l3lab/ntp-mathlib huggingface dataset:
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  

Proof state:
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
⊢ ∀ (a₁ a₂ : ↑(Ω^ N X x)), Setoid.r a₁ a₂ ↔ Setoid.r ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)

Next tactic:
intros a₁ a₂
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; 

Proof state:
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
⊢ Setoid.r a₁ a₂ ↔ Setoid.r ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)

Next tactic:
constructor
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> 

Proof state:
case mp
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
⊢ Setoid.r a₁ a₂ → Setoid.r ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)

Next tactic:
rintro ⟨H⟩
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> 

Proof state:
case mpr
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
⊢ Setoid.r ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂) → Setoid.r a₁ a₂

Next tactic:
rintro ⟨H⟩
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · 

Proof state:
case mp.intro
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : ContinuousMap.HomotopyRel (↑a₁) (↑a₂) (Cube.boundary N)
⊢ Setoid.r ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)

Next tactic:
exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  

Proof state:
case mpr.intro
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
⊢ Setoid.r a₁ a₂

Next tactic:
refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · 

Proof state:
case mpr.intro.refine'_1
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
⊢ Continuous fun tx => (tx.1, tx.2 default)

Next tactic:
exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
  · 

Proof state:
case mpr.intro.refine'_2
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
y : N → ↑I
⊢ ContinuousMap.toFun (ContinuousMap.mk fun tx => H (tx.1, tx.2 default)) (0, y) = ↑a₁ y

Next tactic:
exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm)
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
  · exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm)
  · 

Proof state:
case mpr.intro.refine'_3
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
y : N → ↑I
⊢ ContinuousMap.toFun (ContinuousMap.mk fun tx => H (tx.1, tx.2 default)) (1, y) = ↑a₂ y

Next tactic:
exact (H.apply_one _).trans (congr_arg a₂ (eq_const_of_unique y).symm)
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
  · exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm)
  · exact (H.apply_one _).trans (congr_arg a₂ (eq_const_of_unique y).symm)
  · 

Proof state:
case mpr.intro.refine'_4
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
⊢ ∀ (t : ↑I),
    ∀ x_1 ∈ Cube.boundary N,
      (ContinuousMap.mk fun x_2 =>
            ContinuousMap.toFun
              { toContinuousMap := ContinuousMap.mk fun tx => H (tx.1, tx.2 default),
                  map_zero_left := (_ : ∀ (y : N → ↑I), H (0, (0, y).2 default) = ↑a₁ y),
                  map_one_left := (_ : ∀ (y : N → ↑I), H (1, (1, y).2 default) = ↑a₂ y) }.toContinuousMap
              (t, x_2))
          x_1 =
        ↑a₁ x_1

Next tactic:
rintro t y ⟨i, iH⟩
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
  · exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm)
  · exact (H.apply_one _).trans (congr_arg a₂ (eq_const_of_unique y).symm)
  · rintro t y ⟨i, iH⟩
    

Proof state:
case mpr.intro.refine'_4.intro
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
t : ↑I
y : N → ↑I
i : N
iH : y i = 0 ∨ y i = 1
⊢ (ContinuousMap.mk fun x_1 =>
        ContinuousMap.toFun
          { toContinuousMap := ContinuousMap.mk fun tx => H (tx.1, tx.2 default),
              map_zero_left := (_ : ∀ (y : N → ↑I), H (0, (0, y).2 default) = ↑a₁ y),
              map_one_left := (_ : ∀ (y : N → ↑I), H (1, (1, y).2 default) = ↑a₂ y) }.toContinuousMap
          (t, x_1))
      y =
    ↑a₁ y

Next tactic:
cases Unique.eq_default i
/-- The homotopy group at `x` indexed by a singleton is in bijection with the fundamental group,
  i.e. the loops based at `x` up to homotopy. -/
def homotopyGroupEquivFundamentalGroupOfUnique (N) [Unique N] :
    HomotopyGroup N X x ≃ FundamentalGroup X x := by
  refine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm
  refine' Quotient.congr (genLoopEquivOfUnique N) _
  intros a₁ a₂; constructor <;> rintro ⟨H⟩
  · exact
      ⟨{  toFun := fun tx => H (tx.fst, fun _ => tx.snd)
          map_zero_left := fun _ => H.apply_zero _
          map_one_left := fun _ => H.apply_one _
          prop' := fun t y iH => H.prop' _ _ ⟨default, iH⟩ }⟩
  refine'
    ⟨⟨⟨⟨fun tx => H (tx.fst, tx.snd default), H.continuous.comp _⟩, fun y => _, fun y => _⟩, _⟩⟩
  · exact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)
  · exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm)
  · exact (H.apply_one _).trans (congr_arg a₂ (eq_const_of_unique y).symm)
  · rintro t y ⟨i, iH⟩
    cases Unique.eq_default i
    

Proof state:
case mpr.intro.refine'_4.intro.refl
N✝ : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N✝
N : Type ?u.764964
inst✝ : Unique N
a₁ a₂ : ↑(Ω^ N X x)
H : Path.Homotopy ((genLoopEquivOfUnique N) a₁) ((genLoopEquivOfUnique N) a₂)
t : ↑I
y : N → ↑I
iH : y default = 0 ∨ y default = 1
⊢ (ContinuousMap.mk fun x_1 =>
        ContinuousMap.toFun
          { toContinuousMap := ContinuousMap.mk fun tx => H (tx.1, tx.2 default),
              map_zero_left := (_ : ∀ (y : N → ↑I), H (0, (0, y).2 default) = ↑a₁ y),
              map_one_left := (_ : ∀ (y : N → ↑I), H (1, (1, y).2 default) = ↑a₂ y) }.toContinuousMap
          (t, x_1))
      y =
    ↑a₁ y

Next tactic:
exact (H.eq_fst _ iH).trans (congr_arg a₁ (eq_const_of_unique y).symm)
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
⊢ auxGroup i = auxGroup j

Next tactic:
by_cases h : i = j
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · 

Proof state:
case pos
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i = j
⊢ auxGroup i = auxGroup j

Next tactic:
rw [h]
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · rw [h]
  

Proof state:
case neg
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : ¬i = j
⊢ auxGroup i = auxGroup j

Next tactic:
refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · rw [h]
  refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)
  

Proof state:
case neg
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : ¬i = j
⊢ ∀ (a b c d : HomotopyGroup N X x), Mul.mul (Mul.mul a b) (Mul.mul c d) = Mul.mul (Mul.mul a c) (Mul.mul b d)

Next tactic:
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · rw [h]
  refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)
  rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
  

Proof state:
case neg.mk.mk.mk.mk
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : ¬i = j
a✝ : HomotopyGroup N X x
a : ↑(Ω^ N X x)
b✝ : HomotopyGroup N X x
b : ↑(Ω^ N X x)
c✝ : HomotopyGroup N X x
c : ↑(Ω^ N X x)
d✝ : HomotopyGroup N X x
d : ↑(Ω^ N X x)
⊢ Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r b)) (Mul.mul (Quot.mk Setoid.r c) (Quot.mk Setoid.r d)) =
    Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d))

Next tactic:
change Quotient.mk' _ = _
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · rw [h]
  refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)
  rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
  change Quotient.mk' _ = _
  

Proof state:
case neg.mk.mk.mk.mk
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : ¬i = j
a✝ : HomotopyGroup N X x
a : ↑(Ω^ N X x)
b✝ : HomotopyGroup N X x
b : ↑(Ω^ N X x)
c✝ : HomotopyGroup N X x
c : ↑(Ω^ N X x)
d✝ : HomotopyGroup N X x
d : ↑(Ω^ N X x)
⊢ Quotient.mk'
      ((loopHomeo i).symm
        (Path.trans
          ((loopHomeo i).toEquiv ((loopHomeo j).symm (Path.trans ((loopHomeo j).toEquiv d) ((loopHomeo j).toEquiv c))))
          ((loopHomeo i).toEquiv
            ((loopHomeo j).symm (Path.trans ((loopHomeo j).toEquiv b) ((loopHomeo j).toEquiv a)))))) =
    Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d))

Next tactic:
apply congr_arg Quotient.mk'
theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by
  by_cases h : i = j; · rw [h]
  refine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)
  rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
  change Quotient.mk' _ = _
  apply congr_arg Quotient.mk'
  

Proof state:
case neg.mk.mk.mk.mk
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : ¬i = j
a✝ : HomotopyGroup N X x
a : ↑(Ω^ N X x)
b✝ : HomotopyGroup N X x
b : ↑(Ω^ N X x)
c✝ : HomotopyGroup N X x
c : ↑(Ω^ N X x)
d✝ : HomotopyGroup N X x
d : ↑(Ω^ N X x)
⊢ (loopHomeo i).symm
      (Path.trans
        ((loopHomeo i).toEquiv ((loopHomeo j).symm (Path.trans ((loopHomeo j).toEquiv d) ((loopHomeo j).toEquiv c))))
        ((loopHomeo i).toEquiv ((loopHomeo j).symm (Path.trans ((loopHomeo j).toEquiv b) ((loopHomeo j).toEquiv a))))) =
    (loopHomeo j).symm
      (Path.trans
        ((loopHomeo j).toEquiv ((loopHomeo i).symm (Path.trans ((loopHomeo i).toEquiv d) ((loopHomeo i).toEquiv b))))
        ((loopHomeo j).toEquiv ((loopHomeo i).symm (Path.trans ((loopHomeo i).toEquiv c) ((loopHomeo i).toEquiv a)))))

Next tactic:
simp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply,
    coe_symm_toEquiv, loopHomeo_symm_apply]
theorem transAt_indep {i} (j) (f g : Ω^ N X x) :
    (⟦transAt i f g⟧ : HomotopyGroup N X x) = ⟦transAt j f g⟧ := by
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f g : ↑(Ω^ N X x)
⊢ ⟦transAt i f g⟧ = ⟦transAt j f g⟧

Next tactic:
simp_rw [← fromLoop_trans_toLoop]
theorem transAt_indep {i} (j) (f g : Ω^ N X x) :
    (⟦transAt i f g⟧ : HomotopyGroup N X x) = ⟦transAt j f g⟧ := by
  simp_rw [← fromLoop_trans_toLoop]
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f g : ↑(Ω^ N X x)
⊢ ⟦fromLoop i (Path.trans (toLoop i f) (toLoop i g))⟧ = ⟦fromLoop j (Path.trans (toLoop j f) (toLoop j g))⟧

Next tactic:
let m := fun (G) (_ : Group G) => ((· * ·) : G → G → G)
theorem transAt_indep {i} (j) (f g : Ω^ N X x) :
    (⟦transAt i f g⟧ : HomotopyGroup N X x) = ⟦transAt j f g⟧ := by
  simp_rw [← fromLoop_trans_toLoop]
  let m := fun (G) (_ : Group G) => ((· * ·) : G → G → G)
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f g : ↑(Ω^ N X x)
m : (G : Type ?u.866612) → Group G → G → G → G := fun G x x_1 x_2 => x_1 * x_2
⊢ ⟦fromLoop i (Path.trans (toLoop i f) (toLoop i g))⟧ = ⟦fromLoop j (Path.trans (toLoop j f) (toLoop j g))⟧

Next tactic:
exact congr_fun₂ (congr_arg (m <| HomotopyGroup N X x) <| auxGroup_indep i j) ⟦g⟧ ⟦f⟧
theorem symmAt_indep {i} (j) (f : Ω^ N X x) :
    (⟦symmAt i f⟧ : HomotopyGroup N X x) = ⟦symmAt j f⟧ := by
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f : ↑(Ω^ N X x)
⊢ ⟦symmAt i f⟧ = ⟦symmAt j f⟧

Next tactic:
simp_rw [← fromLoop_symm_toLoop]
theorem symmAt_indep {i} (j) (f : Ω^ N X x) :
    (⟦symmAt i f⟧ : HomotopyGroup N X x) = ⟦symmAt j f⟧ := by
  simp_rw [← fromLoop_symm_toLoop]
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f : ↑(Ω^ N X x)
⊢ ⟦fromLoop i (Path.symm (toLoop i f))⟧ = ⟦fromLoop j (Path.symm (toLoop j f))⟧

Next tactic:
let inv := fun (G) (_ : Group G) => ((·⁻¹) : G → G)
theorem symmAt_indep {i} (j) (f : Ω^ N X x) :
    (⟦symmAt i f⟧ : HomotopyGroup N X x) = ⟦symmAt j f⟧ := by
  simp_rw [← fromLoop_symm_toLoop]
  let inv := fun (G) (_ : Group G) => ((·⁻¹) : G → G)
  

Proof state:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f : ↑(Ω^ N X x)
inv : (G : Type ?u.868698) → Group G → G → G := fun G x x_1 => x_1⁻¹
⊢ ⟦fromLoop i (Path.symm (toLoop i f))⟧ = ⟦fromLoop j (Path.symm (toLoop j f))⟧

Next tactic:
exact congr_fun (congr_arg (inv <| HomotopyGroup N X x) <| auxGroup_indep i j) ⟦f⟧
/-- Characterization of multiplication -/
theorem mul_spec [Nonempty N] {i} {p q : Ω^ N X x} :
    -- porting note: TODO: introduce `HomotopyGroup.mk` and remove defeq abuse.
    ((· * ·) : _ → _ → HomotopyGroup N X x) ⟦p⟧ ⟦q⟧ = ⟦transAt i q p⟧ := by
  

Proof state:
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p q : ↑(Ω^ N X x)
⊢ (fun x_1 x_2 => x_1 * x_2) ⟦p⟧ ⟦q⟧ = ⟦transAt i q p⟧

Next tactic:
rw [transAt_indep _ q, ← fromLoop_trans_toLoop]
/-- Characterization of multiplication -/
theorem mul_spec [Nonempty N] {i} {p q : Ω^ N X x} :
    -- porting note: TODO: introduce `HomotopyGroup.mk` and remove defeq abuse.
    ((· * ·) : _ → _ → HomotopyGroup N X x) ⟦p⟧ ⟦q⟧ = ⟦transAt i q p⟧ := by
  rw [transAt_indep _ q, ← fromLoop_trans_toLoop]; 

Proof state:
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p q : ↑(Ω^ N X x)
⊢ (fun x_1 x_2 => x_1 * x_2) ⟦p⟧ ⟦q⟧ = ⟦fromLoop ?m.904245 (Path.trans (toLoop ?m.904245 q) (toLoop ?m.904245 p))⟧
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p q : ↑(Ω^ N X x)
⊢ N

Next tactic:
apply Quotient.sound
/-- Characterization of multiplication -/
theorem mul_spec [Nonempty N] {i} {p q : Ω^ N X x} :
    -- porting note: TODO: introduce `HomotopyGroup.mk` and remove defeq abuse.
    ((· * ·) : _ → _ → HomotopyGroup N X x) ⟦p⟧ ⟦q⟧ = ⟦transAt i q p⟧ := by
  rw [transAt_indep _ q, ← fromLoop_trans_toLoop]; apply Quotient.sound; 

Proof state:
case a
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p q : ↑(Ω^ N X x)
⊢ (loopHomeo (Classical.arbitrary N)).symm
      (Path.trans ((loopHomeo (Classical.arbitrary N)).toEquiv q) ((loopHomeo (Classical.arbitrary N)).toEquiv p)) ≈
    fromLoop ?m.904245 (Path.trans (toLoop ?m.904245 q) (toLoop ?m.904245 p))
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p q : ↑(Ω^ N X x)
⊢ N

Next tactic:
rfl
/-- Characterization of multiplicative inverse -/
theorem inv_spec [Nonempty N] {i} {p : Ω^ N X x} : ((⟦p⟧)⁻¹ : HomotopyGroup N X x) = ⟦symmAt i p⟧ :=
  by 

Proof state:
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p : ↑(Ω^ N X x)
⊢ (⟦p⟧)⁻¹ = ⟦symmAt i p⟧

Next tactic:
rw [symmAt_indep _ p, ← fromLoop_symm_toLoop]
/-- Characterization of multiplicative inverse -/
theorem inv_spec [Nonempty N] {i} {p : Ω^ N X x} : ((⟦p⟧)⁻¹ : HomotopyGroup N X x) = ⟦symmAt i p⟧ :=
  by rw [symmAt_indep _ p, ← fromLoop_symm_toLoop]; 

Proof state:
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p : ↑(Ω^ N X x)
⊢ (⟦p⟧)⁻¹ = ⟦fromLoop ?m.905414 (Path.symm (toLoop ?m.905414 p))⟧
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p : ↑(Ω^ N X x)
⊢ N

Next tactic:
apply Quotient.sound