l3lab/ntp-mathlib huggingface dataset:
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · 

Proof state:
case unique_from.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
⊢ G ⟶ F

Next tactic:
refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
    

Proof state:
case unique_from.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
⊢ ∀ ⦃X Y : C⦄ (f : X ⟶ Y),
    G.map f ≫ (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y =
      (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X ≫ F.map f

Next tactic:
intros
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
    intros
    

Proof state:
case unique_from.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
⊢ G.map f✝ ≫ (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y✝ =
    (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X✝ ≫ F.map f✝

Next tactic:
exact (hF _).eq_of_tgt _ _
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_tgt _ _
  · 

Proof state:
case unique_from.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
⊢ ∀ (a : G ⟶ F), a = default

Next tactic:
intro f
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_tgt _ _
  · intro f
    

Proof state:
case unique_from.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
f : G ⟶ F
⊢ f = default

Next tactic:
ext
theorem Functor.isZero (F : C ⥤ D) (hF : ∀ X, IsZero (F.obj X)) : IsZero F := by
  constructor <;> intro G <;> refine' ⟨⟨⟨_⟩, _⟩⟩
  · refine'
      { app := fun X => (hF _).to_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_src _ _
  · intro f
    ext
    apply (hF _).eq_of_src _ _
  · refine'
      { app := fun X => (hF _).from_ _
        naturality := _ }
    intros
    exact (hF _).eq_of_tgt _ _
  · intro f
    ext
    

Proof state:
case unique_from.refine'_2.w.h
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F : C ⥤ D
hF : ∀ (X : C), IsZero (F.obj X)
G : C ⥤ D
f : G ⟶ F
x✝ : C
⊢ f.app x✝ = default.app x✝

Next tactic:
apply (hF _).eq_of_tgt _ _
theorem IsZero.obj [HasZeroObject D] {F : C ⥤ D} (hF : IsZero F) (X : C) : IsZero (F.obj X) := by
  

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject D
F : C ⥤ D
hF : IsZero F
X : C
⊢ IsZero (F.obj X)

Next tactic:
let G : C ⥤ D := (CategoryTheory.Functor.const C).obj 0
theorem IsZero.obj [HasZeroObject D] {F : C ⥤ D} (hF : IsZero F) (X : C) : IsZero (F.obj X) := by
  let G : C ⥤ D := (CategoryTheory.Functor.const C).obj 0
  

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject D
F : C ⥤ D
hF : IsZero F
X : C
G : C ⥤ D := (Functor.const C).obj 0
⊢ IsZero (F.obj X)

Next tactic:
have hG : IsZero G := Functor.isZero _ fun _ => isZero_zero _
theorem IsZero.obj [HasZeroObject D] {F : C ⥤ D} (hF : IsZero F) (X : C) : IsZero (F.obj X) := by
  let G : C ⥤ D := (CategoryTheory.Functor.const C).obj 0
  have hG : IsZero G := Functor.isZero _ fun _ => isZero_zero _
  

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject D
F : C ⥤ D
hF : IsZero F
X : C
G : C ⥤ D := (Functor.const C).obj 0
hG : IsZero G
⊢ IsZero (F.obj X)

Next tactic:
let e : F ≅ G := hF.iso hG
theorem IsZero.obj [HasZeroObject D] {F : C ⥤ D} (hF : IsZero F) (X : C) : IsZero (F.obj X) := by
  let G : C ⥤ D := (CategoryTheory.Functor.const C).obj 0
  have hG : IsZero G := Functor.isZero _ fun _ => isZero_zero _
  let e : F ≅ G := hF.iso hG
  

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject D
F : C ⥤ D
hF : IsZero F
X : C
G : C ⥤ D := (Functor.const C).obj 0
hG : IsZero G
e : F ≅ G := iso hF hG
⊢ IsZero (F.obj X)

Next tactic:
exact (isZero_zero _).of_iso (e.app X)
instance (X : C) : Subsingleton (X ≅ 0) := ⟨fun f g => by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
X : C
f g : X ≅ 0
⊢ f = g

Next tactic:
ext
instance {X : C} (f : 0 ⟶ X) : Mono f where right_cancellation g h _ := by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
X : C
f : 0 ⟶ X
Z✝ : C
g h : Z✝ ⟶ 0
x✝ : g ≫ f = h ≫ f
⊢ g = h

Next tactic:
ext
instance {X : C} (f : X ⟶ 0) : Epi f where left_cancellation g h _ := by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
X : C
f : X ⟶ 0
Z✝ : C
g h : 0 ⟶ Z✝
x✝ : f ≫ g = f ≫ h
⊢ g = h

Next tactic:
ext
instance zero_to_zero_isIso (f : (0 : C) ⟶ 0) : IsIso f := by
  

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
f : 0 ⟶ 0
⊢ IsIso f

Next tactic:
convert show IsIso (𝟙 (0 : C)) by infer_instance
instance zero_to_zero_isIso (f : (0 : C) ⟶ 0) : IsIso f := by
  convert show IsIso (𝟙 (0 : C)) by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
f : 0 ⟶ 0
⊢ IsIso (𝟙 0)

Next tactic:
infer_instance
instance (priority := 100) initialMonoClass : InitialMonoClass C :=
  InitialMonoClass.of_isInitial zeroIsInitial fun X => by 

Proof state:
C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroObject C
X : C
⊢ Mono (IsInitial.to zeroIsInitial X)

Next tactic:
infer_instance
theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by
  

Proof state:
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
⊢ Injective boxes

Next tactic:
rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂)
theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by
  rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂)
  

Proof state:
case mk.mk
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
s₁ : Finset (Box ι)
h₁ : ∀ J ∈ s₁, J ≤ I
h₁' : Set.Pairwise (↑s₁) (Disjoint on Box.toSet)
h₂ : ∀ J ∈ s₁, J ≤ I
h₂' : Set.Pairwise (↑s₁) (Disjoint on Box.toSet)
⊢ { boxes := s₁, le_of_mem' := h₁, pairwiseDisjoint := h₁' } =
    { boxes := s₁, le_of_mem' := h₂, pairwiseDisjoint := h₂' }

Next tactic:
rfl
/-- The singleton prepartition `{J}`, `J ≤ I`. -/
@[simps]
def single (I J : Box ι) (h : J ≤ I) : Prepartition I :=
  ⟨{J}, by 

Proof state:
ι : Type u_1
I✝ J✝ J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I✝
x : ι → ℝ
I J : Box ι
h : J ≤ I
⊢ ∀ J_1 ∈ {J}, J_1 ≤ I

Next tactic:
simpa
/-- The singleton prepartition `{J}`, `J ≤ I`. -/
@[simps]
def single (I J : Box ι) (h : J ≤ I) : Prepartition I :=
  ⟨{J}, by simpa, by 

Proof state:
ι : Type u_1
I✝ J✝ J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I✝
x : ι → ℝ
I J : Box ι
h : J ≤ I
⊢ Set.Pairwise (↑{J}) (Disjoint on Box.toSet)

Next tactic:
simp
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    

Proof state:
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
⊢ ∀ (a b : Prepartition I), a ≤ b → b ≤ a → a = b

Next tactic:
suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    

Proof state:
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
this : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
⊢ ∀ (a b : Prepartition I), a ≤ b → b ≤ a → a = b
case this
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
⊢ ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes

Next tactic:
exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    

Proof state:
case this
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
⊢ ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes

Next tactic:
intro π₁ π₂ h₁ h₂ J hJ
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    

Proof state:
case this
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
⊢ J ∈ π₂.boxes

Next tactic:
rcases h₁ hJ with ⟨J', hJ', hle⟩
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; 

Proof state:
case this.intro.intro
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
⊢ J ∈ π₂.boxes

Next tactic:
rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
    

Proof state:
case this.intro.intro.intro.intro
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
J'' : Box ι
hJ'' : J'' ∈ π₁
hle' : J' ≤ J''
⊢ J ∈ π₂.boxes

Next tactic:
obtain rfl : J = J''
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
    obtain rfl : J = J''; 

Proof state:
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
J'' : Box ι
hJ'' : J'' ∈ π₁
hle' : J' ≤ J''
⊢ J = J''
case this.intro.intro.intro.intro
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
hJ'' : J ∈ π₁
hle' : J' ≤ J
⊢ J ∈ π₂.boxes

Next tactic:
exact π₁.eq_of_le hJ hJ'' (hle.trans hle')
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
    obtain rfl : J = J''; exact π₁.eq_of_le hJ hJ'' (hle.trans hle')
    

Proof state:
case this.intro.intro.intro.intro
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
hJ'' : J ∈ π₁
hle' : J' ≤ J
⊢ J ∈ π₂.boxes

Next tactic:
obtain rfl : J' = J
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
    obtain rfl : J = J''; exact π₁.eq_of_le hJ hJ'' (hle.trans hle')
    obtain rfl : J' = J; 

Proof state:
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J : Box ι
hJ : J ∈ π₁.boxes
J' : Box ι
hJ' : J' ∈ π₂
hle : J ≤ J'
hJ'' : J ∈ π₁
hle' : J' ≤ J
⊢ J' = J
case this.intro.intro.intro.intro
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J' : Box ι
hJ' : J' ∈ π₂
hJ : J' ∈ π₁.boxes
hle : J' ≤ J'
hJ'' : J' ∈ π₁
hle' : J' ≤ J'
⊢ J' ∈ π₂.boxes

Next tactic:
exact le_antisymm ‹_› ‹_›
instance partialOrder : PartialOrder (Prepartition I) where
  le := (· ≤ ·)
  le_refl π I hI := ⟨I, hI, le_rfl⟩
  le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
    let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
    let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
    ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
  le_antisymm := by
    suffices : ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes
    exact fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
    intro π₁ π₂ h₁ h₂ J hJ
    rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
    obtain rfl : J = J''; exact π₁.eq_of_le hJ hJ'' (hle.trans hle')
    obtain rfl : J' = J; exact le_antisymm ‹_› ‹_›
    

Proof state:
case this.intro.intro.intro.intro
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁✝ π₂✝ : Prepartition I
x : ι → ℝ
π₁ π₂ : Prepartition I
h₁ : π₁ ≤ π₂
h₂ : π₂ ≤ π₁
J' : Box ι
hJ' : J' ∈ π₂
hJ : J' ∈ π₁.boxes
hle : J' ≤ J'
hJ'' : J' ∈ π₁
hle' : J' ≤ J'
⊢ J' ∈ π₂.boxes

Next tactic:
assumption
instance : OrderTop (Prepartition I) where
  top := single I I le_rfl
  le_top π J hJ := ⟨I, by 

Proof state:
ι : Type u_1
I J✝ J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
π : Prepartition I
J : Box ι
hJ : J ∈ π
⊢ I ∈ ⊤

Next tactic:
simp
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  

Proof state:
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
⊢ InjOn (fun J => {i | Box.lower J i = x i}) {J | J ∈ π ∧ x ∈ Box.Icc J}

Next tactic:
rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  

Proof state:
case intro.intro
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | Box.lower J₁ i = x i} = {i | Box.lower J₂ i = x i}
⊢ J₁ = J₂

Next tactic:
suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    

Proof state:
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | Box.lower J₁ i = x i} = {i | Box.lower J₂ i = x i}
this : ∀ (i : ι), Set.Nonempty (Set.Ioc (Box.lower J₁ i) (Box.upper J₁ i) ∩ Set.Ioc (Box.lower J₂ i) (Box.upper J₂ i))
⊢ J₁ = J₂

Next tactic:
choose y hy₁ hy₂ using this
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    

Proof state:
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | Box.lower J₁ i = x i} = {i | Box.lower J₂ i = x i}
y : ι → ℝ
hy₁ : ∀ (i : ι), y i ∈ Set.Ioc (Box.lower J₁ i) (Box.upper J₁ i)
hy₂ : ∀ (i : ι), y i ∈ Set.Ioc (Box.lower J₂ i) (Box.upper J₂ i)
⊢ J₁ = J₂

Next tactic:
exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
  

Proof state:
case intro.intro
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | Box.lower J₁ i = x i} = {i | Box.lower J₂ i = x i}
⊢ ∀ (i : ι), Set.Nonempty (Set.Ioc (Box.lower J₁ i) (Box.upper J₁ i) ∩ Set.Ioc (Box.lower J₂ i) (Box.upper J₂ i))

Next tactic:
intro i
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
  intro i
  

Proof state:
case intro.intro
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | Box.lower J₁ i = x i} = {i | Box.lower J₂ i = x i}
i : ι
⊢ Set.Nonempty (Set.Ioc (Box.lower J₁ i) (Box.upper J₁ i) ∩ Set.Ioc (Box.lower J₂ i) (Box.upper J₂ i))

Next tactic:
simp only [Set.ext_iff, mem_setOf] at H
/-- An auxiliary lemma used to prove that the same point can't belong to more than
`2 ^ Fintype.card ι` closed boxes of a prepartition. -/
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
    InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
  rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
  suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
  intro i
  simp only [Set.ext_iff, mem_setOf] at H
  

Proof state:
case intro.intro
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x✝ x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
i : ι
H : ∀ (x_1 : ι), Box.lower J₁ x_1 = x x_1 ↔ Box.lower J₂ x_1 = x x_1
⊢ Set.Nonempty (Set.Ioc (Box.lower J₁ i) (Box.upper J₁ i) ∩ Set.Ioc (Box.lower J₂ i) (Box.upper J₂ i))

Next tactic:
rcases (hx₁.1 i).eq_or_lt with hi₁ | hi₁