[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.GroupAction.FixedPoints).

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

```lean
theorem set_mem_fixedBy_of_subset_fixedBy {s : Set α} {g : G} (s_ss_fixedBy : s ⊆ fixedBy α g) :
    s ∈ fixedBy (Set α) g
:= by thisIsWrongLemmaNameXYZ
rw [← fixedBy_inv]
  ext x
  rw [Set.mem_inv_smul_set_iff]
  refine ⟨fun gxs => ?xs, fun xs => (s_ss_fixedBy xs).symm ▸ xs⟩
  rw [← fixedBy_inv] at s_ss_fixedBy
  rwa [← s_ss_fixedBy gxs, inv_smul_smul] at gxs
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  fixedBy\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `fixedBy` 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=82), pos_end=Position(line=1, column=93), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  fixedBy\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (Set α)\n\nHint: The identifier `fixedBy` 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=8), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα : Type u_1\nG : Sort u_2\nx✝ : Sort u_3\nfixedBy : x✝\ns : Set α\ng : G\ns_ss_fixedBy : s ⊆ sorry\n⊢ s ∈ sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=381, 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.GroupTheory.GroupAction.FixedPoints).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem set_mem_fixedBy_of_subset_fixedBy {s : Set α} {g : G} (s_ss_fixedBy : s ⊆ fixedBy α g) :
    s ∈ fixedBy (Set α) g
:= by thisIsWrongLemmaNameXYZ
rw [← fixedBy_inv]
  ext x
  rw [Set.mem_inv_smul_set_iff]
  refine ⟨fun gxs => ?xs, fun xs => (s_ss_fixedBy xs).symm ▸ xs⟩
  rw [← fixedBy_inv] at s_ss_fixedBy
  rwa [← s_ss_fixedBy gxs, inv_smul_smul] at gxs
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  fixedBy\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `fixedBy` 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=82), pos_end=Position(line=1, column=93), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  fixedBy\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (Set α)\n\nHint: The identifier `fixedBy` 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=8), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα : Type u_1\nG : Sort u_2\nx✝ : Sort u_3\nfixedBy : x✝\ns : Set α\ng : G\ns_ss_fixedBy : s ⊆ sorry\n⊢ s ∈ sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=381, 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.RingTheory.SimpleModule.Isotypic).

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

```lean
theorem LinearEquiv.isIsotypicOfType_iff (e : M ≃ₗ[R] N) :
    IsIsotypicOfType R M S ↔ IsIsotypicOfType R N S
  frobnicateTacticThatDoesNotExist :=
⟨(·.of_injective _ e.symm.injective), (·.of_injective _ e.injective)⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=230, messages=[Message(data='`LinearEquiv.isIsotypicOfType_iff` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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

```lean
theorem LinearEquiv.isIsotypicOfType_iff (e : M ≃ₗ[R] N) :
    IsIsotypicOfType R M S ↔ IsIsotypicOfType R N S
  frobnicateTacticThatDoesNotExist :=
⟨(·.of_injective _ e.symm.injective), (·.of_injective _ e.injective)⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=230, messages=[Message(data='`LinearEquiv.isIsotypicOfType_iff` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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.LinearAlgebra.Dimension.FreeAndStrongRankCondition).

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

```lean
theorem Submodule.rank_le_one_iff_isPrincipal (W : Submodule K V) [Module.Free K W] :
    Module.rank K W ≤ 1 ↔ W.IsPrincipal
:= by thisIsWrongLemmaNameXYZ
simp only [rank_le_one_iff, Submodule.isPrincipal_iff, le_antisymm_iff, le_span_singleton_iff,
    span_singleton_le_iff_mem]
  constructor
  · rintro ⟨⟨m, hm⟩, hm'⟩
    choose f hf using hm'
    exact ⟨m, ⟨fun v hv => ⟨f ⟨v, hv⟩, congr_arg ((↑) : W → V) (hf ⟨v, hv⟩)⟩, hm⟩⟩
  · rintro ⟨a, ⟨h, ha⟩⟩
    choose f hf using h
    exact ⟨⟨a, ha⟩, fun v => ⟨f v.1 v.2, Subtype.ext (hf v.1 v.2)⟩⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=604, 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.Dimension.FreeAndStrongRankCondition).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Submodule.rank_le_one_iff_isPrincipal (W : Submodule K V) [Module.Free K W] :
    Module.rank K W ≤ 1 ↔ W.IsPrincipal
:= by thisIsWrongLemmaNameXYZ
simp only [rank_le_one_iff, Submodule.isPrincipal_iff, le_antisymm_iff, le_span_singleton_iff,
    span_singleton_le_iff_mem]
  constructor
  · rintro ⟨⟨m, hm⟩, hm'⟩
    choose f hf using hm'
    exact ⟨m, ⟨fun v hv => ⟨f ⟨v, hv⟩, congr_arg ((↑) : W → V) (hf ⟨v, hv⟩)⟩, hm⟩⟩
  · rintro ⟨a, ⟨h, ha⟩⟩
    choose f hf using h
    exact ⟨⟨a, ha⟩, fun v => ⟨f v.1 v.2, Subtype.ext (hf v.1 v.2)⟩⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=604, 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.LegendreSymbol.AddCharacter).

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

```lean
theorem IsPrimitive.of_ne_one {F : Type u} [Field F] {ψ : AddChar F R'} (hψ : ψ ≠ 1) :
    IsPrimitive ψ
:= ⟨⟩⟨⟩
fun a ha h ↦ hψ <| by simpa [mulShift_mulShift, ha] using congr_arg (mulShift · a⁻¹) h

/-- If `r` is not a unit, then `e.mulShift r` is not primitive. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data="failed to synthesize instance of type class\n  Monoid 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=58), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  IsPrimitive\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  ψ\n\nHint: The identifier `IsPrimitive` 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=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=288, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=3, column=5), pos_end=Position(line=3, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=66), 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.LegendreSymbol.AddCharacter).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsPrimitive.of_ne_one {F : Type u} [Field F] {ψ : AddChar F R'} (hψ : ψ ≠ 1) :
    IsPrimitive ψ
:= ⟨⟩⟨⟩
fun a ha h ↦ hψ <| by simpa [mulShift_mulShift, ha] using congr_arg (mulShift · a⁻¹) h

/-- If `r` is not a unit, then `e.mulShift r` is not primitive. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data="failed to synthesize instance of type class\n  Monoid 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=58), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  IsPrimitive\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  ψ\n\nHint: The identifier `IsPrimitive` 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=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=288, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=3, column=5), pos_end=Position(line=3, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=66), 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.Algebra.CubicDiscriminant).

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

```lean
theorem ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly ≠ 0
:= thisIsWrongLemmaNameXYZ
(or_imp.mp ne_zero).1 ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=114, messages=[Message(data='Unknown identifier `P.a`', pos=Position(line=1, column=35), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `P.toPoly`', pos=Position(line=1, column=46), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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.Algebra.CubicDiscriminant).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly ≠ 0
:= thisIsWrongLemmaNameXYZ
(or_imp.mp ne_zero).1 ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=114, messages=[Message(data='Unknown identifier `P.a`', pos=Position(line=1, column=35), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `P.toPoly`', pos=Position(line=1, column=46), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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.Data.Set.Basic).

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

```lean
theorem union_empty (a : Set α) : a ∪ ∅ = a
:= thisIsWrongLemmaNameXYZ
ext fun _ => iff_of_eq (or_false _)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=111, messages=[Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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.Data.Set.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem union_empty (a : Set α) : a ∪ ∅ = a
:= thisIsWrongLemmaNameXYZ
ext fun _ => iff_of_eq (or_false _)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=111, messages=[Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, 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.Order.LiminfLimsup).

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

```lean
theorem limsSup_eq_iInf_sSup {f : Filter α} : limsSup f = ⨅ s ∈ f, sSup s
:= .someMissingNamespaceXYZ
f.basis_sets.limsSup_eq_iInf_sSup
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=140, messages=[Message(data='failed to synthesize instance of type class\n  SupSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=67), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  InfSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=58), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  InfSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=58), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  limsSup\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `limsSup` 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=46), pos_end=Position(line=1, column=55), 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.Order.LiminfLimsup).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem limsSup_eq_iInf_sSup {f : Filter α} : limsSup f = ⨅ s ∈ f, sSup s
:= .someMissingNamespaceXYZ
f.basis_sets.limsSup_eq_iInf_sSup
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=140, messages=[Message(data='failed to synthesize instance of type class\n  SupSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=67), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  InfSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=58), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  InfSet α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=58), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  limsSup\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `limsSup` 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=46), pos_end=Position(line=1, column=55), 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.Topology.Algebra.GroupCompletion).

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

```lean
instance instAddCommGroup : AddCommGroup (Completion α)
:= _unbalancedParenXYZ
{ (inferInstance : AddGroup <| Completion α) with
    add_comm a b :=
      Completion.induction_on₂ a b
        (isClosed_eq (by fun_prop) (by fun_prop))
        fun x y ↦ by
        change (x : Completion α) + ↑y = ↑y + ↑x
        rw [← coe_add, ← coe_add, add_comm] }
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=366, messages=[Message(data="Function expected at\n  Completion\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 `Completion` 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=42), pos_end=Position(line=1, column=54), 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)]
```

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

```lean
instance instAddCommGroup : AddCommGroup (Completion α)
:= _unbalancedParenXYZ
{ (inferInstance : AddGroup <| Completion α) with
    add_comm a b :=
      Completion.induction_on₂ a b
        (isClosed_eq (by fun_prop) (by fun_prop))
        fun x y ↦ by
        change (x : Completion α) + ↑y = ↑y + ↑x
        rw [← coe_add, ← coe_add, add_comm] }
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=366, messages=[Message(data="Function expected at\n  Completion\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 `Completion` 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=42), pos_end=Position(line=1, column=54), 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)]
```

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.Calculus.DerivativeTest).

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

```lean
theorem isLocalMax_of_deriv_deriv_neg (hf : deriv (deriv f) x₀ < 0) (hd : deriv f x₀ = 0)
    (hc : ContinuousAt f x₀) : IsLocalMax f x₀
:= by thisIsWrongLemmaNameXYZ
simpa using isLocalMin_of_deriv_deriv_pos (by simpa) (by simpa) hc.neg |>.neg

end SecondDeriv
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=152, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=152, i_end=269, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=6, column=0), pos_end=Position(line=6, column=15), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem isLocalMax_of_deriv_deriv_neg (hf : deriv (deriv f) x₀ < 0) (hd : deriv f x₀ = 0)
    (hc : ContinuousAt f x₀) : IsLocalMax f x₀
:= by thisIsWrongLemmaNameXYZ
simpa using isLocalMin_of_deriv_deriv_pos (by simpa) (by simpa) hc.neg |>.neg

end SecondDeriv
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=152, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=152, i_end=269, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=6, column=0), pos_end=Position(line=6, column=15), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma connected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :
    H.Connected ↔
      H.verts.Nonempty ∧
        ∀ {u v}, u ∈ H.verts → v ∈ H.verts → ∃ p : G.Walk u v, p.toSubgraph ≤ H
:= by _unbalancedParenXYZ
rw [H.connected_iff, preconnected_iff_forall_exists_walk_subgraph, and_comm]

end Subgraph

section induced_subgraphs

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=215, messages=[Message(data='unknown tactic', pos=Position(line=5, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `G.Subgraph`', pos=Position(line=1, column=53), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `G.Walk`', pos=Position(line=4, column=51), pos_end=Position(line=4, column=57), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nH : sorry\n⊢ sorry ↔ sorry ∧ ∀ {u : ?m.18 H} {v : ?m.19 H}, u ∈ sorry → v ∈ sorry → ∃ p, sorry', pos=Position(line=5, column=3), pos_end=Position(line=5, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=215, i_end=326, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=353, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=353, i_end=409, messages=[Message(data='unexpected end of input', pos=Position(line=12, 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.Combinatorics.SimpleGraph.Connectivity.Subgraph).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma connected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :
    H.Connected ↔
      H.verts.Nonempty ∧
        ∀ {u v}, u ∈ H.verts → v ∈ H.verts → ∃ p : G.Walk u v, p.toSubgraph ≤ H
:= by _unbalancedParenXYZ
rw [H.connected_iff, preconnected_iff_forall_exists_walk_subgraph, and_comm]

end Subgraph

section induced_subgraphs

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=215, messages=[Message(data='unknown tactic', pos=Position(line=5, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `G.Subgraph`', pos=Position(line=1, column=53), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `G.Walk`', pos=Position(line=4, column=51), pos_end=Position(line=4, column=57), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nH : sorry\n⊢ sorry ↔ sorry ∧ ∀ {u : ?m.18 H} {v : ?m.19 H}, u ∈ sorry → v ∈ sorry → ∃ p, sorry', pos=Position(line=5, column=3), pos_end=Position(line=5, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=215, i_end=326, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=353, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=353, i_end=409, messages=[Message(data='unexpected end of input', pos=Position(line=12, 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.GroupTheory.Subgroup.Centralizer).

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

```lean
theorem le_centralizer_iff_isMulCommutative : K ≤ centralizer K ↔ IsMulCommutative K
:= _unbalancedParenXYZ
⟨fun h => ⟨⟨fun x y => Subtype.ext (h y.2 x x.2)⟩⟩,
    fun h x hx y hy => congr_arg Subtype.val (h.1.1 ⟨y, hy⟩ ⟨x, hx⟩)⟩

variable (H)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=255, messages=[Message(data='failed to synthesize instance of type class\n  Mul 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=66), pos_end=Position(line=1, column=84), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Type ?u.335)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  centralizer\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `centralizer` 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=50), pos_end=Position(line=1, column=63), 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=255, i_end=267, 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.GroupTheory.Subgroup.Centralizer).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem le_centralizer_iff_isMulCommutative : K ≤ centralizer K ↔ IsMulCommutative K
:= _unbalancedParenXYZ
⟨fun h => ⟨⟨fun x y => Subtype.ext (h y.2 x x.2)⟩⟩,
    fun h x hx y hy => congr_arg Subtype.val (h.1.1 ⟨y, hy⟩ ⟨x, hx⟩)⟩

variable (H)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=255, messages=[Message(data='failed to synthesize instance of type class\n  Mul 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=66), pos_end=Position(line=1, column=84), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Type ?u.335)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=63), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  centralizer\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `centralizer` 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=50), pos_end=Position(line=1, column=63), 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=255, i_end=267, 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.RingTheory.Int.Basic).

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

```lean
theorem prime_iff_natAbs_prime {k : ℤ} : Prime k ↔ Nat.Prime k.natAbs
  frobnicateTacticThatDoesNotExist :=
(Int.associated_natAbs k).prime_iff.trans Nat.prime_iff_prime_int.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=182, messages=[Message(data='Function expected at\n  Nat.Prime k.natAbs\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=51), pos_end=Position(line=2, column=34), 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.RingTheory.Int.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem prime_iff_natAbs_prime {k : ℤ} : Prime k ↔ Nat.Prime k.natAbs
  frobnicateTacticThatDoesNotExist :=
(Int.associated_natAbs k).prime_iff.trans Nat.prime_iff_prime_int.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=182, messages=[Message(data='Function expected at\n  Nat.Prime k.natAbs\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=51), pos_end=Position(line=2, column=34), 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.LinearAlgebra.Projection).

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

```lean
theorem ofIsCompl_smul {R : Type*} [CommRing R] {E : Type*} [AddCommGroup E] [Module R E]
    {F : Type*} [AddCommGroup F] [Module R F] {p q : Submodule R E} (h : IsCompl p q)
    {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (c : R) : ofIsCompl h (c • φ) (c • ψ) = c • ofIsCompl h φ ψ
  frobnicateTacticThatDoesNotExist :=
ofIsCompl_eq _ (by simp) (by simp)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=364, messages=[Message(data="Function expected at\n  ofIsCompl\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  h\n\nHint: The identifier `ofIsCompl` 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=46), pos_end=Position(line=3, column=73), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ofIsCompl\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  h\n\nHint: The identifier `ofIsCompl` 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=80), pos_end=Position(line=4, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `ofIsCompl_eq`', pos=Position(line=5, column=0), pos_end=Position(line=5, column=12), 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.LinearAlgebra.Projection).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem ofIsCompl_smul {R : Type*} [CommRing R] {E : Type*} [AddCommGroup E] [Module R E]
    {F : Type*} [AddCommGroup F] [Module R F] {p q : Submodule R E} (h : IsCompl p q)
    {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (c : R) : ofIsCompl h (c • φ) (c • ψ) = c • ofIsCompl h φ ψ
  frobnicateTacticThatDoesNotExist :=
ofIsCompl_eq _ (by simp) (by simp)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=364, messages=[Message(data="Function expected at\n  ofIsCompl\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  h\n\nHint: The identifier `ofIsCompl` 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=46), pos_end=Position(line=3, column=73), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ofIsCompl\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  h\n\nHint: The identifier `ofIsCompl` 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=80), pos_end=Position(line=4, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `ofIsCompl_eq`', pos=Position(line=5, column=0), pos_end=Position(line=5, column=12), 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.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody).

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

```lean
theorem minkowskiBound_lt_top : minkowskiBound K I < ⊤
:= by .someMissingNamespaceXYZ
classical
  -- FIXME: Make `finiteness` work here
  exact ENNReal.mul_lt_top (fundamentalDomain_isBounded _).measure_lt_top <|
    ENNReal.pow_lt_top ENNReal.ofNat_lt_top
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=65, 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  minkowskiBound\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 `minkowskiBound` 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=32), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nminkowskiBound : x✝\n⊢ sorry < ⊤', 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=65, i_end=258, 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.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem minkowskiBound_lt_top : minkowskiBound K I < ⊤
:= by .someMissingNamespaceXYZ
classical
  -- FIXME: Make `finiteness` work here
  exact ENNReal.mul_lt_top (fundamentalDomain_isBounded _).measure_lt_top <|
    ENNReal.pow_lt_top ENNReal.ofNat_lt_top
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=65, 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  minkowskiBound\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 `minkowskiBound` 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=32), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nminkowskiBound : x✝\n⊢ sorry < ⊤', 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=65, i_end=258, 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.Algebra.Order.GroupWithZero.Unbundled.Basic).

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

```lean
lemma inv_lt_one_iff₀ : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a
:= by thisIsWrongLemmaNameXYZ
simp_rw [← not_le, one_le_inv_iff₀, not_and_or, not_lt]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=66, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=66, i_end=148, 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.Algebra.Order.GroupWithZero.Unbundled.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma inv_lt_one_iff₀ : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a
:= by thisIsWrongLemmaNameXYZ
simp_rw [← not_le, one_le_inv_iff₀, not_and_or, not_lt]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=66, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=66, i_end=148, 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.Data.Finsupp.Single).

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

```lean
theorem zipWith_single_single (f : M → N → P) (hf : f 0 0 = 0) (a : α) (m : M) (n : N) :
    zipWith f hf (single a m) (single a n) = single a (f m n)
:= by ⟨⟩⟨⟩
classical
  grind

end ZipWith
end Finsupp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=3, column=5), pos_end=Position(line=3, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  OfNat M 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  M\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=54), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat N 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  N\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=56), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat P 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  P\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=60), pos_end=Position(line=1, column=61), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  zipWith\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 `zipWith` 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=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  single\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `single` 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=45), pos_end=Position(line=2, column=61), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Sort u_4\nx✝¹ : Sort u_5\nzipWith : x✝¹\nx✝ : Sort u_6\nsingle : x✝\nf : M → N → P\nhf : f sorry sorry = 0\na : α\nm : M\nn : N\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=162, i_end=206, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=206, i_end=217, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem zipWith_single_single (f : M → N → P) (hf : f 0 0 = 0) (a : α) (m : M) (n : N) :
    zipWith f hf (single a m) (single a n) = single a (f m n)
:= by ⟨⟩⟨⟩
classical
  grind

end ZipWith
end Finsupp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=3, column=5), pos_end=Position(line=3, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  OfNat M 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  M\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=54), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat N 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  N\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=56), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat P 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  P\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=60), pos_end=Position(line=1, column=61), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  zipWith\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 `zipWith` 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=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  single\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `single` 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=45), pos_end=Position(line=2, column=61), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Sort u_4\nx✝¹ : Sort u_5\nzipWith : x✝¹\nx✝ : Sort u_6\nsingle : x✝\nf : M → N → P\nhf : f sorry sorry = 0\na : α\nm : M\nn : N\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=162, i_end=206, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=206, i_end=217, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem ordConnectedComponent_subset : ordConnectedComponent s x ⊆ s
:= fun _ hy => .someMissingNamespaceXYZ
hy right_mem_uIcc
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data="Function expected at\n  ordConnectedComponent\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `ordConnectedComponent` 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=39), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid dotted identifier notation: The expected type of `.someMissingNamespaceXYZ` could not be determined', pos=Position(line=2, column=15), pos_end=Position(line=2, column=39), severity=<Severity.ERROR: 3>, kind='lean.invalidDottedIdent._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.Interval.Set.OrdConnectedComponent).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem ordConnectedComponent_subset : ordConnectedComponent s x ⊆ s
:= fun _ hy => .someMissingNamespaceXYZ
hy right_mem_uIcc
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data="Function expected at\n  ordConnectedComponent\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `ordConnectedComponent` 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=39), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid dotted identifier notation: The expected type of `.someMissingNamespaceXYZ` could not be determined', pos=Position(line=2, column=15), pos_end=Position(line=2, column=39), severity=<Severity.ERROR: 3>, kind='lean.invalidDottedIdent._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.Separation.SeparatedNhds).

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

```lean
theorem preimage [TopologicalSpace Y] {f : X → Y} {s t : Set Y} (h : SeparatedNhds s t)
    (hf : Continuous f) : SeparatedNhds (f ⁻¹' s) (f ⁻¹' t)
:= thisIsWrongLemmaNameXYZ
let ⟨U, V, oU, oV, sU, tV, UV⟩ := h
  ⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
    UV.preimage f⟩

protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t :=
  let ⟨_, _, _, _, hsU, htV, hd⟩ := h; hd.mono hsU htV
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[Message(data='failed to synthesize instance of type class\n  TopologicalSpace X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=10), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  TopologicalSpace X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=26), pos_end=Position(line=2, column=59), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._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), CompilationUnit(i_begin=183, i_end=469, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data='protected declarations must be in a namespace', pos=Position(line=8, column=18), pos_end=Position(line=8, column=26), 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.Separation.SeparatedNhds).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem preimage [TopologicalSpace Y] {f : X → Y} {s t : Set Y} (h : SeparatedNhds s t)
    (hf : Continuous f) : SeparatedNhds (f ⁻¹' s) (f ⁻¹' t)
:= thisIsWrongLemmaNameXYZ
let ⟨U, V, oU, oV, sU, tV, UV⟩ := h
  ⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
    UV.preimage f⟩

protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t :=
  let ⟨_, _, _, _, hsU, htV, hd⟩ := h; hd.mono hsU htV
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[Message(data='failed to synthesize instance of type class\n  TopologicalSpace X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=10), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  TopologicalSpace X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=26), pos_end=Position(line=2, column=59), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._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), CompilationUnit(i_begin=183, i_end=469, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data='protected declarations must be in a namespace', pos=Position(line=8, column=18), pos_end=Position(line=8, column=26), 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.Convex.Function).

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

```lean
theorem StrictConvexOn.sub_concaveOn (hf : StrictConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) :
    StrictConvexOn 𝕜 s (f - g)
:= thisIsWrongLemmaNameXYZ
(sub_eq_add_neg f g).symm ▸ hf.add_convexOn hg.neg
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=209, messages=[Message(data='`StrictConvexOn.sub_concaveOn` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=36), 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.Convex.Function).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem StrictConvexOn.sub_concaveOn (hf : StrictConvexOn 𝕜 s f) (hg : ConcaveOn 𝕜 s g) :
    StrictConvexOn 𝕜 s (f - g)
:= thisIsWrongLemmaNameXYZ
(sub_eq_add_neg f g).symm ▸ hf.add_convexOn hg.neg
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=209, messages=[Message(data='`StrictConvexOn.sub_concaveOn` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=36), 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.Additive.Convolution).

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

```lean
lemma convolution_op_smul_eq_convolution_mul_inv (A B : Finset G) (s x : G) :
    A.convolution (B <• s) x = A.convolution B (x * s⁻¹)
:= by thisIsWrongLemmaNameXYZ
nth_rw 2 [← inv_inv B]
  rw [← inv_inv (B <• s), inv_op_smul_finset_distrib, ← card_inter_smul, ← card_inter_smul,
    smul_smul]

end Finset
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=100, messages=[Message(data="unexpected token '•'; expected term", pos=Position(line=2, column=22), pos_end=Position(line=2, column=23), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=100, i_end=321, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=10), 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.Combinatorics.Additive.Convolution).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma convolution_op_smul_eq_convolution_mul_inv (A B : Finset G) (s x : G) :
    A.convolution (B <• s) x = A.convolution B (x * s⁻¹)
:= by thisIsWrongLemmaNameXYZ
nth_rw 2 [← inv_inv B]
  rw [← inv_inv (B <• s), inv_op_smul_finset_distrib, ← card_inter_smul, ← card_inter_smul,
    smul_smul]

end Finset
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=100, messages=[Message(data="unexpected token '•'; expected term", pos=Position(line=2, column=22), pos_end=Position(line=2, column=23), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=100, i_end=321, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=10), 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.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups).

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

```lean
theorem isCoatom_stabilizer {s : Set α}
    (h0 : s.Nonempty) (h1 : sᶜ.Nonempty) (hs : Nat.card α ≠ 2 * ncard s) :
    IsCoatom (stabilizer (alternatingGroup α) s)
:= by ⟨⟩⟨⟩
rw [← ncard_add_ncard_compl s, two_mul, ne_eq, Nat.add_left_cancel_iff] at hs
  wlog hs' : ncard s < ncard sᶜ
  · rw [← stabilizer_compl]
    apply this h1 <;> rw [compl_compl] <;> grind
  · by_cases h0' : s.Nontrivial
    · apply isCoatom_stabilizer_of_ncard_lt_ncard_compl h0' hs'
    · simp only [not_nontrivial_iff] at h0'
      apply isCoatom_stabilizer_singleton _ h0 h0'
      rw [← ncard_add_ncard_compl s]
      rw [← ncard_pos] at h0 h1
      grind

end alternatingGroup
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=177, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=4, column=5), pos_end=Position(line=4, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ncard\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `ncard` 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="Function expected at\n  stabilizer\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (alternatingGroup α)\n\nHint: The identifier `stabilizer` 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=14), pos_end=Position(line=3, column=47), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα : Type u_1\nx✝¹ : Sort u_2\nncard : x✝¹\nx✝ : Sort u_3\nstabilizer : x✝\ns : Set α\nh0 : s.Nonempty\nh1 : sᶜ.Nonempty\nhs : Nat.card α ≠ 2 * sorry\n⊢ IsCoatom sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=177, i_end=684, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=17, column=0), pos_end=Position(line=17, column=20), 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.SpecificGroups.Alternating.MaximalSubgroups).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem isCoatom_stabilizer {s : Set α}
    (h0 : s.Nonempty) (h1 : sᶜ.Nonempty) (hs : Nat.card α ≠ 2 * ncard s) :
    IsCoatom (stabilizer (alternatingGroup α) s)
:= by ⟨⟩⟨⟩
rw [← ncard_add_ncard_compl s, two_mul, ne_eq, Nat.add_left_cancel_iff] at hs
  wlog hs' : ncard s < ncard sᶜ
  · rw [← stabilizer_compl]
    apply this h1 <;> rw [compl_compl] <;> grind
  · by_cases h0' : s.Nontrivial
    · apply isCoatom_stabilizer_of_ncard_lt_ncard_compl h0' hs'
    · simp only [not_nontrivial_iff] at h0'
      apply isCoatom_stabilizer_singleton _ h0 h0'
      rw [← ncard_add_ncard_compl s]
      rw [← ncard_pos] at h0 h1
      grind

end alternatingGroup
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=177, messages=[Message(data="unexpected token '⟨'; expected '{' or tactic", pos=Position(line=4, column=5), pos_end=Position(line=4, column=7), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ncard\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `ncard` 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="Function expected at\n  stabilizer\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (alternatingGroup α)\n\nHint: The identifier `stabilizer` 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=14), pos_end=Position(line=3, column=47), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα : Type u_1\nx✝¹ : Sort u_2\nncard : x✝¹\nx✝ : Sort u_3\nstabilizer : x✝\ns : Set α\nh0 : s.Nonempty\nh1 : sᶜ.Nonempty\nhs : Nat.card α ≠ 2 * sorry\n⊢ IsCoatom sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=5), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=177, i_end=684, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=17, column=0), pos_end=Position(line=17, column=20), 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.RootsOfUnity.PrimitiveRoots).

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

```lean
theorem pow_sub_one_eq [IsDomain R] {ζ : R} (hζ : IsPrimitiveRoot ζ k) (hk : 1 < k) :
    ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
  frobnicateTacticThatDoesNotExist := by
rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]

set_option backward.isDefEq.respectTransparency false in
/-- The (additive) monoid equivalence between `ZMod k`
and the powers of a primitive root of unity `ζ`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=390, messages=[Message(data="unexpected token '/--'; expected '{' or tactic", pos=Position(line=7, column=56), pos_end=Position(line=8, column=3), severity=<Severity.ERROR: 3>, kind=None), 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=1, column=24), pos_end=Position(line=1, column=34), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=390, i_end=497, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=51), 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.RingTheory.RootsOfUnity.PrimitiveRoots).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem pow_sub_one_eq [IsDomain R] {ζ : R} (hζ : IsPrimitiveRoot ζ k) (hk : 1 < k) :
    ζ ^ k.pred = -∑ i ∈ range k.pred, ζ ^ i
  frobnicateTacticThatDoesNotExist := by
rw [eq_neg_iff_add_eq_zero, add_comm, ← sum_range_succ, ← Nat.succ_eq_add_one,
    Nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk]

set_option backward.isDefEq.respectTransparency false in
/-- The (additive) monoid equivalence between `ZMod k`
and the powers of a primitive root of unity `ζ`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=390, messages=[Message(data="unexpected token '/--'; expected '{' or tactic", pos=Position(line=7, column=56), pos_end=Position(line=8, column=3), severity=<Severity.ERROR: 3>, kind=None), 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=1, column=24), pos_end=Position(line=1, column=34), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=390, i_end=497, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=51), 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.LinearAlgebra.Matrix.IsDiag).

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

```lean
theorem IsDiag.transpose [Zero α] {A : Matrix n n α} (ha : A.IsDiag) : Aᵀ.IsDiag
  frobnicateTacticThatDoesNotExist := fun _ _ h =>
ha h.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=74, messages=[Message(data="unexpected token 'ᵀ'; expected ':=', 'where' or '|'", pos=Position(line=1, column=72), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=74, i_end=145, 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.Matrix.IsDiag).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsDiag.transpose [Zero α] {A : Matrix n n α} (ha : A.IsDiag) : Aᵀ.IsDiag
  frobnicateTacticThatDoesNotExist := fun _ _ h =>
ha h.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=74, messages=[Message(data="unexpected token 'ᵀ'; expected ':=', 'where' or '|'", pos=Position(line=1, column=72), pos_end=Position(line=1, column=73), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=74, i_end=145, 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.ModularForms.JacobiTheta.Bounds).

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

```lean
def f_nat (k : ℕ) (a t : ℝ) (n : ℕ) : ℝ
:= (n + a) ^ k * exp (-π * (n + a) ^ 2 * t) ⟨⟩⟨⟩
/-- An upper bound for the summand when `0 ≤ a`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=99, messages=[Message(data='Unknown identifier `exp`', pos=Position(line=2, column=17), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=99, i_end=159, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=46), pos_end=Position(line=2, column=47), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=3, column=51), 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.ModularForms.JacobiTheta.Bounds).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def f_nat (k : ℕ) (a t : ℝ) (n : ℕ) : ℝ
:= (n + a) ^ k * exp (-π * (n + a) ^ 2 * t) ⟨⟩⟨⟩
/-- An upper bound for the summand when `0 ≤ a`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=99, messages=[Message(data='Unknown identifier `exp`', pos=Position(line=2, column=17), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=99, i_end=159, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=46), pos_end=Position(line=2, column=47), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=3, column=51), 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.Algebra.CharP.Frobenius).

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

```lean
lemma frobenius_mul : frobenius R p (x * y) = frobenius R p x * frobenius R p y
:= _unbalancedParenXYZ
map_mul (frobenius R p) x y
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data='`frobenius_mul` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=19), 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.CharP.Frobenius).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma frobenius_mul : frobenius R p (x * y) = frobenius R p x * frobenius R p y
:= _unbalancedParenXYZ
map_mul (frobenius R p) x y
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data='`frobenius_mul` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=19), 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.FinEnum).

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

```lean
theorem Finset.mem_enum [DecidableEq α] (s : Finset α) (xs : List α) :
    s ∈ Finset.enum xs ↔ ∀ x ∈ s, x ∈ xs
:= by thisIsWrongLemmaNameXYZ
induction xs generalizing s with
  | nil => simp [enum, eq_empty_iff_forall_notMem]
  | cons x xs ih =>
      simp only [enum, List.bind_eq_flatMap, List.mem_flatMap, List.mem_cons,
        List.not_mem_nil, or_false, ih]
      refine ⟨by aesop, fun hs => ⟨s.erase x, ?_⟩⟩
      simp only [or_iff_not_imp_left] at hs
      simp +contextual [eq_comm (a := s), or_iff_not_imp_left, hs]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=132, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown constant `Finset.enum`', pos=Position(line=2, column=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nxs : List α\n⊢ s ∈ sorry ↔ ∀ x ∈ s, x ∈ xs', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=132, i_end=546, 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.Data.FinEnum).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Finset.mem_enum [DecidableEq α] (s : Finset α) (xs : List α) :
    s ∈ Finset.enum xs ↔ ∀ x ∈ s, x ∈ xs
:= by thisIsWrongLemmaNameXYZ
induction xs generalizing s with
  | nil => simp [enum, eq_empty_iff_forall_notMem]
  | cons x xs ih =>
      simp only [enum, List.bind_eq_flatMap, List.mem_flatMap, List.mem_cons,
        List.not_mem_nil, or_false, ih]
      refine ⟨by aesop, fun hs => ⟨s.erase x, ?_⟩⟩
      simp only [or_iff_not_imp_left] at hs
      simp +contextual [eq_comm (a := s), or_iff_not_imp_left, hs]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=132, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown constant `Finset.enum`', pos=Position(line=2, column=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nxs : List α\n⊢ s ∈ sorry ↔ ∀ x ∈ s, x ∈ xs', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=132, i_end=546, 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.Order.Disjoint).

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

```lean
theorem Disjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Disjoint b d → Disjoint a c
:= .someMissingNamespaceXYZ
fun h _ ha hc ↦ h (ha.trans h₁) (hc.trans h₂)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=167, messages=[Message(data='`Disjoint.mono` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=21), 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.Order.Disjoint).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Disjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Disjoint b d → Disjoint a c
:= .someMissingNamespaceXYZ
fun h _ ha hc ↦ h (ha.trans h₁) (hc.trans h₂)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=167, messages=[Message(data='`Disjoint.mono` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=21), 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]