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
Tactic: refine hausdorffDist_triangle <| hausdorffEDist_ne_top_of_nonempty_of_bounded
          (range_nonempty _) (range_nonempty _) ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
refine_1 : Bornology.IsBounded (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
⊢ hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
      (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) ≤
    hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
        (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
      hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
        (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z))
have refine_2 : Bornology.IsBounded (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
Tactic: refine hausdorffDist_triangle <| hausdorffEDist_ne_top_of_nonempty_of_bounded
          (range_nonempty _) (range_nonempty _) ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
⊢ hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
        (toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
      hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
        (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) =
    hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
have rw : hausdorffDist (α := γ1) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
    hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
      (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) =
  hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
    hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
Tactic: rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
          hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw :
  hausdorffDist (α := γ1) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
        (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) =
    hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
⊢ hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
        (toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
      hausdorffDist (α := GlueSpace hΦ hΨ) (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
        (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) =
    hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
have rw₁ : hausdorffDist (α := γ1) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
    hausdorffDist (α := γ2) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
  hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
    hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
Tactic: rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
          hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
have rw : ghDist X Y + hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
  dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
Tactic: rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw :
  ghDist X Y + hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
have rw₁ : ghDist X Y + ghDist Y Z = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
Tactic: rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw :
  ghDist X Y + hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
rw₁ : ghDist X Y + ghDist Y Z = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
have rw₂ : dist (toGHSpace X) (toGHSpace Y) + ghDist Y Z = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
Tactic: rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw :
  ghDist X Y + hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
rw₁ : ghDist X Y + ghDist Y Z = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
rw₂ :
  dist (toGHSpace X) (toGHSpace Y) + ghDist Y Z = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (α := OptimalGHCoupling Y Z) (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
have rw₃ : dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) =
  dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z)
Tactic: rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
⊢ dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
have rw : dist x (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
Tactic: rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw : dist x (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
⊢ dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
have rw₁ : dist x y + dist y (toGHSpace Z) = dist x y + dist y z
Tactic: rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
rw : dist x (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
rw₁ : dist x y + dist y (toGHSpace Z) = dist x y + dist y z
⊢ dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) = dist x y + dist y z
have rw₂ : dist x y + dist y z = dist x y + dist y z
Tactic: rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
⊢ dist x z ≤ dist x y + dist y z
have dist : dist x z = dist (toGHSpace X) (toGHSpace Z)
Tactic: calc
      dist x z = dist (toGHSpace X) (toGHSpace Z) := by
        rw [x.toGHSpace_rep, z.toGHSpace_rep]
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
            (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) :=
        (ghDist_le_hausdorffDist ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y))
          ((toGlueR_isometry hΦ hΨ).comp (isometry_optimalGHInjr Y Z)))
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
              (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
            hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
              (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) := by
        refine hausdorffDist_triangle <| hausdorffEDist_ne_top_of_nonempty_of_bounded
          (range_nonempty _) (range_nonempty _) ?_ ?_
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y)))).isBounded
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjr X Y)))).isBounded
      _ = hausdorffDist (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
              (toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
            hausdorffDist (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
              (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) := by
        simp only [← range_comp, Comm]
      _ = hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
            hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) := by
        rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
          hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
      _ = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) := by
        rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
      _ = dist x y + dist y z := by rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : Eq (α := Y → GlueSpace hΦ hΨ) (toGlueL hΦ hΨ ∘ optimalGHInjr X Y) (toGlueR hΦ hΨ ∘ optimalGHInjl Y Z)
dist : Dist.dist x z = Dist.dist (toGHSpace X) (toGHSpace Z)
⊢ Dist.dist x z ≤ Dist.dist x y + Dist.dist y z
have dist₁ : hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
    (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) ≤
  hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
      (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
    hausdorffDist (α := GlueSpace hΦ hΨ) (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
      (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z))
Tactic: calc
      dist x z = dist (toGHSpace X) (toGHSpace Z) := by
        rw [x.toGHSpace_rep, z.toGHSpace_rep]
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
            (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) :=
        (ghDist_le_hausdorffDist ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y))
          ((toGlueR_isometry hΦ hΨ).comp (isometry_optimalGHInjr Y Z)))
      _ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
              (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
            hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
              (range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) := by
        refine hausdorffDist_triangle <| hausdorffEDist_ne_top_of_nonempty_of_bounded
          (range_nonempty _) (range_nonempty _) ?_ ?_
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y)))).isBounded
        · exact (isCompact_range (Isometry.continuous
            ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjr X Y)))).isBounded
      _ = hausdorffDist (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
              (toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
            hausdorffDist (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
              (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) := by
        simp only [← range_comp, Comm]
      _ = hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
            hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) := by
        rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
          hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
      _ = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) := by
        rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
      _ = dist x y + dist y z := by rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
⊢ x = y
have rcases : ∀ (Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)) (Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)) (Φisom : Isometry Φ)
  (Ψisom : Isometry Ψ) (DΦΨ : ghDist x.Rep y.Rep = hausdorffDist (range Φ) (range Ψ)), x = y
Tactic: rcases ghDist_eq_hausdorffDist x.Rep y.Rep with ⟨Φ, Ψ, Φisom, Ψisom, DΦΨ⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : ghDist x.Rep y.Rep = hausdorffDist (range Φ) (range Ψ)
⊢ x = y
have DΦΨ₁ : 0 = hausdorffDist (range Φ) (range Ψ)
Tactic: rw [← dist_ghDist, hxy] at DΦΨ
Cleared hypotheses: DΦΨ
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
⊢ range Φ = range Ψ
have hΦ : IsCompact (range Φ)
Tactic: have hΦ : IsCompact (range Φ) := isCompact_range Φisom.continuous
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
⊢ range Φ = range Ψ
have hΨ : IsCompact (range Ψ)
Tactic: have hΨ : IsCompact (range Ψ) := isCompact_range Ψisom.continuous
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ range Φ = range Ψ
have apply : IsClosed (range Φ)
Tactic: apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
apply : IsClosed (range Φ)
⊢ range Φ = range Ψ
have apply₁ : IsClosed (range Ψ)
Tactic: apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
apply : IsClosed (range Φ)
apply₁ : IsClosed (range Ψ)
⊢ range Φ = range Ψ
have apply₂ : hausdorffEDist (range Φ) (range Ψ) ≠ ∞
Tactic: apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
⊢ x = y
have this : range Φ = range Ψ
Tactic: have : range Φ = range Ψ := by
      have hΦ : IsCompact (range Φ) := isCompact_range Φisom.continuous
      have hΨ : IsCompact (range Ψ) := isCompact_range Ψisom.continuous
      apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
      · exact hΦ.isClosed
      · exact hΨ.isClosed
      · exact hausdorffEDist_ne_top_of_nonempty_of_bounded (range_nonempty _) (range_nonempty _)
          hΦ.isBounded hΨ.isBounded
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
⊢ ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Φ) : Type) ≃ᵢ y.Rep)
have rw : ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Ψ) : Type) ≃ᵢ y.Rep)
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
⊢ x = y
have T : ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Φ) : Type) ≃ᵢ y.Rep)
Tactic: have T : (range Ψ ≃ᵢ y.Rep) = (range Φ ≃ᵢ y.Rep) := by rw [this]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
T : ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Φ) : Type) ≃ᵢ y.Rep)
⊢ x = y
have have : ∀ (eΨ : (↑(range Φ) : Type) ≃ᵢ y.Rep), x = y
Tactic: have eΨ := cast T Ψisom.isometryEquivOnRange.symm
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
T : ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Φ) : Type) ≃ᵢ y.Rep)
eΨ : (↑(range Φ) : Type) ≃ᵢ y.Rep
⊢ x = y
have have : ∀ (e : x.Rep ≃ᵢ y.Rep), x = y
Tactic: have e := Φisom.isometryEquivOnRange.trans eΨ
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
Goal:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun (n : ℕ) ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
T : ((↑(range Ψ) : Type) ≃ᵢ y.Rep) = ((↑(range Φ) : Type) ≃ᵢ y.Rep)
eΨ : (↑(range Φ) : Type) ≃ᵢ y.Rep
e : x.Rep ≃ᵢ y.Rep
⊢ x = y
have rw : Nonempty (x.Rep ≃ᵢ y.Rep)
Tactic: rw [← x.toGHSpace_rep, ← y.toGHSpace_rep, toGHSpace_eq_toGHSpace_iff_isometryEquiv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
Goal:
X : Type u
inst : MetricSpace X
p q : NonemptyCompacts X
⊢ dist p.toGHSpace q.toGHSpace ≤ dist p q
have ha : Isometry.{u, u} (α := ↥p) (Subtype.val (p := fun (x : X) ↦ x ∈ p))
Tactic: have ha : Isometry ((↑) : p → X) := isometry_subtype_coe