l3lab/ntp-mathlib huggingface dataset:
/-- Each connected component of the category is connected. -/
instance (j : ConnectedComponents J) : IsConnected (Component j) := by
  -- Show it's connected by constructing a zigzag (in `Component j`) between any two objects
  apply isConnected_of_zigzag
  rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
  -- We know that the underlying objects j₁ j₂ have some zigzag between them in `J`
  have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
  -- Get an explicit zigzag as a list
  rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
  -- Everything which has a zigzag to j₂ can be lifted to the same component as `j₂`.
  let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' h⟩
  -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
  have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
    intro i hi
    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
    · intro j k
      apply Relation.ReflTransGen.head
    · apply Relation.ReflTransGen.refl
  -- Now lift the zigzag from `j₁` to `j₂` in `J` to the same thing in `component j`.
  refine' ⟨l.pmap f hf, _, _⟩
  · 

Proof state:
case h.mk.mk.intro.intro.refine'_1
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j₁ j₂ : J
hj₁ : Quotient.mk'' j₁ = Quotient.mk'' j₂
h₁₂ : Zigzag j₁ j₂
l : List J
hl₁ : List.Chain Zag j₁ l
hl₂ : List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂
f : (x : J) → Zigzag x j₂ → Component (Quotient.mk'' j₂) :=
  fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j₂) }
hf : ∀ (a : J), a ∈ l → Zigzag a j₂
⊢ List.Chain Zag { obj := j₁, property := hj₁ } (List.pmap f l hf)

Next tactic:
refine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl₁ h₁₂ _
/-- Each connected component of the category is connected. -/
instance (j : ConnectedComponents J) : IsConnected (Component j) := by
  -- Show it's connected by constructing a zigzag (in `Component j`) between any two objects
  apply isConnected_of_zigzag
  rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
  -- We know that the underlying objects j₁ j₂ have some zigzag between them in `J`
  have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
  -- Get an explicit zigzag as a list
  rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
  -- Everything which has a zigzag to j₂ can be lifted to the same component as `j₂`.
  let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' h⟩
  -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
  have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
    intro i hi
    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
    · intro j k
      apply Relation.ReflTransGen.head
    · apply Relation.ReflTransGen.refl
  -- Now lift the zigzag from `j₁` to `j₂` in `J` to the same thing in `component j`.
  refine' ⟨l.pmap f hf, _, _⟩
  · refine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl₁ h₁₂ _
    

Proof state:
case h.mk.mk.intro.intro.refine'_1
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j₁ j₂ : J
hj₁ : Quotient.mk'' j₁ = Quotient.mk'' j₂
h₁₂ : Zigzag j₁ j₂
l : List J
hl₁ : List.Chain Zag j₁ l
hl₂ : List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂
f : (x : J) → Zigzag x j₂ → Component (Quotient.mk'' j₂) :=
  fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j₂) }
hf : ∀ (a : J), a ∈ l → Zigzag a j₂
x y : J
x✝¹ : Zigzag x j₂
x✝ : Zigzag y j₂
h : Zag x y
⊢ Zag (f x x✝¹) (f y x✝)

Next tactic:
exact zag_of_zag_obj (Component.ι _) h
/-- Each connected component of the category is connected. -/
instance (j : ConnectedComponents J) : IsConnected (Component j) := by
  -- Show it's connected by constructing a zigzag (in `Component j`) between any two objects
  apply isConnected_of_zigzag
  rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
  -- We know that the underlying objects j₁ j₂ have some zigzag between them in `J`
  have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
  -- Get an explicit zigzag as a list
  rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
  -- Everything which has a zigzag to j₂ can be lifted to the same component as `j₂`.
  let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' h⟩
  -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
  have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
    intro i hi
    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
    · intro j k
      apply Relation.ReflTransGen.head
    · apply Relation.ReflTransGen.refl
  -- Now lift the zigzag from `j₁` to `j₂` in `J` to the same thing in `component j`.
  refine' ⟨l.pmap f hf, _, _⟩
  · refine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl₁ h₁₂ _
    exact zag_of_zag_obj (Component.ι _) h
  · 

Proof state:
case h.mk.mk.intro.intro.refine'_2
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j₁ j₂ : J
hj₁ : Quotient.mk'' j₁ = Quotient.mk'' j₂
h₁₂ : Zigzag j₁ j₂
l : List J
hl₁ : List.Chain Zag j₁ l
hl₂ : List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂
f : (x : J) → Zigzag x j₂ → Component (Quotient.mk'' j₂) :=
  fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j₂) }
hf : ∀ (a : J), a ∈ l → Zigzag a j₂
⊢ List.getLast ({ obj := j₁, property := hj₁ } :: List.pmap f l hf)
      (_ : { obj := j₁, property := hj₁ } :: List.pmap f l hf ≠ []) =
    { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
erw [List.getLast_pmap _ f (j₁ :: l) (by simpa [h₁₂] using hf) (List.cons_ne_nil _ _)]
/-- Each connected component of the category is connected. -/
instance (j : ConnectedComponents J) : IsConnected (Component j) := by
  -- Show it's connected by constructing a zigzag (in `Component j`) between any two objects
  apply isConnected_of_zigzag
  rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
  -- We know that the underlying objects j₁ j₂ have some zigzag between them in `J`
  have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
  -- Get an explicit zigzag as a list
  rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
  -- Everything which has a zigzag to j₂ can be lifted to the same component as `j₂`.
  let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' h⟩
  -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
  have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
    intro i hi
    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
    · intro j k
      apply Relation.ReflTransGen.head
    · apply Relation.ReflTransGen.refl
  -- Now lift the zigzag from `j₁` to `j₂` in `J` to the same thing in `component j`.
  refine' ⟨l.pmap f hf, _, _⟩
  · refine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl₁ h₁₂ _
    exact zag_of_zag_obj (Component.ι _) h
  · erw [List.getLast_pmap _ f (j₁ :: l) (by 

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j₁ j₂ : J
hj₁ : Quotient.mk'' j₁ = Quotient.mk'' j₂
h₁₂ : Zigzag j₁ j₂
l : List J
hl₁ : List.Chain Zag j₁ l
hl₂ : List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂
f : (x : J) → Zigzag x j₂ → Component (Quotient.mk'' j₂) :=
  fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j₂) }
hf : ∀ (a : J), a ∈ l → Zigzag a j₂
⊢ ∀ (a : J), a ∈ j₁ :: l → Zigzag a j₂

Next tactic:
simpa [h₁₂] using hf
/-- Each connected component of the category is connected. -/
instance (j : ConnectedComponents J) : IsConnected (Component j) := by
  -- Show it's connected by constructing a zigzag (in `Component j`) between any two objects
  apply isConnected_of_zigzag
  rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
  -- We know that the underlying objects j₁ j₂ have some zigzag between them in `J`
  have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
  -- Get an explicit zigzag as a list
  rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
  -- Everything which has a zigzag to j₂ can be lifted to the same component as `j₂`.
  let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' h⟩
  -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
  have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
    intro i hi
    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
    · intro j k
      apply Relation.ReflTransGen.head
    · apply Relation.ReflTransGen.refl
  -- Now lift the zigzag from `j₁` to `j₂` in `J` to the same thing in `component j`.
  refine' ⟨l.pmap f hf, _, _⟩
  · refine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl₁ h₁₂ _
    exact zag_of_zag_obj (Component.ι _) h
  · erw [List.getLast_pmap _ f (j₁ :: l) (by simpa [h₁₂] using hf) (List.cons_ne_nil _ _)]
    

Proof state:
case h.mk.mk.intro.intro.refine'_2
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j₁ j₂ : J
hj₁ : Quotient.mk'' j₁ = Quotient.mk'' j₂
h₁₂ : Zigzag j₁ j₂
l : List J
hl₁ : List.Chain Zag j₁ l
hl₂ : List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂
f : (x : J) → Zigzag x j₂ → Component (Quotient.mk'' j₂) :=
  fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j₂) }
hf : ∀ (a : J), a ∈ l → Zigzag a j₂
⊢ f (List.getLast (j₁ :: l) (_ : j₁ :: l ≠ [])) (_ : Zigzag (List.getLast (j₁ :: l) (_ : j₁ :: l ≠ [])) j₂) =
    { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
exact FullSubcategory.ext _ _ hl₂
instance : Full (decomposedTo J)
    where
  preimage := by
    

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
⊢ {X Y : Decomposed J} → ((decomposedTo J).obj X ⟶ (decomposedTo J).obj Y) → (X ⟶ Y)

Next tactic:
rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
k' : ConnectedComponents J
Y : J
hY : Quotient.mk'' Y = k'
f :
  (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj { fst := k', snd := { obj := Y, property := hY } }
⊢ { fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := { obj := Y, property := hY } }

Next tactic:
dsimp at f
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
k' : ConnectedComponents J
Y : J
hY : Quotient.mk'' Y = k'
f : X ⟶ Y
⊢ { fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := { obj := Y, property := hY } }

Next tactic:
have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
k' : ConnectedComponents J
Y : J
hY : Quotient.mk'' Y = k'
f : X ⟶ Y
⊢ j' = k'

Next tactic:
rw [← hX, ← hY, Quotient.eq'']
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
k' : ConnectedComponents J
Y : J
hY : Quotient.mk'' Y = k'
f : X ⟶ Y
⊢ Setoid.r X Y

Next tactic:
exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
k' : ConnectedComponents J
Y : J
hY : Quotient.mk'' Y = k'
f : X ⟶ Y
this : j' = k'
⊢ { fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := { obj := Y, property := hY } }

Next tactic:
subst this
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
Y : J
f : X ⟶ Y
hY : Quotient.mk'' Y = j'
⊢ { fst := j', snd := { obj := X, property := hX } } ⟶ { fst := j', snd := { obj := Y, property := hY } }

Next tactic:
exact Sigma.SigmaHom.mk f
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
⊢ ∀ {X Y : Decomposed J} (f : (decomposedTo J).obj X ⟶ (decomposedTo J).obj Y),
    (decomposedTo J).map
        (Sigma.casesOn (motive := fun x =>
          {Y : Decomposed J} → ((decomposedTo J).obj x ⟶ (decomposedTo J).obj Y) → (x ⟶ Y)) X
          (fun j' snd =>
            @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')
              (fun x =>
                {Y : Decomposed J} →
                  ((decomposedTo J).obj { fst := j', snd := x } ⟶ (decomposedTo J).obj Y) →
                    ({ fst := j', snd := x } ⟶ Y))
              snd fun X hX {Y} =>
              Sigma.casesOn (motive := fun x =>
                ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶ (decomposedTo J).obj x) →
                  ({ fst := j', snd := { obj := X, property := hX } } ⟶ x))
                Y fun k' snd =>
                FullSubcategory.casesOn (motive := fun x =>
                  ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
                      (decomposedTo J).obj { fst := k', snd := x }) →
                    ({ fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := x }))
                  snd fun Y hY f =>
                  let_fun this := (_ : j' = k');
                  Eq.ndrec (motive := fun k' =>
                    (hY : Quotient.mk'' Y = k') →
                      { fst := j', snd := { obj := X, property := hX } } ⟶
                        { fst := k', snd := { obj := Y, property := hY } })
                    (fun hY => Sigma.SigmaHom.mk f) this hY)
          f) =
      f

Next tactic:
rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
Y : J
f :
  (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj
      { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }
⊢ (decomposedTo J).map
      (Sigma.casesOn (motive := fun x =>
        {Y : Decomposed J} → ((decomposedTo J).obj x ⟶ (decomposedTo J).obj Y) → (x ⟶ Y))
        { fst := j', snd := { obj := X, property := hX } }
        (fun j' snd =>
          @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')
            (fun x =>
              {Y : Decomposed J} →
                ((decomposedTo J).obj { fst := j', snd := x } ⟶ (decomposedTo J).obj Y) → ({ fst := j', snd := x } ⟶ Y))
            snd fun X hX {Y} =>
            Sigma.casesOn (motive := fun x =>
              ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶ (decomposedTo J).obj x) →
                ({ fst := j', snd := { obj := X, property := hX } } ⟶ x))
              Y fun k' snd =>
              FullSubcategory.casesOn (motive := fun x =>
                ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
                    (decomposedTo J).obj { fst := k', snd := x }) →
                  ({ fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := x }))
                snd fun Y hY f =>
                let_fun this := (_ : j' = k');
                Eq.ndrec (motive := fun k' =>
                  (hY : Quotient.mk'' Y = k') →
                    { fst := j', snd := { obj := X, property := hX } } ⟶
                      { fst := k', snd := { obj := Y, property := hY } })
                  (fun hY => Sigma.SigmaHom.mk f) this hY)
        f) =
    f

Next tactic:
have : Quotient.mk'' Y = j' := by
      rw [← hX, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inr ⟨f⟩)
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
    have : Quotient.mk'' Y = j' := by
      

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
Y : J
f :
  (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj
      { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }
⊢ Quotient.mk'' Y = j'

Next tactic:
rw [← hX, Quotient.eq'']
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
    have : Quotient.mk'' Y = j' := by
      rw [← hX, Quotient.eq'']
      

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
Y : J
f :
  (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj
      { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }
⊢ Setoid.r Y X

Next tactic:
exact Relation.ReflTransGen.single (Or.inr ⟨f⟩)
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
    have : Quotient.mk'' Y = j' := by
      rw [← hX, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inr ⟨f⟩)
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j' : ConnectedComponents J
X : J
hX : Quotient.mk'' X = j'
Y : J
f :
  (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj
      { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }
this : Quotient.mk'' Y = j'
⊢ (decomposedTo J).map
      (Sigma.casesOn (motive := fun x =>
        {Y : Decomposed J} → ((decomposedTo J).obj x ⟶ (decomposedTo J).obj Y) → (x ⟶ Y))
        { fst := j', snd := { obj := X, property := hX } }
        (fun j' snd =>
          @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')
            (fun x =>
              {Y : Decomposed J} →
                ((decomposedTo J).obj { fst := j', snd := x } ⟶ (decomposedTo J).obj Y) → ({ fst := j', snd := x } ⟶ Y))
            snd fun X hX {Y} =>
            Sigma.casesOn (motive := fun x =>
              ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶ (decomposedTo J).obj x) →
                ({ fst := j', snd := { obj := X, property := hX } } ⟶ x))
              Y fun k' snd =>
              FullSubcategory.casesOn (motive := fun x =>
                ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
                    (decomposedTo J).obj { fst := k', snd := x }) →
                  ({ fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := x }))
                snd fun Y hY f =>
                let_fun this := (_ : j' = k');
                Eq.ndrec (motive := fun k' =>
                  (hY : Quotient.mk'' Y = k') →
                    { fst := j', snd := { obj := X, property := hX } } ⟶
                      { fst := k', snd := { obj := Y, property := hY } })
                  (fun hY => Sigma.SigmaHom.mk f) this hY)
        f) =
    f

Next tactic:
subst this
instance : Full (decomposedTo J)
    where
  preimage := by
    rintro ⟨j', X, hX⟩ ⟨k', Y, hY⟩ f
    dsimp at f
    have : j' = k' := by
      rw [← hX, ← hY, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inl ⟨f⟩)
    subst this
    exact Sigma.SigmaHom.mk f
  witness := by
    rintro ⟨j', X, hX⟩ ⟨_, Y, rfl⟩ f
    have : Quotient.mk'' Y = j' := by
      rw [← hX, Quotient.eq'']
      exact Relation.ReflTransGen.single (Or.inr ⟨f⟩)
    subst this
    

Proof state:
case mk.mk.mk.mk
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
X Y : J
hX : Quotient.mk'' X = Quotient.mk'' Y
f :
  (decomposedTo J).obj { fst := Quotient.mk'' Y, snd := { obj := X, property := hX } } ⟶
    (decomposedTo J).obj
      { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }
⊢ (decomposedTo J).map
      (Sigma.casesOn (motive := fun x =>
        {Y : Decomposed J} → ((decomposedTo J).obj x ⟶ (decomposedTo J).obj Y) → (x ⟶ Y))
        { fst := Quotient.mk'' Y, snd := { obj := X, property := hX } }
        (fun j' snd =>
          @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')
            (fun x =>
              {Y : Decomposed J} →
                ((decomposedTo J).obj { fst := j', snd := x } ⟶ (decomposedTo J).obj Y) → ({ fst := j', snd := x } ⟶ Y))
            snd fun X hX {Y} =>
            Sigma.casesOn (motive := fun x =>
              ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶ (decomposedTo J).obj x) →
                ({ fst := j', snd := { obj := X, property := hX } } ⟶ x))
              Y fun k' snd =>
              FullSubcategory.casesOn (motive := fun x =>
                ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } ⟶
                    (decomposedTo J).obj { fst := k', snd := x }) →
                  ({ fst := j', snd := { obj := X, property := hX } } ⟶ { fst := k', snd := x }))
                snd fun Y hY f =>
                let_fun this := (_ : j' = k');
                Eq.ndrec (motive := fun k' =>
                  (hY : Quotient.mk'' Y = k') →
                    { fst := j', snd := { obj := X, property := hX } } ⟶
                      { fst := k', snd := { obj := Y, property := hY } })
                  (fun hY => Sigma.SigmaHom.mk f) this hY)
        f) =
    f

Next tactic:
rfl
instance : Faithful (decomposedTo J) where
  map_injective := by
    

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
⊢ ∀ {X Y : Decomposed J}, Function.Injective (decomposedTo J).map

Next tactic:
rintro ⟨_, j, rfl⟩ ⟨_, k, hY⟩ ⟨f⟩ ⟨_⟩ rfl