[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.AlternatingFaceMapComplex).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
def inclusionOfMooreComplexMap (X : SimplicialObject A) :
    (normalizedMooreComplex A).obj X ⟶ (alternatingFaceMapComplex A).obj X
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.objD (L66-L72)
def objD (n : ℕ) : X _⦋n + 1⦌ ⟶ X _⦋n⦌ :=
∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i

/-!
## The chain complex relation `d ≫ d`
-/
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.d_squared (L73-L121)
theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by
-- we start by expanding d ≫ d as a double sum
  dsimp
  simp only [comp_sum, sum_comp, ← Finset.sum_product']
  -- then, we decompose the index set P into a subset S and its complement Sᶜ
  let P := Fin (n + 2) × Fin (n + 3)
  let S : Finset P := {ij : P | (ij.2 : ℕ) ≤ (ij.1 : ℕ)}
  rw [Finset.univ_product_univ, ← Finset.sum_add_sum_compl S, ← eq_neg_iff_add_eq_zero,
    ← Finset.sum_neg_distrib]
  /- we are reduced to showing that two sums are equal, and this is obtained
    by constructing a bijection φ : S -> Sᶜ, which maps (i,j) to (j,i+1),
    and by comparing the terms -/
  let φ : ∀ ij : P, ij ∈ S → P := fun ij hij =>
    (Fin.castLT ij.2 (lt_of_le_of_lt (Finset.mem_filter.mp hij).right (Fin.is_lt ij.1)), ij.1.succ)
  apply Finset.sum_bij φ
  · -- φ(S) is contained in Sᶜ
    intro ij hij
    simp_rw [S, φ, Finset.compl_filter, Finset.mem_filter_univ, Fin.val_succ,
      Fin.val_castLT] at hij ⊢
    lia
  · -- φ : S → Sᶜ is injective
    rintro ⟨i, j⟩ hij ⟨i', j'⟩ hij' h
    rw [Prod.mk_inj]
    exact ⟨by simpa [φ] using congr_arg Prod.snd h,
      by simpa [φ, Fin.castSucc_castLT] using congr_arg Fin.castSucc (congr_arg Prod.fst h)⟩
  · -- φ : S → Sᶜ is surjective
    rintro ⟨i', j'⟩ hij'
    simp_rw [S, Finset.compl_filter, Finset.mem_filter_univ, not_le] at hij'
    refine ⟨(j'.pred <| ?_, Fin.castSucc i'), ?_, ?_⟩
    · rintro rfl
      simp only [Fin.val_zero, not_lt_zero'] at hij'
    · simpa [S] using Nat.le_sub_one_of_lt hij'
    · simp only [φ, Fin.castLT_castSucc, Fin.succ_pred]
  · -- identification of corresponding terms in both sums
    rintro ⟨i, j⟩ hij
    dsimp
    simp only [zsmul_comp, comp_zsmul, smul_smul, ← neg_smul]
    congr 1
    · simp only [φ, Fin.val_succ, pow_add, pow_one, mul_neg, neg_neg, mul_one]
      apply mul_comm
    · rw [CategoryTheory.SimplicialObject.δ_comp_δ'']
      simpa [S] using hij

/-!
## Construction of the alternating face map complex functor
-/

/-- The alternating face map complex, on objects -/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj (L122-L124)
def obj : ChainComplex C ℕ :=
ChainComplex.of (fun n => X _⦋n⦌) (objD X) (d_squared X)
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj_X (L126-L128)
theorem obj_X (X : SimplicialObject C) (n : ℕ) : (AlternatingFaceMapComplex.obj X).X n = X _⦋n⦌ :=
rfl
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq (L130-L137)
theorem obj_d_eq (X : SimplicialObject C) (n : ℕ) :
    (AlternatingFaceMapComplex.obj X).d (n + 1) n
      = ∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i := by
apply ChainComplex.of_d

variable {X} {Y}

/-- The alternating face map complex, on morphisms -/
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map (L138-L147)
def map (f : X ⟶ Y) : obj X ⟶ obj Y :=
ChainComplex.ofHom _ _ _ _ _ _ (fun n => f.app (op ⦋n⦌)) fun n => by
    dsimp
    rw [comp_sum, sum_comp]
    refine Finset.sum_congr rfl fun _ _ => ?_
    rw [comp_zsmul, zsmul_comp]
    congr 1
    symm
    apply f.naturality
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map_f (L149-L156)
theorem map_f (f : X ⟶ Y) (n : ℕ) : (map f).f n = f.app (op ⦋n⦌) :=
rfl

end AlternatingFaceMapComplex

variable (C : Type*) [Category* C] [Preadditive C]

/-- The alternating face map complex, as a functor -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex (L157-L162)
def alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := AlternatingFaceMapComplex.obj
map f := AlternatingFaceMapComplex.map f

variable {C}
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_obj_X (L164-L167)
theorem alternatingFaceMapComplex_obj_X (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).X n = X _⦋n⦌ :=
rfl
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_obj_d (L169-L173)
theorem alternatingFaceMapComplex_obj_d (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n := by
dsimp only [alternatingFaceMapComplex, AlternatingFaceMapComplex.obj]
  apply ChainComplex.of_d
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_map_f (L175-L179)
theorem alternatingFaceMapComplex_map_f {X Y : SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
    ((alternatingFaceMapComplex C).map f).f n = f.app (op ⦋n⦌) :=
rfl

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map_alternatingFaceMapComplex (L180-L200)
theorem map_alternatingFaceMapComplex {D : Type*} [Category* D] [Preadditive D] (F : C ⥤ D)
    [F.Additive] :
    alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
      (SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D := by
apply CategoryTheory.Functor.ext
  · intro X Y f
    ext n
    simp only [Functor.comp_map, HomologicalComplex.comp_f, alternatingFaceMapComplex_map_f,
      Functor.mapHomologicalComplex_map_f, HomologicalComplex.eqToHom_f, eqToHom_refl, comp_id,
      id_comp, SimplicialObject.whiskering_obj_map_app]
  · intro X
    apply HomologicalComplex.ext
    · rintro i j (rfl : j + 1 = i)
      dsimp only [Functor.comp_obj]
      simp only [Functor.mapHomologicalComplex_obj_d, alternatingFaceMapComplex_obj_d,
        eqToHom_refl, id_comp, comp_id, AlternatingFaceMapComplex.objD, Functor.map_sum,
        Functor.map_zsmul]
      rfl
    · ext n
      rfl
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.karoubi_alternatingFaceMapComplex_d (L201-L213)
theorem karoubi_alternatingFaceMapComplex_d (P : Karoubi (SimplicialObject C)) (n : ℕ) :
    ((AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)).d (n + 1) n).f =
      P.p.app (op ⦋n + 1⦌) ≫ (AlternatingFaceMapComplex.obj P.X).d (n + 1) n := by
dsimp
  simp only [AlternatingFaceMapComplex.obj_d_eq, Karoubi.sum_hom, Preadditive.comp_sum,
    Karoubi.zsmul_hom, Preadditive.comp_zsmul]
  rfl

namespace AlternatingFaceMapComplex

set_option backward.isDefEq.respectTransparency false in
/-- The natural transformation which gives the augmentation of the alternating face map
complex attached to an augmented simplicial object. -/
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.ε (L214-L232)
def ε [Limits.HasZeroObject C] :
    SimplicialObject.Augmented.drop ⋙ AlgebraicTopology.alternatingFaceMapComplex C ⟶
      SimplicialObject.Augmented.point ⋙ ChainComplex.single₀ C where
  app X := by
refine (ChainComplex.toSingle₀Equiv _ _).symm ?_
    refine ⟨X.hom.app (op ⦋0⦌), ?_⟩
    dsimp
    rw [alternatingFaceMapComplex_obj_d, objD, Fin.sum_univ_two, Fin.val_zero,
      pow_zero, one_smul, Fin.val_one, pow_one, neg_smul, one_smul, add_comp,
      neg_comp, SimplicialObject.δ_naturality, SimplicialObject.δ_naturality]
    apply add_neg_cancel
  naturality X Y f := by
    apply HomologicalComplex.to_single_hom_ext
    dsimp
    erw [ChainComplex.toSingle₀Equiv_symm_apply_f_zero,
      ChainComplex.toSingle₀Equiv_symm_apply_f_zero]
    simp only [ChainComplex.single₀_map_f_zero]
    exact congr_app f.w _
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero (L234-L237)
lemma ε_app_f_zero [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) :
    (ε.app X).f 0 = X.hom.app (op ⦋0⦌) :=
ChainComplex.toSingle₀Equiv_symm_apply_f_zero _ _
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.AlternatingFaceMapComplex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
def inclusionOfMooreComplexMap (X : SimplicialObject A) :
    (normalizedMooreComplex A).obj X ⟶ (alternatingFaceMapComplex A).obj X
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.objD (L66-L72)
def objD (n : ℕ) : X _⦋n + 1⦌ ⟶ X _⦋n⦌ :=
∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i

/-!
## The chain complex relation `d ≫ d`
-/
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.d_squared (L73-L121)
theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by
-- we start by expanding d ≫ d as a double sum
  dsimp
  simp only [comp_sum, sum_comp, ← Finset.sum_product']
  -- then, we decompose the index set P into a subset S and its complement Sᶜ
  let P := Fin (n + 2) × Fin (n + 3)
  let S : Finset P := {ij : P | (ij.2 : ℕ) ≤ (ij.1 : ℕ)}
  rw [Finset.univ_product_univ, ← Finset.sum_add_sum_compl S, ← eq_neg_iff_add_eq_zero,
    ← Finset.sum_neg_distrib]
  /- we are reduced to showing that two sums are equal, and this is obtained
    by constructing a bijection φ : S -> Sᶜ, which maps (i,j) to (j,i+1),
    and by comparing the terms -/
  let φ : ∀ ij : P, ij ∈ S → P := fun ij hij =>
    (Fin.castLT ij.2 (lt_of_le_of_lt (Finset.mem_filter.mp hij).right (Fin.is_lt ij.1)), ij.1.succ)
  apply Finset.sum_bij φ
  · -- φ(S) is contained in Sᶜ
    intro ij hij
    simp_rw [S, φ, Finset.compl_filter, Finset.mem_filter_univ, Fin.val_succ,
      Fin.val_castLT] at hij ⊢
    lia
  · -- φ : S → Sᶜ is injective
    rintro ⟨i, j⟩ hij ⟨i', j'⟩ hij' h
    rw [Prod.mk_inj]
    exact ⟨by simpa [φ] using congr_arg Prod.snd h,
      by simpa [φ, Fin.castSucc_castLT] using congr_arg Fin.castSucc (congr_arg Prod.fst h)⟩
  · -- φ : S → Sᶜ is surjective
    rintro ⟨i', j'⟩ hij'
    simp_rw [S, Finset.compl_filter, Finset.mem_filter_univ, not_le] at hij'
    refine ⟨(j'.pred <| ?_, Fin.castSucc i'), ?_, ?_⟩
    · rintro rfl
      simp only [Fin.val_zero, not_lt_zero'] at hij'
    · simpa [S] using Nat.le_sub_one_of_lt hij'
    · simp only [φ, Fin.castLT_castSucc, Fin.succ_pred]
  · -- identification of corresponding terms in both sums
    rintro ⟨i, j⟩ hij
    dsimp
    simp only [zsmul_comp, comp_zsmul, smul_smul, ← neg_smul]
    congr 1
    · simp only [φ, Fin.val_succ, pow_add, pow_one, mul_neg, neg_neg, mul_one]
      apply mul_comm
    · rw [CategoryTheory.SimplicialObject.δ_comp_δ'']
      simpa [S] using hij

/-!
## Construction of the alternating face map complex functor
-/

/-- The alternating face map complex, on objects -/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj (L122-L124)
def obj : ChainComplex C ℕ :=
ChainComplex.of (fun n => X _⦋n⦌) (objD X) (d_squared X)
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj_X (L126-L128)
theorem obj_X (X : SimplicialObject C) (n : ℕ) : (AlternatingFaceMapComplex.obj X).X n = X _⦋n⦌ :=
rfl
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq (L130-L137)
theorem obj_d_eq (X : SimplicialObject C) (n : ℕ) :
    (AlternatingFaceMapComplex.obj X).d (n + 1) n
      = ∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i := by
apply ChainComplex.of_d

variable {X} {Y}

/-- The alternating face map complex, on morphisms -/
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map (L138-L147)
def map (f : X ⟶ Y) : obj X ⟶ obj Y :=
ChainComplex.ofHom _ _ _ _ _ _ (fun n => f.app (op ⦋n⦌)) fun n => by
    dsimp
    rw [comp_sum, sum_comp]
    refine Finset.sum_congr rfl fun _ _ => ?_
    rw [comp_zsmul, zsmul_comp]
    congr 1
    symm
    apply f.naturality
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map_f (L149-L156)
theorem map_f (f : X ⟶ Y) (n : ℕ) : (map f).f n = f.app (op ⦋n⦌) :=
rfl

end AlternatingFaceMapComplex

variable (C : Type*) [Category* C] [Preadditive C]

/-- The alternating face map complex, as a functor -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex (L157-L162)
def alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
  obj := AlternatingFaceMapComplex.obj
map f := AlternatingFaceMapComplex.map f

variable {C}
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_obj_X (L164-L167)
theorem alternatingFaceMapComplex_obj_X (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).X n = X _⦋n⦌ :=
rfl
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_obj_d (L169-L173)
theorem alternatingFaceMapComplex_obj_d (X : SimplicialObject C) (n : ℕ) :
    ((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n := by
dsimp only [alternatingFaceMapComplex, AlternatingFaceMapComplex.obj]
  apply ChainComplex.of_d
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.alternatingFaceMapComplex_map_f (L175-L179)
theorem alternatingFaceMapComplex_map_f {X Y : SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
    ((alternatingFaceMapComplex C).map f).f n = f.app (op ⦋n⦌) :=
rfl

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.map_alternatingFaceMapComplex (L180-L200)
theorem map_alternatingFaceMapComplex {D : Type*} [Category* D] [Preadditive D] (F : C ⥤ D)
    [F.Additive] :
    alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
      (SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D := by
apply CategoryTheory.Functor.ext
  · intro X Y f
    ext n
    simp only [Functor.comp_map, HomologicalComplex.comp_f, alternatingFaceMapComplex_map_f,
      Functor.mapHomologicalComplex_map_f, HomologicalComplex.eqToHom_f, eqToHom_refl, comp_id,
      id_comp, SimplicialObject.whiskering_obj_map_app]
  · intro X
    apply HomologicalComplex.ext
    · rintro i j (rfl : j + 1 = i)
      dsimp only [Functor.comp_obj]
      simp only [Functor.mapHomologicalComplex_obj_d, alternatingFaceMapComplex_obj_d,
        eqToHom_refl, id_comp, comp_id, AlternatingFaceMapComplex.objD, Functor.map_sum,
        Functor.map_zsmul]
      rfl
    · ext n
      rfl
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.karoubi_alternatingFaceMapComplex_d (L201-L213)
theorem karoubi_alternatingFaceMapComplex_d (P : Karoubi (SimplicialObject C)) (n : ℕ) :
    ((AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)).d (n + 1) n).f =
      P.p.app (op ⦋n + 1⦌) ≫ (AlternatingFaceMapComplex.obj P.X).d (n + 1) n := by
dsimp
  simp only [AlternatingFaceMapComplex.obj_d_eq, Karoubi.sum_hom, Preadditive.comp_sum,
    Karoubi.zsmul_hom, Preadditive.comp_zsmul]
  rfl

namespace AlternatingFaceMapComplex

set_option backward.isDefEq.respectTransparency false in
/-- The natural transformation which gives the augmentation of the alternating face map
complex attached to an augmented simplicial object. -/
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.ε (L214-L232)
def ε [Limits.HasZeroObject C] :
    SimplicialObject.Augmented.drop ⋙ AlgebraicTopology.alternatingFaceMapComplex C ⟶
      SimplicialObject.Augmented.point ⋙ ChainComplex.single₀ C where
  app X := by
refine (ChainComplex.toSingle₀Equiv _ _).symm ?_
    refine ⟨X.hom.app (op ⦋0⦌), ?_⟩
    dsimp
    rw [alternatingFaceMapComplex_obj_d, objD, Fin.sum_univ_two, Fin.val_zero,
      pow_zero, one_smul, Fin.val_one, pow_one, neg_smul, one_smul, add_comp,
      neg_comp, SimplicialObject.δ_naturality, SimplicialObject.δ_naturality]
    apply add_neg_cancel
  naturality X Y f := by
    apply HomologicalComplex.to_single_hom_ext
    dsimp
    erw [ChainComplex.toSingle₀Equiv_symm_apply_f_zero,
      ChainComplex.toSingle₀Equiv_symm_apply_f_zero]
    simp only [ChainComplex.single₀_map_f_zero]
    exact congr_app f.w _
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero (L234-L237)
lemma ε_app_f_zero [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) :
    (ε.app X).f 0 = X.hom.app (op ⦋0⦌) :=
ChainComplex.toSingle₀Equiv_symm_apply_f_zero _ _
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Computability.Partrec).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem comp₂ {f : γ → δ →. σ} {g : α → β → γ} {h : α → β → δ} (hf : Partrec₂ f)
    (hg : Computable₂ g) (hh : Computable₂ h) : Partrec₂ fun a b => f (g a b) (h a b)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Partrec.rfindX (L65-L90)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun _ => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        rcases h.lt_or_eq_dec with h | h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩
      · exact ⟨m, ⟨_, e⟩, al⟩)

end Rfind

/-- Find the smallest `n` satisfying `p n`, where all `p k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 2
```lean
-- Mathlib.Computability.Partrec.rfind (L91-L93)
def rfind (p : ℕ →. Bool) : Part ℕ :=
⟨_, fun h => (rfindX p h).1⟩
```

-- snippet 3
```lean
-- Mathlib.Computability.Partrec.rfind_spec (L94-L96)
theorem rfind_spec {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : true ∈ p n :=
h.snd ▸ (rfindX p h.fst).2.1
```

-- snippet 4
```lean
-- Mathlib.Computability.Partrec.rfind_min (L97-L99)
theorem rfind_min {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : ∀ {m : ℕ}, m < n → false ∈ p m :=
@(h.snd ▸ @((rfindX p h.fst).2.2))
```

-- snippet 5
```lean
-- Mathlib.Computability.Partrec.rfind_dom (L101-L104)
theorem rfind_dom {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m < n → (p m).Dom :=
Iff.rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Partrec.rfind_dom' (L105-L111)
theorem rfind_dom' {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m ≤ n → (p m).Dom :=
exists_congr fun _ =>
    and_congr_right fun pn =>
      ⟨fun H _ h => (Decidable.eq_or_lt_of_le h).elim (fun e => e.symm ▸ pn.fst) (H _), fun H _ h =>
        H (le_of_lt h)⟩
```

-- snippet 7
```lean
-- Mathlib.Computability.Partrec.mem_rfind (L113-L121)
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    rcases lt_trichotomy m n with (h | h | h)
    · injection mem_unique (h₂ h) (rfind_spec hm)
    · rwa [← h]
    · injection mem_unique h₁ (rfind_min hm h)⟩
```

-- snippet 8
```lean
-- Mathlib.Computability.Partrec.rfind_min' (L122-L126)
theorem rfind_min' {p : ℕ → Bool} {m : ℕ} (pm : p m) : ∃ n ∈ rfind p, n ≤ m :=
have : true ∈ (p : ℕ →. Bool) m := ⟨trivial, pm⟩
  let ⟨n, hn⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨m, this, fun {_} _ => ⟨⟩⟩
  ⟨n, hn, not_lt.1 fun h => by injection mem_unique this (rfind_min hn h)⟩
```

-- snippet 9
```lean
-- Mathlib.Computability.Partrec.rfind_zero_none (L127-L133)
theorem rfind_zero_none (p : ℕ →. Bool) (p0 : p 0 = Part.none) : rfind p = Part.none :=
eq_none_iff.2 fun _ h =>
    let ⟨_, _, h₂⟩ := rfind_dom'.1 h.fst
    (p0 ▸ h₂ (zero_le _) : (@Part.none Bool).Dom)

/-- Find the smallest `n` satisfying `f n`, where all `f k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 10
```lean
-- Mathlib.Computability.Partrec.rfindOpt (L134-L136)
def rfindOpt {α} (f : ℕ → Option α) : Part α :=
(rfind fun n => (f n).isSome).bind fun n => f n
```

-- snippet 11
```lean
-- Mathlib.Computability.Partrec.rfindOpt_spec (L137-L140)
theorem rfindOpt_spec {α} {f : ℕ → Option α} {a} (h : a ∈ rfindOpt f) : ∃ n, a ∈ f n :=
let ⟨n, _, h₂⟩ := mem_bind_iff.1 h
  ⟨n, mem_coe.1 h₂⟩
```

-- snippet 12
```lean
-- Mathlib.Computability.Partrec.rfindOpt_dom (L141-L150)
theorem rfindOpt_dom {α} {f : ℕ → Option α} : (rfindOpt f).Dom ↔ ∃ n a, a ∈ f n :=
⟨fun h => (rfindOpt_spec ⟨h, rfl⟩).imp fun _ h => ⟨_, h⟩, fun h => by
    have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2
    have s := Nat.find_spec h'
    have fd : (rfind fun n => (f n).isSome).Dom :=
      ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩
    refine ⟨fd, ?_⟩
    have := rfind_spec (get_mem fd)
    simpa using this⟩
```

-- snippet 13
```lean
-- Mathlib.Computability.Partrec.rfindOpt_mono (L151-L173)
theorem rfindOpt_mono {α} {f : ℕ → Option α} (H : ∀ {a m n}, m ≤ n → a ∈ f m → a ∈ f n) {a} :
    a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
    have h' := rfindOpt_dom.2 ⟨_, _, h⟩
    obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
    have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
    simp at this; simp [this, get_mem]⟩

/-- `Nat.Partrec f` means that the partial function `f : ℕ →. ℕ` is partially recursive. -/
protected inductive Partrec : (ℕ →. ℕ) → Prop
  | zero : Nat.Partrec (pure 0)
  | succ : Nat.Partrec succ
  | left : Nat.Partrec ↑fun n : ℕ => n.unpair.1
  | right : Nat.Partrec ↑fun n : ℕ => n.unpair.2
  | pair {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => pair <$> f n <*> g n
  | comp {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => g n >>= f
  | prec {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec (unpaired fun a n =>
      n.rec (f a) fun y IH => do let i ← IH; g (pair a (pair y i)))
  | rfind {f} : Nat.Partrec f →
    Nat.Partrec fun a => rfind fun n => (fun m => m = 0) <$> f (pair a n)

namespace Partrec
```

-- snippet 14
```lean
-- Mathlib.Computability.Partrec.of_eq (L174-L176)
theorem of_eq {f g : ℕ →. ℕ} (hf : Nat.Partrec f) (H : ∀ n, f n = g n) : Nat.Partrec g :=
(funext H : f = g) ▸ hf
```

-- snippet 15
```lean
-- Mathlib.Computability.Partrec.of_eq_tot (L177-L180)
theorem of_eq_tot {f : ℕ →. ℕ} {g : ℕ → ℕ} (hf : Nat.Partrec f) (H : ∀ n, g n ∈ f n) :
    Nat.Partrec g :=
hf.of_eq fun n => eq_some_iff.2 (H n)
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Computability.Partrec).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem comp₂ {f : γ → δ →. σ} {g : α → β → γ} {h : α → β → δ} (hf : Partrec₂ f)
    (hg : Computable₂ g) (hh : Computable₂ h) : Partrec₂ fun a b => f (g a b) (h a b)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Partrec.rfindX (L65-L90)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun _ => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        rcases h.lt_or_eq_dec with h | h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩
      · exact ⟨m, ⟨_, e⟩, al⟩)

end Rfind

/-- Find the smallest `n` satisfying `p n`, where all `p k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 2
```lean
-- Mathlib.Computability.Partrec.rfind (L91-L93)
def rfind (p : ℕ →. Bool) : Part ℕ :=
⟨_, fun h => (rfindX p h).1⟩
```

-- snippet 3
```lean
-- Mathlib.Computability.Partrec.rfind_spec (L94-L96)
theorem rfind_spec {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : true ∈ p n :=
h.snd ▸ (rfindX p h.fst).2.1
```

-- snippet 4
```lean
-- Mathlib.Computability.Partrec.rfind_min (L97-L99)
theorem rfind_min {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : ∀ {m : ℕ}, m < n → false ∈ p m :=
@(h.snd ▸ @((rfindX p h.fst).2.2))
```

-- snippet 5
```lean
-- Mathlib.Computability.Partrec.rfind_dom (L101-L104)
theorem rfind_dom {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m < n → (p m).Dom :=
Iff.rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Partrec.rfind_dom' (L105-L111)
theorem rfind_dom' {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m ≤ n → (p m).Dom :=
exists_congr fun _ =>
    and_congr_right fun pn =>
      ⟨fun H _ h => (Decidable.eq_or_lt_of_le h).elim (fun e => e.symm ▸ pn.fst) (H _), fun H _ h =>
        H (le_of_lt h)⟩
```

-- snippet 7
```lean
-- Mathlib.Computability.Partrec.mem_rfind (L113-L121)
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    rcases lt_trichotomy m n with (h | h | h)
    · injection mem_unique (h₂ h) (rfind_spec hm)
    · rwa [← h]
    · injection mem_unique h₁ (rfind_min hm h)⟩
```

-- snippet 8
```lean
-- Mathlib.Computability.Partrec.rfind_min' (L122-L126)
theorem rfind_min' {p : ℕ → Bool} {m : ℕ} (pm : p m) : ∃ n ∈ rfind p, n ≤ m :=
have : true ∈ (p : ℕ →. Bool) m := ⟨trivial, pm⟩
  let ⟨n, hn⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨m, this, fun {_} _ => ⟨⟩⟩
  ⟨n, hn, not_lt.1 fun h => by injection mem_unique this (rfind_min hn h)⟩
```

-- snippet 9
```lean
-- Mathlib.Computability.Partrec.rfind_zero_none (L127-L133)
theorem rfind_zero_none (p : ℕ →. Bool) (p0 : p 0 = Part.none) : rfind p = Part.none :=
eq_none_iff.2 fun _ h =>
    let ⟨_, _, h₂⟩ := rfind_dom'.1 h.fst
    (p0 ▸ h₂ (zero_le _) : (@Part.none Bool).Dom)

/-- Find the smallest `n` satisfying `f n`, where all `f k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 10
```lean
-- Mathlib.Computability.Partrec.rfindOpt (L134-L136)
def rfindOpt {α} (f : ℕ → Option α) : Part α :=
(rfind fun n => (f n).isSome).bind fun n => f n
```

-- snippet 11
```lean
-- Mathlib.Computability.Partrec.rfindOpt_spec (L137-L140)
theorem rfindOpt_spec {α} {f : ℕ → Option α} {a} (h : a ∈ rfindOpt f) : ∃ n, a ∈ f n :=
let ⟨n, _, h₂⟩ := mem_bind_iff.1 h
  ⟨n, mem_coe.1 h₂⟩
```

-- snippet 12
```lean
-- Mathlib.Computability.Partrec.rfindOpt_dom (L141-L150)
theorem rfindOpt_dom {α} {f : ℕ → Option α} : (rfindOpt f).Dom ↔ ∃ n a, a ∈ f n :=
⟨fun h => (rfindOpt_spec ⟨h, rfl⟩).imp fun _ h => ⟨_, h⟩, fun h => by
    have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2
    have s := Nat.find_spec h'
    have fd : (rfind fun n => (f n).isSome).Dom :=
      ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩
    refine ⟨fd, ?_⟩
    have := rfind_spec (get_mem fd)
    simpa using this⟩
```

-- snippet 13
```lean
-- Mathlib.Computability.Partrec.rfindOpt_mono (L151-L173)
theorem rfindOpt_mono {α} {f : ℕ → Option α} (H : ∀ {a m n}, m ≤ n → a ∈ f m → a ∈ f n) {a} :
    a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
    have h' := rfindOpt_dom.2 ⟨_, _, h⟩
    obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
    have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
    simp at this; simp [this, get_mem]⟩

/-- `Nat.Partrec f` means that the partial function `f : ℕ →. ℕ` is partially recursive. -/
protected inductive Partrec : (ℕ →. ℕ) → Prop
  | zero : Nat.Partrec (pure 0)
  | succ : Nat.Partrec succ
  | left : Nat.Partrec ↑fun n : ℕ => n.unpair.1
  | right : Nat.Partrec ↑fun n : ℕ => n.unpair.2
  | pair {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => pair <$> f n <*> g n
  | comp {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => g n >>= f
  | prec {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec (unpaired fun a n =>
      n.rec (f a) fun y IH => do let i ← IH; g (pair a (pair y i)))
  | rfind {f} : Nat.Partrec f →
    Nat.Partrec fun a => rfind fun n => (fun m => m = 0) <$> f (pair a n)

namespace Partrec
```

-- snippet 14
```lean
-- Mathlib.Computability.Partrec.of_eq (L174-L176)
theorem of_eq {f g : ℕ →. ℕ} (hf : Nat.Partrec f) (H : ∀ n, f n = g n) : Nat.Partrec g :=
(funext H : f = g) ▸ hf
```

-- snippet 15
```lean
-- Mathlib.Computability.Partrec.of_eq_tot (L177-L180)
theorem of_eq_tot {f : ℕ →. ℕ} {g : ℕ → ℕ} (hf : Nat.Partrec f) (H : ∀ n, g n ∈ f n) :
    Nat.Partrec g :=
hf.of_eq fun n => eq_some_iff.2 (H n)
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.Light.TopCatAdjunction).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Condensed.Light.TopCatAdjunction.lightCondSetToSequential`:
State 0: ⊢ def lightCondSetToSequential : LightCondSet.{u} ⥤ Sequential.{u} where
  obj X
Tactic: apply SeqContinuous.continuous  unfold SeqContinuous
State 1: ⊢ <goal after apply SeqContinuous.continuous>
Tactic: intro f p h  let g := (topCatAdjunctionCounitEquiv X).invFun ∘ (OnePoint.continuousMapMkNat f p h)  change Filter.Tendsto (fun n : ℕ ↦ g n) _ _  erw [← OnePoint.continuous_iff_from_nat]  let x : X.toLightCondSet.obj.obj ⟨(ℕ∪{∞})⟩ := OnePoint.continuousMapMkNat f p h
State 2: ⊢ <goal after intro f p h>
Tactic: exact continuous_coinducingCoprod X.toLightCondSet x  /--  The counit of the adjunction `lightCondSetToSequential ⊣ sequentialToLightCondSet`  is an isomorphism.  Note: for now, we only have `ℕ∪{∞}` as a light profinite set at universe level 0, which is why we  can only prove this for `X : Sequential.{0}`.  -/  noncomputable def sequentialAdjunctionCounitIso (X : Sequential.{0}) :  lightCondSetToSequential.obj (sequentialToLightCondSet.obj X) ≅ X :=  isoOfHomeo (sequentialAdjunctionHomeo X.toTop)
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.Light.TopCatAdjunction).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Condensed.Light.TopCatAdjunction.lightCondSetToSequential`:
State 0: ⊢ def lightCondSetToSequential : LightCondSet.{u} ⥤ Sequential.{u} where
  obj X
Tactic: apply SeqContinuous.continuous  unfold SeqContinuous
State 1: ⊢ <goal after apply SeqContinuous.continuous>
Tactic: intro f p h  let g := (topCatAdjunctionCounitEquiv X).invFun ∘ (OnePoint.continuousMapMkNat f p h)  change Filter.Tendsto (fun n : ℕ ↦ g n) _ _  erw [← OnePoint.continuous_iff_from_nat]  let x : X.toLightCondSet.obj.obj ⟨(ℕ∪{∞})⟩ := OnePoint.continuousMapMkNat f p h
State 2: ⊢ <goal after intro f p h>
Tactic: exact continuous_coinducingCoprod X.toLightCondSet x  /--  The counit of the adjunction `lightCondSetToSequential ⊣ sequentialToLightCondSet`  is an isomorphism.  Note: for now, we only have `ℕ∪{∞}` as a light profinite set at universe level 0, which is why we  can only prove this for `X : Sequential.{0}`.  -/  noncomputable def sequentialAdjunctionCounitIso (X : Sequential.{0}) :  lightCondSetToSequential.obj (sequentialToLightCondSet.obj X) ≅ X :=  isoOfHomeo (sequentialAdjunctionHomeo X.toTop)
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.TopCatAdjunction).

[User]
The following Lean code fails to compile. Fix it.

```lean
lemma topCatAdjunctionCounit_bijective (X : TopCat.{u + 1}) :
    Function.Bijective (topCatAdjunctionCounit X)
:= _unbalancedParenXYZ
(topCatAdjunctionCounitEquiv X).bijective

/-- The unit of the adjunction `condensedSetToTopCat ⊣ topCatToCondensedSet` -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=178, messages=[Message(data="Function expected at\n  topCatAdjunctionCounit\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  X\n\nHint: The identifier `topCatAdjunctionCounit` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=24), pos_end=Position(line=2, column=48), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=178, i_end=259, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=79), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.TopCatAdjunction).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma topCatAdjunctionCounit_bijective (X : TopCat.{u + 1}) :
    Function.Bijective (topCatAdjunctionCounit X)
:= _unbalancedParenXYZ
(topCatAdjunctionCounitEquiv X).bijective

/-- The unit of the adjunction `condensedSetToTopCat ⊣ topCatToCondensedSet` -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=178, messages=[Message(data="Function expected at\n  topCatAdjunctionCounit\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  X\n\nHint: The identifier `topCatAdjunctionCounit` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=24), pos_end=Position(line=2, column=48), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=178, i_end=259, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=79), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
abbrev toArrow {x : X} (p : FundamentalGroup X x) :
    FundamentalGroupoid.mk x ⟶ FundamentalGroupoid.mk x
```

Declaration: `Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup.toArrow` (module Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
abbrev toArrow {x : X} (p : FundamentalGroup X x) :
    FundamentalGroupoid.mk x ⟶ FundamentalGroupoid.mk x
```

Declaration: `Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup.toArrow` (module Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup).[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.Discrete.Colimit).

[User]
The following Lean code fails to compile. Fix it.

```lean
def lanSheafProfinite (X : Type (u + 1)) :
    Sheaf (coherentTopology Profinite.{u}) (Type (u + 1)) where
  obj
  frobnicateTacticThatDoesNotExist := lanPresheaf (locallyConstantPresheaf X)
property := by
    rw [Presheaf.isSheaf_of_iso_iff (lanPresheafNatIso
      fun _ ↦ isColimitLocallyConstantPresheafDiagram _ _)]
    exact ((CompHausLike.LocallyConstant.functor.{u, u + 1}
      (hs := fun _ _ _ ↦ ((Profinite.effectiveEpi_tfae _).out 0 2).mp)).obj X).property

/-- `lanPresheaf (locallyConstantPresheaf X)` as a condensed set. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=191, messages=[Message(data="Function expected at\n  Sheaf\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (coherentTopology Profinite.{u})\n\nHint: The identifier `Sheaf` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=191, i_end=542, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=5, column=0), pos_end=Position(line=5, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=11, column=68), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.Discrete.Colimit).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def lanSheafProfinite (X : Type (u + 1)) :
    Sheaf (coherentTopology Profinite.{u}) (Type (u + 1)) where
  obj
  frobnicateTacticThatDoesNotExist := lanPresheaf (locallyConstantPresheaf X)
property := by
    rw [Presheaf.isSheaf_of_iso_iff (lanPresheafNatIso
      fun _ ↦ isColimitLocallyConstantPresheafDiagram _ _)]
    exact ((CompHausLike.LocallyConstant.functor.{u, u + 1}
      (hs := fun _ _ _ ↦ ((Profinite.effectiveEpi_tfae _).out 0 2).mp)).obj X).property

/-- `lanPresheaf (locallyConstantPresheaf X)` as a condensed set. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=191, messages=[Message(data="Function expected at\n  Sheaf\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (coherentTopology Profinite.{u})\n\nHint: The identifier `Sheaf` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=191, i_end=542, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=5, column=0), pos_end=Position(line=5, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=11, column=68), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


Instruction: Edit `eraseProofs` so it still compiles and the proof is simplified where possible. Return the full updated `eraseProofs` block.

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Lean.Expr.Basic).

[User]
You are editing a file in `Mathlib/Lean/Expr/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Lean.Expr.Basic.eraseProofs`, L215-L223)
```lean
def eraseProofs (e : Expr) : MetaM Expr := Meta.transform (skipConstInApp := true) e
    (pre := fun e => do
      if (← Meta.isProof e) then
        return .continue (← mkSorry (← inferType e) true)
      else
        return .continue)

/-- If an `Expr` has the form `Type u`, then return `some u`, otherwise `none`. -/
```

## Edit instruction
Edit `eraseProofs` so it still compiles and the proof is simplified where possible. Return the full updated `eraseProofs` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Lean.Expr.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Lean/Expr/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Lean.Expr.Basic.eraseProofs`, L215-L223)
```lean
def eraseProofs (e : Expr) : MetaM Expr := Meta.transform (skipConstInApp := true) e
    (pre := fun e => do
      if (← Meta.isProof e) then
        return .continue (← mkSorry (← inferType e) true)
      else
        return .continue)

/-- If an `Expr` has the form `Type u`, then return `some u`, otherwise `none`. -/
```

## Edit instruction
Edit `eraseProofs` so it still compiles and the proof is simplified where possible. Return the full updated `eraseProofs` block.
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Bicategory.Normalize).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem evalWhiskerRightAux_of {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) :
    η ▷ h = (Iso.refl _).hom ≫ η ▷ h ≫ (Iso.refl _).hom
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_nil_nil (L37-L40)
theorem evalComp_nil_nil (α : f ≅ g) (β : g ≅ h) :
    (α ≪≫ β).hom = (α ≪≫ β).hom := by
simp
```

-- snippet 2
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_nil_cons (L41-L44)
theorem evalComp_nil_cons (α : f ≅ g) (β : g ≅ h) (η : h ⟶ i) (ηs : i ⟶ j) :
    α.hom ≫ (β.hom ≫ η ≫ ηs) = (α ≪≫ β).hom ≫ η ≫ ηs := by
simp
```

-- snippet 3
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_cons (L45-L49)
theorem evalComp_cons (α : f ≅ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : i ⟶ j} {ι : h ⟶ j}
    (e_ι : ηs ≫ θ = ι) :
    (α.hom ≫ η ≫ ηs) ≫ θ = α.hom ≫ η ≫ ι := by
simp [e_ι]
```

-- snippet 4
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_comp (L50-L55)
theorem eval_comp
    {η η' : f ⟶ g} {θ θ' : g ⟶ h} {ι : f ⟶ h}
    (e_η : η = η') (e_θ : θ = θ') (e_ηθ : η' ≫ θ' = ι) :
    η ≫ θ = ι := by
simp [e_η, e_θ, e_ηθ]
```

-- snippet 5
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_of (L56-L59)
theorem eval_of (η : f ⟶ g) :
    η = (Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom := by
simp
```

-- snippet 6
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_monoidalComp (L60-L65)
theorem eval_monoidalComp
    {η η' : f ⟶ g} {α : g ≅ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
    (e_η : η = η') (e_θ : θ = θ') (e_αθ : α.hom ≫ θ' = αθ) (e_ηαθ : η' ≫ αθ = ηαθ) :
    η ≫ α.hom ≫ θ = ηαθ := by
simp [e_η, e_θ, e_αθ, e_ηαθ]
```

-- snippet 7
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_nil (L67-L70)
theorem evalWhiskerLeft_nil (f : a ⟶ b) {g h : b ⟶ c} (α : g ≅ h) :
    (whiskerLeftIso f α).hom = (whiskerLeftIso f α).hom := by
simp
```

-- snippet 8
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_of_cons (L71-L76)
theorem evalWhiskerLeft_of_cons
    {f : a ⟶ b} {g h i j : b ⟶ c}
    (α : g ≅ h) (η : h ⟶ i) {ηs : i ⟶ j} {θ : f ≫ i ⟶ f ≫ j} (e_θ : f ◁ ηs = θ) :
    f ◁ (α.hom ≫ η ≫ ηs) = (whiskerLeftIso f α).hom ≫ f ◁ η ≫ θ := by
simp [e_θ]
```

-- snippet 9
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_comp (L77-L85)
theorem evalWhiskerLeft_comp
    {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d}
    {η : h ⟶ i} {η₁ : g ≫ h ⟶ g ≫ i} {η₂ : f ≫ g ≫ h ⟶ f ≫ g ≫ i}
    {η₃ : f ≫ g ≫ h ⟶ (f ≫ g) ≫ i} {η₄ : (f ≫ g) ≫ h ⟶ (f ≫ g) ≫ i}
    (e_η₁ : g ◁ η = η₁) (e_η₂ : f ◁ η₁ = η₂)
    (e_η₃ : η₂ ≫ (α_ _ _ _).inv = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄) :
    (f ≫ g) ◁ η = η₄ := by
simp [e_η₁, e_η₂, e_η₃, e_η₄]
```

-- snippet 10
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_id (L86-L91)
theorem evalWhiskerLeft_id {η : f ⟶ g}
    {η₁ : f ⟶ 𝟙 a ≫ g} {η₂ : 𝟙 a ≫ f ⟶ 𝟙 a ≫ g}
    (e_η₁ : η ≫ (λ_ _).inv = η₁) (e_η₂ : (λ_ _).hom ≫ η₁ = η₂) :
    𝟙 a ◁ η = η₂ := by
simp [e_η₁, e_η₂]
```

-- snippet 11
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_whiskerLeft (L92-L98)
theorem eval_whiskerLeft
    {f : a ⟶ b} {g h : b ⟶ c}
    {η η' : g ⟶ h} {θ : f ≫ g ⟶ f ≫ h}
    (e_η : η = η') (e_θ : f ◁ η' = θ) :
    f ◁ η = θ := by
simp [e_η, e_θ]
```

-- snippet 12
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_whiskerRight (L99-L105)
theorem eval_whiskerRight
    {f g : a ⟶ b} {h : b ⟶ c}
    {η η' : f ⟶ g} {θ : f ≫ h ⟶ g ≫ h}
    (e_η : η = η') (e_θ : η' ▷ h = θ) :
    η ▷ h = θ := by
simp [e_η, e_θ]
```

-- snippet 13
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_nil (L107-L110)
theorem evalWhiskerRight_nil (α : f ≅ g) (h : b ⟶ c) :
    α.hom ▷ h = α.hom ▷ h := by
simp
```

-- snippet 14
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_cons_of_of (L115-L123)
theorem evalWhiskerRight_cons_of_of
    {f g h i : a ⟶ b} {j : b ⟶ c}
    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {ηs₁ : h ≫ j ⟶ i ≫ j}
    {η₁ : g ≫ j ⟶ h ≫ j} {η₂ : g ≫ j ⟶ i ≫ j} {η₃ : f ≫ j ⟶ i ≫ j}
    (e_ηs₁ : ηs ▷ j = ηs₁) (e_η₁ : η ▷ j = η₁)
    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃) :
    (α.hom ≫ η ≫ ηs) ▷ j = η₃ := by
simp_all
```

-- snippet 15
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_cons_whisker (L124-L137)
theorem evalWhiskerRight_cons_whisker
    {f : a ⟶ b} {g : a ⟶ c} {h i : b ⟶ c} {j : a ⟶ c} {k : c ⟶ d}
    {α : g ≅ f ≫ h} {η : h ⟶ i} {ηs : f ≫ i ⟶ j}
    {η₁ : h ≫ k ⟶ i ≫ k} {η₂ : f ≫ (h ≫ k) ⟶ f ≫ (i ≫ k)} {ηs₁ : (f ≫ i) ≫ k ⟶ j ≫ k}
    {ηs₂ : f ≫ (i ≫ k) ⟶ j ≫ k} {η₃ : f ≫ (h ≫ k) ⟶ j ≫ k} {η₄ : (f ≫ h) ≫ k ⟶ j ≫ k}
    {η₅ : g ≫ k ⟶ j ≫ k}
    (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ k = η₁) (e_η₂ : f ◁ η₁ = η₂)
    (e_ηs₁ : ηs ▷ k = ηs₁) (e_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
    (e_η₃ : η₂ ≫ ηs₂ = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄)
    (e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅) :
    (α.hom ≫ (f ◁ η) ≫ ηs) ▷ k = η₅ := by
simp at e_η₁ e_η₅
  simp [e_η₁, e_η₂, e_ηs₁, e_ηs₂, e_η₃, e_η₄, e_η₅]
```

-- snippet 16
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Basic.bicategoryNf (L35-L37)
def bicategoryNf (mvarId : MVarId) : MetaM (List MVarId) := do
BicategoryLike.normalForm Bicategory.Context `bicategory mvarId
```

-- snippet 17
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.left' (L96-L99)
instance left' (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence f (𝟙 a ≫ g) :=
⟨⊗𝟙 ≪≫ (λ_ g).symm⟩
```

-- snippet 18
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.mkContext (L68-L85)
def mkContext? (e : Expr) : MetaM (Option Context) := do
let e ← instantiateMVars e
  let type ← instantiateMVars <| ← inferType e
  match (← whnfR (← inferType e)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, f, _]) =>
    let fType ← instantiateMVars <| ← inferType f
    match (← whnfR fType).getAppFnArgs with
    | (``Quiver.Hom, #[_, _, a, _]) =>
      let B ← inferType a
      let .succ level₀ ← getLevel B | return none
      let .succ level₁ ← getLevel fType | return none
      let .succ level₂ ← getLevel type | return none
      let some instBicategory ← synthInstance?
        (mkAppN (.const ``Bicategory [level₂, level₁, level₀]) #[B]) | return none
      return some ⟨level₂, level₁, level₀, B, instBicategory⟩
    | _ => return none
  | _ => return none
```

-- snippet 19
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalComp_refl (L125-L128)
theorem bicategoricalComp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : η ⊗≫ θ = η ≫ θ := by
dsimp [bicategoricalComp]; simp

end CategoryTheory
```

-- snippet 20
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalIso (L41-L45)
abbrev bicategoricalIso (f g : a ⟶ b) [BicategoricalCoherence f g] : f ≅ g :=
⊗𝟙

/-- Compose two morphisms in a bicategorical category,
inserting unitors and associators between as necessary. -/
```

-- snippet 21
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.structuralIso_inv (L113-L116)
theorem structuralIso_inv {f g : a ⟶ b} (η : f ≅ g) :
    η.symm.hom = η.inv := by
simp only [Iso.symm_hom]
```

-- snippet 22
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalIsoComp (L56-L59)
def bicategoricalIsoComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
    (η : f ≅ g) (θ : h ≅ i) : f ≅ i :=
η ≪≫ ⊗𝟙 ≪≫ θ
```

-- snippet 23
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.tensorRight' (L86-L89)
instance tensorRight' (f : a ⟶ b) (g : b ⟶ b)
    [BicategoricalCoherence g (𝟙 b)] : BicategoricalCoherence (f ≫ g) f :=
⟨whiskerLeftIso f ⊗𝟙 ≪≫ (ρ_ f)⟩
```

-- snippet 24
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.whiskerRight (L76-L79)
instance whiskerRight (f g : a ⟶ b) (h : b ⟶ c)
    [BicategoricalCoherence f g] : BicategoricalCoherence (f ≫ h) (g ≫ h) :=
⟨whiskerRightIso ⊗𝟙 h⟩
```

-- snippet 25
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.tgtExprOfIso (L47-L67)
def tgtExprOfIso (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Iso, #[_, _, _, g]) => return g
  | _ => throwError m!"{η} is not a morphism"

initialize registerTraceClass `bicategory

/-- The context for evaluating expressions. -/
structure Context where
  /-- The level for 2-morphisms. -/
  level₂ : Level
  /-- The level for 1-morphisms. -/
  level₁ : Level
  /-- The level for objects. -/
  level₀ : Level
  /-- The expression for the underlying category. -/
  B : Q(Type level₀)
  /-- The bicategory instance. -/
  instBicategory : Q(Bicategory.{level₂, level₁} $B)

/-- Populate a `context` object for evaluating `e`. -/
```

-- snippet 26
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.tensorRight (L81-L84)
instance tensorRight (f : a ⟶ b) (g : b ⟶ b)
    [BicategoricalCoherence (𝟙 b) g] : BicategoricalCoherence f (f ≫ g) :=
⟨(ρ_ f).symm ≪≫ (whiskerLeftIso f ⊗𝟙)⟩
```

-- snippet 27
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.tgtExpr (L35-L40)
def tgtExpr (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, _, g]) => return g
  | _ => throwError m!"{η} is not a morphism"

/-- The domain of an isomorphism. -/
```

-- snippet 28
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalComp (L46-L50)
def bicategoricalComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
    (η : f ⟶ g) (θ : h ⟶ i) : f ⟶ i :=
η ≫ ⊗𝟙.hom ≫ θ

-- type as \ot \gg
```

-- snippet 29
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.whiskerLeft (L71-L74)
instance whiskerLeft (f : a ⟶ b) (g h : b ⟶ c)
    [BicategoricalCoherence g h] : BicategoricalCoherence (f ≫ g) (f ≫ h) :=
⟨whiskerLeftIso f ⊗𝟙⟩
```

-- snippet 30
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.left (L91-L94)
instance left (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence (𝟙 a ≫ f) g :=
⟨λ_ f ≪≫ ⊗𝟙⟩
```

-- snippet 31
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.right' (L106-L109)
instance right' (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence f (g ≫ 𝟙 b) :=
⟨⊗𝟙 ≪≫ (ρ_ g).symm⟩
```

-- snippet 32
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.BicategoryM (L90-L91)
abbrev BicategoryM := CoherenceM Context
```

-- snippet 33
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Basic.bicategory (L52-L54)
def bicategory (mvarId : MVarId) : MetaM (List MVarId) :=
BicategoryLike.main  Bicategory.Context `bicategory mvarId
```

-- snippet 34
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.srcExpr (L29-L34)
def srcExpr (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, f, _]) => return f
  | _ => throwError m!"{η} is not a morphism"

/-- The codomain of a morphism. -/
```

-- snippet 35
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.srcExprOfIso (L41-L46)
def srcExprOfIso (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Iso, #[_, _, f, _]) => return f
  | _ => throwError m!"{η} is not a morphism"

/-- The codomain of an isomorphism. -/
```

-- snippet 36
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.structuralIsoOfExpr_comp (L117-L122)
theorem structuralIsoOfExpr_comp {f g h : a ⟶ b}
    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)
    (θ : g ⟶ h) (θ' : g ≅ h) (ih_θ : θ'.hom = θ) :
    (η' ≪≫ θ').hom  = η ≫ θ := by
simp [ih_η, ih_θ]
```

-- snippet 37
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.refl (L67-L69)
instance refl (f : a ⟶ b) : BicategoricalCoherence f f :=
⟨Iso.refl _⟩
```

-- snippet 38
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.assoc (L111-L115)
instance assoc (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
    [BicategoricalCoherence (f ≫ g ≫ h) i] :
    BicategoricalCoherence ((f ≫ g) ≫ h) i :=
⟨α_ f g h ≪≫ ⊗𝟙⟩
```

-- snippet 39
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.assoc' (L117-L123)
instance assoc' (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
    [BicategoricalCoherence i (f ≫ g ≫ h)] :
    BicategoricalCoherence i ((f ≫ g) ≫ h) :=
⟨⊗𝟙 ≪≫ (α_ f g h).symm⟩

end BicategoricalCoherence
```

-- snippet 40
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.right (L101-L104)
instance right (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence (f ≫ 𝟙 b) g :=
⟨ρ_ f ≪≫ ⊗𝟙⟩
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Bicategory.Normalize).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem evalWhiskerRightAux_of {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) :
    η ▷ h = (Iso.refl _).hom ≫ η ▷ h ≫ (Iso.refl _).hom
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_nil_nil (L37-L40)
theorem evalComp_nil_nil (α : f ≅ g) (β : g ≅ h) :
    (α ≪≫ β).hom = (α ≪≫ β).hom := by
simp
```

-- snippet 2
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_nil_cons (L41-L44)
theorem evalComp_nil_cons (α : f ≅ g) (β : g ≅ h) (η : h ⟶ i) (ηs : i ⟶ j) :
    α.hom ≫ (β.hom ≫ η ≫ ηs) = (α ≪≫ β).hom ≫ η ≫ ηs := by
simp
```

-- snippet 3
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalComp_cons (L45-L49)
theorem evalComp_cons (α : f ≅ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : i ⟶ j} {ι : h ⟶ j}
    (e_ι : ηs ≫ θ = ι) :
    (α.hom ≫ η ≫ ηs) ≫ θ = α.hom ≫ η ≫ ι := by
simp [e_ι]
```

-- snippet 4
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_comp (L50-L55)
theorem eval_comp
    {η η' : f ⟶ g} {θ θ' : g ⟶ h} {ι : f ⟶ h}
    (e_η : η = η') (e_θ : θ = θ') (e_ηθ : η' ≫ θ' = ι) :
    η ≫ θ = ι := by
simp [e_η, e_θ, e_ηθ]
```

-- snippet 5
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_of (L56-L59)
theorem eval_of (η : f ⟶ g) :
    η = (Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom := by
simp
```

-- snippet 6
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_monoidalComp (L60-L65)
theorem eval_monoidalComp
    {η η' : f ⟶ g} {α : g ≅ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
    (e_η : η = η') (e_θ : θ = θ') (e_αθ : α.hom ≫ θ' = αθ) (e_ηαθ : η' ≫ αθ = ηαθ) :
    η ≫ α.hom ≫ θ = ηαθ := by
simp [e_η, e_θ, e_αθ, e_ηαθ]
```

-- snippet 7
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_nil (L67-L70)
theorem evalWhiskerLeft_nil (f : a ⟶ b) {g h : b ⟶ c} (α : g ≅ h) :
    (whiskerLeftIso f α).hom = (whiskerLeftIso f α).hom := by
simp
```

-- snippet 8
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_of_cons (L71-L76)
theorem evalWhiskerLeft_of_cons
    {f : a ⟶ b} {g h i j : b ⟶ c}
    (α : g ≅ h) (η : h ⟶ i) {ηs : i ⟶ j} {θ : f ≫ i ⟶ f ≫ j} (e_θ : f ◁ ηs = θ) :
    f ◁ (α.hom ≫ η ≫ ηs) = (whiskerLeftIso f α).hom ≫ f ◁ η ≫ θ := by
simp [e_θ]
```

-- snippet 9
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_comp (L77-L85)
theorem evalWhiskerLeft_comp
    {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d}
    {η : h ⟶ i} {η₁ : g ≫ h ⟶ g ≫ i} {η₂ : f ≫ g ≫ h ⟶ f ≫ g ≫ i}
    {η₃ : f ≫ g ≫ h ⟶ (f ≫ g) ≫ i} {η₄ : (f ≫ g) ≫ h ⟶ (f ≫ g) ≫ i}
    (e_η₁ : g ◁ η = η₁) (e_η₂ : f ◁ η₁ = η₂)
    (e_η₃ : η₂ ≫ (α_ _ _ _).inv = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄) :
    (f ≫ g) ◁ η = η₄ := by
simp [e_η₁, e_η₂, e_η₃, e_η₄]
```

-- snippet 10
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerLeft_id (L86-L91)
theorem evalWhiskerLeft_id {η : f ⟶ g}
    {η₁ : f ⟶ 𝟙 a ≫ g} {η₂ : 𝟙 a ≫ f ⟶ 𝟙 a ≫ g}
    (e_η₁ : η ≫ (λ_ _).inv = η₁) (e_η₂ : (λ_ _).hom ≫ η₁ = η₂) :
    𝟙 a ◁ η = η₂ := by
simp [e_η₁, e_η₂]
```

-- snippet 11
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_whiskerLeft (L92-L98)
theorem eval_whiskerLeft
    {f : a ⟶ b} {g h : b ⟶ c}
    {η η' : g ⟶ h} {θ : f ≫ g ⟶ f ≫ h}
    (e_η : η = η') (e_θ : f ◁ η' = θ) :
    f ◁ η = θ := by
simp [e_η, e_θ]
```

-- snippet 12
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_whiskerRight (L99-L105)
theorem eval_whiskerRight
    {f g : a ⟶ b} {h : b ⟶ c}
    {η η' : f ⟶ g} {θ : f ≫ h ⟶ g ≫ h}
    (e_η : η = η') (e_θ : η' ▷ h = θ) :
    η ▷ h = θ := by
simp [e_η, e_θ]
```

-- snippet 13
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_nil (L107-L110)
theorem evalWhiskerRight_nil (α : f ≅ g) (h : b ⟶ c) :
    α.hom ▷ h = α.hom ▷ h := by
simp
```

-- snippet 14
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_cons_of_of (L115-L123)
theorem evalWhiskerRight_cons_of_of
    {f g h i : a ⟶ b} {j : b ⟶ c}
    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {ηs₁ : h ≫ j ⟶ i ≫ j}
    {η₁ : g ≫ j ⟶ h ≫ j} {η₂ : g ≫ j ⟶ i ≫ j} {η₃ : f ≫ j ⟶ i ≫ j}
    (e_ηs₁ : ηs ▷ j = ηs₁) (e_η₁ : η ▷ j = η₁)
    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃) :
    (α.hom ≫ η ≫ ηs) ▷ j = η₃ := by
simp_all
```

-- snippet 15
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.evalWhiskerRight_cons_whisker (L124-L137)
theorem evalWhiskerRight_cons_whisker
    {f : a ⟶ b} {g : a ⟶ c} {h i : b ⟶ c} {j : a ⟶ c} {k : c ⟶ d}
    {α : g ≅ f ≫ h} {η : h ⟶ i} {ηs : f ≫ i ⟶ j}
    {η₁ : h ≫ k ⟶ i ≫ k} {η₂ : f ≫ (h ≫ k) ⟶ f ≫ (i ≫ k)} {ηs₁ : (f ≫ i) ≫ k ⟶ j ≫ k}
    {ηs₂ : f ≫ (i ≫ k) ⟶ j ≫ k} {η₃ : f ≫ (h ≫ k) ⟶ j ≫ k} {η₄ : (f ≫ h) ≫ k ⟶ j ≫ k}
    {η₅ : g ≫ k ⟶ j ≫ k}
    (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ k = η₁) (e_η₂ : f ◁ η₁ = η₂)
    (e_ηs₁ : ηs ▷ k = ηs₁) (e_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
    (e_η₃ : η₂ ≫ ηs₂ = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄)
    (e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅) :
    (α.hom ≫ (f ◁ η) ≫ ηs) ▷ k = η₅ := by
simp at e_η₁ e_η₅
  simp [e_η₁, e_η₂, e_ηs₁, e_ηs₂, e_η₃, e_η₄, e_η₅]
```

-- snippet 16
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Basic.bicategoryNf (L35-L37)
def bicategoryNf (mvarId : MVarId) : MetaM (List MVarId) := do
BicategoryLike.normalForm Bicategory.Context `bicategory mvarId
```

-- snippet 17
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.left' (L96-L99)
instance left' (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence f (𝟙 a ≫ g) :=
⟨⊗𝟙 ≪≫ (λ_ g).symm⟩
```

-- snippet 18
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.mkContext (L68-L85)
def mkContext? (e : Expr) : MetaM (Option Context) := do
let e ← instantiateMVars e
  let type ← instantiateMVars <| ← inferType e
  match (← whnfR (← inferType e)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, f, _]) =>
    let fType ← instantiateMVars <| ← inferType f
    match (← whnfR fType).getAppFnArgs with
    | (``Quiver.Hom, #[_, _, a, _]) =>
      let B ← inferType a
      let .succ level₀ ← getLevel B | return none
      let .succ level₁ ← getLevel fType | return none
      let .succ level₂ ← getLevel type | return none
      let some instBicategory ← synthInstance?
        (mkAppN (.const ``Bicategory [level₂, level₁, level₀]) #[B]) | return none
      return some ⟨level₂, level₁, level₀, B, instBicategory⟩
    | _ => return none
  | _ => return none
```

-- snippet 19
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalComp_refl (L125-L128)
theorem bicategoricalComp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : η ⊗≫ θ = η ≫ θ := by
dsimp [bicategoricalComp]; simp

end CategoryTheory
```

-- snippet 20
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalIso (L41-L45)
abbrev bicategoricalIso (f g : a ⟶ b) [BicategoricalCoherence f g] : f ≅ g :=
⊗𝟙

/-- Compose two morphisms in a bicategorical category,
inserting unitors and associators between as necessary. -/
```

-- snippet 21
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.structuralIso_inv (L113-L116)
theorem structuralIso_inv {f g : a ⟶ b} (η : f ≅ g) :
    η.symm.hom = η.inv := by
simp only [Iso.symm_hom]
```

-- snippet 22
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalIsoComp (L56-L59)
def bicategoricalIsoComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
    (η : f ≅ g) (θ : h ≅ i) : f ≅ i :=
η ≪≫ ⊗𝟙 ≪≫ θ
```

-- snippet 23
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.tensorRight' (L86-L89)
instance tensorRight' (f : a ⟶ b) (g : b ⟶ b)
    [BicategoricalCoherence g (𝟙 b)] : BicategoricalCoherence (f ≫ g) f :=
⟨whiskerLeftIso f ⊗𝟙 ≪≫ (ρ_ f)⟩
```

-- snippet 24
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.whiskerRight (L76-L79)
instance whiskerRight (f g : a ⟶ b) (h : b ⟶ c)
    [BicategoricalCoherence f g] : BicategoricalCoherence (f ≫ h) (g ≫ h) :=
⟨whiskerRightIso ⊗𝟙 h⟩
```

-- snippet 25
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.tgtExprOfIso (L47-L67)
def tgtExprOfIso (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Iso, #[_, _, _, g]) => return g
  | _ => throwError m!"{η} is not a morphism"

initialize registerTraceClass `bicategory

/-- The context for evaluating expressions. -/
structure Context where
  /-- The level for 2-morphisms. -/
  level₂ : Level
  /-- The level for 1-morphisms. -/
  level₁ : Level
  /-- The level for objects. -/
  level₀ : Level
  /-- The expression for the underlying category. -/
  B : Q(Type level₀)
  /-- The bicategory instance. -/
  instBicategory : Q(Bicategory.{level₂, level₁} $B)

/-- Populate a `context` object for evaluating `e`. -/
```

-- snippet 26
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.tensorRight (L81-L84)
instance tensorRight (f : a ⟶ b) (g : b ⟶ b)
    [BicategoricalCoherence (𝟙 b) g] : BicategoricalCoherence f (f ≫ g) :=
⟨(ρ_ f).symm ≪≫ (whiskerLeftIso f ⊗𝟙)⟩
```

-- snippet 27
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.tgtExpr (L35-L40)
def tgtExpr (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, _, g]) => return g
  | _ => throwError m!"{η} is not a morphism"

/-- The domain of an isomorphism. -/
```

-- snippet 28
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.bicategoricalComp (L46-L50)
def bicategoricalComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
    (η : f ⟶ g) (θ : h ⟶ i) : f ⟶ i :=
η ≫ ⊗𝟙.hom ≫ θ

-- type as \ot \gg
```

-- snippet 29
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.whiskerLeft (L71-L74)
instance whiskerLeft (f : a ⟶ b) (g h : b ⟶ c)
    [BicategoricalCoherence g h] : BicategoricalCoherence (f ≫ g) (f ≫ h) :=
⟨whiskerLeftIso f ⊗𝟙⟩
```

-- snippet 30
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.left (L91-L94)
instance left (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence (𝟙 a ≫ f) g :=
⟨λ_ f ≪≫ ⊗𝟙⟩
```

-- snippet 31
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.right' (L106-L109)
instance right' (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence f (g ≫ 𝟙 b) :=
⟨⊗𝟙 ≪≫ (ρ_ g).symm⟩
```

-- snippet 32
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.BicategoryM (L90-L91)
abbrev BicategoryM := CoherenceM Context
```

-- snippet 33
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Basic.bicategory (L52-L54)
def bicategory (mvarId : MVarId) : MetaM (List MVarId) :=
BicategoryLike.main  Bicategory.Context `bicategory mvarId
```

-- snippet 34
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.srcExpr (L29-L34)
def srcExpr (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Quiver.Hom, #[_, _, f, _]) => return f
  | _ => throwError m!"{η} is not a morphism"

/-- The codomain of a morphism. -/
```

-- snippet 35
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.srcExprOfIso (L41-L46)
def srcExprOfIso (η : Expr) : MetaM Expr := do
match (← whnfR (← inferType η)).getAppFnArgs with
  | (``Iso, #[_, _, f, _]) => return f
  | _ => throwError m!"{η} is not a morphism"

/-- The codomain of an isomorphism. -/
```

-- snippet 36
```lean
-- Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes.structuralIsoOfExpr_comp (L117-L122)
theorem structuralIsoOfExpr_comp {f g h : a ⟶ b}
    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)
    (θ : g ⟶ h) (θ' : g ≅ h) (ih_θ : θ'.hom = θ) :
    (η' ≪≫ θ').hom  = η ≫ θ := by
simp [ih_η, ih_θ]
```

-- snippet 37
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.refl (L67-L69)
instance refl (f : a ⟶ b) : BicategoricalCoherence f f :=
⟨Iso.refl _⟩
```

-- snippet 38
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.assoc (L111-L115)
instance assoc (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
    [BicategoricalCoherence (f ≫ g ≫ h) i] :
    BicategoricalCoherence ((f ≫ g) ≫ h) i :=
⟨α_ f g h ≪≫ ⊗𝟙⟩
```

-- snippet 39
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.assoc' (L117-L123)
instance assoc' (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
    [BicategoricalCoherence i (f ≫ g ≫ h)] :
    BicategoricalCoherence i ((f ≫ g) ≫ h) :=
⟨⊗𝟙 ≪≫ (α_ f g h).symm⟩

end BicategoricalCoherence
```

-- snippet 40
```lean
-- Mathlib.Tactic.CategoryTheory.BicategoricalComp.right (L101-L104)
instance right (f g : a ⟶ b) [BicategoricalCoherence f g] :
    BicategoricalCoherence (f ≫ 𝟙 b) g :=
⟨ρ_ f ≪≫ ⊗𝟙⟩
```
[/INST]


Instruction: Edit `punitProd` so it still compiles and the proof is simplified where possible. Return the full updated `punitProd` block.

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Equiv.Prod).

[User]
You are editing a file in `Mathlib/Logic/Equiv/Prod.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.Equiv.Prod.punitProd`, L134-L139)
```lean
def punitProd (α) : PUnit × α ≃ α := calc
    PUnit × α ≃ α × PUnit := prodComm _ _
    _ ≃ α := prodPUnit _

/-- `PUnit` is a right identity for dependent type product up to an equivalence. -/
```

## Edit instruction
Edit `punitProd` so it still compiles and the proof is simplified where possible. Return the full updated `punitProd` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Equiv.Prod).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Logic/Equiv/Prod.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.Equiv.Prod.punitProd`, L134-L139)
```lean
def punitProd (α) : PUnit × α ≃ α := calc
    PUnit × α ≃ α × PUnit := prodComm _ _
    _ ≃ α := prodPUnit _

/-- `PUnit` is a right identity for dependent type product up to an equivalence. -/
```

## Edit instruction
Edit `punitProd` so it still compiles and the proof is simplified where possible. Return the full updated `punitProd` block.
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.SimplicialObject.Op).

[User]
The following Lean code fails to compile. Fix it.

```lean
lemma opFunctorCompOpFunctorIso_hom_app_app (X : SimplicialObject C) (n : SimplexCategoryᵒᵖ) :
    (opFunctorCompOpFunctorIso.hom.app X).app n = opObjIso.hom ≫ opObjIso.hom
  frobnicateTacticThatDoesNotExist := by
simp [opFunctorCompOpFunctorIso, opObjIso, opFunctor]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='expected token', pos=Position(line=2, column=63), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `opObjIso.hom`', pos=Position(line=2, column=50), pos_end=Position(line=2, column=62), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  SimplicialObject\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  C\n\nHint: The identifier `SimplicialObject` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=49), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `opFunctorCompOpFunctorIso.hom.app`', pos=Position(line=2, column=5), pos_end=Position(line=2, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=162, i_end=273, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.SimplicialObject.Op).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma opFunctorCompOpFunctorIso_hom_app_app (X : SimplicialObject C) (n : SimplexCategoryᵒᵖ) :
    (opFunctorCompOpFunctorIso.hom.app X).app n = opObjIso.hom ≫ opObjIso.hom
  frobnicateTacticThatDoesNotExist := by
simp [opFunctorCompOpFunctorIso, opObjIso, opFunctor]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='expected token', pos=Position(line=2, column=63), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `opObjIso.hom`', pos=Position(line=2, column=50), pos_end=Position(line=2, column=62), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  SimplicialObject\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  C\n\nHint: The identifier `SimplicialObject` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=49), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `opFunctorCompOpFunctorIso.hom.app`', pos=Position(line=2, column=5), pos_end=Position(line=2, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=162, i_end=273, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Function.Basic).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
    g = update f a b ↔ g a = b ∧ ∀ x ≠ a, g x = f x
```

Declaration: `Mathlib.Logic.Function.Basic.eq_update_iff` (module Mathlib.Logic.Function.Basic).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Function.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
    g = update f a b ↔ g a = b ∧ ∀ x ≠ a, g x = f x
```

Declaration: `Mathlib.Logic.Function.Basic.eq_update_iff` (module Mathlib.Logic.Function.Basic).[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Relation).

[User]
Complete the following proof.

```lean
theorem Reflexive.ne_imp_iff (h : Reflexive r) {x y : α} : x ≠ y → r x y ↔ r x y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Relation).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Reflexive.ne_imp_iff (h : Reflexive r) {x y : α} : x ≠ y → r x y ↔ r x y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.NormNum.Ineq).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Tactic.NormNum.Ineq.isRat_le_true`:
State 0: ⊢ theorem isRat_le_true [Ring α] [LinearOrder α] [IsStrictOrderedRing α] :
    {a b : α} → {na nb : ℤ} → {da db : ℕ} →
    IsRat a na da → IsRat b nb db →
    decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
  | _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
    have h
Tactic: have ha : 0 ≤ ⅟(da : α) := invOf_nonneg.mpr <| Nat.cast_nonneg da
State 1: ⊢ <goal after have ha : 0 ≤ ⅟(da : α) := invOf_nonneg.mpr <| Nat.cast_nonneg da>
Tactic: have hb : 0 ≤ ⅟(db : α) := invOf_nonneg.mpr <| Nat.cast_nonneg db
State 2: ⊢ <goal after have hb : 0 ≤ ⅟(db : α) := invOf_nonneg.mpr <| Nat.cast_nonneg db>
Tactic: have h := (mul_le_mul_of_nonneg_left · hb) <| mul_le_mul_of_nonneg_right h ha
State 3: ⊢ <goal after have h := (mul_le_mul_of_nonneg_left · hb) <| mul_le_mul_of_nonneg_right h ha>
Tactic: rw [← mul_assoc, Int.commute_cast] at h
State 4: ⊢ <goal after rw [← mul_assoc, Int.commute_cast] at h>
Tactic: simp only [Int.ofNat_eq_natCast, Int.mul_def, Int.cast_mul, Int.cast_natCast,  mul_invOf_cancel_right'] at h  rwa [Int.commute_cast] at h
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.NormNum.Ineq).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Tactic.NormNum.Ineq.isRat_le_true`:
State 0: ⊢ theorem isRat_le_true [Ring α] [LinearOrder α] [IsStrictOrderedRing α] :
    {a b : α} → {na nb : ℤ} → {da db : ℕ} →
    IsRat a na da → IsRat b nb db →
    decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
  | _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
    have h
Tactic: have ha : 0 ≤ ⅟(da : α) := invOf_nonneg.mpr <| Nat.cast_nonneg da
State 1: ⊢ <goal after have ha : 0 ≤ ⅟(da : α) := invOf_nonneg.mpr <| Nat.cast_nonneg da>
Tactic: have hb : 0 ≤ ⅟(db : α) := invOf_nonneg.mpr <| Nat.cast_nonneg db
State 2: ⊢ <goal after have hb : 0 ≤ ⅟(db : α) := invOf_nonneg.mpr <| Nat.cast_nonneg db>
Tactic: have h := (mul_le_mul_of_nonneg_left · hb) <| mul_le_mul_of_nonneg_right h ha
State 3: ⊢ <goal after have h := (mul_le_mul_of_nonneg_left · hb) <| mul_le_mul_of_nonneg_right h ha>
Tactic: rw [← mul_assoc, Int.commute_cast] at h
State 4: ⊢ <goal after rw [← mul_assoc, Int.commute_cast] at h>
Tactic: simp only [Int.ofNat_eq_natCast, Int.mul_def, Int.cast_mul, Int.cast_natCast,  mul_invOf_cancel_right'] at h  rwa [Int.commute_cast] at h
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.Basic).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def List.mapAccumLM (f : β' → α → m' (β' × γ')) : β' → List α → m' (β' × List γ')
  | a, [] => pure (a, [])
  | a, x :: xs => do
    let (a', y) ← f a x
    let (a'', ys) ← List.mapAccumLM f a' xs
    pure (a'', y :: ys)

end Monad
```

Declaration: `Mathlib.Control.Basic.List` (module Mathlib.Control.Basic).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def List.mapAccumLM (f : β' → α → m' (β' × γ')) : β' → List α → m' (β' × List γ')
  | a, [] => pure (a, [])
  | a, x :: xs => do
    let (a', y) ← f a x
    let (a'', ys) ← List.mapAccumLM f a' xs
    pure (a'', y :: ys)

end Monad
```

Declaration: `Mathlib.Control.Basic.List` (module Mathlib.Control.Basic).[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.LawfulFix).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Control.LawfulFix.exists_fix_le_approx`:
State 0: ⊢ theorem exists_fix_le_approx (x : α) : ∃ i, Part.fix f x ≤ approx f i x
Tactic: intro b' h'
State 1: ⊢ <goal after intro b' h'>
Tactic: have hb' := approx_le_fix f i _ _ hb
State 2: ⊢ <goal after have hb' := approx_le_fix f i _ _ hb>
Tactic: obtain rfl := Part.mem_unique h' hb'
State 3: ⊢ <goal after obtain rfl := Part.mem_unique h' hb'>
Tactic: exact hb  · exists 0
State 4: ⊢ <goal after exact hb>
Tactic: intro b' h'
State 5: ⊢ <goal after intro b' h'>
Tactic: simp only [mem_iff f] at h'
State 6: ⊢ <goal after simp only [mem_iff f] at h'>
Tactic: obtain ⟨i, h'⟩ := h'
State 7: ⊢ <goal after obtain ⟨i, h'⟩ := h'>
Tactic: cases hh _ _ h'  /-- The series of approximations of `fix f` (see `approx`) as a `Chain` -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.LawfulFix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Control.LawfulFix.exists_fix_le_approx`:
State 0: ⊢ theorem exists_fix_le_approx (x : α) : ∃ i, Part.fix f x ≤ approx f i x
Tactic: intro b' h'
State 1: ⊢ <goal after intro b' h'>
Tactic: have hb' := approx_le_fix f i _ _ hb
State 2: ⊢ <goal after have hb' := approx_le_fix f i _ _ hb>
Tactic: obtain rfl := Part.mem_unique h' hb'
State 3: ⊢ <goal after obtain rfl := Part.mem_unique h' hb'>
Tactic: exact hb  · exists 0
State 4: ⊢ <goal after exact hb>
Tactic: intro b' h'
State 5: ⊢ <goal after intro b' h'>
Tactic: simp only [mem_iff f] at h'
State 6: ⊢ <goal after simp only [mem_iff f] at h'>
Tactic: obtain ⟨i, h'⟩ := h'
State 7: ⊢ <goal after obtain ⟨i, h'⟩ := h'>
Tactic: cases hh _ _ h'  /-- The series of approximations of `fix f` (see `approx`) as a `Chain` -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.TopCatAdjunction).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Condensed.TopCatAdjunction.toTopCatMap`:
State 0: ⊢ def toTopCatMap : X.toTopCat ⟶ Y.toTopCat
Tactic: rw [continuous_coinduced_dom]
State 1: ⊢ <goal after rw [continuous_coinduced_dom]>
Tactic: apply continuous_sigma
State 2: ⊢ <goal after apply continuous_sigma>
Tactic: intro ⟨S, x⟩
State 3: ⊢ <goal after intro ⟨S, x⟩>
Tactic: simp only [Function.comp_apply, coinducingCoprod]
State 4: ⊢ <goal after simp only [Function.comp_apply, coinducingCoprod]>
Tactic: rw [show (fun (a : S) ↦  f.hom.app ⟨of PUnit⟩ (X.obj.map ((of PUnit.{u + 1}).const a).op x)) = _  from funext fun a ↦ NatTrans.naturality_apply f.hom ((of PUnit.{u + 1}).const a).op x]
State 5: ⊢ <goal after rw [show (fun (a : S) ↦>
Tactic: exact continuous_coinducingCoprod Y _ }  end CondensedSet  /-- The functor `CondensedSet ⥤ TopCat` -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.TopCatAdjunction).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Condensed.TopCatAdjunction.toTopCatMap`:
State 0: ⊢ def toTopCatMap : X.toTopCat ⟶ Y.toTopCat
Tactic: rw [continuous_coinduced_dom]
State 1: ⊢ <goal after rw [continuous_coinduced_dom]>
Tactic: apply continuous_sigma
State 2: ⊢ <goal after apply continuous_sigma>
Tactic: intro ⟨S, x⟩
State 3: ⊢ <goal after intro ⟨S, x⟩>
Tactic: simp only [Function.comp_apply, coinducingCoprod]
State 4: ⊢ <goal after simp only [Function.comp_apply, coinducingCoprod]>
Tactic: rw [show (fun (a : S) ↦  f.hom.app ⟨of PUnit⟩ (X.obj.map ((of PUnit.{u + 1}).const a).op x)) = _  from funext fun a ↦ NatTrans.naturality_apply f.hom ((of PUnit.{u + 1}).const a).op x]
State 5: ⊢ <goal after rw [show (fun (a : S) ↦>
Tactic: exact continuous_coinducingCoprod Y _ }  end CondensedSet  /-- The functor `CondensedSet ⥤ TopCat` -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


Instruction: Edit `NormalizedHom` so it still compiles and the proof is simplified where possible. Return the full updated `NormalizedHom` block.

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Coherence.Datatypes).

[User]
You are editing a file in `Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Tactic.CategoryTheory.Coherence.Datatypes.NormalizedHom`, L436-L449)
```lean
def NormalizedHom.consM [MonadMor₁ m] (p : NormalizedHom) (f : Atom₁) :
    m NormalizedHom := do
return NormalizedHom.cons (← comp₁M p.e (.of f)) p f

/-- `Context ρ` provides the context for manipulating 2-morphisms in a monoidal category or
bicategory. In particular, we will store `MonoidalCategory` or `Bicategory` instance in a context,
and use this through a reader monad when we construct the lean expressions for 2-morphisms. -/
class Context (ρ : Type) where
  /-- Construct a context from a lean expression for a 2-morphism. -/
  mkContext? : Expr → MetaM (Option ρ)

export Context (mkContext?)

/-- Construct a context from a lean expression for a 2-morphism. -/
```

## Edit instruction
Edit `NormalizedHom` so it still compiles and the proof is simplified where possible. Return the full updated `NormalizedHom` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Coherence.Datatypes).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Tactic.CategoryTheory.Coherence.Datatypes.NormalizedHom`, L436-L449)
```lean
def NormalizedHom.consM [MonadMor₁ m] (p : NormalizedHom) (f : Atom₁) :
    m NormalizedHom := do
return NormalizedHom.cons (← comp₁M p.e (.of f)) p f

/-- `Context ρ` provides the context for manipulating 2-morphisms in a monoidal category or
bicategory. In particular, we will store `MonoidalCategory` or `Bicategory` instance in a context,
and use this through a reader monad when we construct the lean expressions for 2-morphisms. -/
class Context (ρ : Type) where
  /-- Construct a context from a lean expression for a 2-morphism. -/
  mkContext? : Expr → MetaM (Option ρ)

export Context (mkContext?)

/-- Construct a context from a lean expression for a 2-morphism. -/
```

## Edit instruction
Edit `NormalizedHom` so it still compiles and the proof is simplified where possible. Return the full updated `NormalizedHom` block.
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Coherence.Normalize).

[User]
The following Lean code fails to compile. Fix it.

```lean
def associatorInvM (f g h : Mor₁) : m NormalExpr
:= do _unbalancedParenXYZ
MonadNormalExpr.nilM <| ← MonadMor₂Iso.symmM <|
    .structuralAtom <| ← MonadMor₂Iso.associatorM f g h

/-- The left unitor as a term of `normalExpr`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=77, messages=[Message(data="Function expected at\n  m\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  NormalExpr\n\nHint: The identifier `m` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=36), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=77, i_end=239, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=0), pos_end=Position(line=3, column=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=49), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Coherence.Normalize).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def associatorInvM (f g h : Mor₁) : m NormalExpr
:= do _unbalancedParenXYZ
MonadNormalExpr.nilM <| ← MonadMor₂Iso.symmM <|
    .structuralAtom <| ← MonadMor₂Iso.associatorM f g h

/-- The left unitor as a term of `normalExpr`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=77, messages=[Message(data="Function expected at\n  m\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  NormalExpr\n\nHint: The identifier `m` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=36), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=77, i_end=239, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=0), pos_end=Position(line=3, column=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=49), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.Widget.CongrM).

[User]
Complete the following proof.

```lean
def makeCongrMString (pos : Array Lean.SubExpr.GoalsLocation) (goalType : Expr)
    (_ : SelectInsertParams) :
    MetaM (String × String × Option (String.Pos.Raw × String.Pos.Raw))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.Widget.CongrM).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def makeCongrMString (pos : Array Lean.SubExpr.GoalsLocation) (goalType : Expr)
    (_ : SelectInsertParams) :
    MetaM (String × String × Option (String.Pos.Raw × String.Pos.Raw))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]