l3lab/ntp-mathlib huggingface dataset:
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    r { val := 2 * (2 * (2 * ↑x) - 1) - 1, property := (_ : (fun x => x ∈ I) (2 * (2 * (2 * ↑x) - 1) - 1)) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case neg.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝¹ : ¬↑x ≤ 1 / 4
h✝ : 1 / 2 * (↑x + 1) ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    p { val := 2 * (1 / 2 * (↑x + 1)), property := (_ : (fun x => x ∈ I) (2 * (1 / 2 * (↑x + 1)))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝¹ : ¬↑x ≤ 1 / 4
h✝ : 1 / 2 * (↑x + 1) ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬1 / 2 * (↑x + 1) ≤ 1 / 2
h✝ : 2 * (1 / 2 * (↑x + 1)) - 1 ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    q { val := 2 * (2 * (1 / 2 * (↑x + 1)) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (1 / 2 * (↑x + 1)) - 1))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬1 / 2 * (↑x + 1) ≤ 1 / 2
h✝ : 2 * (1 / 2 * (↑x + 1)) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬1 / 2 * (↑x + 1) ≤ 1 / 2
h✝ : ¬2 * (1 / 2 * (↑x + 1)) - 1 ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    r
      { val := 2 * (2 * (1 / 2 * (↑x + 1)) - 1) - 1,
        property := (_ : (fun x => x ∈ I) (2 * (2 * (1 / 2 * (↑x + 1)) - 1) - 1)) }

Next tactic:
congr
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · congr
    

Proof state:
case neg.h.e_6.h.e_val.e_a.e_a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬1 / 2 * (↑x + 1) ≤ 1 / 2
h✝ : ¬2 * (1 / 2 * (↑x + 1)) - 1 ≤ 1 / 2
⊢ ↑x = 2 * (1 / 2 * (↑x + 1)) - 1

Next tactic:
ring
/-- For paths `p q r`, we have a homotopy from `(p.trans q).trans r` to `p.trans (q.trans r)`. -/
def transAssoc {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    Homotopy ((p.trans q).trans r) (p.trans (q.trans r)) :=
  ((Homotopy.reparam (p.trans (q.trans r))
          (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
⊢ Continuous fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t ∈ I) }

Next tactic:
continuity
instance : CategoryTheory.Groupoid (FundamentalGroupoid X) where
  Hom x y := Path.Homotopic.Quotient x.as y.as
  id x := ⟦Path.refl x.as⟧
  comp {x y z} := Path.Homotopic.Quotient.comp
  id_comp {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦(Path.refl x.as).trans a⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.reflTrans a⟩
  comp_id {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦a.trans (Path.refl y.as)⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.transRefl a⟩
  assoc {w x y z} f g h :=
    Quotient.inductionOn₃ f g h fun p q r =>
      show ⟦(p.trans q).trans r⟧ = ⟦p.trans (q.trans r)⟧ from
        Quotient.sound ⟨Path.Homotopy.transAssoc p q r⟩
  inv {x y} p :=
    Quotient.lift (fun l : Path x.as y.as => ⟦l.symm⟧)
      (by
        

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
x y : FundamentalGroupoid X
p : x ⟶ y
⊢ ∀ (a b : Path x.as y.as), a ≈ b → (fun l => ⟦Path.symm l⟧) a = (fun l => ⟦Path.symm l⟧) b

Next tactic:
rintro a b ⟨h⟩
instance : CategoryTheory.Groupoid (FundamentalGroupoid X) where
  Hom x y := Path.Homotopic.Quotient x.as y.as
  id x := ⟦Path.refl x.as⟧
  comp {x y z} := Path.Homotopic.Quotient.comp
  id_comp {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦(Path.refl x.as).trans a⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.reflTrans a⟩
  comp_id {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦a.trans (Path.refl y.as)⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.transRefl a⟩
  assoc {w x y z} f g h :=
    Quotient.inductionOn₃ f g h fun p q r =>
      show ⟦(p.trans q).trans r⟧ = ⟦p.trans (q.trans r)⟧ from
        Quotient.sound ⟨Path.Homotopy.transAssoc p q r⟩
  inv {x y} p :=
    Quotient.lift (fun l : Path x.as y.as => ⟦l.symm⟧)
      (by
        rintro a b ⟨h⟩
        

Proof state:
case intro
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
x y : FundamentalGroupoid X
p : x ⟶ y
a b : Path x.as y.as
h : Path.Homotopy a b
⊢ (fun l => ⟦Path.symm l⟧) a = (fun l => ⟦Path.symm l⟧) b

Next tactic:
simp only
instance : CategoryTheory.Groupoid (FundamentalGroupoid X) where
  Hom x y := Path.Homotopic.Quotient x.as y.as
  id x := ⟦Path.refl x.as⟧
  comp {x y z} := Path.Homotopic.Quotient.comp
  id_comp {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦(Path.refl x.as).trans a⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.reflTrans a⟩
  comp_id {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦a.trans (Path.refl y.as)⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.transRefl a⟩
  assoc {w x y z} f g h :=
    Quotient.inductionOn₃ f g h fun p q r =>
      show ⟦(p.trans q).trans r⟧ = ⟦p.trans (q.trans r)⟧ from
        Quotient.sound ⟨Path.Homotopy.transAssoc p q r⟩
  inv {x y} p :=
    Quotient.lift (fun l : Path x.as y.as => ⟦l.symm⟧)
      (by
        rintro a b ⟨h⟩
        simp only
        

Proof state:
case intro
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
x y : FundamentalGroupoid X
p : x ⟶ y
a b : Path x.as y.as
h : Path.Homotopy a b
⊢ ⟦Path.symm a⟧ = ⟦Path.symm b⟧

Next tactic:
rw [Quotient.eq]
instance : CategoryTheory.Groupoid (FundamentalGroupoid X) where
  Hom x y := Path.Homotopic.Quotient x.as y.as
  id x := ⟦Path.refl x.as⟧
  comp {x y z} := Path.Homotopic.Quotient.comp
  id_comp {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦(Path.refl x.as).trans a⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.reflTrans a⟩
  comp_id {x y} f :=
    Quotient.inductionOn f fun a =>
      show ⟦a.trans (Path.refl y.as)⟧ = ⟦a⟧ from Quotient.sound ⟨Path.Homotopy.transRefl a⟩
  assoc {w x y z} f g h :=
    Quotient.inductionOn₃ f g h fun p q r =>
      show ⟦(p.trans q).trans r⟧ = ⟦p.trans (q.trans r)⟧ from
        Quotient.sound ⟨Path.Homotopy.transAssoc p q r⟩
  inv {x y} p :=
    Quotient.lift (fun l : Path x.as y.as => ⟦l.symm⟧)
      (by
        rintro a b ⟨h⟩
        simp only
        rw [Quotient.eq]
        

Proof state:
case intro
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
x y : FundamentalGroupoid X
p : x ⟶ y
a b : Path x.as y.as
h : Path.Homotopy a b
⊢ Path.symm a ≈ Path.symm b

Next tactic:
exact ⟨h.symm₂⟩
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by 

Proof state:
X✝¹ : Type u
Y✝¹ : Type v
inst✝¹ : TopologicalSpace X✝¹
inst✝ : TopologicalSpace Y✝¹
x₀ x₁ : X✝¹
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
X Y : ↑((fun X => Bundled.mk (FundamentalGroupoid ↑X)) X✝)
p : X ⟶ Y
⊢ (fun x => { as := f x.as }) X ⟶ (fun x => { as := f x.as }) Y

Next tactic:
exact Path.Homotopic.Quotient.mapFn p f
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x y z : ↑((fun X => Bundled.mk (FundamentalGroupoid ↑X)) X✝)
p : x ⟶ y
q : y ⟶ z
⊢ { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map (p ≫ q) =
    { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map p ≫
      { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map q

Next tactic:
refine Quotient.inductionOn₂ p q fun a b => ?_
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        refine Quotient.inductionOn₂ p q fun a b => ?_
        

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x y z : ↑((fun X => Bundled.mk (FundamentalGroupoid ↑X)) X✝)
p : x ⟶ y
q : y ⟶ z
a : Path x.as y.as
b : Path y.as z.as
⊢ { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map (⟦a⟧ ≫ ⟦b⟧) =
    { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map ⟦a⟧ ≫
      { obj := fun x => { as := f x.as }, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map ⟦b⟧

Next tactic:
simp only [comp_eq, ← Path.Homotopic.map_lift, ← Path.Homotopic.comp_lift, Path.map_trans]
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        refine Quotient.inductionOn₂ p q fun a b => ?_
        simp only [comp_eq, ← Path.Homotopic.map_lift, ← Path.Homotopic.comp_lift, Path.map_trans]
        -- This was not needed before leanprover/lean4#2644
        

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x y z : ↑((fun X => Bundled.mk (FundamentalGroupoid ↑X)) X✝)
p : x ⟶ y
q : y ⟶ z
a : Path x.as y.as
b : Path y.as z.as
⊢ ⟦Path.trans (Path.map a (_ : Continuous ⇑f)) (Path.map b (_ : Continuous ⇑f))⟧ =
    Path.Homotopic.Quotient.comp ⟦Path.map a (_ : Continuous ⇑f)⟧ ⟦Path.map b (_ : Continuous ⇑f)⟧

Next tactic:
erw [ ← Path.Homotopic.comp_lift]
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        refine Quotient.inductionOn₂ p q fun a b => ?_
        simp only [comp_eq, ← Path.Homotopic.map_lift, ← Path.Homotopic.comp_lift, Path.map_trans]
        -- This was not needed before leanprover/lean4#2644
        erw [ ← Path.Homotopic.comp_lift]; 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
X✝ Y✝ : TopCat
f : X✝ ⟶ Y✝
x y z : ↑((fun X => Bundled.mk (FundamentalGroupoid ↑X)) X✝)
p : x ⟶ y
q : y ⟶ z
a : Path x.as y.as
b : Path y.as z.as
⊢ ⟦Path.trans (Path.map a (_ : Continuous ⇑f)) (Path.map b (_ : Continuous ⇑f))⟧ =
    ⟦Path.trans (Path.map a (_ : Continuous ⇑f)) (Path.map b (_ : Continuous ⇑f))⟧

Next tactic:
rfl
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        refine Quotient.inductionOn₂ p q fun a b => ?_
        simp only [comp_eq, ← Path.Homotopic.map_lift, ← Path.Homotopic.comp_lift, Path.map_trans]
        -- This was not needed before leanprover/lean4#2644
        erw [ ← Path.Homotopic.comp_lift]; rfl}
  map_id X := by
    

Proof state:
X✝ : Type u
Y : Type v
inst✝¹ : TopologicalSpace X✝
inst✝ : TopologicalSpace Y
x₀ x₁ : X✝
X : TopCat
⊢ { obj := fun X => Bundled.mk (FundamentalGroupoid ↑X),
          map := fun {X Y} f =>
            CategoryTheory.Functor.mk
              { obj := fun x => { as := f x.as }, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.map
      (𝟙 X) =
    𝟙
      ({ obj := fun X => Bundled.mk (FundamentalGroupoid ↑X),
            map := fun {X Y} f =>
              CategoryTheory.Functor.mk
                { obj := fun x => { as := f x.as }, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.obj
        X)

Next tactic:
simp only
/-- The functor sending a topological space `X` to its fundamental groupoid. -/
def fundamentalGroupoidFunctor : TopCat ⥤ CategoryTheory.Grpd where
  obj X := { α := FundamentalGroupoid X }
  map f :=
    { obj := fun x => ⟨f x.as⟩
      map := fun {X Y} p => by exact Path.Homotopic.Quotient.mapFn p f
      map_id := fun X => rfl
      map_comp := fun {x y z} p q => by
        refine Quotient.inductionOn₂ p q fun a b => ?_
        simp only [comp_eq, ← Path.Homotopic.map_lift, ← Path.Homotopic.comp_lift, Path.map_trans]
        -- This was not needed before leanprover/lean4#2644
        erw [ ← Path.Homotopic.comp_lift]; rfl}
  map_id X := by
    simp only
    

Proof state:
X✝ : Type u
Y : Type v
inst✝¹ : TopologicalSpace X✝
inst✝ : TopologicalSpace Y
x₀ x₁ : X✝
X : TopCat
⊢ CategoryTheory.Functor.mk
      { obj := fun x => { as := (𝟙 X) x.as }, map := fun {X_1 Y} p => Path.Homotopic.Quotient.mapFn p (𝟙 X) } =
    𝟙 (Bundled.mk (FundamentalGroupoid ↑X))

Next tactic:
change _ = (⟨_, _, _⟩ : FundamentalGroupoid X ⥤ FundamentalGroupoid X)