[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.CancelDenoms.Core).

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

```lean
theorem cancel_factors_ne {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
    (hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
    (a ≠ b) = (1 / gcd * (bd * a') ≠ 1 / gcd * (ad * b'))
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CancelDenoms.Core.mul_subst (L44-L48)
theorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}
    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2
```

-- snippet 2
```lean
-- Mathlib.Tactic.CancelDenoms.Core.div_subst (L49-L52)
theorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}
    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1
```

-- snippet 3
```lean
-- Mathlib.Tactic.CancelDenoms.Core.cancel_factors_eq_div (L53-L56)
theorem cancel_factors_eq_div {α} [Field α] {n e e' : α}
    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n
```

-- snippet 4
```lean
-- Mathlib.Tactic.CancelDenoms.Core.add_subst (L57-L59)
theorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
    n * (e1 + e2) = t1 + t2
```

-- snippet 5
```lean
-- Mathlib.Tactic.CancelDenoms.Core.sub_subst (L60-L62)
theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
    n * (e1 - e2) = t1 - t2
```

-- snippet 6
```lean
-- Mathlib.Tactic.CancelDenoms.Core.neg_subst (L63-L64)
theorem neg_subst {α} [Ring α] {n e t : α} (h1 : n * e = t) : n * -e = -t
```

-- snippet 7
```lean
-- Mathlib.Tactic.CancelDenoms.Core.pow_subst (L65-L68)
theorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}
    (h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2
```

-- snippet 8
```lean
-- Mathlib.Tactic.CancelDenoms.Core.inv_subst (L69-L71)
theorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) :
    k * (e ⁻¹) = 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.Tactic.CancelDenoms.Core).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem cancel_factors_ne {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
    (hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
    (a ≠ b) = (1 / gcd * (bd * a') ≠ 1 / gcd * (ad * b'))
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CancelDenoms.Core.mul_subst (L44-L48)
theorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}
    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2
```

-- snippet 2
```lean
-- Mathlib.Tactic.CancelDenoms.Core.div_subst (L49-L52)
theorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}
    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1
```

-- snippet 3
```lean
-- Mathlib.Tactic.CancelDenoms.Core.cancel_factors_eq_div (L53-L56)
theorem cancel_factors_eq_div {α} [Field α] {n e e' : α}
    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n
```

-- snippet 4
```lean
-- Mathlib.Tactic.CancelDenoms.Core.add_subst (L57-L59)
theorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
    n * (e1 + e2) = t1 + t2
```

-- snippet 5
```lean
-- Mathlib.Tactic.CancelDenoms.Core.sub_subst (L60-L62)
theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
    n * (e1 - e2) = t1 - t2
```

-- snippet 6
```lean
-- Mathlib.Tactic.CancelDenoms.Core.neg_subst (L63-L64)
theorem neg_subst {α} [Ring α] {n e t : α} (h1 : n * e = t) : n * -e = -t
```

-- snippet 7
```lean
-- Mathlib.Tactic.CancelDenoms.Core.pow_subst (L65-L68)
theorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}
    (h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2
```

-- snippet 8
```lean
-- Mathlib.Tactic.CancelDenoms.Core.inv_subst (L69-L71)
theorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) :
    k * (e ⁻¹) = 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.Tactic.Explode.Pretty).

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

```lean
def entriesToMessageData (entries : Entries) : MetaM MessageData
:= do .someMissingNamespaceXYZ
-- ['1', '2', '3']
  let paddedLines ← padRight <| entries.l.map fun entry => m!"{entry.line!}"
  -- ['   ', '1,2', '1  ']
  let paddedDeps ← padRight <| entries.l.map fun entry =>
    String.intercalate "," <| entry.deps.map (fun dep => (dep.map toString).getD "_")
  -- ['p  ', 'hP ', '∀I ']
  let paddedThms ← padRight <| entries.l.map (·.thm)

  rowToMessageData paddedLines paddedDeps paddedThms entries.l

end Explode

end Mathlib
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=117, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  MessageData\n\nHint: The identifier `MetaM` 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=47), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=117, i_end=530, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=2), pos_end=Position(line=4, column=5), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=13, column=0), pos_end=Position(line=13, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=530, i_end=541, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=15, column=0), pos_end=Position(line=15, column=11), 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.Explode.Pretty).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def entriesToMessageData (entries : Entries) : MetaM MessageData
:= do .someMissingNamespaceXYZ
-- ['1', '2', '3']
  let paddedLines ← padRight <| entries.l.map fun entry => m!"{entry.line!}"
  -- ['   ', '1,2', '1  ']
  let paddedDeps ← padRight <| entries.l.map fun entry =>
    String.intercalate "," <| entry.deps.map (fun dep => (dep.map toString).getD "_")
  -- ['p  ', 'hP ', '∀I ']
  let paddedThms ← padRight <| entries.l.map (·.thm)

  rowToMessageData paddedLines paddedDeps paddedThms entries.l

end Explode

end Mathlib
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=117, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  MessageData\n\nHint: The identifier `MetaM` 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=47), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=117, i_end=530, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=2), pos_end=Position(line=4, column=5), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=13, column=0), pos_end=Position(line=13, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=530, i_end=541, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=15, column=0), pos_end=Position(line=15, column=11), 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.Logic.ExistsUnique).

[User]
Complete the following proof.

```lean
theorem ExistsUnique.choose_eq_iff {p : α → Prop} {a : α} (h : ∃! x, p x) :
    h.choose = a ↔ p a
```

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.ExistsUnique).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem ExistsUnique.choose_eq_iff {p : α → Prop} {a : α} (h : ∃! x, p x) :
    h.choose = a ↔ p a
```

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.Control.EquivFunctor).

[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.EquivFunctor.mapEquiv`:
State 0: ⊢ def mapEquiv : f α ≃ f β where
  toFun
Tactic: convert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm
State 1: ⊢ <goal after convert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm>
Tactic: simp  right_inv y := by
State 2: ⊢ <goal after simp>
Tactic: convert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm
State 3: ⊢ <goal after convert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm>
Tactic: simp
```

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.EquivFunctor).[/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.EquivFunctor.mapEquiv`:
State 0: ⊢ def mapEquiv : f α ≃ f β where
  toFun
Tactic: convert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm
State 1: ⊢ <goal after convert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm>
Tactic: simp  right_inv y := by
State 2: ⊢ <goal after simp>
Tactic: convert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm
State 3: ⊢ <goal after convert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm>
Tactic: simp
```

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.AlgebraicTopology.ModelCategory.Homotopy).

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

```lean
lemma whitehead [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]
    (f : X ⟶ Y) [WeakEquivalence f] :
    ∃ (g : Y ⟶ X), RightHomotopyRel (f ≫ g) (𝟙 X) ∧ RightHomotopyRel (g ≫ f) (𝟙 Y)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.rightHomotopyRel (L63-L91)
lemma rightHomotopyRel (h : LeftHomotopyRel f g) : RightHomotopyRel f g
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.leftHomotopyRel (L92-L97)
lemma leftHomotopyRel (h : RightHomotopyRel f g) : LeftHomotopyRel f g
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.leftHomotopyRel_iff_rightHomotopyRel (L102-L121)
lemma leftHomotopyRel_iff_rightHomotopyRel :
    LeftHomotopyRel f g ↔ RightHomotopyRel f g
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.postcomp_bijective_of_fibration_of_weakEquivalence (L129-L152)
lemma postcomp_bijective_of_fibration_of_weakEquivalence
    [IsCofibrant X] (g : Y ⟶ Z) [Fibration g] [WeakEquivalence g] :
    Function.Bijective (fun (f : LeftHomotopyClass X Y) ↦ f.postcomp g)
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.postcomp_bijective_of_weakEquivalence (L153-L175)
lemma postcomp_bijective_of_weakEquivalence
    [IsCofibrant X] (g : Y ⟶ Z) [IsFibrant Y] [IsFibrant Z] [WeakEquivalence g] :
    Function.Bijective (fun (f : LeftHomotopyClass X Y) ↦ f.postcomp g)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.precomp_bijective_of_cofibration_of_weakEquivalence (L176-L199)
lemma precomp_bijective_of_cofibration_of_weakEquivalence
    [IsFibrant Z] (f : X ⟶ Y) [Cofibration f] [WeakEquivalence f] :
    Function.Bijective (fun (g : RightHomotopyClass Y Z) ↦ g.precomp f)
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.precomp_bijective_of_weakEquivalence (L200-L215)
lemma precomp_bijective_of_weakEquivalence
    [IsFibrant Z] (f : X ⟶ Y) [IsCofibrant X] [IsCofibrant Y] [WeakEquivalence f] :
    Function.Bijective (fun (g : RightHomotopyClass Y Z) ↦ g.precomp f)
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.LeftHomotopyClass (L231-L236)
lemma LeftHomotopyClass.whitehead [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]
    (f : X ⟶ Y) [WeakEquivalence f] :
    ∃ (g : Y ⟶ X), LeftHomotopyRel (f ≫ g) (𝟙 X) ∧ LeftHomotopyRel (g ≫ f) (𝟙 Y)
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.CofibrantObject (L43-L47)
abbrev CofibrantObject : Type u
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.homMk (L55-L57)
abbrev homMk {X Y : C} [IsCofibrant X] [IsCofibrant Y] (f : X ⟶ Y) :
    mk X ⟶ mk Y
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Basic.mk' (L126-L163)
def mk' [CategoryWithFibrations C] [CategoryWithCofibrations C]
    [CategoryWithWeakEquivalences C] [HasFiniteLimits C] [HasFiniteColimits C]
    [(weakEquivalences C).HasTwoOutOfThreeProperty]
    [IsWeakFactorizationSystem (cofibrations C) (trivialFibrations C)]
    [IsWeakFactorizationSystem (trivialCofibrations C) (fibrations C)] :
    ModelCategory C where
  cm3a
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.mk (L48-L50)
abbrev mk (X : C) [IsCofibrant X] : CofibrantObject C
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.cofibrantObjects (L39-L42)
def cofibrantObjects : ObjectProperty C
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.mk_surjective (L51-L54)
lemma mk_surjective (X : CofibrantObject C) :
    ∃ (Y : C) (_ : IsCofibrant Y), X = mk Y
```

<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.ModelCategory.Homotopy).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
lemma whitehead [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]
    (f : X ⟶ Y) [WeakEquivalence f] :
    ∃ (g : Y ⟶ X), RightHomotopyRel (f ≫ g) (𝟙 X) ∧ RightHomotopyRel (g ≫ f) (𝟙 Y)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.rightHomotopyRel (L63-L91)
lemma rightHomotopyRel (h : LeftHomotopyRel f g) : RightHomotopyRel f g
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.leftHomotopyRel (L92-L97)
lemma leftHomotopyRel (h : RightHomotopyRel f g) : LeftHomotopyRel f g
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.leftHomotopyRel_iff_rightHomotopyRel (L102-L121)
lemma leftHomotopyRel_iff_rightHomotopyRel :
    LeftHomotopyRel f g ↔ RightHomotopyRel f g
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.postcomp_bijective_of_fibration_of_weakEquivalence (L129-L152)
lemma postcomp_bijective_of_fibration_of_weakEquivalence
    [IsCofibrant X] (g : Y ⟶ Z) [Fibration g] [WeakEquivalence g] :
    Function.Bijective (fun (f : LeftHomotopyClass X Y) ↦ f.postcomp g)
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.postcomp_bijective_of_weakEquivalence (L153-L175)
lemma postcomp_bijective_of_weakEquivalence
    [IsCofibrant X] (g : Y ⟶ Z) [IsFibrant Y] [IsFibrant Z] [WeakEquivalence g] :
    Function.Bijective (fun (f : LeftHomotopyClass X Y) ↦ f.postcomp g)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.precomp_bijective_of_cofibration_of_weakEquivalence (L176-L199)
lemma precomp_bijective_of_cofibration_of_weakEquivalence
    [IsFibrant Z] (f : X ⟶ Y) [Cofibration f] [WeakEquivalence f] :
    Function.Bijective (fun (g : RightHomotopyClass Y Z) ↦ g.precomp f)
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.precomp_bijective_of_weakEquivalence (L200-L215)
lemma precomp_bijective_of_weakEquivalence
    [IsFibrant Z] (f : X ⟶ Y) [IsCofibrant X] [IsCofibrant Y] [WeakEquivalence f] :
    Function.Bijective (fun (g : RightHomotopyClass Y Z) ↦ g.precomp f)
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Homotopy.LeftHomotopyClass (L231-L236)
lemma LeftHomotopyClass.whitehead [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]
    (f : X ⟶ Y) [WeakEquivalence f] :
    ∃ (g : Y ⟶ X), LeftHomotopyRel (f ≫ g) (𝟙 X) ∧ LeftHomotopyRel (g ≫ f) (𝟙 Y)
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.CofibrantObject (L43-L47)
abbrev CofibrantObject : Type u
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.homMk (L55-L57)
abbrev homMk {X Y : C} [IsCofibrant X] [IsCofibrant Y] (f : X ⟶ Y) :
    mk X ⟶ mk Y
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Basic.mk' (L126-L163)
def mk' [CategoryWithFibrations C] [CategoryWithCofibrations C]
    [CategoryWithWeakEquivalences C] [HasFiniteLimits C] [HasFiniteColimits C]
    [(weakEquivalences C).HasTwoOutOfThreeProperty]
    [IsWeakFactorizationSystem (cofibrations C) (trivialFibrations C)]
    [IsWeakFactorizationSystem (trivialCofibrations C) (fibrations C)] :
    ModelCategory C where
  cm3a
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.mk (L48-L50)
abbrev mk (X : C) [IsCofibrant X] : CofibrantObject C
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.cofibrantObjects (L39-L42)
def cofibrantObjects : ObjectProperty C
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.ModelCategory.Bifibrant.mk_surjective (L51-L54)
lemma mk_surjective (X : CofibrantObject C) :
    ∃ (Y : C) (_ : IsCofibrant Y), X = mk Y
```
[/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.LinearCombination.Lemmas).

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

```lean
theorem div_lt_const [Semifield α] [LinearOrder α] [IsStrictOrderedRing α]
    (p : b < c) {a : α} (ha : 0 < a) : b / a < c / a
:= _unbalancedParenXYZ
div_lt_div_of_pos_right p ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[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)]
```

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.LinearCombination.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem div_lt_const [Semifield α] [LinearOrder α] [IsStrictOrderedRing α]
    (p : b < c) {a : α} (ha : 0 < a) : b / a < c / a
:= _unbalancedParenXYZ
div_lt_div_of_pos_right p ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[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)]
```

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


Instruction: Edit `ExProd` so it still compiles and the proof is simplified where possible. Return the full updated `ExProd` 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.Ring.Common).

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

## Current excerpt (around `Mathlib.Tactic.Ring.Common.ExProd`, L271-L279)
```lean
def ExProd.mkNegNat (_ : Q(Ring $α)) (n : ℕ) : (e : Q($α)) × ExProd sα e := let lit : Q(ℕ) := mkRawNatLit n
  ⟨q((Int.negOfNat $lit).rawCast : $α), .const (-n) none⟩

/--
Constructs the expression corresponding to `.const q h` for `q = n / d`
and `h` a proof that `(d : α) ≠ 0`.
(The `.const` constructor does not check that the expression is correct.)
-/
```

## Edit instruction
Edit `ExProd` so it still compiles and the proof is simplified where possible. Return the full updated `ExProd` 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.Ring.Common).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Tactic/Ring/Common.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Tactic.Ring.Common.ExProd`, L271-L279)
```lean
def ExProd.mkNegNat (_ : Q(Ring $α)) (n : ℕ) : (e : Q($α)) × ExProd sα e := let lit : Q(ℕ) := mkRawNatLit n
  ⟨q((Int.negOfNat $lit).rawCast : $α), .const (-n) none⟩

/--
Constructs the expression corresponding to `.const q h` for `q = n / d`
and `h` a proof that `(d : α) ≠ 0`.
(The `.const` constructor does not check that the expression is correct.)
-/
```

## Edit instruction
Edit `ExProd` so it still compiles and the proof is simplified where possible. Return the full updated `ExProd` 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.Dynamics.Ergodic.Function).

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

```lean
theorem Ergodic.ae_eq_const_of_ae_eq_comp₀ (h : Ergodic f μ) (hgm : NullMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c
:= .someMissingNamespaceXYZ
h.quasiErgodic.ae_eq_const_of_ae_eq_comp₀ hgm hg_eq

end CountableSeparatingOnUniv

variable [TopologicalSpace X] [MetrizableSpace X] [Nonempty X] {f : α → α}

namespace QuasiErgodic

/-- Let `f : α → α` be a quasi-ergodic map.
Let `g : α → X` be an a.e. strongly measurable function
from `α` to a nonempty metrizable topological space.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=239, messages=[Message(data='`Ergodic.ae_eq_const_of_ae_eq_comp₀` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=239, i_end=270, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=6, column=0), pos_end=Position(line=6, column=29), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=270, i_end=350, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.3\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=8, column=31), pos_end=Position(line=8, column=48), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=350, i_end=374, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=374, i_end=600, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=15, column=65), 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.Dynamics.Ergodic.Function).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Ergodic.ae_eq_const_of_ae_eq_comp₀ (h : Ergodic f μ) (hgm : NullMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c
:= .someMissingNamespaceXYZ
h.quasiErgodic.ae_eq_const_of_ae_eq_comp₀ hgm hg_eq

end CountableSeparatingOnUniv

variable [TopologicalSpace X] [MetrizableSpace X] [Nonempty X] {f : α → α}

namespace QuasiErgodic

/-- Let `f : α → α` be a quasi-ergodic map.
Let `g : α → X` be an a.e. strongly measurable function
from `α` to a nonempty metrizable topological space.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=239, messages=[Message(data='`Ergodic.ae_eq_const_of_ae_eq_comp₀` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=239, i_end=270, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=6, column=0), pos_end=Position(line=6, column=29), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=270, i_end=350, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.3\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=8, column=31), pos_end=Position(line=8, column=48), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=350, i_end=374, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=374, i_end=600, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=15, column=65), 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.SimplicialSet.CategoryWithFibrations).

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

```lean
lemma cofibrations_eq : cofibrations SSet.{u} = monomorphisms _
```

Declaration: `Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations.cofibrations_eq` (module Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations).[/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
lemma cofibrations_eq : cofibrations SSet.{u} = monomorphisms _
```

Declaration: `Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations.cofibrations_eq` (module Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations).[/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.DFA).

[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.Computability.DFA.pumping_lemma`:
State 0: ⊢ theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
    (hlen : Fintype.card σ ≤ List.length x) :
    ∃ a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts
Tactic: obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl
State 1: ⊢ <goal after obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl>
Tactic: use a, b, c, hx, hlen, hnil
State 2: ⊢ <goal after use a, b, c, hx, hlen, hnil>
Tactic: intro y hy
State 3: ⊢ <goal after intro y hy>
Tactic: rw [Language.mem_mul] at hy  rcases hy with ⟨ab, hab, c', hc', rfl⟩
State 4: ⊢ <goal after rw [Language.mem_mul] at hy>
Tactic: rw [Language.mem_mul] at hab  rcases hab with ⟨a', ha', b', hb', rfl⟩
State 5: ⊢ <goal after rw [Language.mem_mul] at hab>
Tactic: rw [Set.mem_singleton_iff] at ha' hc'  substs ha' hc'
State 6: ⊢ <goal after rw [Set.mem_singleton_iff] at ha' hc'>
Tactic: have h := M.evalFrom_of_pow hb hb'  rwa [mem_accepts, eval, evalFrom_of_append, evalFrom_of_append, h, hc]  section Maps  variable {α' σ' : Type*}  /--  `M.comap f` pulls back the alphabet of `M` along `f`. In other words, it applies `f` to the input  before passing it to `M`.  -/
```

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.Computability.DFA).[/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.Computability.DFA.pumping_lemma`:
State 0: ⊢ theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
    (hlen : Fintype.card σ ≤ List.length x) :
    ∃ a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts
Tactic: obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl
State 1: ⊢ <goal after obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl>
Tactic: use a, b, c, hx, hlen, hnil
State 2: ⊢ <goal after use a, b, c, hx, hlen, hnil>
Tactic: intro y hy
State 3: ⊢ <goal after intro y hy>
Tactic: rw [Language.mem_mul] at hy  rcases hy with ⟨ab, hab, c', hc', rfl⟩
State 4: ⊢ <goal after rw [Language.mem_mul] at hy>
Tactic: rw [Language.mem_mul] at hab  rcases hab with ⟨a', ha', b', hb', rfl⟩
State 5: ⊢ <goal after rw [Language.mem_mul] at hab>
Tactic: rw [Set.mem_singleton_iff] at ha' hc'  substs ha' hc'
State 6: ⊢ <goal after rw [Set.mem_singleton_iff] at ha' hc'>
Tactic: have h := M.evalFrom_of_pow hb hb'  rwa [mem_accepts, eval, evalFrom_of_append, evalFrom_of_append, h, hc]  section Maps  variable {α' σ' : Type*}  /--  `M.comap f` pulls back the alphabet of `M` along `f`. In other words, it applies `f` to the input  before passing it to `M`.  -/
```

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


Instruction: Edit `eqv` so it still compiles and the proof is simplified where possible. Return the full updated `eqv` 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.Denumerable).

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

## Current excerpt (around `Mathlib.Logic.Denumerable.eqv`, L71-L74)
```lean
def eqv (α) [Denumerable α] : α ≃ ℕ := ⟨encode, ofNat α, ofNat_encode, encode_ofNat⟩

-- See Note [lower instance priority]
```

## Edit instruction
Edit `eqv` so it still compiles and the proof is simplified where possible. Return the full updated `eqv` 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.Denumerable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Logic/Denumerable.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.Denumerable.eqv`, L71-L74)
```lean
def eqv (α) [Denumerable α] : α ≃ ℕ := ⟨encode, ofNat α, ofNat_encode, encode_ofNat⟩

-- See Note [lower instance priority]
```

## Edit instruction
Edit `eqv` so it still compiles and the proof is simplified where possible. Return the full updated `eqv` 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.Lean.Linter).

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

```lean
def whenLinterActivated (opt : Lean.Option Bool) (x : CommandElab) (breakOnError
:= true) : thisIsWrongLemmaNameXYZ
CommandElab :=
  withSetOptionIn fun stx => whenLinterOption opt do
    unless ← pure breakOnError <&&> MonadLog.hasErrors do
      x stx

end Lean.Linter
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=257, messages=[Message(data="Function expected at\n  thisIsWrongLemmaNameXYZ\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  CommandElab\n\nHint: The identifier `thisIsWrongLemmaNameXYZ` 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=11), pos_end=Position(line=3, column=11), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `withSetOptionIn`', pos=Position(line=4, column=2), pos_end=Position(line=4, column=17), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=257, i_end=272, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=15), 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.Lean.Linter).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def whenLinterActivated (opt : Lean.Option Bool) (x : CommandElab) (breakOnError
:= true) : thisIsWrongLemmaNameXYZ
CommandElab :=
  withSetOptionIn fun stx => whenLinterOption opt do
    unless ← pure breakOnError <&&> MonadLog.hasErrors do
      x stx

end Lean.Linter
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=257, messages=[Message(data="Function expected at\n  thisIsWrongLemmaNameXYZ\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  CommandElab\n\nHint: The identifier `thisIsWrongLemmaNameXYZ` 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=11), pos_end=Position(line=3, column=11), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `withSetOptionIn`', pos=Position(line=4, column=2), pos_end=Position(line=4, column=17), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=257, i_end=272, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=15), 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.Dynamics.BirkhoffSum.Basic).

[User]
Complete the following proof.

```lean
theorem birkhoffSum_succ' (f : α → α) (g : α → M) (n : ℕ) (x : α) :
    birkhoffSum f g (n + 1) x = g x + birkhoffSum f g n (f x)
```

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.Dynamics.BirkhoffSum.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem birkhoffSum_succ' (f : α → α) (g : α → M) (n : ℕ) (x : α) :
    birkhoffSum f g (n + 1) x = g x + birkhoffSum f g n (f x)
```

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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Equiv.Fin.Basic).

[User]
Complete the following proof.

```lean
theorem finSumFinEquiv_symm_last : finSumFinEquiv.symm (Fin.last n) = Sum.inr 0
```

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.Equiv.Fin.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem finSumFinEquiv_symm_last : finSumFinEquiv.symm (Fin.last n) = Sum.inr 0
```

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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber).

[User]
Complete the following proof.

```lean
theorem commute_int_add (n : ℤ) : Function.Commute f (n + ·)
```

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.Dynamics.Circle.RotationNumber.TranslationNumber).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem commute_int_add (n : ℤ) : Function.Commute f (n + ·)
```

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


Instruction: Edit `wellFounded_of_isEmpty` so it still compiles and the proof is simplified where possible. Return the full updated `wellFounded_of_isEmpty` 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.IsEmpty.Basic).

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

## Current excerpt (around `Mathlib.Logic.IsEmpty.Basic.wellFounded_of_isEmpty`, L78-L82)
```lean
theorem wellFounded_of_isEmpty {α} [IsEmpty α] (r : α → α → Prop) : WellFounded r := ⟨isEmptyElim⟩

variable (α)
```

## Edit instruction
Edit `wellFounded_of_isEmpty` so it still compiles and the proof is simplified where possible. Return the full updated `wellFounded_of_isEmpty` 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.IsEmpty.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Logic/IsEmpty/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.IsEmpty.Basic.wellFounded_of_isEmpty`, L78-L82)
```lean
theorem wellFounded_of_isEmpty {α} [IsEmpty α] (r : α → α → Prop) : WellFounded r := ⟨isEmptyElim⟩

variable (α)
```

## Edit instruction
Edit `wellFounded_of_isEmpty` so it still compiles and the proof is simplified where possible. Return the full updated `wellFounded_of_isEmpty` block.
[/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.Tactic.ComputeAsymptotics.Multiseries.Corecursion).

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

```lean
theorem FriendlyOperationClass.comp (F : γ → Seq α → Seq α) (g : γ' → γ)
    [h : FriendlyOperationClass F] : FriendlyOperationClass (fun c ↦ F (g c))
```

Declaration: `Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion.FriendlyOperationClass` (module Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion).[/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 FriendlyOperationClass.comp (F : γ → Seq α → Seq α) (g : γ' → γ)
    [h : FriendlyOperationClass F] : FriendlyOperationClass (fun c ↦ F (g c))
```

Declaration: `Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion.FriendlyOperationClass` (module Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion).[/INST]


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

[User]
Complete the following proof.

```lean
theorem mul_eq_eval₁ [CommGroupWithZero M] (a₁ : ℤ × M) {a₂ : ℤ × M} {l₁ l₂ l : NF M}
    (h : l₁.eval * (a₂ ::ᵣ l₂).eval = l.eval) :
    (a₁ ::ᵣ l₁).eval * (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval
```

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.FieldSimp.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mul_eq_eval₁ [CommGroupWithZero M] (a₁ : ℤ × M) {a₂ : ℤ × M} {l₁ l₂ l : NF M}
    (h : l₁.eval * (a₂ ::ᵣ l₂).eval = l.eval) :
    (a₁ ::ᵣ l₁).eval * (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval
```

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


Instruction: Edit `refl_trans` so it still compiles and the proof is simplified where possible. Return the full updated `refl_trans` 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.PartialEquiv).

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

## Current excerpt (around `Mathlib.Logic.Equiv.PartialEquiv.refl_trans`, L641-L643)
```lean
theorem refl_trans : (PartialEquiv.refl α).trans e = e := PartialEquiv.ext (fun _ => rfl) (fun _ => rfl) (by simp [trans_source, preimage_id])
```

## Edit instruction
Edit `refl_trans` so it still compiles and the proof is simplified where possible. Return the full updated `refl_trans` 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.PartialEquiv).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Logic/Equiv/PartialEquiv.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.Equiv.PartialEquiv.refl_trans`, L641-L643)
```lean
theorem refl_trans : (PartialEquiv.refl α).trans e = e := PartialEquiv.ext (fun _ => rfl) (fun _ => rfl) (by simp [trans_source, preimage_id])
```

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


Instruction: Edit `eval_bicategoricalComp` so it still compiles and the proof is simplified where possible. Return the full updated `eval_bicategoricalComp` 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.Bicategory.Normalize).

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

## Current excerpt (around `Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_bicategoricalComp`, L153-L158)
```lean
theorem eval_bicategoricalComp
    {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]
```

## Edit instruction
Edit `eval_bicategoricalComp` so it still compiles and the proof is simplified where possible. Return the full updated `eval_bicategoricalComp` 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.Bicategory.Normalize).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Tactic.CategoryTheory.Bicategory.Normalize.eval_bicategoricalComp`, L153-L158)
```lean
theorem eval_bicategoricalComp
    {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]
```

## Edit instruction
Edit `eval_bicategoricalComp` so it still compiles and the proof is simplified where possible. Return the full updated `eval_bicategoricalComp` 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.Computability.PartrecCode).

[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.Computability.PartrecCode.rfind'`:
State 0: ⊢ theorem rfind' {f} (hf : Nat.Partrec f) :
    Nat.Partrec
      (Nat.unpaired fun a m =>
        (Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m))
Tactic: refine  Partrec.map  ((@Partrec₂.unpaired' fun a b : ℕ =>  Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + b))).1  ?_)  (Primrec.nat_add.comp Primrec.snd <| Primrec.snd.comp Primrec.fst).to_comp.to₂
State 1: ⊢ <goal after refine>
Tactic: have : Nat.Partrec (fun a => Nat.rfind (fun n => (fun m => decide (m = 0)) <$>  Nat.unpaired (fun a b => f (Nat.pair (Nat.unpair a).1 (b + (Nat.unpair a).2)))  (Nat.pair a n))) :=  rfind  (Partrec₂.unpaired'.2  ((Partrec.nat_iff.2 hf).comp  (Primrec₂.pair.comp (Primrec.fst.comp <| Primrec.unpair.comp Primrec.fst)  (Primrec.nat_add.comp Primrec.snd  (Primrec.snd.comp <| Primrec.unpair.comp Primrec.fst))).to_comp))
State 2: ⊢ <goal after have : Nat.Partrec (fun a => Nat.rfind (fun n => (fun m => decide (m = 0)) <$>>
Tactic: simpa  /-- Code for partial recursive functions from ℕ to ℕ.  See `Nat.Partrec.Code.eval` for the interpretation of these constructors.  -/  inductive Code : Type  | zero : Code  | succ : Code  | left : Code  | right : Code  | pair : Code → Code → Code  | comp : Code → Code → Code  | prec : Code → Code → Code  | rfind' : Code → Code  compile_inductive% Code  end Nat.Partrec  namespace Nat.Partrec.Code
```

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.Computability.PartrecCode).[/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.Computability.PartrecCode.rfind'`:
State 0: ⊢ theorem rfind' {f} (hf : Nat.Partrec f) :
    Nat.Partrec
      (Nat.unpaired fun a m =>
        (Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + m))).map (· + m))
Tactic: refine  Partrec.map  ((@Partrec₂.unpaired' fun a b : ℕ =>  Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + b))).1  ?_)  (Primrec.nat_add.comp Primrec.snd <| Primrec.snd.comp Primrec.fst).to_comp.to₂
State 1: ⊢ <goal after refine>
Tactic: have : Nat.Partrec (fun a => Nat.rfind (fun n => (fun m => decide (m = 0)) <$>  Nat.unpaired (fun a b => f (Nat.pair (Nat.unpair a).1 (b + (Nat.unpair a).2)))  (Nat.pair a n))) :=  rfind  (Partrec₂.unpaired'.2  ((Partrec.nat_iff.2 hf).comp  (Primrec₂.pair.comp (Primrec.fst.comp <| Primrec.unpair.comp Primrec.fst)  (Primrec.nat_add.comp Primrec.snd  (Primrec.snd.comp <| Primrec.unpair.comp Primrec.fst))).to_comp))
State 2: ⊢ <goal after have : Nat.Partrec (fun a => Nat.rfind (fun n => (fun m => decide (m = 0)) <$>>
Tactic: simpa  /-- Code for partial recursive functions from ℕ to ℕ.  See `Nat.Partrec.Code.eval` for the interpretation of these constructors.  -/  inductive Code : Type  | zero : Code  | succ : Code  | left : Code  | right : Code  | pair : Code → Code → Code  | comp : Code → Code → Code  | prec : Code → Code → Code  | rfind' : Code → Code  compile_inductive% Code  end Nat.Partrec  namespace Nat.Partrec.Code
```

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.Discrete.LocallyConstant).

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

```lean
abbrev functor : Type (u + 1) ⥤ CondensedSet.{u}
:= thisIsWrongLemmaNameXYZ
CompHausLike.LocallyConstant.functor.{u, u + 1} (P := fun _ ↦ True)
    (hs := fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)

/--
`CondensedSet.LocallyConstant.functor` is isomorphic to `Condensed.discrete`
(by uniqueness of adjoints).
-/
noncomputable def iso : functor ≅ discrete (Type (u + 1)) :=
  (LocallyConstant.adjunction _ _).leftAdjointUniq (discreteUnderlyingAdj _)

/-- `CondensedSet.LocallyConstant.functor` is fully faithful. -/
noncomputable def functorFullyFaithful : functor.FullyFaithful :=
  (LocallyConstant.adjunction.{u, u + 1} _ _).fullyFaithfulLOfIsIsoUnit

noncomputable instance : functor.Faithful := functorFullyFaithful.faithful

noncomputable instance : functor.Full := functorFullyFaithful.full
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=30, messages=[Message(data='expected token', pos=Position(line=1, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=30, i_end=473, messages=[Message(data="Function expected at\n  discrete\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (Type (u + 1))\n\nHint: The identifier `discrete` 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=10, column=34), pos_end=Position(line=10, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown constant `LocallyConstant.adjunction`', pos=Position(line=11, column=3), pos_end=Position(line=11, column=29), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=473, i_end=677, messages=[Message(data='Unknown constant `functor.FullyFaithful`', pos=Position(line=14, column=41), pos_end=Position(line=14, column=62), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unknown universe level `u`', pos=Position(line=15, column=34), pos_end=Position(line=15, column=35), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=677, i_end=753, messages=[Message(data='Unknown constant `functor.Faithful`', pos=Position(line=17, column=25), pos_end=Position(line=17, column=41), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=753, i_end=819, messages=[Message(data='Unknown constant `functor.Full`', pos=Position(line=19, column=25), pos_end=Position(line=19, column=37), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], 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.LocallyConstant).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
abbrev functor : Type (u + 1) ⥤ CondensedSet.{u}
:= thisIsWrongLemmaNameXYZ
CompHausLike.LocallyConstant.functor.{u, u + 1} (P := fun _ ↦ True)
    (hs := fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)

/--
`CondensedSet.LocallyConstant.functor` is isomorphic to `Condensed.discrete`
(by uniqueness of adjoints).
-/
noncomputable def iso : functor ≅ discrete (Type (u + 1)) :=
  (LocallyConstant.adjunction _ _).leftAdjointUniq (discreteUnderlyingAdj _)

/-- `CondensedSet.LocallyConstant.functor` is fully faithful. -/
noncomputable def functorFullyFaithful : functor.FullyFaithful :=
  (LocallyConstant.adjunction.{u, u + 1} _ _).fullyFaithfulLOfIsIsoUnit

noncomputable instance : functor.Faithful := functorFullyFaithful.faithful

noncomputable instance : functor.Full := functorFullyFaithful.full
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=30, messages=[Message(data='expected token', pos=Position(line=1, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=30, i_end=473, messages=[Message(data="Function expected at\n  discrete\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (Type (u + 1))\n\nHint: The identifier `discrete` 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=10, column=34), pos_end=Position(line=10, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown constant `LocallyConstant.adjunction`', pos=Position(line=11, column=3), pos_end=Position(line=11, column=29), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=473, i_end=677, messages=[Message(data='Unknown constant `functor.FullyFaithful`', pos=Position(line=14, column=41), pos_end=Position(line=14, column=62), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unknown universe level `u`', pos=Position(line=15, column=34), pos_end=Position(line=15, column=35), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=677, i_end=753, messages=[Message(data='Unknown constant `functor.Faithful`', pos=Position(line=17, column=25), pos_end=Position(line=17, column=41), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=753, i_end=819, messages=[Message(data='Unknown constant `functor.Full`', pos=Position(line=19, column=25), pos_end=Position(line=19, column=37), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], 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.AlgebraicTopology.SimplicialObject.Basic).

[User]
Complete the following proof.

```lean
def truncation (n : ℕ) : SimplicialObject C ⥤ SimplicialObject.Truncated C n
```

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.AlgebraicTopology.SimplicialObject.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def truncation (n : ℕ) : SimplicialObject C ⥤ SimplicialObject.Truncated C n
```

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.Control.Functor).

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

```lean
theorem functor_comp_id {F} [AF : Functor F] [LawfulFunctor F] :
    Comp.functor (G
:= Id) = AF := thisIsWrongLemmaNameXYZ
@Functor.ext F _ AF (Comp.lawfulFunctor (G := Id)) _ fun _ _ _ _ => rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=195, messages=[Message(data='Unknown identifier `Comp.functor`', pos=Position(line=2, column=4), pos_end=Position(line=2, column=16), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=15), pos_end=Position(line=3, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], 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.Control.Functor).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem functor_comp_id {F} [AF : Functor F] [LawfulFunctor F] :
    Comp.functor (G
:= Id) = AF := thisIsWrongLemmaNameXYZ
@Functor.ext F _ AF (Comp.lawfulFunctor (G := Id)) _ fun _ _ _ _ => rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=195, messages=[Message(data='Unknown identifier `Comp.functor`', pos=Position(line=2, column=4), pos_end=Position(line=2, column=16), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=15), pos_end=Position(line=3, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


Instruction: Edit `N₂Γ₂_inv_app_f_f` so it still compiles and the proof is simplified where possible. Return the full updated `N₂Γ₂_inv_app_f_f` 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.AlgebraicTopology.DoldKan.GammaCompN).

[User]
You are editing a file in `Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.AlgebraicTopology.DoldKan.GammaCompN.N₂Γ₂_inv_app_f_f`, L160-L171)
```lean
theorem N₂Γ₂_inv_app_f_f (X : Karoubi (ChainComplex C ℕ)) (n : ℕ) :
    (N₂Γ₂.inv.app X).f.f n =
      X.p.f n ≫ ((Γ₀.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op ⦋n⦌)) := by
dsimp [N₂Γ₂]
  simp only [whiskeringLeft_obj_preimage_app, NatTrans.comp_app, Functor.comp_map,
    Karoubi.comp_f, N₂Γ₂ToKaroubiIso_inv_app, HomologicalComplex.comp_f,
    N₁Γ₀_inv_app_f_f, toKaroubi_obj_X, Splitting.toKaroubiNondegComplexIsoN₁_hom_f_f,
    PInfty_on_Γ₀_splitting_summand_eq_self, N₂_map_f_f, Γ₂_map_f_app, unop_op, Karoubi.decompId_p_f,
    PInfty_on_Γ₀_splitting_summand_eq_self_assoc, Splitting.IndexSet.id_fst, SimplexCategory.len_mk,
    Splitting.ι_desc]
  apply Karoubi.HomologicalComplex.p_idem_assoc
```

## Edit instruction
Edit `N₂Γ₂_inv_app_f_f` so it still compiles and the proof is simplified where possible. Return the full updated `N₂Γ₂_inv_app_f_f` 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.AlgebraicTopology.DoldKan.GammaCompN).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.AlgebraicTopology.DoldKan.GammaCompN.N₂Γ₂_inv_app_f_f`, L160-L171)
```lean
theorem N₂Γ₂_inv_app_f_f (X : Karoubi (ChainComplex C ℕ)) (n : ℕ) :
    (N₂Γ₂.inv.app X).f.f n =
      X.p.f n ≫ ((Γ₀.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op ⦋n⦌)) := by
dsimp [N₂Γ₂]
  simp only [whiskeringLeft_obj_preimage_app, NatTrans.comp_app, Functor.comp_map,
    Karoubi.comp_f, N₂Γ₂ToKaroubiIso_inv_app, HomologicalComplex.comp_f,
    N₁Γ₀_inv_app_f_f, toKaroubi_obj_X, Splitting.toKaroubiNondegComplexIsoN₁_hom_f_f,
    PInfty_on_Γ₀_splitting_summand_eq_self, N₂_map_f_f, Γ₂_map_f_app, unop_op, Karoubi.decompId_p_f,
    PInfty_on_Γ₀_splitting_summand_eq_self_assoc, Splitting.IndexSet.id_fst, SimplexCategory.len_mk,
    Splitting.ι_desc]
  apply Karoubi.HomologicalComplex.p_idem_assoc
```

## Edit instruction
Edit `N₂Γ₂_inv_app_f_f` so it still compiles and the proof is simplified where possible. Return the full updated `N₂Γ₂_inv_app_f_f` block.
[/INST]


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

[User]
Complete the following proof.

```lean
def pureCoherence (mvarId : MVarId) : MetaM (List MVarId)
```

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.CategoryTheory.Monoidal.PureCoherence).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def pureCoherence (mvarId : MVarId) : MetaM (List MVarId)
```

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.Linter.OldObtain).

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

```lean
theorem foo : True
:= by ⟨⟩⟨⟩
-- These cases are fine.
  obtain := trivial
  obtain h := trivial
  obtain : True := trivial
  obtain h : True := trivial
  -- These are linted against.
  obtain : True
  · trivial
  obtain h : True
  · trivial
```
We allow the first four (since an explicit proof is provided), but lint against the last two.

## Why is this bad?

This is similar to removing all uses of `Tactic.Replace` and `Tactic.Have`
from mathlib: in summary,
- this version is a Lean3-ism, which can be unlearned now
- the syntax `obtain foo : type := proof` is slightly shorter;
  particularly so when the first tactic of the proof is `exact`
- when using the old syntax as `obtain foo : type; · proof`, there is an intermediate state with
  multiple goals right before the focusing dot. This can be confusing.
  (This gets amplified with the in-flight "multiple goal linter", which seems generally desired ---
  for many reasons, including teachability. Granted, the linter could be tweaked to not lint in this
  case... but by now, the "old" syntax is not clearly better.)
- the old syntax *could* be slightly nicer when deferring goals: however, this is rare.
  In the 30 replacements of the last PR, this occurred twice. In both cases, the `suffices` tactic
  could also be used, as was in fact clearer.
-/

meta section

open Lean Elab Linter

namespace Mathlib.Linter.Style

/-- Whether a syntax element is an `obtain` tactic call without a provided proof. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=25, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=2, column=5), pos_end=Position(line=2, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n⊢ True', pos=Position(line=2, column=3), pos_end=Position(line=2, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=25, i_end=544, messages=[Message(data='unexpected token; expected identifier or stx', pos=Position(line=21, column=12), pos_end=Position(line=21, column=20), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=544, i_end=689, messages=[Message(data="unexpected token; expected ':' or ':='", pos=Position(line=23, column=30), pos_end=Position(line=23, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=689, i_end=1088, messages=[Message(data="unexpected token '.'; expected ':'", pos=Position(line=27, column=60), pos_end=Position(line=27, column=61), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1088, i_end=1108, messages=[Message(data="unexpected token '*'; expected identifier or stx", pos=Position(line=28, column=16), pos_end=Position(line=28, column=18), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1108, i_end=1342, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1342, i_end=1365, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1365, i_end=1397, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1397, i_end=1481, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=39, column=84), 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.Linter.OldObtain).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem foo : True
:= by ⟨⟩⟨⟩
-- These cases are fine.
  obtain := trivial
  obtain h := trivial
  obtain : True := trivial
  obtain h : True := trivial
  -- These are linted against.
  obtain : True
  · trivial
  obtain h : True
  · trivial
```
We allow the first four (since an explicit proof is provided), but lint against the last two.

## Why is this bad?

This is similar to removing all uses of `Tactic.Replace` and `Tactic.Have`
from mathlib: in summary,
- this version is a Lean3-ism, which can be unlearned now
- the syntax `obtain foo : type := proof` is slightly shorter;
  particularly so when the first tactic of the proof is `exact`
- when using the old syntax as `obtain foo : type; · proof`, there is an intermediate state with
  multiple goals right before the focusing dot. This can be confusing.
  (This gets amplified with the in-flight "multiple goal linter", which seems generally desired ---
  for many reasons, including teachability. Granted, the linter could be tweaked to not lint in this
  case... but by now, the "old" syntax is not clearly better.)
- the old syntax *could* be slightly nicer when deferring goals: however, this is rare.
  In the 30 replacements of the last PR, this occurred twice. In both cases, the `suffices` tactic
  could also be used, as was in fact clearer.
-/

meta section

open Lean Elab Linter

namespace Mathlib.Linter.Style

/-- Whether a syntax element is an `obtain` tactic call without a provided proof. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=25, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=2, column=5), pos_end=Position(line=2, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n⊢ True', pos=Position(line=2, column=3), pos_end=Position(line=2, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=25, i_end=544, messages=[Message(data='unexpected token; expected identifier or stx', pos=Position(line=21, column=12), pos_end=Position(line=21, column=20), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=544, i_end=689, messages=[Message(data="unexpected token; expected ':' or ':='", pos=Position(line=23, column=30), pos_end=Position(line=23, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=689, i_end=1088, messages=[Message(data="unexpected token '.'; expected ':'", pos=Position(line=27, column=60), pos_end=Position(line=27, column=61), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1088, i_end=1108, messages=[Message(data="unexpected token '*'; expected identifier or stx", pos=Position(line=28, column=16), pos_end=Position(line=28, column=18), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1108, i_end=1342, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1342, i_end=1365, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1365, i_end=1397, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1397, i_end=1481, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=39, column=84), 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 `optionToList` so it still compiles and the proof is simplified where possible. Return the full updated `optionToList` 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.Computability.Primrec.List).

[User]
You are editing a file in `Mathlib/Computability/Primrec/List.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Computability.Primrec.List.optionToList`, L243-L247)
```lean
theorem optionToList : Primrec (Option.toList : Option α → List α) := (option_casesOn Primrec.id (const [])
    ((list_cons.comp Primrec.id (const [])).comp₂ Primrec₂.right)).of_eq
  (fun o => by rcases o <;> simp)
```

## Edit instruction
Edit `optionToList` so it still compiles and the proof is simplified where possible. Return the full updated `optionToList` 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.Computability.Primrec.List).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Computability/Primrec/List.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Computability.Primrec.List.optionToList`, L243-L247)
```lean
theorem optionToList : Primrec (Option.toList : Option α → List α) := (option_casesOn Primrec.id (const [])
    ((list_cons.comp Primrec.id (const [])).comp₂ Primrec₂.right)).of_eq
  (fun o => by rcases o <;> simp)
```

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


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

[User]
Complete the following proof.

```lean
theorem evalFrom_cons (S : Set σ) (a : α) (x : List α) :
    M.evalFrom S (a :: x) = M.evalFrom (M.stepSet S a) x
```

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.Computability.NFA).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem evalFrom_cons (S : Set σ) (a : α) (x : List α) :
    M.evalFrom S (a :: x) = M.evalFrom (M.stepSet S a) x
```

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.Dynamics.Minimal).

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

```lean
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s
:= .someMissingNamespaceXYZ
let ⟨x, hx⟩ := hne
  (MulAction.dense_orbit M x).mono (range_subset_iff.2 fun c ↦ hsmul c ⟨x, hx, rfl⟩)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=167, messages=[Message(data='`dense_of_nonempty_smul_invariant` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=167, i_end=280, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=3), 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.Dynamics.Minimal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s
:= .someMissingNamespaceXYZ
let ⟨x, hx⟩ := hne
  (MulAction.dense_orbit M x).mono (range_subset_iff.2 fun c ↦ hsmul c ⟨x, hx, rfl⟩)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=167, messages=[Message(data='`dense_of_nonempty_smul_invariant` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=167, i_end=280, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=3), 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.DoldKan.Normalized).

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

```lean
def PInftyToNormalizedMooreComplex (X : SimplicialObject A) : K[X] ⟶ N[X]
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.HigherFacesVanish (L47-L52)
theorem HigherFacesVanish.inclusionOfMooreComplexMap (n : ℕ) :
    HigherFacesVanish (n + 1) ((inclusionOfMooreComplexMap X).f (n + 1))
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.factors_normalizedMooreComplex_PInfty (L53-L63)
theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
    Subobject.Factors (NormalizedMooreComplex.objX X n) (PInfty.f n)
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (L75-L78)
theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
    PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInftyToNormalizedMooreComplex_naturality (L80-L85)
theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X ⟶ Y) :
    AlternatingFaceMapComplex.map f ≫ PInftyToNormalizedMooreComplex Y =
      PInftyToNormalizedMooreComplex X ≫ NormalizedMooreComplex.map f
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInfty_comp_PInftyToNormalizedMooreComplex (L87-L89)
theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
    PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.inclusionOfMooreComplexMap_comp_PInfty (L91-L97)
theorem inclusionOfMooreComplexMap_comp_PInfty (X : SimplicialObject A) :
    inclusionOfMooreComplexMap X ≫ PInfty = inclusionOfMooreComplexMap X
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.splitMonoInclusionOfMooreComplexMap (L107-L121)
def splitMonoInclusionOfMooreComplexMap (X : SimplicialObject A) :
    SplitMono (inclusionOfMooreComplexMap X) where
  retraction
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.N₁_iso_normalizedMooreComplex_comp_toKaroubi (L122-L142)
def N₁_iso_normalizedMooreComplex_comp_toKaroubi : N₁ ≅ normalizedMooreComplex A ⋙ toKaroubi _ where
  hom
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso (L92-L102)
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq (L86-L90)
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso (L75-L85)
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B'
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁ (L68-L69)
def equivalence₁ : A ≌ B'
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₀ (L60-L66)
def equivalence₀ : A ≌ B'
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse (L70-L73)
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse
```

<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.DoldKan.Normalized).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
def PInftyToNormalizedMooreComplex (X : SimplicialObject A) : K[X] ⟶ N[X]
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.HigherFacesVanish (L47-L52)
theorem HigherFacesVanish.inclusionOfMooreComplexMap (n : ℕ) :
    HigherFacesVanish (n + 1) ((inclusionOfMooreComplexMap X).f (n + 1))
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.factors_normalizedMooreComplex_PInfty (L53-L63)
theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
    Subobject.Factors (NormalizedMooreComplex.objX X n) (PInfty.f n)
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (L75-L78)
theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
    PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInftyToNormalizedMooreComplex_naturality (L80-L85)
theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X ⟶ Y) :
    AlternatingFaceMapComplex.map f ≫ PInftyToNormalizedMooreComplex Y =
      PInftyToNormalizedMooreComplex X ≫ NormalizedMooreComplex.map f
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.PInfty_comp_PInftyToNormalizedMooreComplex (L87-L89)
theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
    PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.inclusionOfMooreComplexMap_comp_PInfty (L91-L97)
theorem inclusionOfMooreComplexMap_comp_PInfty (X : SimplicialObject A) :
    inclusionOfMooreComplexMap X ≫ PInfty = inclusionOfMooreComplexMap X
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.splitMonoInclusionOfMooreComplexMap (L107-L121)
def splitMonoInclusionOfMooreComplexMap (X : SimplicialObject A) :
    SplitMono (inclusionOfMooreComplexMap X) where
  retraction
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Normalized.N₁_iso_normalizedMooreComplex_comp_toKaroubi (L122-L142)
def N₁_iso_normalizedMooreComplex_comp_toKaroubi : N₁ ≅ normalizedMooreComplex A ⋙ toKaroubi _ where
  hom
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso (L92-L102)
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq (L86-L90)
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso (L75-L85)
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B'
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁ (L68-L69)
def equivalence₁ : A ≌ B'
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₀ (L60-L66)
def equivalence₀ : A ≌ B'
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse (L70-L73)
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse
```
[/INST]


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

[User]
Complete the following proof.

```lean
abbrev ι₃ : Δ[2] ⟶ Λ[3, 2]
```

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.AlgebraicTopology.SimplicialSet.HornColimits).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
abbrev ι₃ : Δ[2] ⟶ Λ[3, 2]
```

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.AlgebraicTopology.SimplicialSet.HomotopyCat).

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

```lean
instance subsingleton_hom (X : Truncated.{u} 2) [Unique (X _⦋0⦌₂)] [Subsingleton (X _⦋1⦌₂)]
    (x y : X.HomotopyCategory) :
    Subsingleton (x ⟶ y)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.OneTruncation₂ (L51-L52)
def OneTruncation₂ (S : SSet.Truncated 2)
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.reflQuiver (L59-L62)
instance reflQuiver (S : SSet.Truncated 2) : ReflQuiver (OneTruncation₂ S) where
  Hom
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.hom_ext (L64-L68)
lemma hom_ext
    {S : SSet.Truncated 2} {x y : OneTruncation₂ S} {f g : x ⟶ y}
    (h : f.edge = g.edge) : f = g
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.id_edge (L70-L75)
lemma id_edge {S : SSet.Truncated 2} (x : OneTruncation₂ S) :
    Truncated.Edge.edge (𝟙rq x) = S.map (σ₂ 0).op x
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.map (L77-L85)
def map {S T : SSet.Truncated 2} (f : S ⟶ T) :
    OneTruncation₂ S ⥤rq OneTruncation₂ T where
  obj x
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.oneTruncation₂ (L87-L92)
def oneTruncation₂ : SSet.Truncated.{u} 2 ⥤ ReflQuiv.{u, u} where
  obj S
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.homOfEq_edge (L94-L100)
lemma homOfEq_edge
    {X : SSet.Truncated.{u} 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X}
    (f : x₁ ⟶ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :
    (Quiver.homOfEq f hx hy).edge = f.edge
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.nerveEquiv (L107-L110)
def nerveEquiv : OneTruncation₂ ((SSet.truncation 2).obj (nerve C)) ≃ C
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.IsUniquelyCodimOneFace (L37-L41)
def IsUniquelyCodimOneFace : Prop
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.δ_index (L80-L83)
lemma δ_index :
    X.δ (hxy.index hd) (y.cast (by rw [hxy.dim_eq, hd])).simplex = (x.cast hd).simplex
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.cast (L65-L67)
lemma cast : IsUniquelyCodimOneFace (x.cast hd) (y.cast (by rw [hxy.dim_eq, hd]))
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.dim_eq (L58-L59)
lemma dim_eq : y.dim = x.dim + 1
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.existsUnique_δ_cast_simplex (L68-L79)
lemma existsUnique_δ_cast_simplex :
    ∃! (i : Fin (d + 2)), X.δ i (y.cast (by rw [hxy.dim_eq, hd])).simplex =
      (x.cast hd).simplex
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.iff (L42-L57)
lemma iff {d : ℕ} (x : X _⦋d⦌) (y : X _⦋d + 1⦌) :
    IsUniquelyCodimOneFace (S.mk x) (S.mk y) ↔
      ∃! (i : Fin (d + 2)), X.δ i y = x
```

<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.SimplicialSet.HomotopyCat).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
instance subsingleton_hom (X : Truncated.{u} 2) [Unique (X _⦋0⦌₂)] [Subsingleton (X _⦋1⦌₂)]
    (x y : X.HomotopyCategory) :
    Subsingleton (x ⟶ y)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.OneTruncation₂ (L51-L52)
def OneTruncation₂ (S : SSet.Truncated 2)
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.reflQuiver (L59-L62)
instance reflQuiver (S : SSet.Truncated 2) : ReflQuiver (OneTruncation₂ S) where
  Hom
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.hom_ext (L64-L68)
lemma hom_ext
    {S : SSet.Truncated 2} {x y : OneTruncation₂ S} {f g : x ⟶ y}
    (h : f.edge = g.edge) : f = g
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.id_edge (L70-L75)
lemma id_edge {S : SSet.Truncated 2} (x : OneTruncation₂ S) :
    Truncated.Edge.edge (𝟙rq x) = S.map (σ₂ 0).op x
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.map (L77-L85)
def map {S T : SSet.Truncated 2} (f : S ⟶ T) :
    OneTruncation₂ S ⥤rq OneTruncation₂ T where
  obj x
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.oneTruncation₂ (L87-L92)
def oneTruncation₂ : SSet.Truncated.{u} 2 ⥤ ReflQuiv.{u, u} where
  obj S
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.homOfEq_edge (L94-L100)
lemma homOfEq_edge
    {X : SSet.Truncated.{u} 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X}
    (f : x₁ ⟶ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :
    (Quiver.homOfEq f hx hy).edge = f.edge
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat.nerveEquiv (L107-L110)
def nerveEquiv : OneTruncation₂ ((SSet.truncation 2).obj (nerve C)) ≃ C
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.IsUniquelyCodimOneFace (L37-L41)
def IsUniquelyCodimOneFace : Prop
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.δ_index (L80-L83)
lemma δ_index :
    X.δ (hxy.index hd) (y.cast (by rw [hxy.dim_eq, hd])).simplex = (x.cast hd).simplex
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.cast (L65-L67)
lemma cast : IsUniquelyCodimOneFace (x.cast hd) (y.cast (by rw [hxy.dim_eq, hd]))
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.dim_eq (L58-L59)
lemma dim_eq : y.dim = x.dim + 1
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.existsUnique_δ_cast_simplex (L68-L79)
lemma existsUnique_δ_cast_simplex :
    ∃! (i : Fin (d + 2)), X.δ i (y.cast (by rw [hxy.dim_eq, hd])).simplex =
      (x.cast hd).simplex
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace.iff (L42-L57)
lemma iff {d : ℕ} (x : X _⦋d⦌) (y : X _⦋d + 1⦌) :
    IsUniquelyCodimOneFace (S.mk x) (S.mk y) ↔
      ∃! (i : Fin (d + 2)), X.δ i y = x
```
[/INST]


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

[User]
Complete the following proof.

```lean
def sumArrowEquivProdArrow (α β γ : Type*) : (α ⊕ β → γ) ≃ (α → γ) × (β → γ)
```

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.Equiv.Prod).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def sumArrowEquivProdArrow (α β γ : Type*) : (α ⊕ β → γ) ≃ (α → γ) × (β → γ)
```

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