awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
⊢ IsCompact (X := Cb X Y) (candidatesB X Y)
have refine_1 : ∀ (f : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) (x : (X ⊕ Y) × (X ⊕ Y)),
  Membership.mem (γ := Set ((X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ)) (candidatesB X Y) f →
    Membership.mem (α := ℝ) (Icc 0 (↑(maxVar X Y) : ℝ)) ((f : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) x : ℝ)
Tactic: refine arzela_ascoli₂
      (Icc 0 (maxVar X Y) : Set ℝ) isCompact_Icc (candidatesB X Y) closed_candidatesB ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
refine_1 :
  ∀ (f : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) (x : (X ⊕ Y) × (X ⊕ Y)),
    Membership.mem (γ := Set ((X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ)) (candidatesB X Y) f →
      Membership.mem (α := ℝ) (Icc 0 (↑(maxVar X Y) : ℝ)) ((f : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) x : ℝ)
⊢ IsCompact (X := Cb X Y) (candidatesB X Y)
have refine_2 : Equicontinuous (X := (X ⊕ Y) × (X ⊕ Y)) (α := ℝ) fun (x : (↑(candidatesB X Y) : Type (max u v))) ↦
  (⇑(↑x : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ)
Tactic: refine arzela_ascoli₂
      (Icc 0 (maxVar X Y) : Set ℝ) isCompact_Icc (candidatesB X Y) closed_candidatesB ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
⊢ Equicontinuous (X := (X ⊕ Y) × (X ⊕ Y)) (α := ℝ) fun (x : (↑(candidatesB X Y) : Type (max u v))) ↦
    (⇑(↑x : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ)
have refine_2 : Tendsto (β := ℝ) (fun (t : ℝ) ↦ 2 * (↑(maxVar X Y) : ℝ) * t) (𝓝 0) (𝓝 0)
Tactic: refine equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) ?_ _ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
refine_2 : Tendsto (β := ℝ) (fun (t : ℝ) ↦ 2 * (↑(maxVar X Y) : ℝ) * t) (𝓝 0) (𝓝 0)
⊢ Equicontinuous (X := (X ⊕ Y) × (X ⊕ Y)) (α := ℝ) fun (x : (↑(candidatesB X Y) : Type (max u v))) ↦
    (⇑(↑x : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ)
have refine_2₁ : ∀ (x y : (X ⊕ Y) × (X ⊕ Y)) (i : (↑(candidatesB X Y) : Type (max u v))),
  dist (α := ℝ) (((↑i : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) x : ℝ)
      (((↑i : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) y : ℝ) ≤
    (fun (t : ℝ) ↦ 2 * (↑(maxVar X Y) : ℝ) * t) (dist x y)
Tactic: refine equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) ?_ _ ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
⊢ Tendsto (β := ℝ) (fun (t : ℝ) ↦ 2 * (↑(maxVar X Y) : ℝ) * t) (𝓝 0) (𝓝 0)
have this : Tendsto (fun (t : ℝ) ↦ 2 * (↑(maxVar X Y) : ℝ) * t) (𝓝 0) (𝓝 (2 * (↑(maxVar X Y) : ℝ) * 0))
Tactic: have : Tendsto (fun t : ℝ => 2 * (maxVar X Y : ℝ) * t) (𝓝 0) (𝓝 (2 * maxVar X Y * 0)) :=
        tendsto_const_nhds.mul tendsto_id
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
⊢ (fun (p : (X ⊕ Y) × (X ⊕ Y)) ↦ dist p.1 p.2) ∈ candidates X Y
have simp_rw : (∀ (x y : X), dist (α := X ⊕ Y) (inl x) (inl y) = dist x y) ∧ ∀ (x y : Y), dist (α := X ⊕ Y) (inr x) (inr y) = dist x y
Tactic: simp_rw [candidates, Set.mem_setOf_eq, dist_comm, dist_triangle, dist_self, maxVar_bound,
    forall_const, and_true]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
⊢ HD (candidatesBDist X Y) ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have refine_1 : ∀ (x : X),
  ⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
Tactic: refine max_le (ciSup_le fun x => ?_) (ciSup_le fun y => ?_)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
refine_1 :
  ∀ (x : X),
    ⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
      diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
⊢ HD (candidatesBDist X Y) ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have refine_2 : ∀ (y : Y),
  ⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
Tactic: refine max_le (ciSup_le fun x => ?_) (ciSup_le fun y => ?_)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
x : X
⊢ ⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have A : LE.le (α := ℝ) (⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
  ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr default) : ℝ)
Tactic: have A : ⨅ y, candidatesBDist X Y (inl x, inr y) ≤ candidatesBDist X Y (inl x, inr default) :=
      ciInf_le (by simpa using HD_below_aux1 0) default
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
x : X
A :
  LE.le (α := ℝ) (⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr default) : ℝ)
⊢ dist x default + 1 + dist.{v} (α := Y) default default ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have h₁ : dist x default ≤ diam.{u} (α := X) univ
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
x : X
A :
  LE.le (α := ℝ) (⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr default) : ℝ)
h₁ : dist x default ≤ diam.{u} (α := X) univ
⊢ dist x default + 1 + dist.{v} (α := Y) default default ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have h₂ : dist.{v} (α := Y) default default ≤ diam.{v} (α := Y) univ
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
x : X
A :
  LE.le (α := ℝ) (⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr default) : ℝ)
⊢ ⨅ (y : Y), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have B : dist (α := X ⊕ Y) (inl x) (inr default) ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
Tactic: have B : dist (inl x) (inr default) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) :=
      calc
        dist (inl x) (inr (default : Y)) = dist x (default : X) + 1 + dist default default := rfl
        _ ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := by
          gcongr <;>
            exact dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
y : Y
⊢ ⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have A : LE.le (α := ℝ) (⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
  ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl default, inr y) : ℝ)
Tactic: have A : ⨅ x, candidatesBDist X Y (inl x, inr y) ≤ candidatesBDist X Y (inl default, inr y) :=
      ciInf_le (by simpa using HD_below_aux2 0) default
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
y : Y
A :
  LE.le (α := ℝ) (⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl default, inr y) : ℝ)
⊢ dist.{u} (α := X) default default + 1 + dist default y ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have h₁ : dist.{u} (α := X) default default ≤ diam.{u} (α := X) univ
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
y : Y
A :
  LE.le (α := ℝ) (⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl default, inr y) : ℝ)
h₁ : dist.{u} (α := X) default default ≤ diam.{u} (α := X) univ
⊢ dist.{u} (α := X) default default + 1 + dist default y ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have h₂ : dist default y ≤ diam.{v} (α := Y) univ
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
inst_2 : CompactSpace X
inst_3 : CompactSpace Y
inst_4 : Nonempty X
inst_5 : Nonempty Y
y : Y
A :
  LE.le (α := ℝ) (⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) : ℝ)
    ((candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl default, inr y) : ℝ)
⊢ ⨅ (x : X), (candidatesBDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) ≤
    diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
have B : dist (α := X ⊕ Y) (inl default) (inr y) ≤ diam.{u} (α := X) univ + 1 + diam.{v} (α := Y) univ
Tactic: have B : dist (inl default) (inr y) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) :=
      calc
        dist (inl (default : X)) (inr y) = dist default default + 1 + dist default y := rfl
        _ ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := by
          gcongr <;>
            exact dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
⊢ Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) (optimalGHDist X Y)
have left : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y)
  (Classical.choose (p := fun (x : Cb X Y) ↦
    Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) x ∧
      ∀ (g : Cb X Y), Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) g → HD x ≤ HD g)
    sorry)
Tactic: cases Classical.choose_spec (exists_minimizer X Y)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
left :
  Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y)
    (Classical.choose (p := fun (x : Cb X Y) ↦
      Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) x ∧
        ∀ (g : Cb X Y), Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) g → HD x ≤ HD g)
      ⋯)
⊢ Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) (optimalGHDist X Y)
have right : ∀ (g : Cb X Y),
  Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) g →
    HD
        (Classical.choose (p := fun (x : Cb X Y) ↦
          Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) x ∧
            ∀ (g : Cb X Y), Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) g → HD x ≤ HD g)
          sorry) ≤
      HD g
Tactic: cases Classical.choose_spec (exists_minimizer X Y)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
⊢ IsCompact (X := OptimalGHCoupling X Y) univ
have rw : IsCompact (X := OptimalGHCoupling X Y) (range Quotient.mk')
Tactic: rw [← range_quotient_mk']
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f
have refine : ∀ (r : ℝ) (hr : HD (optimalGHDist X Y) < r),
  hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ r
Tactic: refine le_trans (le_of_forall_gt_imp_ge_of_dense fun r hr => ?_) (HD_optimalGHDist_le X Y f h)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
z : X
⊢ ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist (optimalGHInjl X Y z) y ≤ r
have I1 : ⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) < r
Tactic: have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
z : X
I1 : ⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) < r
⊢ ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist (optimalGHInjl X Y z) y ≤ r
have I2 : LE.le (α := ℝ) (⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y))
  (⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y))
Tactic: have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
z : X
I1 : ⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) < r
I2 :
  LE.le (α := ℝ) (⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y))
    (⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y))
⊢ ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist (optimalGHInjl X Y z) y ≤ r
have I : ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y) < r
Tactic: have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
z : X
I1 : ⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) < r
I2 :
  LE.le (α := ℝ) (⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y))
    (⨆ (x : X), ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y))
I : ⨅ (y : Y), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y) < r
⊢ ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist (optimalGHInjl X Y z) y ≤ r
have rcases : ∀ (z' : Y) (hr' : (fun (y : Y) ↦ ((optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl z, inr y) : ℝ)) z' < r),
  ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist (optimalGHInjl X Y z) y ≤ r
Tactic: rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ r
have A : ∀ (x : OptimalGHCoupling X Y),
  Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
    ∃ (y : OptimalGHCoupling X Y),
      Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
Tactic: have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
    rintro _ ⟨z, rfl⟩
    have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_left _ _) hr
    have I2 :
        ⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
    have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
A :
  ∀ (x : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
      ∃ (y : OptimalGHCoupling X Y),
        Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ r
have refine_1 : 0 ≤ r
Tactic: refine hausdorffDist_le_of_mem_dist ?_ A ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
A :
  ∀ (x : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
      ∃ (y : OptimalGHCoupling X Y),
        Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
refine_1 : 0 ≤ r
⊢ hausdorffDist (α := OptimalGHCoupling X Y) (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ r
have refine_2 : ∀ (x : OptimalGHCoupling X Y),
  Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) x →
    ∃ (y : OptimalGHCoupling X Y),
      Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) y ∧ dist x y ≤ r
Tactic: refine hausdorffDist_le_of_mem_dist ?_ A ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
A :
  ∀ (x : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
      ∃ (y : OptimalGHCoupling X Y),
        Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
⊢ 0 ≤ r
have refine_1 : ∀ (inhabited_h : Inhabited X), 0 ≤ r
Tactic: inhabit X
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (goal)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
A :
  ∀ (x : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
      ∃ (y : OptimalGHCoupling X Y),
        Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
inhabited_h : Inhabited X
⊢ 0 ≤ r
have refine_1 : ∀ (y : OptimalGHCoupling X Y) (hy : dist (optimalGHInjl X Y default) y ≤ r), 0 ≤ r
Tactic: rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorffRealized (hypothesis)
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : CompactSpace X
inst_2 : Nonempty X
inst_3 : MetricSpace Y
inst_4 : CompactSpace Y
inst_5 : Nonempty Y
f : Cb X Y
h : Membership.mem (γ := Set (Cb X Y)) (candidatesB X Y) f
r : ℝ
hr : HD (optimalGHDist X Y) < r
A :
  ∀ (x : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) x →
      ∃ (y : OptimalGHCoupling X Y),
        Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjr X Y)) y ∧ dist x y ≤ r
z : Y
⊢ ∃ (y : OptimalGHCoupling X Y),
    Membership.mem (γ := Set (OptimalGHCoupling X Y)) (range (optimalGHInjl X Y)) y ∧ dist (optimalGHInjr X Y z) y ≤ r
have I1 : ⨆ (y : Y), ⨅ (x : X), (optimalGHDist X Y : (a : (X ⊕ Y) × (X ⊕ Y)) → ℝ) (inl x, inr y) < r
Tactic: have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr