awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (hypothesis)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
heq : g x n = g y n
heq₂ : g x (2 * n) = g y (2 * n)
leq : f x n = f y n
leq₂ : f x (2 * n) = f y (2 * n)
hc : conjugator x.toWord = conjugator y.toWord
⊢ x = y
have heq' : (replicate n (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
  (replicate n (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
Tactic: obtain ⟨hc, heq'⟩ := List.append_inj heq (by grind)
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
heq : g x n = g y n
heq₂ : g x (2 * n) = g y (2 * n)
leq : f x n = f y n
leq₂ : f x (2 * n) = f y (2 * n)
hc : conjugator x.toWord = conjugator y.toWord
heq' :
  (replicate n (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate n (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
⊢ x = y
have obtain : ∀ (n : ℕ) (hn : n + 1 ≠ 0) (heq : g x (n + 1) = g y (n + 1)) (heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1)))
  (leq : f x (n + 1) = f y (n + 1)) (leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1)))
  (heq' :
    (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
      (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)),
  x = y
Tactic: obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
Cleared hypotheses: n, hn, heq, heq₂, leq, leq₂, heq'
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (hypothesis)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
⊢ reduceCyclically x.toWord = reduceCyclically y.toWord
have heq'₁ : reduceCyclically x.toWord ++ ((replicate n (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord)) =
  reduceCyclically y.toWord ++ ((replicate n (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord))
Tactic: simp only [replicate_succ, flatten_cons, append_assoc] at heq'
Cleared hypotheses: heq'
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (hypothesis)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
⊢ x = y
have hm : reduceCyclically x.toWord = reduceCyclically y.toWord
Tactic: have hm : reduceCyclically x.toWord = reduceCyclically y.toWord := by
      simp only [replicate_succ, flatten_cons, append_assoc] at heq'
      exact (List.append_inj heq' <| mul_left_cancel₀ hn <| by grind).1
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (hypothesis)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
⊢ x = y
have this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
Tactic: have := congr_arg mk <| (conj_conjugator_reduceCyclically x.toWord).symm
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
⊢ x = y
have rwa : ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator y.toWord))), x = y
Tactic: rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
rwa :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator y.toWord))), x = y
⊢ x = y
have rwa₁ : ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically y.toWord ++ invRev (conjugator y.toWord))), x = y
Tactic: rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
rwa :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator y.toWord))), x = y
rwa₁ :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically y.toWord ++ invRev (conjugator y.toWord))), x = y
⊢ x = y
have rwa₂ : ∀ (this : mk x.toWord = mk y.toWord), x = y
Tactic: rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
rwa :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator y.toWord))), x = y
rwa₁ :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically y.toWord ++ invRev (conjugator y.toWord))), x = y
rwa₂ : ∀ (this : mk x.toWord = mk y.toWord), x = y
⊢ x = y
have rwa₃ : ∀ (this : x = mk y.toWord), x = y
Tactic: rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
x y : FreeGroup α
f : (a : FreeGroup α) → (n : ℕ) → ℕ :=
  fun (a : FreeGroup α) (n : ℕ) ↦
    (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)
g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) :=
  fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord))
hc : conjugator x.toWord = conjugator y.toWord
n : ℕ
hn : n + 1 ≠ 0
heq : g x (n + 1) = g y (n + 1)
heq₂ : g x (2 * (n + 1)) = g y (2 * (n + 1))
leq : f x (n + 1) = f y (n + 1)
leq₂ : f x (2 * (n + 1)) = f y (2 * (n + 1))
heq' :
  (replicate (n + 1) (reduceCyclically x.toWord)).flatten ++ invRev (conjugator x.toWord) =
    (replicate (n + 1) (reduceCyclically y.toWord)).flatten ++ invRev (conjugator y.toWord)
hm : reduceCyclically x.toWord = reduceCyclically y.toWord
this : mk x.toWord = mk (conjugator x.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator x.toWord))
rwa :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically x.toWord ++ invRev (conjugator y.toWord))), x = y
rwa₁ :
  ∀ (this : mk x.toWord = mk (conjugator y.toWord ++ reduceCyclically y.toWord ++ invRev (conjugator y.toWord))), x = y
rwa₂ : ∀ (this : mk x.toWord = mk y.toWord), x = y
rwa₃ : ∀ (this : x = mk y.toWord), x = y
⊢ x = y
have rwa₄ : ∀ (this : x = y), x = y
Tactic: rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
heq : Eq.{u + 1} (α := FreeGroup α) ((fun (a : FreeGroup α) ↦ a ^ n) x) ((fun (a : FreeGroup α) ↦ a ^ n) y)
let let₁ let₂ let₃ : x = y
⊢ x = y
have let₄ : ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)), x = y
Tactic: classical
    let f (a : FreeGroup α) (n : ℕ) : ℕ :=
        (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length +
          (conjugator a.toWord).length)
    let g (a : FreeGroup α) (k : ℕ) : List (α × Bool) :=
        conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++
          invRev (conjugator a.toWord))
    have heq₂ : x ^ (2 * n) = y ^ (2 * n) := by simp_rw [mul_comm, pow_mul, heq]
    replace heq : g x n = g y n := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq
    replace heq₂ : g x (2 * n) = g y (2 * n) := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq₂
    have leq : f x n = f y n := by simpa [g] using congr_arg List.length heq
    have leq₂ : f x (2 * n) = f y (2 * n) := by simpa [g] using congr_arg List.length heq₂
    obtain ⟨hc, heq'⟩ := List.append_inj heq (by grind)
    obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
    have hm : reduceCyclically x.toWord = reduceCyclically y.toWord := by
      simp only [replicate_succ, flatten_cons, append_assoc] at heq'
      exact (List.append_inj heq' <| mul_left_cancel₀ hn <| by grind).1
    have := congr_arg mk <| (conj_conjugator_reduceCyclically x.toWord).symm
    rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
heq : Eq.{u + 1} (α := FreeGroup α) ((fun (a : FreeGroup α) ↦ a ^ n) x) ((fun (a : FreeGroup α) ↦ a ^ n) y)
let let₁ let₂ let₃ : x = y
let₄ : ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)), x = y
⊢ x = y
have let₅ : let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
  conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)) (heq : g x n = g y n), x = y
Tactic: classical
    let f (a : FreeGroup α) (n : ℕ) : ℕ :=
        (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length +
          (conjugator a.toWord).length)
    let g (a : FreeGroup α) (k : ℕ) : List (α × Bool) :=
        conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++
          invRev (conjugator a.toWord))
    have heq₂ : x ^ (2 * n) = y ^ (2 * n) := by simp_rw [mul_comm, pow_mul, heq]
    replace heq : g x n = g y n := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq
    replace heq₂ : g x (2 * n) = g y (2 * n) := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq₂
    have leq : f x n = f y n := by simpa [g] using congr_arg List.length heq
    have leq₂ : f x (2 * n) = f y (2 * n) := by simpa [g] using congr_arg List.length heq₂
    obtain ⟨hc, heq'⟩ := List.append_inj heq (by grind)
    obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
    have hm : reduceCyclically x.toWord = reduceCyclically y.toWord := by
      simp only [replicate_succ, flatten_cons, append_assoc] at heq'
      exact (List.append_inj heq' <| mul_left_cancel₀ hn <| by grind).1
    have := congr_arg mk <| (conj_conjugator_reduceCyclically x.toWord).symm
    rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: heq
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
heq : Eq.{u + 1} (α := FreeGroup α) ((fun (a : FreeGroup α) ↦ a ^ n) x) ((fun (a : FreeGroup α) ↦ a ^ n) y)
let let₁ let₂ let₃ : x = y
let₄ : ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)), x = y
let₅ :
  let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
  ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)) (heq : g x n = g y n), x = y
⊢ x = y
have let₆ : let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
  conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
∀ (heq : g x n = g y n) (heq₂ : g x (2 * n) = g y (2 * n)), x = y
Tactic: classical
    let f (a : FreeGroup α) (n : ℕ) : ℕ :=
        (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length +
          (conjugator a.toWord).length)
    let g (a : FreeGroup α) (k : ℕ) : List (α × Bool) :=
        conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++
          invRev (conjugator a.toWord))
    have heq₂ : x ^ (2 * n) = y ^ (2 * n) := by simp_rw [mul_comm, pow_mul, heq]
    replace heq : g x n = g y n := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq
    replace heq₂ : g x (2 * n) = g y (2 * n) := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq₂
    have leq : f x n = f y n := by simpa [g] using congr_arg List.length heq
    have leq₂ : f x (2 * n) = f y (2 * n) := by simpa [g] using congr_arg List.length heq₂
    obtain ⟨hc, heq'⟩ := List.append_inj heq (by grind)
    obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
    have hm : reduceCyclically x.toWord = reduceCyclically y.toWord := by
      simp only [replicate_succ, flatten_cons, append_assoc] at heq'
      exact (List.append_inj heq' <| mul_left_cancel₀ hn <| by grind).1
    have := congr_arg mk <| (conj_conjugator_reduceCyclically x.toWord).symm
    rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: heq
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.CyclicallyReduced (goal)
Goal:
α : Type u
L L₁ L₂ L₃ : List (α × Bool)
n : ℕ
hn : n ≠ 0
x y : FreeGroup α
heq : Eq.{u + 1} (α := FreeGroup α) ((fun (a : FreeGroup α) ↦ a ^ n) x) ((fun (a : FreeGroup α) ↦ a ^ n) y)
let let₁ let₂ let₃ : x = y
let₄ : ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)), x = y
let₅ :
  let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
  ∀ (heq₂ : x ^ (2 * n) = y ^ (2 * n)) (heq : g x n = g y n), x = y
let₆ :
  let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
    conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
  ∀ (heq : g x n = g y n) (heq₂ : g x (2 * n) = g y (2 * n)), x = y
⊢ x = y
have let₇ : let f : (a : FreeGroup α) → (n : ℕ) → ℕ := fun (a : FreeGroup α) (n : ℕ) ↦
  (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length);
let g : (a : FreeGroup α) → (k : ℕ) → List (α × Bool) := fun (a : FreeGroup α) (k : ℕ) ↦
  conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++ invRev (conjugator a.toWord));
∀ (heq : g x n = g y n) (heq₂ : g x (2 * n) = g y (2 * n)) (leq : f x n = f y n), x = y
Tactic: classical
    let f (a : FreeGroup α) (n : ℕ) : ℕ :=
        (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length +
          (conjugator a.toWord).length)
    let g (a : FreeGroup α) (k : ℕ) : List (α × Bool) :=
        conjugator a.toWord ++ ((replicate k (reduceCyclically a.toWord)).flatten ++
          invRev (conjugator a.toWord))
    have heq₂ : x ^ (2 * n) = y ^ (2 * n) := by simp_rw [mul_comm, pow_mul, heq]
    replace heq : g x n = g y n := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq
    replace heq₂ : g x (2 * n) = g y (2 * n) := by
      simpa [toWord_pow, reduce_flatten_replicate, isReduced_toWord, hn] using congr_arg toWord heq₂
    have leq : f x n = f y n := by simpa [g] using congr_arg List.length heq
    have leq₂ : f x (2 * n) = f y (2 * n) := by simpa [g] using congr_arg List.length heq₂
    obtain ⟨hc, heq'⟩ := List.append_inj heq (by grind)
    obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
    have hm : reduceCyclically x.toWord = reduceCyclically y.toWord := by
      simp only [replicate_succ, flatten_cons, append_assoc] at heq'
      exact (List.append_inj heq' <| mul_left_cancel₀ hn <| by grind).1
    have := congr_arg mk <| (conj_conjugator_reduceCyclically x.toWord).symm
    rwa [hc, hm, conj_conjugator_reduceCyclically, mk_toWord, mk_toWord] at this
Cleared hypotheses: heq
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.IsFreeGroup (goal)
Goal:
ι : Type u_1
ι' : Type u_2
G : Type u_3
H : Type u_4
inst : Group G
inst_1 : Group H
b b' : MulEquiv.{u_3, u_1} G (FreeGroup ι)
hbb' :
  Eq (α := (a : ι) → G)
    ((fun (b : FreeGroupBasis ι G) (i : ι) ↦ ((b.repr.symm : (a : FreeGroup ι) → G) (FreeGroup.of i) : G))
      { repr := b })
    ((fun (b : FreeGroupBasis ι G) (i : ι) ↦ ((b.repr.symm : (a : FreeGroup ι) → G) (FreeGroup.of i) : G))
      { repr := b' })
⊢ Eq (α := MonoidHom.{u_1, u_3} (FreeGroup ι) G) (↑b.symm : FreeGroup ι →* G) (↑b'.symm : FreeGroup ι →* G)
have h : ∀ (i : ι),
  Eq.{u_3 + 1} (α := G) (((↑b.symm : FreeGroup ι →* G) : (a : FreeGroup ι) → G) (FreeGroup.of i) : G)
    (((↑b'.symm : FreeGroup ι →* G) : (a : FreeGroup ι) → G) (FreeGroup.of i) : G)
Tactic: ext i
-- Lean 4 proof step from Mathlib.GroupTheory.FreeGroup.IsFreeGroup (hypothesis)
Goal:
ι : Type u_1
ι' : Type u_2
G : Type u_3
H : Type u_4
inst : Group G
inst_1 : Group H
b b' : MulEquiv.{u_3, u_1} G (FreeGroup ι)
hbb' :
  Eq (α := (a : ι) → G)
    ((fun (b : FreeGroupBasis ι G) (i : ι) ↦ ((b.repr.symm : (a : FreeGroup ι) → G) (FreeGroup.of i) : G))
      { repr := b })
    ((fun (b : FreeGroupBasis ι G) (i : ι) ↦ ((b.repr.symm : (a : FreeGroup ι) → G) (FreeGroup.of i) : G))
      { repr := b' })
⊢ { repr := b : FreeGroupBasis ι G } = { repr := b' }
have H₁ : Eq (α := MonoidHom.{u_1, u_3} (FreeGroup ι) G) (↑b.symm : FreeGroup ι →* G) (↑b'.symm : FreeGroup ι →* G)
Tactic: have H : (b.symm : FreeGroup ι →* G) = (b'.symm : FreeGroup ι →* G) := by
      ext i; exact congr_fun hbb' i