l3lab/ntp-mathlib huggingface dataset:
/-- Make a `VectorBundle` from a `VectorPrebundle`.  Concretely this means
that, given a `VectorPrebundle` structure for a sigma-type `E` -- which consists of a
number of "pretrivializations" identifying parts of `E` with product spaces `U × F` -- one
establishes that for the topology constructed on the sigma-type using
`VectorPrebundle.totalSpaceTopology`, these "pretrivializations" are actually
"trivializations" (i.e., homeomorphisms with respect to the constructed topology). -/
theorem toVectorBundle : @VectorBundle R _ F E _ _ _ _ _ _ a.totalSpaceTopology _ a.toFiberBundle :=
  letI := a.totalSpaceTopology; letI := a.toFiberBundle
  { trivialization_linear' := by
      rintro _ ⟨e, he, rfl⟩
      apply linear_trivializationOfMemPretrivializationAtlas
    continuousOn_coordChange' := by
      rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
      refine (a.continuousOn_coordChange he he').congr fun b hb ↦ ?_
      ext v
      -- porting note: help `rw` find instances
      haveI h₁ := a.linear_trivializationOfMemPretrivializationAtlas he
      haveI h₂ := a.linear_trivializationOfMemPretrivializationAtlas he'
      rw [trivializationOfMemPretrivializationAtlas] at h₁ h₂
      

Proof state:
case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a
this : FiberBundle F E := toFiberBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ (toFiberPrebundle a).pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)
h₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')
⊢ ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)
          (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)
      v =
    (coordChange a he he' b) v

Next tactic:
rw [a.coordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
        Trivialization.coordChangeL_apply]
/-- Rewrite `ContinuousLinearMap.inCoordinates` using continuous linear equivalences. -/
theorem inCoordinates_eq (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y)
    (hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet) :
    inCoordinates F E F' E' x₀ x y₀ y ϕ =
      ((trivializationAt F' E' y₀).continuousLinearEquivAt 𝕜₂ y hy : E' y →L[𝕜₂] F').comp
        (ϕ.comp <|
          (((trivializationAt F E x₀).continuousLinearEquivAt 𝕜₁ x hx).symm : F →L[𝕜₁] E x)) := by
  

Proof state:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝²² : NontriviallyNormedField R
inst✝²¹ : (x : B) → AddCommMonoid (E x)
inst✝²⁰ : (x : B) → Module R (E x)
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace R F
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : (x : B) → TopologicalSpace (E x)
𝕜₁ : Type u_5
𝕜₂ : Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜₁
inst✝¹⁴ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B' : Type u_7
inst✝¹³ : TopologicalSpace B'
inst✝¹² : NormedSpace 𝕜₁ F
inst✝¹¹ : (x : B) → Module 𝕜₁ (E x)
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
F' : Type u_8
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜₂ F'
E' : B' → Type u_9
inst✝⁷ : (x : B') → AddCommMonoid (E' x)
inst✝⁶ : (x : B') → Module 𝕜₂ (E' x)
inst✝⁵ : TopologicalSpace (TotalSpace F' E')
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜₁ F E
inst✝² : (x : B') → TopologicalSpace (E' x)
inst✝¹ : FiberBundle F' E'
inst✝ : VectorBundle 𝕜₂ F' E'
x₀ x : B
y₀ y : B'
ϕ : E x →SL[σ] E' y
hx : x ∈ (trivializationAt F E x₀).baseSet
hy : y ∈ (trivializationAt F' E' y₀).baseSet
⊢ inCoordinates F E F' E' x₀ x y₀ y ϕ =
    comp (↑(Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F' E' y₀) y hy))
      (comp ϕ ↑(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F E x₀) x hx)))

Next tactic:
ext
/-- Rewrite `ContinuousLinearMap.inCoordinates` using continuous linear equivalences. -/
theorem inCoordinates_eq (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y)
    (hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet) :
    inCoordinates F E F' E' x₀ x y₀ y ϕ =
      ((trivializationAt F' E' y₀).continuousLinearEquivAt 𝕜₂ y hy : E' y →L[𝕜₂] F').comp
        (ϕ.comp <|
          (((trivializationAt F E x₀).continuousLinearEquivAt 𝕜₁ x hx).symm : F →L[𝕜₁] E x)) := by
  ext
  

Proof state:
case h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝²² : NontriviallyNormedField R
inst✝²¹ : (x : B) → AddCommMonoid (E x)
inst✝²⁰ : (x : B) → Module R (E x)
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace R F
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : (x : B) → TopologicalSpace (E x)
𝕜₁ : Type u_5
𝕜₂ : Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜₁
inst✝¹⁴ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B' : Type u_7
inst✝¹³ : TopologicalSpace B'
inst✝¹² : NormedSpace 𝕜₁ F
inst✝¹¹ : (x : B) → Module 𝕜₁ (E x)
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
F' : Type u_8
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜₂ F'
E' : B' → Type u_9
inst✝⁷ : (x : B') → AddCommMonoid (E' x)
inst✝⁶ : (x : B') → Module 𝕜₂ (E' x)
inst✝⁵ : TopologicalSpace (TotalSpace F' E')
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜₁ F E
inst✝² : (x : B') → TopologicalSpace (E' x)
inst✝¹ : FiberBundle F' E'
inst✝ : VectorBundle 𝕜₂ F' E'
x₀ x : B
y₀ y : B'
ϕ : E x →SL[σ] E' y
hx : x ∈ (trivializationAt F E x₀).baseSet
hy : y ∈ (trivializationAt F' E' y₀).baseSet
x✝ : F
⊢ (inCoordinates F E F' E' x₀ x y₀ y ϕ) x✝ =
    (comp (↑(Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F' E' y₀) y hy))
        (comp ϕ
          ↑(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F E x₀) x hx))))
      x✝

Next tactic:
simp_rw [inCoordinates, ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe,
    Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq]
/-- Rewrite `ContinuousLinearMap.inCoordinates` in a `VectorBundleCore`. -/
protected theorem _root_.VectorBundleCore.inCoordinates_eq {ι ι'} (Z : VectorBundleCore 𝕜₁ B F ι)
    (Z' : VectorBundleCore 𝕜₂ B' F' ι') {x₀ x : B} {y₀ y : B'} (ϕ : F →SL[σ] F')
    (hx : x ∈ Z.baseSet (Z.indexAt x₀)) (hy : y ∈ Z'.baseSet (Z'.indexAt y₀)) :
    inCoordinates F Z.Fiber F' Z'.Fiber x₀ x y₀ y ϕ =
      (Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y).comp
        (ϕ.comp <| Z.coordChange (Z.indexAt x₀) (Z.indexAt x) x) := by
  

Proof state:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝²² : NontriviallyNormedField R
inst✝²¹ : (x : B) → AddCommMonoid (E x)
inst✝²⁰ : (x : B) → Module R (E x)
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace R F
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : (x : B) → TopologicalSpace (E x)
𝕜₁ : Type u_5
𝕜₂ : Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜₁
inst✝¹⁴ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B' : Type u_7
inst✝¹³ : TopologicalSpace B'
inst✝¹² : NormedSpace 𝕜₁ F
inst✝¹¹ : (x : B) → Module 𝕜₁ (E x)
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
F' : Type u_8
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜₂ F'
E' : B' → Type u_9
inst✝⁷ : (x : B') → AddCommMonoid (E' x)
inst✝⁶ : (x : B') → Module 𝕜₂ (E' x)
inst✝⁵ : TopologicalSpace (TotalSpace F' E')
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜₁ F E
inst✝² : (x : B') → TopologicalSpace (E' x)
inst✝¹ : FiberBundle F' E'
inst✝ : VectorBundle 𝕜₂ F' E'
ι : Type u_10
ι' : Type u_11
Z : VectorBundleCore 𝕜₁ B F ι
Z' : VectorBundleCore 𝕜₂ B' F' ι'
x₀ x : B
y₀ y : B'
ϕ : F →SL[σ] F'
hx : x ∈ VectorBundleCore.baseSet Z (VectorBundleCore.indexAt Z x₀)
hy : y ∈ VectorBundleCore.baseSet Z' (VectorBundleCore.indexAt Z' y₀)
⊢ inCoordinates F (VectorBundleCore.Fiber Z) F' (VectorBundleCore.Fiber Z') x₀ x y₀ y ϕ =
    comp (VectorBundleCore.coordChange Z' (VectorBundleCore.indexAt Z' y) (VectorBundleCore.indexAt Z' y₀) y)
      (comp ϕ (VectorBundleCore.coordChange Z (VectorBundleCore.indexAt Z x₀) (VectorBundleCore.indexAt Z x) x))

Next tactic:
simp_rw [inCoordinates, Z'.trivializationAt_continuousLinearMapAt hy,
    Z.trivializationAt_symmL hx]
/-- Each connected component of the category is nonempty. -/
instance (j : ConnectedComponents J) : Nonempty (Component j) := by
  

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j : ConnectedComponents J
⊢ Nonempty (Component j)

Next tactic:
induction j using Quotient.inductionOn'
/-- Each connected component of the category is nonempty. -/
instance (j : ConnectedComponents J) : Nonempty (Component j) := by
  induction j using Quotient.inductionOn'
  

Proof state:
case h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
a✝ : J
⊢ Nonempty (Component (Quotient.mk'' a✝))

Next tactic:
exact ⟨⟨_, rfl⟩⟩
/-- 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
  

Proof state:
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j : ConnectedComponents J
⊢ IsConnected (Component j)

Next tactic:
apply isConnected_of_zigzag
/-- 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
  

Proof state:
case h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₂
inst✝ : Category.{u₁, u₂} C
j : ConnectedComponents J
⊢ ∀ (j₁ j₂ : Component j), ∃ l, List.Chain Zag j₁ l ∧ List.getLast (j₁ :: l) (_ : j₁ :: l ≠ []) = j₂

Next tactic:
rintro ⟨j₁, hj₁⟩ ⟨j₂, rfl⟩
/-- 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`
  

Proof state:
case h.mk.mk
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
⊢ ∃ l,
    List.Chain Zag { obj := j₁, property := hj₁ } l ∧
      List.getLast ({ obj := j₁, property := hj₁ } :: l) (_ : { obj := j₁, property := hj₁ } :: l ≠ []) =
        { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
have h₁₂ : Zigzag j₁ j₂ := Quotient.exact' hj₁
/-- 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
  

Proof state:
case h.mk.mk
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.Chain Zag { obj := j₁, property := hj₁ } l ∧
      List.getLast ({ obj := j₁, property := hj₁ } :: l) (_ : { obj := j₁, property := hj₁ } :: l ≠ []) =
        { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
rcases List.exists_chain_of_relationReflTransGen h₁₂ with ⟨l, hl₁, hl₂⟩
/-- 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`.
  

Proof state:
case h.mk.mk.intro.intro
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
⊢ ∃ l,
    List.Chain Zag { obj := j₁, property := hj₁ } l ∧
      List.getLast ({ obj := j₁, property := hj₁ } :: l) (_ : { obj := j₁, property := hj₁ } :: l ≠ []) =
        { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
let f : ∀ x, Zigzag x j₂ → Component (Quotient.mk'' j₂) := fun x h => ⟨x, Quotient.sound' 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`.
  

Proof state:
case h.mk.mk.intro.intro
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) }
⊢ ∃ l,
    List.Chain Zag { obj := j₁, property := hj₁ } l ∧
      List.getLast ({ obj := j₁, property := hj₁ } :: l) (_ : { obj := j₁, property := hj₁ } :: l ≠ []) =
        { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
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
/-- 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
    

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) }
⊢ ∀ (a : J), a ∈ l → Zigzag a j₂

Next tactic:
intro i hi
/-- 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
    

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) }
i : J
hi : i ∈ l
⊢ Zigzag i j₂

Next tactic:
apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
/-- 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)
    · 

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) }
i : J
hi : i ∈ l
⊢ ∀ ⦃x y : J⦄, Zag x y → (fun t => Zigzag t j₂) y → (fun t => Zigzag t j₂) x

Next tactic:
intro j k
/-- 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
      

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) }
i : J
hi : i ∈ l
j k : J
⊢ Zag j k → (fun t => Zigzag t j₂) k → (fun t => Zigzag t j₂) j

Next tactic:
apply Relation.ReflTransGen.head
/-- 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
    · 

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) }
i : J
hi : i ∈ l
⊢ Zigzag j₂ j₂

Next tactic:
apply Relation.ReflTransGen.refl
/-- 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`.
  

Proof state:
case h.mk.mk.intro.intro
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
⊢ ∃ l,
    List.Chain Zag { obj := j₁, property := hj₁ } l ∧
      List.getLast ({ obj := j₁, property := hj₁ } :: l) (_ : { obj := j₁, property := hj₁ } :: l ≠ []) =
        { obj := j₂, property := (_ : Quotient.mk'' j₂ = Quotient.mk'' j₂) }

Next tactic:
refine' ⟨l.pmap f hf, _, _⟩