-- Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean
/-- Given a free groupoid and an arborescence of its generating quiver, the vertex
    group at the root is freely generated by loops coming from generating arrows
    in the complement of the tree. -/
lemma endIsFree : IsFreeGroup (End (root' T)) :=
  IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubquiverSymmetrify T)ᶜ : Set _)
    (fun e => loopOfHom T (of e.val.hom))
    (by
      intro X _ f
      let f' : Labelling (Generators G) X := fun a b e =>
        if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨⟨a, b, e⟩, h⟩
      rcases unique_lift f' with ⟨F', hF', uF'⟩
      refine ⟨F'.mapEnd _, ?_, ?_⟩
      · suffices ∀ {x y} (q : x ⟶ y), F'.map (loopOfHom T q) = (F'.map q : X) by
          rintro ⟨⟨a, b, e⟩, h⟩
          erw [Functor.mapEnd_apply, this, hF']
          exact dif_neg h
        intros x y q
        suffices ∀ {a} (p : Path (root T) a), F'.map (homOfPath T p) = 1 by
          simp only [this, treeHom, comp_as_mul, inv_as_inv, loopOfHom, inv_one, mul_one,
            one_mul, Functor.map_inv, Functor.map_comp]
        intro a p
        induction' p with b c p e ih
        · rw [homOfPath, F'.map_id, id_as_one]
        rw [homOfPath, F'.map_comp, comp_as_mul, ih, mul_one]
        rcases e with ⟨e | e, eT⟩
        · rw [hF']
          exact dif_pos (Or.inl eT)
        · rw [F'.map_inv, inv_as_inv, inv_eq_one, hF']
          exact dif_pos (Or.inr eT)
      · intro E hE
        ext x
        suffices (functorOfMonoidHom T E).map x = F'.map x by
          simpa only [loopOfHom, functorOfMonoidHom, IsIso.inv_id, treeHom_root,
            Category.id_comp, Category.comp_id] using this
        congr
        apply uF'
        intro a b e
        change E (loopOfHom T _) = dite _ _ _
        split_ifs with h
        · rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one]
        · exact hE ⟨⟨a, b, e⟩, h⟩)

Proof state:
case intro.intro.refine_2.h.h.e_5.h.e_self
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X), (fun F => ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
E : End (IsFreeGroupoid.SpanningTree.root' T) →* X
hE :
  ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), E ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
x : End (IsFreeGroupoid.SpanningTree.root' T)
⊢ functorOfMonoidHom T E = F'

Tactic:
apply uF'

Result:
case intro.intro.refine_2.h.h.e_5.h.e_self.a
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X), (fun F => ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
E : End (IsFreeGroupoid.SpanningTree.root' T) →* X
hE :
  ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), E ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
x : End (IsFreeGroupoid.SpanningTree.root' T)
⊢ ∀ (a b : Generators G) (g : a ⟶ b), (functorOfMonoidHom T E).map (of g) = f' g

Proof state:
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
⊢ ∃! F,
    ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a

Tactic:
rcases unique_lift f' with ⟨F', hF', uF'⟩

Result:
case intro.intro
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X), (fun F => ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
⊢ ∃! F,
    ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a

Proof state:
case intro.intro.refine_2
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X), (fun F => ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
E : End (IsFreeGroupoid.SpanningTree.root' T) →* X
hE :
  ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), E ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
⊢ E = Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F'

Tactic:
ext x

Result:
case intro.intro.refine_2.h
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
  fun a b e => if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X), (fun F => ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
E : End (IsFreeGroupoid.SpanningTree.root' T) →* X
hE :
  ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), E ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
x : End (IsFreeGroupoid.SpanningTree.root' T)
⊢ E x = (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F') x