awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : (↑(y j) : ℝ) ≤ 1 / 2
x_1 : N
⊢ Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) j
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ))) x_1 =
    Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ)))) i
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)) x_1
have pos : (if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ))
  else if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1) else y x_1) =
  if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)
  else if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ)) else y x_1
Tactic: simp only [Function.update, eq_rec_constant, dite_eq_ite]
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : (↑(y j) : ℝ) ≤ 1 / 2
x_1 : N
⊢ (if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ))
    else if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1) else y x_1) =
    if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)
    else if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ)) else y x_1
have pos : x_1 = j → ¬x_1 = i
Tactic: apply ite_ite_comm
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : ¬(↑(y j) : ℝ) ≤ 1 / 2
⊢ Eq.{u_2 + 1} (α := X)
    ((d : (a : N → (↑I : Type)) → X)
        (Function.update (β := fun (a : N) ↦ (↑I : Type))
          (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i
            (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)))
          j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1))) :
      X)
    ((d : (a : N → (↑I : Type)) → X)
        (Function.update (β := fun (a : N) ↦ (↑I : Type))
          (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j
            (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)))
          i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) :
      X)
have neg : Function.update (β := fun (a : N) ↦ (↑I : Type))
    (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) j
    (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)) =
  Function.update (β := fun (a : N) ↦ (↑I : Type))
    (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1))) i
    (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : ¬(↑(y j) : ℝ) ≤ 1 / 2
⊢ Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) j
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)) =
    Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1))) i
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))
have neg : ∀ (x : N),
  Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) j
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)) x =
    Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1))) i
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)) x
Tactic: ext1
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : ¬(↑(y j) : ℝ) ≤ 1 / 2
x_1 : N
⊢ Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y i (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1))) j
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)) x_1 =
    Function.update (β := fun (a : N) ↦ (↑I : Type))
      (Function.update (β := fun (a : N) ↦ (↑I : Type)) y j (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1))) i
      (Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)) x_1
have neg : (if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)
  else if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1) else y x_1) =
  if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)
  else if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1) else y x_1
Tactic: simp only [Function.update, eq_rec_constant, dite_eq_ite]
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N
i j : N
h : i ≠ j
a b c d : (↑(Ω^ N X x) : Type (max u_1 u_2))
y : N → (↑I : Type)
h_1 : ¬(↑(y i) : ℝ) ≤ 1 / 2
h_2 : ¬(↑(y j) : ℝ) ≤ 1 / 2
x_1 : N
⊢ (if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1)
    else if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1) else y x_1) =
    if x_1 = i then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y i) : ℝ) - 1)
    else if x_1 = j then Set.projIcc 0 1 transAt._proof_3 (2 * (↑(y j) : ℝ) - 1) else y x_1
have neg : x_1 = j → ¬x_1 = i
Tactic: apply ite_ite_comm
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x_1 : X
inst_1 : DecidableEq N_1
N : Type u_3
x : X
inst_2 : IsEmpty N
f : (↑(Ω^ N X x) : Type (max u_2 u_3))
⊢ (fun (y : X) ↦
        (Subtype.mk (p := fun (x_2 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X) ↦
            Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_2)
            (ContinuousMap.const (N → (↑I : Type)) y) ⋯ :
          { x_2 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X //
            Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_2 }))
      ((fun (f : (↑(Ω^ N X x) : Type (max u_2 u_3))) ↦ ((f : (a : N → (↑I : Type)) → X) 0 : X)) f) =
    f
have H : ∀ (y : N → (↑I : Type)),
  Eq.{u_2 + 1} (α := X)
    (((fun (y : X) ↦
              (Subtype.mk (p := fun (x_2 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X) ↦
                  Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_2)
                  (ContinuousMap.const (N → (↑I : Type)) y) sorry :
                { x_2 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X //
                  Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_2 }))
            ((fun (f : (↑(Ω^ N X x) : Type (max u_2 u_3))) ↦ ((f : (a : N → (↑I : Type)) → X) 0 : X)) f) :
          (a : N → (↑I : Type)) → X)
        y :
      X)
    ((f : (a : N → (↑I : Type)) → X) y : X)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x_1 : X
inst_1 : DecidableEq N_1
N : Type u_3
x : X
inst_2 : IsEmpty N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ ↔
    (pathSetoid X : X → X → Prop)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₁ : X)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₂ : X)
have mp : (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ →
  (pathSetoid X : X → X → Prop)
    ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₁ : X)
    ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₂ : X)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x_1 : X
inst_1 : DecidableEq N_1
N : Type u_3
x : X
inst_2 : IsEmpty N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
mp :
  (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ →
    (pathSetoid X : X → X → Prop)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₁ : X)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₂ : X)
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ ↔
    (pathSetoid X : X → X → Prop)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₁ : X)
      ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₂ : X)
have mpr : (pathSetoid X : X → X → Prop)
    ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₁ : X)
    ((toEquiv (genLoopHomeoOfIsEmpty N x) : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → X) a₂ : X) →
  (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (hypothesis)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
p : Ω X x
y : N → (↑I : Type)
i : N
iH : y i = 0
⊢ ({ toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := ⋯ :
          ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X } :
        (a : N → (↑I : Type)) → X)
      y =
    x
have iH₁ : y default = 0
Tactic: cases Unique.eq_default i
Cleared hypotheses: i, iH
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (hypothesis)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
p : Ω X x
y : N → (↑I : Type)
i : N
iH : y i = 1
⊢ ({ toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := ⋯ :
          ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X } :
        (a : N → (↑I : Type)) → X)
      y =
    x
have iH₁ : y default = 1
Tactic: cases Unique.eq_default i
Cleared hypotheses: i, iH
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
p : Ω X x
y : N → (↑I : Type)
iH : y default = 0
⊢ ({ toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := ⋯ :
          ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X } :
        (a : N → (↑I : Type)) → X)
      y =
    x
have inl : (p : (a : (↑I : Type)) → X) 0 = x
Tactic: apply (congr_arg p iH).trans
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
p : Ω X x
y : N → (↑I : Type)
iH : y default = 1
⊢ ({ toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := ⋯ :
          ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X } :
        (a : N → (↑I : Type)) → X)
      y =
    x
have inr : (p : (a : (↑I : Type)) → X) 1 = x
Tactic: apply (congr_arg p iH).trans
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
p : (↑(Ω^ N X x) : Type (max u_2 u_3))
⊢ (fun (p : Ω X x) ↦
        Subtype.mk (p := fun (x_1 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X) ↦
          Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_1)
          { toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := ⋯ } ⋯)
      ((fun (p : (↑(Ω^ N X x) : Type (max u_2 u_3))) ↦
          { toFun := fun (t : (↑I : Type)) ↦ (p : (a : N → (↑I : Type)) → X) fun (x : N) ↦ t, continuous_toFun := ⋯,
            source' := ⋯, target' := ⋯ : Path x x })
        p) =
    p
have H : ∀ (y : N → (↑I : Type)),
  Eq.{u_2 + 1} (α := X)
    (((fun (p : Ω X x) ↦
              Subtype.mk (p := fun (x_1 : ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X) ↦
                Membership.mem (γ := Set (ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X)) (Ω^ N X x) x_1)
                { toFun := fun (c : N → (↑I : Type)) ↦ (p : (a : (↑I : Type)) → X) (c default), continuous_toFun := sorry }
                sorry)
            ((fun (p : (↑(Ω^ N X x) : Type (max u_2 u_3))) ↦
                { toFun := fun (t : (↑I : Type)) ↦ (p : (a : N → (↑I : Type)) → X) fun (x : N) ↦ t,
                  continuous_toFun := sorry, source' := sorry, target' := sorry : Path x x })
              p) :
          (a : N → (↑I : Type)) → X)
        y :
      X)
    ((p : (a : N → (↑I : Type)) → X) y : X)
Tactic: ext y
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ ↔
    (Path.Homotopic.setoid { as := x }.as { as := x }.as :
        Path { as := x }.as { as := x }.as → Path { as := x }.as { as := x }.as → Prop)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
have mp : (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ →
  (Path.Homotopic.setoid { as := x }.as { as := x }.as :
      Path { as := x }.as { as := x }.as → Path { as := x }.as { as := x }.as → Prop)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁ :
      Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂ :
      Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
mp :
  (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ →
    (Path.Homotopic.setoid { as := x }.as { as := x }.as :
        Path { as := x }.as { as := x }.as → Path { as := x }.as { as := x }.as → Prop)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂ ↔
    (Path.Homotopic.setoid { as := x }.as { as := x }.as :
        Path { as := x }.as { as := x }.as → Path { as := x }.as { as := x }.as → Prop)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
      ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂ :
        Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
have mpr : (Path.Homotopic.setoid { as := x }.as { as := x }.as :
      Path { as := x }.as { as := x }.as → Path { as := x }.as { as := x }.as → Prop)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁ :
      Path { as := x : FundamentalGroupoid X }.as { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂ :
      Path { as := x : FundamentalGroupoid X }.as { as := x }.as) →
  (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
H :
  Path.Homotopy (X := X) (x₀ := { as := x }.as) (x₁ := { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂)
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
have mpr : Continuous (Y := (↑I : Type) × (↑I : Type)) fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦ (tx.1, tx.2 default)
Tactic: refine
      ⟨⟨⟨⟨fun tx ↦ H (tx.fst, tx.snd default), H.continuous.comp ?_⟩, fun y ↦ ?_, fun y ↦ ?_⟩, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
H :
  Path.Homotopy (X := X) (x₀ := { as := x }.as) (x₁ := { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂)
mpr : Continuous (Y := (↑I : Type) × (↑I : Type)) fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦ (tx.1, tx.2 default)
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
have mpr₁ : ∀ (y : N → (↑I : Type)),
  {
          toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
            (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
          continuous_toFun := sorry : ContinuousMap.{u_3, u_2} ((↑I : Type) × (N → (↑I : Type))) X }.toFun
      (0, y) =
    ((↑a₁ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) y
Tactic: refine
      ⟨⟨⟨⟨fun tx ↦ H (tx.fst, tx.snd default), H.continuous.comp ?_⟩, fun y ↦ ?_, fun y ↦ ?_⟩, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
H :
  Path.Homotopy (X := X) (x₀ := { as := x }.as) (x₁ := { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂)
mpr : Continuous (Y := (↑I : Type) × (↑I : Type)) fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦ (tx.1, tx.2 default)
mpr₁ :
  ∀ (y : N → (↑I : Type)),
    {
            toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
              (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
            continuous_toFun := ⋯ : ContinuousMap.{u_3, u_2} ((↑I : Type) × (N → (↑I : Type))) X }.toFun
        (0, y) =
      ((↑a₁ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) y
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
have mpr₂ : ∀ (y : N → (↑I : Type)),
  {
          toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
            (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
          continuous_toFun := sorry : ContinuousMap.{u_3, u_2} ((↑I : Type) × (N → (↑I : Type))) X }.toFun
      (1, y) =
    ((↑a₂ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) y
Tactic: refine
      ⟨⟨⟨⟨fun tx ↦ H (tx.fst, tx.snd default), H.continuous.comp ?_⟩, fun y ↦ ?_, fun y ↦ ?_⟩, ?_⟩⟩
-- Lean 4 proof step from Mathlib.Topology.Homotopy.HomotopyGroup (goal)
Goal:
N_1 : Type u_1
X : Type u_2
inst : TopologicalSpace X
x : X
inst_1 : DecidableEq N_1
N : Type u_3
inst_2 : Unique N
a₁ a₂ : (↑(Ω^ N X x) : Type (max u_2 u_3))
H :
  Path.Homotopy (X := X) (x₀ := { as := x }.as) (x₁ := { as := x }.as)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₁)
    ((genLoopEquivOfUnique N : (a : (↑(Ω^ N X x) : Type (max u_2 u_3))) → Path { as := x }.as { as := x }.as) a₂)
mpr : Continuous (Y := (↑I : Type) × (↑I : Type)) fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦ (tx.1, tx.2 default)
mpr₁ :
  ∀ (y : N → (↑I : Type)),
    {
            toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
              (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
            continuous_toFun := ⋯ : ContinuousMap.{u_3, u_2} ((↑I : Type) × (N → (↑I : Type))) X }.toFun
        (0, y) =
      ((↑a₁ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) y
mpr₂ :
  ∀ (y : N → (↑I : Type)),
    {
            toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
              (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
            continuous_toFun := ⋯ : ContinuousMap.{u_3, u_2} ((↑I : Type) × (N → (↑I : Type))) X }.toFun
        (1, y) =
      ((↑a₂ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) y
⊢ (Homotopic.setoid N x : (↑(Ω^ N X x) : Type (max u_2 u_3)) → (↑(Ω^ N X x) : Type (max u_2 u_3)) → Prop) a₁ a₂
have mpr₃ : ∀ (t : (↑I : Type)),
  ∀ x_1 ∈ Cube.boundary N,
    Eq.{u_2 + 1} (α := X)
      ((({
                toFun := fun (x_2 : N → (↑I : Type)) ↦
                  (ContinuousMap.Homotopy.toContinuousMap (f₀ := (↑a₁ : C(N → (↑I : Type), X))) (f₁ :=
                        (↑a₂ : C(N → (↑I : Type), X)))
                        {
                          toFun := fun (tx : (↑I : Type) × (N → (↑I : Type))) ↦
                            (H : (a : (↑I : Type) × (↑I : Type)) → X) (tx.1, tx.2 default),
                          continuous_toFun := sorry, map_zero_left := sorry, map_one_left := sorry :
                          (↑a₁ : C(N → (↑I : Type), X)).Homotopy (↑a₂ : C(N → (↑I : Type), X)) }).toFun
                    (t, x_2),
                continuous_toFun := sorry } :
              ContinuousMap.{u_3, u_2} (N → (↑I : Type)) X) :
            (a : N → (↑I : Type)) → X)
          x_1 :
        X)
      (((↑a₁ : C(N → (↑I : Type), X)) : (a : N → (↑I : Type)) → X) x_1 : X)
Tactic: refine
      ⟨⟨⟨⟨fun tx ↦ H (tx.fst, tx.snd default), H.continuous.comp ?_⟩, fun y ↦ ?_, fun y ↦ ?_⟩, ?_⟩⟩