[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.Topology.MetricSpace.Kuratowski).

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

```lean
theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x)
  frobnicateTacticThatDoesNotExist := by
refine Isometry.of_dist_eq fun a b => ?_
  refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
  -- First step: find n with dist a (x n) < e
  rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
  -- Second step: use the norm control at index n to conclude
  have C : dist b (x n) - dist a (x n) = embeddingOfSubset x b n - embeddingOfSubset x a n := by
    simp only [embeddingOfSubset_coe, sub_sub_sub_cancel_right]
  have :=
    calc
      dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
      _ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [dist_comm]; ring
      _ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by grw [← le_abs_self]
      _ ≤ 2 * (e / 2) + |embeddingOfSubset x b n - embeddingOfSubset x a n| := by
        rw [C]
        gcongr
      _ ≤ 2 * (e / 2) + dist (embeddingOfSubset x b) (embeddingOfSubset x a) := by
        gcongr
        simp only [dist_eq_norm]
        exact lp.norm_apply_le_norm ENNReal.top_ne_zero
          (embeddingOfSubset x b - embeddingOfSubset x a) n
      _ = dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e := by ring
  simpa [dist_comm] using this

/-- Every separable metric space embeds isometrically in `ℓ^∞(ℕ)`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=380, messages=[Message(data='Function expected at\n  Isometry ?m.12\nbut this term has type\n  Prop\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=56), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  embeddingOfSubset\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `embeddingOfSubset` 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=66), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  ?m.27\nbut this term has type\n  ?m.26\n\nNote: Expected a function because this term is being applied to the argument\n  refine', pos=Position(line=3, column=38), pos_end=Position(line=6, column=69), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=380, i_end=1434, messages=[Message(data="unexpected token 'with'; expected command", pos=Position(line=6, column=70), pos_end=Position(line=6, column=74), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=26, column=69), 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.Topology.MetricSpace.Kuratowski).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x)
  frobnicateTacticThatDoesNotExist := by
refine Isometry.of_dist_eq fun a b => ?_
  refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
  -- First step: find n with dist a (x n) < e
  rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
  -- Second step: use the norm control at index n to conclude
  have C : dist b (x n) - dist a (x n) = embeddingOfSubset x b n - embeddingOfSubset x a n := by
    simp only [embeddingOfSubset_coe, sub_sub_sub_cancel_right]
  have :=
    calc
      dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
      _ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [dist_comm]; ring
      _ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by grw [← le_abs_self]
      _ ≤ 2 * (e / 2) + |embeddingOfSubset x b n - embeddingOfSubset x a n| := by
        rw [C]
        gcongr
      _ ≤ 2 * (e / 2) + dist (embeddingOfSubset x b) (embeddingOfSubset x a) := by
        gcongr
        simp only [dist_eq_norm]
        exact lp.norm_apply_le_norm ENNReal.top_ne_zero
          (embeddingOfSubset x b - embeddingOfSubset x a) n
      _ = dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e := by ring
  simpa [dist_comm] using this

/-- Every separable metric space embeds isometrically in `ℓ^∞(ℕ)`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=380, messages=[Message(data='Function expected at\n  Isometry ?m.12\nbut this term has type\n  Prop\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=56), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  embeddingOfSubset\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `embeddingOfSubset` 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=66), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  ?m.27\nbut this term has type\n  ?m.26\n\nNote: Expected a function because this term is being applied to the argument\n  refine', pos=Position(line=3, column=38), pos_end=Position(line=6, column=69), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=380, i_end=1434, messages=[Message(data="unexpected token 'with'; expected command", pos=Position(line=6, column=70), pos_end=Position(line=6, column=74), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=26, column=69), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem logCounting_add_analyticOn (hf : Meromorphic f) (hg : AnalyticOn 𝕜 g univ) :
    logCounting (f + g) ⊤ = logCounting f ⊤
:= by .someMissingNamespaceXYZ
simp only [logCounting, ↓reduceDIte]
  rw [hf.meromorphicOn.negPart_divisor_add_of_analyticNhdOn_right
    (isOpen_univ.analyticOn_iff_analyticOnNhd.1 hg)]

/--
Special case of `logCounting_add_analyticOn`: Adding a constant does not change the logarithmic
counting function for the poles.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=144, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  NontriviallyNormedField 𝕜\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=62), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  logCounting\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  (f + g)\n\nHint: The identifier `logCounting` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  logCounting\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `logCounting` 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=28), pos_end=Position(line=2, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n𝕜✝ : Type u_1\nE✝ : Type u_2\nf : 𝕜✝ → E✝\n𝕜 : Type u_3\nα✝ : Type u_4\nx✝¹ : Type u_5\ng : α✝ → x✝¹\nuniv : Set α✝\nx✝ : Sort u_6\nlogCounting : x✝\nhf : Meromorphic f\nhg : sorry\n⊢ sorry = sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=144, i_end=461, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=11, column=2), 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.Analysis.Complex.ValueDistribution.LogCounting.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem logCounting_add_analyticOn (hf : Meromorphic f) (hg : AnalyticOn 𝕜 g univ) :
    logCounting (f + g) ⊤ = logCounting f ⊤
:= by .someMissingNamespaceXYZ
simp only [logCounting, ↓reduceDIte]
  rw [hf.meromorphicOn.negPart_divisor_add_of_analyticNhdOn_right
    (isOpen_univ.analyticOn_iff_analyticOnNhd.1 hg)]

/--
Special case of `logCounting_add_analyticOn`: Adding a constant does not change the logarithmic
counting function for the poles.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=144, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  NontriviallyNormedField 𝕜\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=62), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  logCounting\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  (f + g)\n\nHint: The identifier `logCounting` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  logCounting\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `logCounting` 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=28), pos_end=Position(line=2, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n𝕜✝ : Type u_1\nE✝ : Type u_2\nf : 𝕜✝ → E✝\n𝕜 : Type u_3\nα✝ : Type u_4\nx✝¹ : Type u_5\ng : α✝ → x✝¹\nuniv : Set α✝\nx✝ : Sort u_6\nlogCounting : x✝\nhf : Meromorphic f\nhg : sorry\n⊢ sorry = sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=144, i_end=461, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=11, column=2), 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]

mistral-common:Tekken v15 (Mistral chat template, [INST]/[SYSTEM_PROMPT])


[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.Combinatorics.Matroid.Circuit).

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

```lean
lemma IsCircuit.subset_ground (hC : M.IsCircuit C) : C ⊆ M.E
:= _unbalancedParenXYZ
hC.dep.subset_ground
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='Unknown identifier `M.IsCircuit`', pos=Position(line=1, column=36), pos_end=Position(line=1, column=47), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.E`', pos=Position(line=1, column=57), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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.Combinatorics.Matroid.Circuit).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma IsCircuit.subset_ground (hC : M.IsCircuit C) : C ⊆ M.E
:= _unbalancedParenXYZ
hC.dep.subset_ground
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='Unknown identifier `M.IsCircuit`', pos=Position(line=1, column=36), pos_end=Position(line=1, column=47), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.E`', pos=Position(line=1, column=57), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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]


[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.GroupTheory.MonoidLocalization.Basic).

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

```lean
theorem induction_on {p : Localization S → Prop} (x) (H : ∀ y : M × S, p (mk y.1 y.2)) : p x
:= _unbalancedParenXYZ
ind H x

/-- Non-dependent recursion principle for localizations: given elements `f x y : p`
for all `x` and `y`, such that `r S x x'` and `r S y y'` implies `f x y = f x' y'`,
then `f` is defined on the whole `Localization S`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data="Function expected at\n  mk\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  y.1\n\nHint: The identifier `mk` 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=74), pos_end=Position(line=1, column=84), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=130, i_end=351, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=53), 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.GroupTheory.MonoidLocalization.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem induction_on {p : Localization S → Prop} (x) (H : ∀ y : M × S, p (mk y.1 y.2)) : p x
:= _unbalancedParenXYZ
ind H x

/-- Non-dependent recursion principle for localizations: given elements `f x y : p`
for all `x` and `y`, such that `r S x x'` and `r S y y'` implies `f x y = f x' y'`,
then `f` is defined on the whole `Localization S`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data="Function expected at\n  mk\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  y.1\n\nHint: The identifier `mk` 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=74), pos_end=Position(line=1, column=84), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=130, i_end=351, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=53), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem valAux_one : valAux K v O p 1 = 1
  frobnicateTacticThatDoesNotExist :=
(valAux_eq (hv := hv) <| show coeff 0 1 ≠ 0 from one_ne_zero).trans <| by
    rw [pow_zero, pow_one, map_one, ← (Ideal.Quotient.mk _).map_one, ModP.preVal_mk hv,
      map_one, v.map_one]
    change (1 : ModP O p) ≠ 0
    exact one_ne_zero
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=325, messages=[Message(data="Function expected at\n  valAux\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `valAux` 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=21), pos_end=Position(line=1, column=37), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  1\nbut this term has type\n  ?m.8\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=40), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `valAux_eq`', pos=Position(line=3, column=1), pos_end=Position(line=3, column=10), 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.RingTheory.Perfection).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem valAux_one : valAux K v O p 1 = 1
  frobnicateTacticThatDoesNotExist :=
(valAux_eq (hv := hv) <| show coeff 0 1 ≠ 0 from one_ne_zero).trans <| by
    rw [pow_zero, pow_one, map_one, ← (Ideal.Quotient.mk _).map_one, ModP.preVal_mk hv,
      map_one, v.map_one]
    change (1 : ModP O p) ≠ 0
    exact one_ne_zero
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=325, messages=[Message(data="Function expected at\n  valAux\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `valAux` 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=21), pos_end=Position(line=1, column=37), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  1\nbut this term has type\n  ?m.8\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=40), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `valAux_eq`', pos=Position(line=3, column=1), pos_end=Position(line=3, column=10), 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.TensorProduct.Subalgebra).

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

```lean
theorem mulMap_map_comp_eq (f : S →ₐ[R] T) :
    (mulMap (A.map f) (B.map f)).comp
      (Algebra.TensorProduct.map (f.subalgebraMap A) (f.subalgebraMap B))
        = f.comp (mulMap A B)
:= by .someMissingNamespaceXYZ
ext <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=199, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  mulMap\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  (A.map f)\n\nHint: The identifier `mulMap` 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=5), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nS : Type u_2\nT : Type u_3\nx✝ : Sort u_4\nmulMap : x✝\nf : sorry\n⊢ sorry = sorry', pos=Position(line=5, column=6), pos_end=Position(line=5, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=199, i_end=234, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem mulMap_map_comp_eq (f : S →ₐ[R] T) :
    (mulMap (A.map f) (B.map f)).comp
      (Algebra.TensorProduct.map (f.subalgebraMap A) (f.subalgebraMap B))
        = f.comp (mulMap A B)
:= by .someMissingNamespaceXYZ
ext <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=199, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  mulMap\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  (A.map f)\n\nHint: The identifier `mulMap` 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=5), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nS : Type u_2\nT : Type u_3\nx✝ : Sort u_4\nmulMap : x✝\nf : sorry\n⊢ sorry = sorry', pos=Position(line=5, column=6), pos_end=Position(line=5, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=199, i_end=234, messages=[], 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.NumberTheory.LSeries.Basic).

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

```lean
lemma LSeriesHasSum_iff {f : ℕ → ℂ} {s a : ℂ} :
    LSeriesHasSum f s a ↔ LSeriesSummable f s ∧ LSeries f s = a
:= thisIsWrongLemmaNameXYZ
⟨fun H ↦ ⟨H.LSeriesSummable, H.LSeries_eq⟩, fun ⟨H₁, H₂⟩ ↦ H₂ ▸ H₁.LSeriesHasSum⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=258, messages=[Message(data='`LSeriesHasSum_iff` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=23), 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.NumberTheory.LSeries.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma LSeriesHasSum_iff {f : ℕ → ℂ} {s a : ℂ} :
    LSeriesHasSum f s a ↔ LSeriesSummable f s ∧ LSeries f s = a
:= thisIsWrongLemmaNameXYZ
⟨fun H ↦ ⟨H.LSeriesSummable, H.LSeries_eq⟩, fun ⟨H₁, H₂⟩ ↦ H₂ ▸ H₁.LSeriesHasSum⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=258, messages=[Message(data='`LSeriesHasSum_iff` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=23), 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.Algebra.Polynomial.Derivation).

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

```lean
theorem derivation_ext {D₁ D₂ : Derivation R R[X] A} (h : D₁ X = D₂ X) : D₁ = D₂
:= .someMissingNamespaceXYZ
Derivation.ext fun f => Derivation.eqOn_adjoin (Set.eqOn_singleton.2 h) <| by
    simp only [adjoin_X, Algebra.coe_top, Set.mem_univ]

variable [IsScalarTower R (Polynomial R) A]
variable (R)

/-- The derivation on `R[X]` that takes the value `a` on `X`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=256, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="failed to prove index is valid, possible solutions:\n  - Use `have`-expressions to prove the index is valid\n  - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n  - Use `a[i]?` notation instead, result is an `Option` type\n  - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\nR : Type ?u.92\nX : ?m.2\nA : Type ?u.90\n⊢ ?m.7 R X", pos=Position(line=1, column=45), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data="failed to prove index is valid, possible solutions:\n  - Use `have`-expressions to prove the index is valid\n  - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n  - Use `a[i]?` notation instead, result is an `Option` type\n  - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\nR : Type ?u.92\nX : ?m.2\nA : Type ?u.90\nD₁ : sorry\n⊢ ?m.19 R X", pos=Position(line=1, column=45), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  max (?u.90+1) (?u.91+1) =?= max (?u.90+1) (?u.1582+1)\nwhile trying to unify\n  CoeT.{max (?u.90 + 1) (?u.1582 + 1), max (?u.90 + 1) (?u.91 + 1)}\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53))\n    x\n    (sorryAx.{(max (?u.90 + 1) (?u.91 + 1)) + 1} (Unit → Type (max ?u.90 ?u.91)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.39)) : Type (max 0 ?u.90 ?u.91)\nwith\n  CoeT.{max (?u.90 + 1) (?u.1582 + 1), max (?u.90 + 1) (?u.1582 + 1)}\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53))\n    x\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53)) : Type (max 0 ?u.90 ?u.1582)', pos=Position(line=1, column=0), pos_end=Position(line=1, column=80), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=256, i_end=300, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=6, column=27), pos_end=Position(line=6, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=300, i_end=314, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=314, i_end=378, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=64), 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.Algebra.Polynomial.Derivation).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem derivation_ext {D₁ D₂ : Derivation R R[X] A} (h : D₁ X = D₂ X) : D₁ = D₂
:= .someMissingNamespaceXYZ
Derivation.ext fun f => Derivation.eqOn_adjoin (Set.eqOn_singleton.2 h) <| by
    simp only [adjoin_X, Algebra.coe_top, Set.mem_univ]

variable [IsScalarTower R (Polynomial R) A]
variable (R)

/-- The derivation on `R[X]` that takes the value `a` on `X`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=256, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=32), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="failed to prove index is valid, possible solutions:\n  - Use `have`-expressions to prove the index is valid\n  - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n  - Use `a[i]?` notation instead, result is an `Option` type\n  - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\nR : Type ?u.92\nX : ?m.2\nA : Type ?u.90\n⊢ ?m.7 R X", pos=Position(line=1, column=45), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data="failed to prove index is valid, possible solutions:\n  - Use `have`-expressions to prove the index is valid\n  - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n  - Use `a[i]?` notation instead, result is an `Option` type\n  - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\nR : Type ?u.92\nX : ?m.2\nA : Type ?u.90\nD₁ : sorry\n⊢ ?m.19 R X", pos=Position(line=1, column=45), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  max (?u.90+1) (?u.91+1) =?= max (?u.90+1) (?u.1582+1)\nwhile trying to unify\n  CoeT.{max (?u.90 + 1) (?u.1582 + 1), max (?u.90 + 1) (?u.91 + 1)}\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53))\n    x\n    (sorryAx.{(max (?u.90 + 1) (?u.91 + 1)) + 1} (Unit → Type (max ?u.90 ?u.91)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.39)) : Type (max 0 ?u.90 ?u.91)\nwith\n  CoeT.{max (?u.90 + 1) (?u.1582 + 1), max (?u.90 + 1) (?u.1582 + 1)}\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53))\n    x\n    (sorryAx.{(max (?u.90 + 1) (?u.1582 + 1)) + 1} (Unit → Type (max ?u.90 ?u.1582)) true\n      (Function.const Lean.Name () `1.32.1.51.32.51._sorry._@.1886321366._hygCtx._hyg.53)) : Type (max 0 ?u.90 ?u.1582)', pos=Position(line=1, column=0), pos_end=Position(line=1, column=80), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=256, i_end=300, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=6, column=27), pos_end=Position(line=6, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=300, i_end=314, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=314, i_end=378, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=64), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem cast_listProd (l : List ℚ≥0) : (l.prod : K) = (l.map (↑)).prod
:= .someMissingNamespaceXYZ
map_list_prod (castHom _) _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=132, messages=[Message(data='Type mismatch\n  l.prod\nhas type\n  ℚ≥0\nbut is expected to have type\n  K', pos=Position(line=1, column=39), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind=None), Message(data='cannot coerce\n  x\nto type\n  ?m.12', pos=Position(line=1, column=61), 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)]
```

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

```lean
theorem cast_listProd (l : List ℚ≥0) : (l.prod : K) = (l.map (↑)).prod
:= .someMissingNamespaceXYZ
map_list_prod (castHom _) _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=132, messages=[Message(data='Type mismatch\n  l.prod\nhas type\n  ℚ≥0\nbut is expected to have type\n  K', pos=Position(line=1, column=39), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind=None), Message(data='cannot coerce\n  x\nto type\n  ?m.12', pos=Position(line=1, column=61), 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)]
```

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.Order.WithBot).

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

```lean
theorem toDual_map (f : α → β) (a : WithBot α) :
    WithBot.toDual (map f a) = WithTop.map (toDual ∘ f ∘ ofDual) (WithBot.toDual a)
:= thisIsWrongLemmaNameXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=172, messages=[Message(data="Function expected at\n  map\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `map` 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=20), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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.Order.WithBot).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem toDual_map (f : α → β) (a : WithBot α) :
    WithBot.toDual (map f a) = WithTop.map (toDual ∘ f ∘ ofDual) (WithBot.toDual a)
:= thisIsWrongLemmaNameXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=172, messages=[Message(data="Function expected at\n  map\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `map` 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=20), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Order).

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

```lean
theorem continuous_bot {t : TopologicalSpace β} : Continuous[⊥, t] f
:= _unbalancedParenXYZ
continuous_iff_le_induced.2 bot_le

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=65, messages=[Message(data="unexpected token ','; expected ':', ']' or ']''", pos=Position(line=1, column=62), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=65, i_end=187, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=56), 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.Topology.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem continuous_bot {t : TopologicalSpace β} : Continuous[⊥, t] f
:= _unbalancedParenXYZ
continuous_iff_le_induced.2 bot_le

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=65, messages=[Message(data="unexpected token ','; expected ':', ']' or ']''", pos=Position(line=1, column=62), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=65, i_end=187, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem biUnionTagged_memBaseSet {π : Prepartition I} {πi : ∀ J, TaggedPrepartition J}
    (h : ∀ J ∈ π, l.MemBaseSet J c r (πi J)) (hp : ∀ J ∈ π, (πi J).IsPartition)
    (hc : l.bDistortion → π.compl.distortion ≤ c) : l.MemBaseSet I c r (π.biUnionTagged πi)
  frobnicateTacticThatDoesNotExist := by
refine ⟨TaggedPrepartition.isSubordinate_biUnionTagged.2 fun J hJ => (h J hJ).1,
    fun hH => TaggedPrepartition.isHenstock_biUnionTagged.2 fun J hJ => (h J hJ).2 hH,
    fun hD => ?_, fun hD => ?_⟩
  · rw [Prepartition.distortion_biUnionTagged, Finset.sup_le_iff]
    exact fun J hJ => (h J hJ).3 hD
  · refine ⟨_, ?_, hc hD⟩
    rw [π.iUnion_compl, ← π.iUnion_biUnion_partition hp]
    rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=729, messages=[Message(data='Unknown identifier `l.MemBaseSet`', pos=Position(line=2, column=18), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `l.bDistortion`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=23), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `l.MemBaseSet`', pos=Position(line=3, column=52), pos_end=Position(line=3, column=64), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  Prepartition\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `Prepartition` 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=38), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  TaggedPrepartition\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  J\n\nHint: The identifier `TaggedPrepartition` 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=65), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `⟨...⟩` notation: The expected type of this term could not be determined', pos=Position(line=5, column=7), pos_end=Position(line=7, column=31), 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.Analysis.BoxIntegral.Partition.Filter).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem biUnionTagged_memBaseSet {π : Prepartition I} {πi : ∀ J, TaggedPrepartition J}
    (h : ∀ J ∈ π, l.MemBaseSet J c r (πi J)) (hp : ∀ J ∈ π, (πi J).IsPartition)
    (hc : l.bDistortion → π.compl.distortion ≤ c) : l.MemBaseSet I c r (π.biUnionTagged πi)
  frobnicateTacticThatDoesNotExist := by
refine ⟨TaggedPrepartition.isSubordinate_biUnionTagged.2 fun J hJ => (h J hJ).1,
    fun hH => TaggedPrepartition.isHenstock_biUnionTagged.2 fun J hJ => (h J hJ).2 hH,
    fun hD => ?_, fun hD => ?_⟩
  · rw [Prepartition.distortion_biUnionTagged, Finset.sup_le_iff]
    exact fun J hJ => (h J hJ).3 hD
  · refine ⟨_, ?_, hc hD⟩
    rw [π.iUnion_compl, ← π.iUnion_biUnion_partition hp]
    rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=729, messages=[Message(data='Unknown identifier `l.MemBaseSet`', pos=Position(line=2, column=18), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `l.bDistortion`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=23), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `l.MemBaseSet`', pos=Position(line=3, column=52), pos_end=Position(line=3, column=64), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  Prepartition\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `Prepartition` 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=38), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  TaggedPrepartition\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  J\n\nHint: The identifier `TaggedPrepartition` 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=65), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `⟨...⟩` notation: The expected type of this term could not be determined', pos=Position(line=5, column=7), pos_end=Position(line=7, column=31), 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.Combinatorics.Enumerative.IncidenceAlgebra).

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

```lean
lemma zeta_mul_zeta [NonAssocSemiring 𝕜] [Preorder α] [LocallyFiniteOrder α] [DecidableLE α]
    (a b : α) : (zeta 𝕜 * zeta 𝕜 : IncidenceAlgebra 𝕜 α) a b = (Icc a b).card
:= by .someMissingNamespaceXYZ
rw [mul_apply, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one]
  refine sum_congr rfl fun x hx ↦ ?_
  rw [mem_Icc] at hx
  rw [zeta_of_le hx.1, zeta_of_le hx.2, one_mul]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=196, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  zeta\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  𝕜\n\nHint: The identifier `zeta` 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=17), pos_end=Position(line=2, column=23), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  zeta\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  𝕜\n\nHint: The identifier `zeta` 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=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Icc\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Icc` 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=64), pos_end=Position(line=2, column=71), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n𝕜 : Type u_1\nα : Type u_2\nx✝¹ : Sort u_3\nzeta : x✝¹\nx✝ : Sort u_4\nIcc : x✝\ninst✝³ : NonAssocSemiring 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableLE α\na b : α\n⊢ (sorry * sorry) a b = sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=196, i_end=388, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma zeta_mul_zeta [NonAssocSemiring 𝕜] [Preorder α] [LocallyFiniteOrder α] [DecidableLE α]
    (a b : α) : (zeta 𝕜 * zeta 𝕜 : IncidenceAlgebra 𝕜 α) a b = (Icc a b).card
:= by .someMissingNamespaceXYZ
rw [mul_apply, card_eq_sum_ones, Nat.cast_sum, Nat.cast_one]
  refine sum_congr rfl fun x hx ↦ ?_
  rw [mem_Icc] at hx
  rw [zeta_of_le hx.1, zeta_of_le hx.2, one_mul]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=196, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  zeta\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  𝕜\n\nHint: The identifier `zeta` 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=17), pos_end=Position(line=2, column=23), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  zeta\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  𝕜\n\nHint: The identifier `zeta` 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=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Icc\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Icc` 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=64), pos_end=Position(line=2, column=71), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\n𝕜 : Type u_1\nα : Type u_2\nx✝¹ : Sort u_3\nzeta : x✝¹\nx✝ : Sort u_4\nIcc : x✝\ninst✝³ : NonAssocSemiring 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableLE α\na b : α\n⊢ (sorry * sorry) a b = sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=196, i_end=388, messages=[], 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.GroupTheory.Perm.Subgroup).

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

```lean
theorem sumCongrHom.card_range {α β : Type*} [Fintype (sumCongrHom α β).range]
    [Fintype (Perm α × Perm β)] :
    Fintype.card (sumCongrHom α β).range = Fintype.card (Perm α × Perm β)
:= _unbalancedParenXYZ
Fintype.card_eq.mpr ⟨(ofInjective (sumCongrHom α β) sumCongrHom_injective).symm⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=308, messages=[Message(data="Function expected at\n  sumCongrHom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `sumCongrHom` 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=55), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` 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=14), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  β\n\nHint: The identifier `Perm` 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=23), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  sumCongrHom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `sumCongrHom` 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=3, column=18), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` 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=3, column=57), pos_end=Position(line=3, column=63), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  β\n\nHint: The identifier `Perm` 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=3, column=66), pos_end=Position(line=3, column=72), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=4, column=3), pos_end=Position(line=4, 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.GroupTheory.Perm.Subgroup).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem sumCongrHom.card_range {α β : Type*} [Fintype (sumCongrHom α β).range]
    [Fintype (Perm α × Perm β)] :
    Fintype.card (sumCongrHom α β).range = Fintype.card (Perm α × Perm β)
:= _unbalancedParenXYZ
Fintype.card_eq.mpr ⟨(ofInjective (sumCongrHom α β) sumCongrHom_injective).symm⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=308, messages=[Message(data="Function expected at\n  sumCongrHom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `sumCongrHom` 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=55), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` 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=14), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  β\n\nHint: The identifier `Perm` 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=23), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  sumCongrHom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `sumCongrHom` 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=3, column=18), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` 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=3, column=57), pos_end=Position(line=3, column=63), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  β\n\nHint: The identifier `Perm` 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=3, column=66), pos_end=Position(line=3, column=72), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=4, column=3), pos_end=Position(line=4, 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]


[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.RingTheory.Ideal.Quotient.Operations).

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

```lean
def quotientMapₐ (f : A →ₐ[R₁] B) (hIJ : I ≤ J.comap f) :
    A ⧸ I →ₐ[R₁] B ⧸ J
:= thisIsWrongLemmaNameXYZ
{ quotientMap J (f : A →+* B) hIJ with commutes' := fun r => by simp only [RingHom.toFun_eq_coe,
    quotientMap_algebraMap, AlgHom.coe_toRingHom, AlgHom.commutes, Quotient.mk_algebraMap] }
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=319, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  CommSemiring R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Invalid field notation: Type of\n  J\nis not known; cannot resolve field `comap`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddCon.comap`\n  • `AddSubgroup.comap`\n  • `AddSubmonoid.comap`\n  • `AddSubsemigroup.comap`\n  • `AddValuation.comap`\n  • `AffineSubspace.comap`\n  • `Archimedean.comap`\n  • `BooleanSubalgebra.comap`\n  • `Cauchy.comap`\n  • `ClosedSubmodule.comap`\n  • `CompleteSublattice.comap`\n  • `Con.comap`\n  • `ConvexCone.comap`\n  • `DFA.comap`\n  • `DiscreteQuotient.comap`\n  • `Equivalence.comap`\n  • `Filter.comap`\n  • `FiniteIndexNormalAddSubgroup.comap`\n  • `FiniteIndexNormalSubgroup.comap`\n  • `HomogeneousIdeal.comap`\n  • `Ideal.comap`\n  • `IntermediateField.comap`\n  • `IsPiSystem.comap`\n  • `IsTrans.comap`\n  • `IsUltraUniformity.comap`\n  • `IsUniformAddGroup.comap`\n  • `IsUniformGroup.comap`\n  • `LTSeries.comap`\n  • `LieIdeal.comap`\n  • `LieSubalgebra.comap`\n  • `LieSubmodule.comap`\n  • `LocallyConstant.comap`\n  • `Matroid.comap`\n  • `MeasurableSpace.comap`\n  • `MulArchimedean.comap`\n  • `MvPolynomial.comap`\n  • `NonUnitalStarSubalgebra.comap`\n  • `NonUnitalSubalgebra.comap`\n  • `NonUnitalSubring.comap`\n  • `NonUnitalSubsemiring.comap`\n  • `OpenAddSubgroup.comap`\n  • `OpenSubgroup.comap`\n  • `PointedCone.comap`\n  • `PrimeSpectrum.comap`\n  • `ProperCone.comap`\n  • `Reflexive.comap`\n  • `RingCon.comap`\n  • `SModEq.comap`\n  • `Setoid.comap`\n  • `SimpleGraph.comap`\n  • `SingularManifold.comap`\n  • `StarSubalgebra.comap`\n  • `Subalgebra.comap`\n  • `Subfield.comap`\n  • `Subgroup.comap`\n  • `Sublattice.comap`\n  • `Submodule.comap`\n  • `Submonoid.comap`\n  • `Subring.comap`\n  • `Subsemigroup.comap`\n  • `Subsemiring.comap`\n  • `SummationFilter.comap`\n  • `Symmetric.comap`\n  • `Transitive.comap`\n  • `TwoSidedIdeal.comap`\n  • `Ultrafilter.comap`\n  • `UniformSpace.comap`\n  • `Valuation.comap`\n  • `ValuationSubring.comap`\n  • `ZLattice.comap`\n  • `AddSubgroup.IsSubnormal.comap`\n  • `AddSubgroup.Normal.comap`\n  • `AddValuation.IsEquiv.comap`\n  • `AlgebraicGeometry.ProjectiveSpectrum.comap`\n  • `AlgebraicGeometry.StructureSheaf.comap`\n  • `CategoryTheory.GradedObject.comap`\n  • `CategoryTheory.Mod_.comap`\n  • `CategoryTheory.Pi.comap`\n  • `CategoryTheory.Precoverage.comap`\n  • `CategoryTheory.Subgroupoid.comap`\n  • `Filter.Eventually.comap`\n  • `Filter.HasAntitoneBasis.comap`\n  • `Filter.HasBasis.comap`\n  • `Filter.Realizer.comap`\n  • `Ideal.IsPrimary.comap`\n  • `Ideal.IsPrime.comap`\n  • `Ideal.IsRadical.comap`\n  • `IsDedekindDomain.HeightOneSpectrum.comap`\n  • `MeasurableSpace.CountablyGenerated.comap`\n  • `MeasureTheory.FiniteMeasure.comap`\n  • `MeasureTheory.Measure.comap`\n  • `MeasureTheory.OuterMeasure.comap`\n  • `Module.IsTorsionFree.comap`\n  • `NumberField.InfinitePlace.comap`\n  • `ProbabilityTheory.Kernel.comap`\n  • `SimpleGraph.CliqueFree.comap`\n  • `SimpleGraph.Embedding.comap`\n  • `SimpleGraph.Hom.comap`\n  • `SimpleGraph.IsAcyclic.comap`\n  • `SimpleGraph.IsCompleteMultipartite.comap`\n  • `SimpleGraph.Iso.comap`\n  • `SimpleGraph.Subgraph.comap`\n  • `Subgroup.IsSubnormal.comap`\n  • `Subgroup.Normal.comap`\n  • `Submodule.IsOrtho.comap`\n  • `TopologicalSpace.IsOpenCover.comap`\n  • `TopologicalSpace.OpenNhdsOf.comap`\n  • `TopologicalSpace.Opens.comap`\n  • `Valuation.IsEquiv.comap`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.comap`\n  • `CategoryTheory.GrothendieckTopology.Point.comap`\n  • `FirstOrder.Language.Substructure.comap`\n  • `MeasureTheory.Measure.InnerRegular.comap`\n  • `MeasureTheory.Measure.InnerRegularWRT.comap`\n  • `MeasureTheory.Measure.IsAddHaarMeasure.comap`\n  • `MeasureTheory.Measure.IsAddLeftInvariant.comap`\n  • `MeasureTheory.Measure.IsAddRightInvariant.comap`\n  • `MeasureTheory.Measure.IsFiniteMeasureOnCompacts.comap`\n  • `MeasureTheory.Measure.IsHaarMeasure.comap`\n  • `MeasureTheory.Measure.IsMulLeftInvariant.comap`\n  • `MeasureTheory.Measure.IsMulRightInvariant.comap`\n  • `MeasureTheory.Measure.IsOpenPosMeasure.comap`\n  • `MeasureTheory.Measure.OuterRegular.comap`\n  • `MeasureTheory.Measure.Regular.comap`\n  • `NumberField.InfinitePlace.IsReal.comap`\n  • `NumberField.InfinitePlace.IsUnramified.comap`\n  • `ProbabilityTheory.Kernel.IsFiniteKernel.comap`\n  • `ProbabilityTheory.Kernel.IsMarkovKernel.comap`\n  • `ProbabilityTheory.Kernel.IsSFiniteKernel.comap`\n  • `ProbabilityTheory.Kernel.IsZeroOrMarkovKernel.comap`', pos=Position(line=1, column=45), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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.RingTheory.Ideal.Quotient.Operations).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def quotientMapₐ (f : A →ₐ[R₁] B) (hIJ : I ≤ J.comap f) :
    A ⧸ I →ₐ[R₁] B ⧸ J
:= thisIsWrongLemmaNameXYZ
{ quotientMap J (f : A →+* B) hIJ with commutes' := fun r => by simp only [RingHom.toFun_eq_coe,
    quotientMap_algebraMap, AlgHom.coe_toRingHom, AlgHom.commutes, Quotient.mk_algebraMap] }
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=319, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  CommSemiring R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Invalid field notation: Type of\n  J\nis not known; cannot resolve field `comap`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddCon.comap`\n  • `AddSubgroup.comap`\n  • `AddSubmonoid.comap`\n  • `AddSubsemigroup.comap`\n  • `AddValuation.comap`\n  • `AffineSubspace.comap`\n  • `Archimedean.comap`\n  • `BooleanSubalgebra.comap`\n  • `Cauchy.comap`\n  • `ClosedSubmodule.comap`\n  • `CompleteSublattice.comap`\n  • `Con.comap`\n  • `ConvexCone.comap`\n  • `DFA.comap`\n  • `DiscreteQuotient.comap`\n  • `Equivalence.comap`\n  • `Filter.comap`\n  • `FiniteIndexNormalAddSubgroup.comap`\n  • `FiniteIndexNormalSubgroup.comap`\n  • `HomogeneousIdeal.comap`\n  • `Ideal.comap`\n  • `IntermediateField.comap`\n  • `IsPiSystem.comap`\n  • `IsTrans.comap`\n  • `IsUltraUniformity.comap`\n  • `IsUniformAddGroup.comap`\n  • `IsUniformGroup.comap`\n  • `LTSeries.comap`\n  • `LieIdeal.comap`\n  • `LieSubalgebra.comap`\n  • `LieSubmodule.comap`\n  • `LocallyConstant.comap`\n  • `Matroid.comap`\n  • `MeasurableSpace.comap`\n  • `MulArchimedean.comap`\n  • `MvPolynomial.comap`\n  • `NonUnitalStarSubalgebra.comap`\n  • `NonUnitalSubalgebra.comap`\n  • `NonUnitalSubring.comap`\n  • `NonUnitalSubsemiring.comap`\n  • `OpenAddSubgroup.comap`\n  • `OpenSubgroup.comap`\n  • `PointedCone.comap`\n  • `PrimeSpectrum.comap`\n  • `ProperCone.comap`\n  • `Reflexive.comap`\n  • `RingCon.comap`\n  • `SModEq.comap`\n  • `Setoid.comap`\n  • `SimpleGraph.comap`\n  • `SingularManifold.comap`\n  • `StarSubalgebra.comap`\n  • `Subalgebra.comap`\n  • `Subfield.comap`\n  • `Subgroup.comap`\n  • `Sublattice.comap`\n  • `Submodule.comap`\n  • `Submonoid.comap`\n  • `Subring.comap`\n  • `Subsemigroup.comap`\n  • `Subsemiring.comap`\n  • `SummationFilter.comap`\n  • `Symmetric.comap`\n  • `Transitive.comap`\n  • `TwoSidedIdeal.comap`\n  • `Ultrafilter.comap`\n  • `UniformSpace.comap`\n  • `Valuation.comap`\n  • `ValuationSubring.comap`\n  • `ZLattice.comap`\n  • `AddSubgroup.IsSubnormal.comap`\n  • `AddSubgroup.Normal.comap`\n  • `AddValuation.IsEquiv.comap`\n  • `AlgebraicGeometry.ProjectiveSpectrum.comap`\n  • `AlgebraicGeometry.StructureSheaf.comap`\n  • `CategoryTheory.GradedObject.comap`\n  • `CategoryTheory.Mod_.comap`\n  • `CategoryTheory.Pi.comap`\n  • `CategoryTheory.Precoverage.comap`\n  • `CategoryTheory.Subgroupoid.comap`\n  • `Filter.Eventually.comap`\n  • `Filter.HasAntitoneBasis.comap`\n  • `Filter.HasBasis.comap`\n  • `Filter.Realizer.comap`\n  • `Ideal.IsPrimary.comap`\n  • `Ideal.IsPrime.comap`\n  • `Ideal.IsRadical.comap`\n  • `IsDedekindDomain.HeightOneSpectrum.comap`\n  • `MeasurableSpace.CountablyGenerated.comap`\n  • `MeasureTheory.FiniteMeasure.comap`\n  • `MeasureTheory.Measure.comap`\n  • `MeasureTheory.OuterMeasure.comap`\n  • `Module.IsTorsionFree.comap`\n  • `NumberField.InfinitePlace.comap`\n  • `ProbabilityTheory.Kernel.comap`\n  • `SimpleGraph.CliqueFree.comap`\n  • `SimpleGraph.Embedding.comap`\n  • `SimpleGraph.Hom.comap`\n  • `SimpleGraph.IsAcyclic.comap`\n  • `SimpleGraph.IsCompleteMultipartite.comap`\n  • `SimpleGraph.Iso.comap`\n  • `SimpleGraph.Subgraph.comap`\n  • `Subgroup.IsSubnormal.comap`\n  • `Subgroup.Normal.comap`\n  • `Submodule.IsOrtho.comap`\n  • `TopologicalSpace.IsOpenCover.comap`\n  • `TopologicalSpace.OpenNhdsOf.comap`\n  • `TopologicalSpace.Opens.comap`\n  • `Valuation.IsEquiv.comap`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.comap`\n  • `CategoryTheory.GrothendieckTopology.Point.comap`\n  • `FirstOrder.Language.Substructure.comap`\n  • `MeasureTheory.Measure.InnerRegular.comap`\n  • `MeasureTheory.Measure.InnerRegularWRT.comap`\n  • `MeasureTheory.Measure.IsAddHaarMeasure.comap`\n  • `MeasureTheory.Measure.IsAddLeftInvariant.comap`\n  • `MeasureTheory.Measure.IsAddRightInvariant.comap`\n  • `MeasureTheory.Measure.IsFiniteMeasureOnCompacts.comap`\n  • `MeasureTheory.Measure.IsHaarMeasure.comap`\n  • `MeasureTheory.Measure.IsMulLeftInvariant.comap`\n  • `MeasureTheory.Measure.IsMulRightInvariant.comap`\n  • `MeasureTheory.Measure.IsOpenPosMeasure.comap`\n  • `MeasureTheory.Measure.OuterRegular.comap`\n  • `MeasureTheory.Measure.Regular.comap`\n  • `NumberField.InfinitePlace.IsReal.comap`\n  • `NumberField.InfinitePlace.IsUnramified.comap`\n  • `ProbabilityTheory.Kernel.IsFiniteKernel.comap`\n  • `ProbabilityTheory.Kernel.IsMarkovKernel.comap`\n  • `ProbabilityTheory.Kernel.IsSFiniteKernel.comap`\n  • `ProbabilityTheory.Kernel.IsZeroOrMarkovKernel.comap`', pos=Position(line=1, column=45), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.AffineEquiv).

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

```lean
def arrowCongr : (P₁ →ᵃ[R] P₃) ≃ᵃ[R] (P₂ →ᵃ[R] P₄) where
  __
:= e₁.arrowCongrEquiv e₂ .someMissingNamespaceXYZ
linear := e₁.arrowCongrₗ e₂.linear
  map_vadd' _ _ := by ext; simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=136, messages=[Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=18), pos_end=Position(line=1, column=29), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=17), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `e₁.arrowCongrEquiv`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='expected structure', pos=Position(line=1, column=51), pos_end=Position(line=4, column=0), severity=<Severity.ERROR: 3>, kind=None), Message(data="(kernel) declaration has metavariables 'arrowCongr'", pos=Position(line=1, column=4), pos_end=Position(line=1, column=14), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=136, i_end=208, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), 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.LinearAlgebra.AffineSpace.AffineEquiv).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def arrowCongr : (P₁ →ᵃ[R] P₃) ≃ᵃ[R] (P₂ →ᵃ[R] P₄) where
  __
:= e₁.arrowCongrEquiv e₂ .someMissingNamespaceXYZ
linear := e₁.arrowCongrₗ e₂.linear
  map_vadd' _ _ := by ext; simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=136, messages=[Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=18), pos_end=Position(line=1, column=29), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Ring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=17), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `e₁.arrowCongrEquiv`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='expected structure', pos=Position(line=1, column=51), pos_end=Position(line=4, column=0), severity=<Severity.ERROR: 3>, kind=None), Message(data="(kernel) declaration has metavariables 'arrowCongr'", pos=Position(line=1, column=4), pos_end=Position(line=1, column=14), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=136, i_end=208, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), 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.NumberTheory.Pell).

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

```lean
theorem d_pos_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : 0 < d
:= by .someMissingNamespaceXYZ
refine pos_of_mul_pos_left ?_ (sq_nonneg a.y)
  rw [a.prop_y, sub_pos]
  exact one_lt_pow₀ ha two_ne_zero

/-- If a solution has `x > 1`, then `d` is not a square. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=77, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Solution₁\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  d\n\nHint: The identifier `Solution₁` 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=31), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nSolution₁ : x✝\nd : ℕ\na : sorry\nha : 1 < sorry\n⊢ 0 < d', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=77, i_end=268, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=59), 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.NumberTheory.Pell).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem d_pos_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : 0 < d
:= by .someMissingNamespaceXYZ
refine pos_of_mul_pos_left ?_ (sq_nonneg a.y)
  rw [a.prop_y, sub_pos]
  exact one_lt_pow₀ ha two_ne_zero

/-- If a solution has `x > 1`, then `d` is not a square. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=77, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Solution₁\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  d\n\nHint: The identifier `Solution₁` 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=31), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nSolution₁ : x✝\nd : ℕ\na : sorry\nha : 1 < sorry\n⊢ 0 < d', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=77, i_end=268, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=59), 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]