awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
⊢ coveringNumber ε (f '' A) ≤ coveringNumber ε A
have refine_1 : ∀ (i : Set X) (i_1 : i ⊆ A) (i_2 : IsCover ε A i),
  ⨅ (C : Set Y), ⨅ (_ : C ⊆ f '' A), ⨅ (_ : IsCover ε (f '' A) C), C.encard ≤ i.encard
Tactic: simp only [coveringNumber, le_iInf_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
⊢ ⨅ (C : Set Y), ⨅ (_ : C ⊆ f '' A), ⨅ (_ : IsCover ε (f '' A) C), C.encard ≤ C.encard
have refine_1 : ⨅ (_ : f '' C ⊆ f '' A), ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
Tactic: refine (iInf_le _ (C.image f)).trans ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
⊢ ⨅ (_ : f '' C ⊆ f '' A), ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
have refine_1 : ⨅ (_ : C ⊆ f ⁻¹' f '' A), ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
Tactic: simp only [Set.image_subset_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
⊢ ⨅ (_ : C ⊆ f ⁻¹' f '' A), ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
have this : C ⊆ f ⁻¹' f '' A
Tactic: have : ↑C ⊆ f ⁻¹' f '' A := hC_subset.trans (Set.subset_preimage_image f A)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
this : C ⊆ f ⁻¹' f '' A
⊢ ⨅ (_ : C ⊆ f ⁻¹' f '' A), ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
have refine_1 : ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
Tactic: refine (iInf_le _ this).trans ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
this : C ⊆ f ⁻¹' f '' A
⊢ ⨅ (_ : IsCover ε (f '' A) (f '' C)), (f '' C).encard ≤ C.encard
have refine_1 : ⨅ (_ : IsCover ε A C), (f '' C).encard ≤ C.encard
Tactic: rw [hf.isCover_image_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set X
hC_subset : C ⊆ A
hC_cover : IsCover ε A C
this : C ⊆ f ⁻¹' f '' A
⊢ ⨅ (_ : IsCover ε A C), (f '' C).encard ≤ C.encard
have refine_1 : (f '' C).encard ≤ C.encard
Tactic: refine (iInf_le _ hC_cover).trans ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
⊢ coveringNumber ε A ≤ coveringNumber ε (f '' A)
have refine_2 : ∀ (i : Set Y) (i_1 : i ⊆ f '' A) (i_2 : IsCover ε (f '' A) i),
  ⨅ (C : Set X), ⨅ (_ : C ⊆ A), ⨅ (_ : IsCover ε A C), C.encard ≤ i.encard
Tactic: simp only [coveringNumber, le_iInf_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
⊢ ∃ C' ⊆ A, C = f '' C'
have this : ∀ (x : (↑C : Type u_2)), ∃ y ∈ A, f y = (↑x : Y)
Tactic: have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
this : ∀ (x : (↑C : Type u_2)), ∃ y ∈ A, f y = (↑x : Y)
⊢ ∃ C' ⊆ A, C = f '' C'
have choose : ∀ (g : (x : (↑C : Type u_2)) → X) (hg_mem : ∀ (x : (↑C : Type u_2)), g x ∈ A)
  (hg : ∀ (x : (↑C : Type u_2)), f (g x) = (↑x : Y)), ∃ C' ⊆ A, C = f '' C'
Tactic: choose g hg_mem hg using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : (x : (↑C : Type u_2)) → X
hg_mem : ∀ (x : (↑C : Type u_2)), g x ∈ A
hg : ∀ (x : (↑C : Type u_2)), f (g x) = (↑x : Y)
⊢ ∃ C' ⊆ A, C = f '' C'
have refine_1 : range g ⊆ A
Tactic: refine ⟨Set.range g, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : (x : (↑C : Type u_2)) → X
hg_mem : ∀ (x : (↑C : Type u_2)), g x ∈ A
hg : ∀ (x : (↑C : Type u_2)), f (g x) = (↑x : Y)
refine_1 : range g ⊆ A
⊢ ∃ C' ⊆ A, C = f '' C'
have refine_2 : C = f '' range g
Tactic: refine ⟨Set.range g, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : (x : (↑C : Type u_2)) → X
hg_mem : ∀ (x : (↑C : Type u_2)), g x ∈ A
hg : ∀ (x : (↑C : Type u_2)), f (g x) = (↑x : Y)
⊢ C = f '' range g
have refine_2 : ∀ (x : Y), x ∈ C ↔ x ∈ f '' range g
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
g : (x : (↑C : Type u_2)) → X
hg_mem : ∀ (x : (↑C : Type u_2)), g x ∈ A
hg : ∀ (x : (↑C : Type u_2)), f (g x) = (↑x : Y)
x : Y
⊢ x ∈ C ↔ x ∈ f '' range g
have refine_2 : x ∈ C ↔ ∃ (a : Y) (h : a ∈ C), f (g (Subtype.mk (p := fun (a : Y) ↦ a ∈ C) a sorry)) = x
Tactic: simp
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
⊢ ⨅ (C : Set X), ⨅ (_ : C ⊆ A), ⨅ (_ : IsCover ε A C), C.encard ≤ C.encard
have refine_2 : ∀ (C' : Set X) (hC'_subset : C' ⊆ A) (hC_subset : f '' C' ⊆ f '' A) (hC_cover : IsCover ε (f '' A) (f '' C')),
  ⨅ (C : Set X), ⨅ (_ : C ⊆ A), ⨅ (_ : IsCover ε A C), C.encard ≤ (f '' C').encard
Tactic: obtain ⟨C', hC'_subset, rfl⟩ : ∃ C', C' ⊆ A ∧ C = C'.image f := by
      have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2
      choose g hg_mem hg using this
      refine ⟨Set.range g, ?_, ?_⟩
      · rwa [Set.range_subset_iff]
      · ext
        simp
        grind
Cleared hypotheses: C, hC_subset, hC_cover
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C' : Set X
hC'_subset : C' ⊆ A
hC_subset : f '' C' ⊆ f '' A
hC_cover : IsCover ε (f '' A) (f '' C')
⊢ ⨅ (C : Set X), ⨅ (_ : C ⊆ A), ⨅ (_ : IsCover ε A C), C.encard ≤ (f '' C').encard
have refine_2 : ⨅ (_ : IsCover ε A C'), C'.encard ≤ (f '' C').encard
Tactic: refine (iInf_le _ C').trans <| (iInf_le _ hC'_subset).trans ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (hypothesis)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C' : Set X
hC'_subset : C' ⊆ A
hC_subset : f '' C' ⊆ f '' A
hC_cover : IsCover ε (f '' A) (f '' C')
⊢ ⨅ (_ : IsCover ε A C'), C'.encard ≤ (f '' C').encard
have hC_cover₁ : IsCover ε A C'
Tactic: simp only [hf.isCover_image_iff] at hC_cover
Cleared hypotheses: hC_cover
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C' : Set X
hC'_subset : C' ⊆ A
hC_subset : f '' C' ⊆ f '' A
hC_cover : IsCover ε A C'
⊢ ⨅ (_ : IsCover ε A C'), C'.encard ≤ (f '' C').encard
have refine_2 : C'.encard ≤ (f '' C').encard
Tactic: refine (iInf_le _ hC_cover).trans ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C' : Set X
hC'_subset : C' ⊆ A
hC_subset : f '' C' ⊆ f '' A
hC_cover : IsCover ε A C'
⊢ C'.encard ≤ (f '' C').encard
have refine_2 : InjOn f C'
Tactic: rw [InjOn.encard_image]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
⊢ coveringNumber ε (f '' A) = coveringNumber ε A
have refine_1 : coveringNumber ε (f '' A) ≤ coveringNumber ε A
Tactic: classical
  refine le_antisymm ?_ ?_
  · simp only [coveringNumber, le_iInf_iff]
    intro C hC_subset hC_cover
    refine (iInf_le _ (C.image f)).trans ?_
    simp only [Set.image_subset_iff]
    have : ↑C ⊆ f ⁻¹' f '' A := hC_subset.trans (Set.subset_preimage_image f A)
    refine (iInf_le _ this).trans ?_
    rw [hf.isCover_image_iff]
    refine (iInf_le _ hC_cover).trans ?_
    exact encard_image_le f C
  · simp only [coveringNumber, le_iInf_iff]
    intro C hC_subset hC_cover
    obtain ⟨C', hC'_subset, rfl⟩ : ∃ C', C' ⊆ A ∧ C = C'.image f := by
      have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2
      choose g hg_mem hg using this
      refine ⟨Set.range g, ?_, ?_⟩
      · rwa [Set.range_subset_iff]
      · ext
        simp
        grind
    refine (iInf_le _ C').trans <| (iInf_le _ hC'_subset).trans ?_
    simp only [hf.isCover_image_iff] at hC_cover
    refine (iInf_le _ hC_cover).trans ?_
    rw [InjOn.encard_image]
    exact hf_inj.mono hC'_subset
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.CoveringNumbers (goal)
Goal:
X : Type u_1
Y : Type u_2
inst : PseudoEMetricSpace X
inst_1 : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
refine_1 : coveringNumber ε (f '' A) ≤ coveringNumber ε A
⊢ coveringNumber ε (f '' A) = coveringNumber ε A
have refine_2 : coveringNumber ε A ≤ coveringNumber ε (f '' A)
Tactic: classical
  refine le_antisymm ?_ ?_
  · simp only [coveringNumber, le_iInf_iff]
    intro C hC_subset hC_cover
    refine (iInf_le _ (C.image f)).trans ?_
    simp only [Set.image_subset_iff]
    have : ↑C ⊆ f ⁻¹' f '' A := hC_subset.trans (Set.subset_preimage_image f A)
    refine (iInf_le _ this).trans ?_
    rw [hf.isCover_image_iff]
    refine (iInf_le _ hC_cover).trans ?_
    exact encard_image_le f C
  · simp only [coveringNumber, le_iInf_iff]
    intro C hC_subset hC_cover
    obtain ⟨C', hC'_subset, rfl⟩ : ∃ C', C' ⊆ A ∧ C = C'.image f := by
      have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2
      choose g hg_mem hg using this
      refine ⟨Set.range g, ?_, ?_⟩
      · rwa [Set.range_subset_iff]
      · ext
        simp
        grind
    refine (iInf_le _ C').trans <| (iInf_le _ hC'_subset).trans ?_
    simp only [hf.isCover_image_iff] at hC_cover
    refine (iInf_le _ hC_cover).trans ?_
    rw [InjOn.encard_image]
    exact hf_inj.mono hC'_subset
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u_3
m m' : MetricSpace α
h : Eq.{u_3 + 1} (α := Dist α) PseudoMetricSpace.toDist PseudoMetricSpace.toDist
⊢ m = m'
have mk : ∀ [toPseudoMetricSpace : PseudoMetricSpace α] (eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y)
  (h : Eq.{u_3 + 1} (α := Dist α) PseudoMetricSpace.toDist PseudoMetricSpace.toDist),
  { toPseudoMetricSpace := toPseudoMetricSpace, eq_of_dist_eq_zero := @eq_of_dist_eq_zero : MetricSpace α } = m'
Tactic: cases m
Cleared hypotheses: m, h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u_3
m' : MetricSpace α
toPseudoMetricSpace : PseudoMetricSpace α
eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y
h : Eq.{u_3 + 1} (α := Dist α) PseudoMetricSpace.toDist PseudoMetricSpace.toDist
⊢ { toPseudoMetricSpace := toPseudoMetricSpace, eq_of_dist_eq_zero := @eq_of_dist_eq_zero : MetricSpace α } = m'
have mk : ∀ [toPseudoMetricSpace_1 : PseudoMetricSpace α] (eq_of_dist_eq_zero_1 : ∀ {x y : α}, dist x y = 0 → x = y)
  (h : Eq.{u_3 + 1} (α := Dist α) PseudoMetricSpace.toDist PseudoMetricSpace.toDist),
  { toPseudoMetricSpace := toPseudoMetricSpace, eq_of_dist_eq_zero := @eq_of_dist_eq_zero : MetricSpace α } =
    { toPseudoMetricSpace := toPseudoMetricSpace_1, eq_of_dist_eq_zero := @eq_of_dist_eq_zero_1 }
Tactic: cases m'
Cleared hypotheses: m', h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u_3
toPseudoMetricSpace : PseudoMetricSpace α
eq_of_dist_eq_zero : ∀ {x y : α}, dist x y = 0 → x = y
toPseudoMetricSpace_1 : PseudoMetricSpace α
eq_of_dist_eq_zero_1 : ∀ {x y : α}, dist x y = 0 → x = y
h : Eq.{u_3 + 1} (α := Dist α) PseudoMetricSpace.toDist PseudoMetricSpace.toDist
⊢ { toPseudoMetricSpace := toPseudoMetricSpace, eq_of_dist_eq_zero := @eq_of_dist_eq_zero : MetricSpace α } =
    { toPseudoMetricSpace := toPseudoMetricSpace_1, eq_of_dist_eq_zero := @eq_of_dist_eq_zero_1 }
have mk : toPseudoMetricSpace = toPseudoMetricSpace_1
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x y : γ
⊢ 0 = dist x y ↔ x = y
have rw : dist x y = 0 ↔ x = y
Tactic: rw [eq_comm, dist_eq_zero]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x y : γ
rw : dist x y = 0 ↔ x = y
⊢ 0 = dist x y ↔ x = y
have rw₁ : x = y ↔ x = y
Tactic: rw [eq_comm, dist_eq_zero]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x : γ
r : ℝ
hr : r ≤ 0
⊢ (closedBall x r).Subsingleton
have inl : ∀ (hr : r < 0), (closedBall x r).Subsingleton
Tactic: rcases hr.lt_or_eq with (hr | rfl)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x : γ
r : ℝ
hr : r ≤ 0
inl : ∀ (hr : r < 0), (closedBall x r).Subsingleton
⊢ (closedBall x r).Subsingleton
have inr : ∀ (hr : LE.le (α := ℝ) 0 0), (closedBall x 0).Subsingleton
Tactic: rcases hr.lt_or_eq with (hr | rfl)
Cleared hypotheses: r, hr
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x : γ
r : ℝ
hr_1 : r ≤ 0
hr : r < 0
⊢ (closedBall x r).Subsingleton
have inl : Set.Subsingleton.{w} (α := γ) ∅
Tactic: rw [closedBall_eq_empty.2 hr]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type w
inst : MetricSpace γ
x : γ
hr : LE.le (α := ℝ) 0 0
⊢ (closedBall x 0).Subsingleton
have inr : Set.Subsingleton (α := γ) {x}
Tactic: rw [closedBall_zero]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type u_3
U : UniformSpace γ
m : MetricSpace γ
H : 𝓤 γ = 𝓤 γ
⊢ m.replaceUniformity H = m
have h : ∀ (x x_1 : γ), dist x x_1 = dist x x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
γ : Type u_3
U : TopologicalSpace γ
m : MetricSpace γ
H : U = UniformSpace.toTopologicalSpace
⊢ m.replaceTopology H = m
have h : ∀ (x x_1 : γ), dist x x_1 = dist x x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u_3
m : MetricSpace α
B : Bornology α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
⊢ m.replaceBornology H = m
have h : ∀ (x x_1 : α), dist x x_1 = dist x x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
γ : Type w
inst_1 : MetricSpace γ
x x_1 x_2 : Empty
⊢ 0 ≤ HAdd.hAdd (α := ℝ) 0 0
have rw : LE.le (α := ℝ) 0 0
Tactic: rw [add_zero]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Defs (goal)
Goal:
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
γ : Type w
inst_1 : MetricSpace γ
x x_1 x_2 : PUnit.{u + 1}
⊢ 0 ≤ HAdd.hAdd (α := ℝ) 0 0
have rw : LE.le (α := ℝ) 0 0
Tactic: rw [add_zero]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
f g : α →ᵈ β
h : f.toFun = g.toFun
⊢ f = g
have mk : ∀ (toFun : α → β) (edist_eq' : ∃ (r : ℝ≥0), r ≠ 0 ∧ ∀ (x y : α), edist (toFun x) (toFun y) = (↑r : ℝ≥0∞) * edist x y)
  (h : { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β }.toFun = g.toFun),
  { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β } = g
Tactic: cases f
Cleared hypotheses: f, h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
g : α →ᵈ β
toFun : α → β
edist_eq' : ∃ (r : ℝ≥0), r ≠ 0 ∧ ∀ (x y : α), edist (toFun x) (toFun y) = (↑r : ℝ≥0∞) * edist x y
h : { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β }.toFun = g.toFun
⊢ { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β } = g
have mk : ∀ (toFun_1 : α → β)
  (edist_eq'_1 : ∃ (r : ℝ≥0), r ≠ 0 ∧ ∀ (x y : α), edist (toFun_1 x) (toFun_1 y) = (↑r : ℝ≥0∞) * edist x y)
  (h :
    { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β }.toFun =
      { toFun := toFun_1, edist_eq' := edist_eq'_1 : α →ᵈ β }.toFun),
  { toFun := toFun, edist_eq' := edist_eq' : α →ᵈ β } = { toFun := toFun_1, edist_eq' := edist_eq'_1 }
Tactic: cases g
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
⊢ (if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1
    else
      Exists.choose (p := fun (r : ℝ≥0) ↦
        r ≠ 0 ∧
          ∀ (x y : α),
            edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
        ⋯) ≠
    0
have pos : (∀ (x y : α), edist x y = 0 ∨ edist x y = ∞) → Ne (α := ℝ≥0) 1 0
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
pos : (∀ (x y : α), edist x y = 0 ∨ edist x y = ∞) → Ne (α := ℝ≥0) 1 0
⊢ (if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1
    else
      Exists.choose (p := fun (r : ℝ≥0) ↦
        r ≠ 0 ∧
          ∀ (x y : α),
            edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
        ⋯) ≠
    0
have neg : (¬∀ (x y : α), edist x y = 0 ∨ edist x y = ∞) →
  Exists.choose (p := fun (r : ℝ≥0) ↦
      r ≠ 0 ∧
        ∀ (x y : α), edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
      sorry ≠
    0
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
x y : α
⊢ edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) =
    (↑(if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1
          else
            Exists.choose (p := fun (r : ℝ≥0) ↦
              r ≠ 0 ∧
                ∀ (x y : α),
                  edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
              ⋯) :
        ℝ≥0∞) *
      edist x y
have pos : ∀ (key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞),
  edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑1 : ℝ≥0∞) * edist x y
Tactic: split_ifs with key
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
x y : α
pos :
  ∀ (key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞),
    edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑1 : ℝ≥0∞) * edist x y
⊢ edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) =
    (↑(if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1
          else
            Exists.choose (p := fun (r : ℝ≥0) ↦
              r ≠ 0 ∧
                ∀ (x y : α),
                  edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
              ⋯) :
        ℝ≥0∞) *
      edist x y
have neg : ∀ (key : ¬∀ (x y : α), edist x y = 0 ∨ edist x y = ∞),
  edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) =
    (↑(Exists.choose (p := fun (r : ℝ≥0) ↦
            r ≠ 0 ∧
              ∀ (x y : α),
                edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y)
            sorry) :
        ℝ≥0∞) *
      edist x y
Tactic: split_ifs with key
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (goal)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
x y : α
key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
⊢ edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑1 : ℝ≥0∞) * edist x y
have pos : ∀ (r : ℝ≥0) (hne : r ≠ 0)
  (hr :
    ∀ (x y : α), edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y),
  edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑1 : ℝ≥0∞) * edist x y
Tactic: rcases DilationClass.edist_eq' f with ⟨r, hne, hr⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Dilation (hypothesis)
Goal:
α : Type u_1
β : Type u_2
F : Type u_4
inst : PseudoEMetricSpace α
inst_1 : PseudoEMetricSpace β
inst_2 : FunLike F α β
inst_3 : DilationClass F α β
f : F
x y : α
key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
r : ℝ≥0
hne : r ≠ 0
hr : ∀ (x y : α), edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y
⊢ edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑1 : ℝ≥0∞) * edist x y
have hr₁ : edist.{u_2} (α := β) ((f : (a : α) → β) x : β) ((f : (a : α) → β) y : β) = (↑r : ℝ≥0∞) * edist x y
Tactic: replace hr := hr x y
Cleared hypotheses: hr