Theorem: IsFreeGroupoid.SpanningTree.endIsFree
File: Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean

tactic: intro X _ f
state_before: G : Type u
inst✝² : Groupoid G
inst✝¹ : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
⊢ ∀ {H : Type u} [inst : Group H] (f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → H),
    ∃! F,
      ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
        F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
state_after: 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,
    ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a

tactic: 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⟩
state_before: 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,
    ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ), F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a
state_after: 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'⟩
state_before: 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
state_after: 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

tactic: refine ⟨F'.mapEnd _, ?_, ?_⟩
state_before: 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
state_after: case intro.intro.refine_1
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'
⊢ (fun F =>
      ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
        F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a)
    (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F')

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'
⊢ ∀ (y : End (IsFreeGroupoid.SpanningTree.root' T) →* X),
    (fun F =>
          ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
            F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a)
        y →
      y = Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F'

tactic: 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
state_before: case intro.intro.refine_1
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'
⊢ (fun F =>
      ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
        F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a)
    (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F')
state_after: case intro.intro.refine_1
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'
⊢ ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q

tactic: intros x y q
state_before: case intro.intro.refine_1
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'
⊢ ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
state_after: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
⊢ F'.map (loopOfHom T q) = F'.map q

tactic: 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]
state_before: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
⊢ F'.map (loopOfHom T q) = F'.map q
state_after: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
⊢ ∀ {a : WideSubquiver.toType (Symmetrify (Generators G)) T}
    (p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a), F'.map (homOfPath T p) = 1

tactic: intro a p
state_before: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
⊢ ∀ {a : WideSubquiver.toType (Symmetrify (Generators G)) T}
    (p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a), F'.map (homOfPath T p) = 1
state_after: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
a : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a
⊢ F'.map (homOfPath T p) = 1

tactic: induction' p with b c p e ih
state_before: case intro.intro.refine_1
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'
x y : G
q : x ⟶ y
a : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a
⊢ F'.map (homOfPath T p) = 1
state_after: case intro.intro.refine_1.nil
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'
x y : G
q : x ⟶ y
a : WideSubquiver.toType (Symmetrify (Generators G)) T
⊢ F'.map (homOfPath T Path.nil) = 1

case intro.intro.refine_1.cons
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
e : b ⟶ c
ih : F'.map (homOfPath T p) = 1
⊢ F'.map (homOfPath T (p.cons e)) = 1

tactic: rw [homOfPath, F'.map_comp, comp_as_mul, ih, mul_one]
state_before: case intro.intro.refine_1.cons
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
e : b ⟶ c
ih : F'.map (homOfPath T p) = 1
⊢ F'.map (homOfPath T (p.cons e)) = 1
state_after: case intro.intro.refine_1.cons
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
e : b ⟶ c
ih : F'.map (homOfPath T p) = 1
⊢ F'.map (Sum.recOn (↑e) (fun e => of e) fun e => inv (of e)) = 1

tactic: rcases e with ⟨e | e, eT⟩
state_before: case intro.intro.refine_1.cons
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
e : b ⟶ c
ih : F'.map (homOfPath T p) = 1
⊢ F'.map (Sum.recOn (↑e) (fun e => of e) fun e => inv (of e)) = 1
state_after: case intro.intro.refine_1.cons.mk.inl
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : b ⟶ c
eT : Sum.inl e ∈ T b c
⊢ F'.map (Sum.recOn (↑⟨Sum.inl e, eT⟩) (fun e => of e) fun e => inv (of e)) = 1

case intro.intro.refine_1.cons.mk.inr
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : c ⟶ b
eT : Sum.inr e ∈ T b c
⊢ F'.map (Sum.recOn (↑⟨Sum.inr e, eT⟩) (fun e => of e) fun e => inv (of e)) = 1

tactic: rintro ⟨⟨a, b, e⟩, h⟩
state_before: 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'
this : ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
⊢ (fun F =>
      ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
        F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a)
    (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F')
state_after: case mk.mk
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'
this : ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
a b : Generators G
e : a ⟶ b
h : { left := a, right := b, hom := e } ∈ (wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ
⊢ (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F')
      ((fun e => loopOfHom T (of (↑e).hom)) ⟨{ left := a, right := b, hom := e }, h⟩) =
    f ⟨{ left := a, right := b, hom := e }, h⟩

tactic: erw [Functor.mapEnd_apply, this, hF']
state_before: case mk.mk
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'
this : ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
a b : Generators G
e : a ⟶ b
h : { left := a, right := b, hom := e } ∈ (wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ
⊢ (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F')
      ((fun e => loopOfHom T (of (↑e).hom)) ⟨{ left := a, right := b, hom := e }, h⟩) =
    f ⟨{ left := a, right := b, hom := e }, h⟩
state_after: case mk.mk
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'
this : ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
a b : Generators G
e : a ⟶ b
h : { left := a, right := b, hom := e } ∈ (wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ
⊢ f' (↑⟨{ left := a, right := b, hom := e }, h⟩).hom = f ⟨{ left := a, right := b, hom := e }, h⟩

tactic: exact dif_neg h
state_before: case mk.mk
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'
this : ∀ {x y : G} (q : x ⟶ y), F'.map (loopOfHom T q) = F'.map q
a b : Generators G
e : a ⟶ b
h : { left := a, right := b, hom := e } ∈ (wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ
⊢ f' (↑⟨{ left := a, right := b, hom := e }, h⟩).hom = f ⟨{ left := a, right := b, hom := e }, h⟩
state_after: no goals

tactic: simp only [this, treeHom, comp_as_mul, inv_as_inv, loopOfHom, inv_one, mul_one,
  one_mul, Functor.map_inv, Functor.map_comp]
state_before: 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'
x y : G
q : x ⟶ y
this :
  ∀ {a : WideSubquiver.toType (Symmetrify (Generators G)) T}
    (p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a), F'.map (homOfPath T p) = 1
⊢ F'.map (loopOfHom T q) = F'.map q
state_after: no goals

tactic: rw [homOfPath, F'.map_id, id_as_one]
state_before: case intro.intro.refine_1.nil
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'
x y : G
q : x ⟶ y
a : WideSubquiver.toType (Symmetrify (Generators G)) T
⊢ F'.map (homOfPath T Path.nil) = 1
state_after: no goals

tactic: rw [hF']
state_before: case intro.intro.refine_1.cons.mk.inl
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : b ⟶ c
eT : Sum.inl e ∈ T b c
⊢ F'.map (Sum.recOn (↑⟨Sum.inl e, eT⟩) (fun e => of e) fun e => inv (of e)) = 1
state_after: case intro.intro.refine_1.cons.mk.inl
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : b ⟶ c
eT : Sum.inl e ∈ T b c
⊢ f' e = 1

tactic: exact dif_pos (Or.inl eT)
state_before: case intro.intro.refine_1.cons.mk.inl
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : b ⟶ c
eT : Sum.inl e ∈ T b c
⊢ f' e = 1
state_after: no goals

tactic: rw [F'.map_inv, inv_as_inv, inv_eq_one, hF']
state_before: case intro.intro.refine_1.cons.mk.inr
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : c ⟶ b
eT : Sum.inr e ∈ T b c
⊢ F'.map (Sum.recOn (↑⟨Sum.inr e, eT⟩) (fun e => of e) fun e => inv (of e)) = 1
state_after: case intro.intro.refine_1.cons.mk.inr
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : c ⟶ b
eT : Sum.inr e ∈ T b c
⊢ f' e = 1

tactic: exact dif_pos (Or.inr eT)
state_before: case intro.intro.refine_1.cons.mk.inr
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'
x y : G
q : x ⟶ y
a b c : WideSubquiver.toType (Symmetrify (Generators G)) T
p : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) b
ih : F'.map (homOfPath T p) = 1
e : c ⟶ b
eT : Sum.inr e ∈ T b c
⊢ f' e = 1
state_after: no goals

tactic: intro E hE
state_before: 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'
⊢ ∀ (y : End (IsFreeGroupoid.SpanningTree.root' T) →* X),
    (fun F =>
          ∀ (a : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ),
            F ((fun e => loopOfHom T (of (↑e).hom)) a) = f a)
        y →
      y = Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F'
state_after: 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
state_before: 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'
state_after: 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

tactic: 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
state_before: 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
state_after: 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)
⊢ (functorOfMonoidHom T E).map x = F'.map x

tactic: congr
state_before: 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)
⊢ (functorOfMonoidHom T E).map x = F'.map x
state_after: 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'
state_before: 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'
state_after: 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

tactic: intro a b e
state_before: 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
state_after: 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
e : a ⟶ b
⊢ (functorOfMonoidHom T E).map (of e) = f' e

tactic: change E (loopOfHom T _) = dite _ _ _
state_before: 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
e : a ⟶ b
⊢ (functorOfMonoidHom T E).map (of e) = f' e
state_after: 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
e : a ⟶ b
⊢ E (loopOfHom T (of e)) =
    if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩

tactic: split_ifs with h
state_before: 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
e : a ⟶ b
⊢ E (loopOfHom T (of e)) =
    if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨{ left := a, right := b, hom := e }, h⟩
state_after: case pos
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
e : a ⟶ b
h : e ∈ wideSubquiverSymmetrify T a b
⊢ E (loopOfHom T (of e)) = 1

case neg
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
e : a ⟶ b
h : e ∉ wideSubquiverSymmetrify T a b
⊢ E (loopOfHom T (of e)) = f ⟨{ left := a, right := b, hom := e }, h⟩

tactic: simpa only [loopOfHom, functorOfMonoidHom, IsIso.inv_id, treeHom_root,
  Category.id_comp, Category.comp_id] using this
state_before: 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)
this : (functorOfMonoidHom T E).map x = F'.map x
⊢ E x = (Functor.mapEnd (IsFreeGroupoid.SpanningTree.root' T) F') x
state_after: no goals

tactic: rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one]
state_before: case pos
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
e : a ⟶ b
h : e ∈ wideSubquiverSymmetrify T a b
⊢ E (loopOfHom T (of e)) = 1
state_after: no goals

tactic: exact hE ⟨⟨a, b, e⟩, h⟩
state_before: case neg
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
e : a ⟶ b
h : e ∉ wideSubquiverSymmetrify T a b
⊢ E (loopOfHom T (of e)) = f ⟨{ left := a, right := b, hom := e }, h⟩
state_after: no goals