[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.ArithmeticFunction.Defs).

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

```lean
def natToArithmeticFunction [AddMonoidWithOne R] :
    (ArithmeticFunction ℕ) → (ArithmeticFunction R)
:= .someMissingNamespaceXYZ
fun f => ⟨fun n => ↑(f n), by simp⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=176, messages=[Message(data='Unknown constant `ArithmeticFunction.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  ArithmeticFunction ℕ → ArithmeticFunction R', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `ZeroHom.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  ArithmeticFunction ℕ → ArithmeticFunction R', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), 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.NumberTheory.ArithmeticFunction.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def natToArithmeticFunction [AddMonoidWithOne R] :
    (ArithmeticFunction ℕ) → (ArithmeticFunction R)
:= .someMissingNamespaceXYZ
fun f => ⟨fun n => ↑(f n), by simp⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=176, messages=[Message(data='Unknown constant `ArithmeticFunction.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  ArithmeticFunction ℕ → ArithmeticFunction R', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `ZeroHom.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  ArithmeticFunction ℕ → ArithmeticFunction R', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), 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.Algebra.Algebra.Unitization).

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

```lean
theorem fst_neg [Neg R] [Neg A] (x : Unitization R A) : (-x).fst = -x.fst
:= .someMissingNamespaceXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=105, messages=[Message(data='Unknown constant `Eq.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  (-x).toProd.1 = -x.toProd.1', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), 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.Algebra.Unitization).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem fst_neg [Neg R] [Neg A] (x : Unitization R A) : (-x).fst = -x.fst
:= .someMissingNamespaceXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=105, messages=[Message(data='Unknown constant `Eq.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  (-x).toProd.1 = -x.toProd.1', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), 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]

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.Data.List.Cycle).

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

```lean
instance fintypeNodupNontrivialCycle [Fintype α] :
    Fintype { s : Cycle α // s.Nodup ∧ s.Nontrivial }
:= _unbalancedParenXYZ
Fintype.subtype
    (((Finset.univ : Finset { s : Cycle α // s.Nodup }).map (Function.Embedding.subtype _)).filter
      Cycle.Nontrivial)
    (by simp)

/-- The `s : Cycle α` as a `Finset α`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=287, messages=[Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=287, i_end=330, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=41), 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.Data.List.Cycle).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
instance fintypeNodupNontrivialCycle [Fintype α] :
    Fintype { s : Cycle α // s.Nodup ∧ s.Nontrivial }
:= _unbalancedParenXYZ
Fintype.subtype
    (((Finset.univ : Finset { s : Cycle α // s.Nodup }).map (Function.Embedding.subtype _)).filter
      Cycle.Nontrivial)
    (by simp)

/-- The `s : Cycle α` as a `Finset α`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=287, messages=[Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=287, i_end=330, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=41), 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.Order.ConditionallyCompleteLattice.Indexed).

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

```lean
theorem ciInf_subtype'' {ι} [Nonempty ι] {s : Set ι} (hs : s.Nonempty) {f : ι → α}
    (hf : BddBelow (Set.range fun i : s ↦ f i)) (hf' : ⨅ i : s, f i ≤ sInf ∅) :
    ⨅ i : s, f i = ⨅ (t : ι) (_ : t ∈ s), f t
:= _unbalancedParenXYZ
haveI : Nonempty s := Set.Nonempty.to_subtype hs
  ciInf_subtype hf hf'
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=325, messages=[Message(data="unexpected end of input; expected ';' or line break", pos=Position(line=6, column=22), 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.Order.ConditionallyCompleteLattice.Indexed).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem ciInf_subtype'' {ι} [Nonempty ι] {s : Set ι} (hs : s.Nonempty) {f : ι → α}
    (hf : BddBelow (Set.range fun i : s ↦ f i)) (hf' : ⨅ i : s, f i ≤ sInf ∅) :
    ⨅ i : s, f i = ⨅ (t : ι) (_ : t ∈ s), f t
:= _unbalancedParenXYZ
haveI : Nonempty s := Set.Nonempty.to_subtype hs
  ciInf_subtype hf hf'
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=325, messages=[Message(data="unexpected end of input; expected ';' or line break", pos=Position(line=6, column=22), 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.Topology.Algebra.SeparationQuotient.Basic).

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

```lean
def mkCLM : M →L[R] SeparationQuotient M where
  toFun
:= mk .someMissingNamespaceXYZ
map_add' := mk_add
  map_smul' := mk_smul

variable {R M}

/-- The lift (as a continuous linear map) of `f` with `f x = f y` for `Inseparable x y`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=88, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=12), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  TopologicalSpace M\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=20), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), 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=12), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=88, i_end=147, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=147, i_end=238, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=91), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def mkCLM : M →L[R] SeparationQuotient M where
  toFun
:= mk .someMissingNamespaceXYZ
map_add' := mk_add
  map_smul' := mk_smul

variable {R M}

/-- The lift (as a continuous linear map) of `f` with `f x = f y` for `Inseparable x y`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=88, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=12), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  TopologicalSpace M\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=20), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), 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=12), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=88, i_end=147, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=147, i_end=238, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=91), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
    (T + S).IsPositive
  frobnicateTacticThatDoesNotExist :=
(isPositive_toLinearMap_iff _).mp (hT.toLinearMap.add hS.toLinearMap)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=218, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `isPositive_toLinearMap_iff`', pos=Position(line=4, column=1), pos_end=Position(line=4, column=27), 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.Analysis.InnerProductSpace.Positive).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
    (T + S).IsPositive
  frobnicateTacticThatDoesNotExist :=
(isPositive_toLinearMap_iff _).mp (hT.toLinearMap.add hS.toLinearMap)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=218, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring 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=30), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `isPositive_toLinearMap_iff`', pos=Position(line=4, column=1), pos_end=Position(line=4, column=27), 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.Combinatorics.Additive.DoublingConst).

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

```lean
lemma divConst_inv_left (A B : Finset G) : δₘ[A⁻¹, B] = σₘ[A, B]
  frobnicateTacticThatDoesNotExist := by
rw [mulConst, divConst, card_inv, ← card_inv, inv_div, div_inv_eq_mul, mul_comm]

/-- If `A` has small difference, then it has small doubling, with the constant squared.

This is a consequence of the Ruzsa triangle inequality. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=55, messages=[Message(data="unexpected token ','; expected ':', ']' or ']''", pos=Position(line=1, column=49), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=55, i_end=346, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=58), 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.Additive.DoublingConst).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma divConst_inv_left (A B : Finset G) : δₘ[A⁻¹, B] = σₘ[A, B]
  frobnicateTacticThatDoesNotExist := by
rw [mulConst, divConst, card_inv, ← card_inv, inv_div, div_inv_eq_mul, mul_comm]

/-- If `A` has small difference, then it has small doubling, with the constant squared.

This is a consequence of the Ruzsa triangle inequality. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=55, messages=[Message(data="unexpected token ','; expected ':', ']' or ']''", pos=Position(line=1, column=49), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=55, i_end=346, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=58), 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.Perm.Cycle.Basic).

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

```lean
theorem subtypePerm_apply_zpow_of_mem {g : Perm α} {s : Finset α}
    (hs : ∀ x : α, g x ∈ s ↔ x ∈ s) {i : ℤ} {x : α} (hx : x ∈ s) :
    ((g.subtypePerm hs ^ i) (⟨x, hx⟩ : s) : α) = (g ^ i) x
:= by _unbalancedParenXYZ
simp only [subtypePerm_zpow, subtypePerm_apply]

variable [Fintype α] [DecidableEq α]

/-- Restrict a permutation to its support -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=220, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\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 `Perm` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=43), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nPerm : x✝\nα : Type u_2\ng : sorry\ns : Finset α\nhs : ∀ (x : α), sorry ∈ s ↔ x ∈ s\ni : ℤ\nx : α\nhx : x ∈ s\n⊢ sorry = sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=220, i_end=328, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=328, i_end=372, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=44), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem subtypePerm_apply_zpow_of_mem {g : Perm α} {s : Finset α}
    (hs : ∀ x : α, g x ∈ s ↔ x ∈ s) {i : ℤ} {x : α} (hx : x ∈ s) :
    ((g.subtypePerm hs ^ i) (⟨x, hx⟩ : s) : α) = (g ^ i) x
:= by _unbalancedParenXYZ
simp only [subtypePerm_zpow, subtypePerm_apply]

variable [Fintype α] [DecidableEq α]

/-- Restrict a permutation to its support -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=220, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Perm\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 `Perm` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=43), pos_end=Position(line=1, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nPerm : x✝\nα : Type u_2\ng : sorry\ns : Finset α\nhs : ∀ (x : α), sorry ∈ s ↔ x ∈ s\ni : ℤ\nx : α\nhx : x ∈ s\n⊢ sorry = sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=220, i_end=328, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=328, i_end=372, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=9, column=44), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem invOneSubPow_add (e : ℕ) :
    invOneSubPow S (d + e) = invOneSubPow S d * invOneSubPow S e
:= by thisIsWrongLemmaNameXYZ
simp_rw [invOneSubPow_eq_inv_one_sub_pow, pow_add]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=109, 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  invOneSubPow\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 `invOneSubPow` 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=26), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invOneSubPow\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 `invOneSubPow` 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=29), pos_end=Position(line=2, column=45), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invOneSubPow\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 `invOneSubPow` 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=48), pos_end=Position(line=2, column=64), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\ninvOneSubPow : x✝\ne : ℕ\n⊢ sorry = sorry * 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=109, i_end=182, 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.RingTheory.PowerSeries.WellKnown).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem invOneSubPow_add (e : ℕ) :
    invOneSubPow S (d + e) = invOneSubPow S d * invOneSubPow S e
:= by thisIsWrongLemmaNameXYZ
simp_rw [invOneSubPow_eq_inv_one_sub_pow, pow_add]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=109, 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  invOneSubPow\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 `invOneSubPow` 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=26), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invOneSubPow\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 `invOneSubPow` 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=29), pos_end=Position(line=2, column=45), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invOneSubPow\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 `invOneSubPow` 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=48), pos_end=Position(line=2, column=64), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\ninvOneSubPow : x✝\ne : ℕ\n⊢ sorry = sorry * 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=109, i_end=182, 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.LinearAlgebra.AffineSpace.FiniteDimensional).

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

```lean
theorem AffineIndependent.finrank_vectorSpan [Fintype ι] {p : ι → P} (hi : AffineIndependent k p)
    {n : ℕ} (hc : Fintype.card ι = n + 1) : finrank k (vectorSpan k (Set.range p)) = n
:= by .someMissingNamespaceXYZ
classical
  rw [← Finset.card_univ] at hc
  rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]
  exact hi.finrank_vectorSpan_image_finset hc

/-- The `vectorSpan` of a finite affinely independent family has dimension one less than its
cardinality. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=200, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=200, i_end=491, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=10, column=15), 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.LinearAlgebra.AffineSpace.FiniteDimensional).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem AffineIndependent.finrank_vectorSpan [Fintype ι] {p : ι → P} (hi : AffineIndependent k p)
    {n : ℕ} (hc : Fintype.card ι = n + 1) : finrank k (vectorSpan k (Set.range p)) = n
:= by .someMissingNamespaceXYZ
classical
  rw [← Finset.card_univ] at hc
  rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]
  exact hi.finrank_vectorSpan_image_finset hc

/-- The `vectorSpan` of a finite affinely independent family has dimension one less than its
cardinality. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=200, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=200, i_end=491, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=10, column=15), 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.NumberTheory.Divisors).

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

```lean
theorem divisors_prime_pow {p : ℕ} (pp : p.Prime) (k : ℕ) :
    divisors (p ^ k) = (Finset.range (k + 1)).map ⟨(p ^ ·), Nat.pow_right_injective pp.two_le⟩
:= by thisIsWrongLemmaNameXYZ
ext a
  rw [mem_divisors_prime_pow pp]
  simp [eq_comm]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=171, 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  divisors\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (p ^ k)\n\nHint: The identifier `divisors` 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=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nx✝ : Sort u_1\ndivisors : x✝\np : ℕ\npp : Nat.Prime p\nk : ℕ\n⊢ sorry = Finset.map { toFun := fun x => p ^ x, inj' := ⋯ } (Finset.range (k + 1))", 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=171, i_end=249, 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.Divisors).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem divisors_prime_pow {p : ℕ} (pp : p.Prime) (k : ℕ) :
    divisors (p ^ k) = (Finset.range (k + 1)).map ⟨(p ^ ·), Nat.pow_right_injective pp.two_le⟩
:= by thisIsWrongLemmaNameXYZ
ext a
  rw [mem_divisors_prime_pow pp]
  simp [eq_comm]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=171, 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  divisors\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (p ^ k)\n\nHint: The identifier `divisors` 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=20), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nx✝ : Sort u_1\ndivisors : x✝\np : ℕ\npp : Nat.Prime p\nk : ℕ\n⊢ sorry = Finset.map { toFun := fun x => p ^ x, inj' := ⋯ } (Finset.range (k + 1))", 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=171, i_end=249, 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.Lie.Sl2).

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

```lean
lemma mem_toLieSubalgebra_iff {x : L} {t : IsSl2Triple h e f} :
    x ∈ t.toLieSubalgebra R ↔ ∃ c₁ c₂ c₃ : R, x = c₁ • e + c₂ • f + c₃ • ⁅e, f⁆
:= by thisIsWrongLemmaNameXYZ
simp_rw [t.lie_e_f, toLieSubalgebra, ← LieSubalgebra.mem_toSubmodule, Submodule.mem_span_triple,
    eq_comm]

namespace HasPrimitiveVectorWith

variable {m : M} {μ : R} {t : IsSl2Triple h e f}
local notation "ψ " n => ((toEnd R L M f) ^ n) m

-- Although this is true by definition, we include this lemma (and the assumption) to mirror the API
-- for `lie_h_pow_toEnd_f` and `lie_e_pow_succ_toEnd_f`.
set_option linter.unusedVariables false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=179, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  CommRing R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=8), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Bracket L L\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=73), pos_end=Position(line=2, column=79), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HSMul R L ?m.31\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=50), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LieRing L\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nL : Type u_1\nh e f : L\nR : Type u_2\nx : L\nt : IsSl2Triple h e f\n⊢ x ∈ sorry ↔ ∃ c₁ c₂ c₃, 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=179, i_end=349, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=349, i_end=399, messages=[Message(data='typeclass instance problem is stuck\n  LieRing ?m.9\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `LieRing` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.', pos=Position(line=9, column=30), pos_end=Position(line=9, column=47), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=399, i_end=608, messages=[Message(data='Unknown identifier `toEnd` at quotation precheck\n\nNote: You can use `set_option quotPrecheck false` to disable this check.', pos=Position(line=10, column=27), pos_end=Position(line=10, column=32), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=608, i_end=650, messages=[Message(data='unexpected end of input', pos=Position(line=14, column=42), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Lie.Sl2).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma mem_toLieSubalgebra_iff {x : L} {t : IsSl2Triple h e f} :
    x ∈ t.toLieSubalgebra R ↔ ∃ c₁ c₂ c₃ : R, x = c₁ • e + c₂ • f + c₃ • ⁅e, f⁆
:= by thisIsWrongLemmaNameXYZ
simp_rw [t.lie_e_f, toLieSubalgebra, ← LieSubalgebra.mem_toSubmodule, Submodule.mem_span_triple,
    eq_comm]

namespace HasPrimitiveVectorWith

variable {m : M} {μ : R} {t : IsSl2Triple h e f}
local notation "ψ " n => ((toEnd R L M f) ^ n) m

-- Although this is true by definition, we include this lemma (and the assumption) to mirror the API
-- for `lie_h_pow_toEnd_f` and `lie_e_pow_succ_toEnd_f`.
set_option linter.unusedVariables false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=179, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  CommRing R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=8), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Bracket L L\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=73), pos_end=Position(line=2, column=79), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HSMul R L ?m.31\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=50), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LieRing L\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nL : Type u_1\nh e f : L\nR : Type u_2\nx : L\nt : IsSl2Triple h e f\n⊢ x ∈ sorry ↔ ∃ c₁ c₂ c₃, 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=179, i_end=349, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=349, i_end=399, messages=[Message(data='typeclass instance problem is stuck\n  LieRing ?m.9\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `LieRing` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.', pos=Position(line=9, column=30), pos_end=Position(line=9, column=47), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=399, i_end=608, messages=[Message(data='Unknown identifier `toEnd` at quotation precheck\n\nNote: You can use `set_option quotPrecheck false` to disable this check.', pos=Position(line=10, column=27), pos_end=Position(line=10, column=32), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=608, i_end=650, messages=[Message(data='unexpected end of input', pos=Position(line=14, column=42), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem M.dest_corec' {α : TypeVec.{u} n} {β : Type v} (g₀ : β → P.A)
    (g₁ : ∀ b : β, P.drop.B (g₀ b) ⟹ α) (g₂ : ∀ b : β, P.last.B (g₀ b) → β) (x : β) :
    M.dest P (M.corec' P g₀ g₁ g₂ x) = ⟨g₀ x, splitFun (g₁ x) (M.corec' P g₀ g₁ g₂ ∘ g₂ x)⟩
:= _unbalancedParenXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=119, messages=[Message(data='expected token', pos=Position(line=2, column=35), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=119, i_end=326, 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.PFunctor.Multivariate.M).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem M.dest_corec' {α : TypeVec.{u} n} {β : Type v} (g₀ : β → P.A)
    (g₁ : ∀ b : β, P.drop.B (g₀ b) ⟹ α) (g₂ : ∀ b : β, P.last.B (g₀ b) → β) (x : β) :
    M.dest P (M.corec' P g₀ g₁ g₂ x) = ⟨g₀ x, splitFun (g₁ x) (M.corec' P g₀ g₁ g₂ ∘ g₂ x)⟩
:= _unbalancedParenXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=119, messages=[Message(data='expected token', pos=Position(line=2, column=35), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=119, i_end=326, 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.Filter.Map).

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

```lean
theorem singleton_mem_pure {a : α} : {a} ∈ (pure a : Filter α)
:= thisIsWrongLemmaNameXYZ
mem_singleton a
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=109, 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.Order.Filter.Map).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem singleton_mem_pure {a : α} : {a} ∈ (pure a : Filter α)
:= thisIsWrongLemmaNameXYZ
mem_singleton a
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=109, 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.Topology.ContinuousMap.Compact).

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

```lean
theorem enorm_eq_iSup_enorm : ‖f‖ₑ = ⨆ x, ‖f x‖ₑ
:= thisIsWrongLemmaNameXYZ
(mkOfCompact f).enorm_eq_iSup_enorm

-- A version with better keys
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=156, messages=[Message(data="Function expected at\n  f\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` 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=43), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind=None), 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.Topology.ContinuousMap.Compact).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem enorm_eq_iSup_enorm : ‖f‖ₑ = ⨆ x, ‖f x‖ₑ
:= thisIsWrongLemmaNameXYZ
(mkOfCompact f).enorm_eq_iSup_enorm

-- A version with better keys
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=156, messages=[Message(data="Function expected at\n  f\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` 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=43), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind=None), 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.Analysis.Convex.Between).

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

```lean
theorem Sbtw.left_ne_right {x y z : P} (h : Sbtw R x y z) : x ≠ z
  frobnicateTacticThatDoesNotExist :=
h.wbtw.left_ne_right_of_ne_left h.2.1

variable (R) in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=145, messages=[Message(data='`Sbtw.left_ne_right` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=145, i_end=160, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=15), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem Sbtw.left_ne_right {x y z : P} (h : Sbtw R x y z) : x ≠ z
  frobnicateTacticThatDoesNotExist :=
h.wbtw.left_ne_right_of_ne_left h.2.1

variable (R) in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=145, messages=[Message(data='`Sbtw.left_ne_right` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=145, i_end=160, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=15), 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.Combinatorics.Enumerative.Composition).

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

```lean
theorem ne_single_iff {n : ℕ} (hn : 0 < n) {c : Composition n} :
    c ≠ single n hn ↔ ∀ i, c.blocksFun i < n
:= by .someMissingNamespaceXYZ
contrapose!
  constructor
  · rintro rfl
    exact ⟨⟨0, by simp⟩, by simp⟩
  · rintro ⟨i, hi⟩
    rw [eq_single_iff_length]
    have : ∀ j : Fin c.length, j = i := by
      intro j
      by_contra ji
      apply lt_irrefl (∑ k, c.blocksFun k)
      calc
        ∑ k, c.blocksFun k ≤ c.blocksFun i := by simp only [c.sum_blocksFun, hi]
        _ < ∑ k, c.blocksFun k :=
          Finset.single_lt_sum ji (Finset.mem_univ _) (Finset.mem_univ _) (c.one_le_blocksFun j)
            fun _ _ _ => zero_le _
    simpa using Fintype.card_eq_one_of_forall_eq this

variable {m : ℕ}

/-- Change `n` in `(c : Composition n)` to a propositionally equal value. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=126, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  single\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  n\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=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nsingle : x✝\nn : ℕ\nhn : 0 < n\nc : Composition n\n⊢ c ≠ sorry ↔ ∀ (i : Fin c.length), c.blocksFun i < n', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=749, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=749, i_end=825, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=23, column=76), 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.Enumerative.Composition).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem ne_single_iff {n : ℕ} (hn : 0 < n) {c : Composition n} :
    c ≠ single n hn ↔ ∀ i, c.blocksFun i < n
:= by .someMissingNamespaceXYZ
contrapose!
  constructor
  · rintro rfl
    exact ⟨⟨0, by simp⟩, by simp⟩
  · rintro ⟨i, hi⟩
    rw [eq_single_iff_length]
    have : ∀ j : Fin c.length, j = i := by
      intro j
      by_contra ji
      apply lt_irrefl (∑ k, c.blocksFun k)
      calc
        ∑ k, c.blocksFun k ≤ c.blocksFun i := by simp only [c.sum_blocksFun, hi]
        _ < ∑ k, c.blocksFun k :=
          Finset.single_lt_sum ji (Finset.mem_univ _) (Finset.mem_univ _) (c.one_le_blocksFun j)
            fun _ _ _ => zero_le _
    simpa using Fintype.card_eq_one_of_forall_eq this

variable {m : ℕ}

/-- Change `n` in `(c : Composition n)` to a propositionally equal value. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=126, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  single\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  n\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=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nsingle : x✝\nn : ℕ\nhn : 0 < n\nc : Composition n\n⊢ c ≠ sorry ↔ ∀ (i : Fin c.length), c.blocksFun i < n', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=749, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=749, i_end=825, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=23, column=76), 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.MonoidLocalization.Away).

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

```lean
abbrev Away
:= .someMissingNamespaceXYZ
Localization (Submonoid.powers x)

/-- Given `x : M`, `invSelf` is `x⁻¹` in the Localization (as a quotient type) of `M` at the
Submonoid generated by `x`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='Invalid dotted identifier notation: The expected type of `.someMissingNamespaceXYZ` could not be determined', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.invalidDottedIdent._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=201, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
abbrev Away
:= .someMissingNamespaceXYZ
Localization (Submonoid.powers x)

/-- Given `x : M`, `invSelf` is `x⁻¹` in the Localization (as a quotient type) of `M` at the
Submonoid generated by `x`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='Invalid dotted identifier notation: The expected type of `.someMissingNamespaceXYZ` could not be determined', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.invalidDottedIdent._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=201, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem IsPrimitiveRoot.norm'_eq_one {ζ : ℂ} {n : ℕ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
    ‖ζ‖ = 1
:= ⟨⟩⟨⟩
Complex.norm_eq_one_of_pow_eq_one h.pow_eq_one hn
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data="`IsPrimitiveRoot.norm'_eq_one` 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), CompilationUnit(i_begin=128, i_end=184, 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)], 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.Complex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsPrimitiveRoot.norm'_eq_one {ζ : ℂ} {n : ℕ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
    ‖ζ‖ = 1
:= ⟨⟩⟨⟩
Complex.norm_eq_one_of_pow_eq_one h.pow_eq_one hn
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data="`IsPrimitiveRoot.norm'_eq_one` 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), CompilationUnit(i_begin=128, i_end=184, 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)], 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.Complex.Module).

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

```lean
lemma mem_unitary_iff_isStarNormal_and_realPart_sq_add_imaginaryPart_sq_eq_one [Ring A]
    [StarRing A] [Module ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] [StarModule ℂ A] {x : A} :
    x ∈ unitary A ↔ IsStarNormal x ∧ ℜ x ^ 2 + ℑ x ^ 2 = (1 : A)
:= by ⟨⟩⟨⟩
rw [Unitary.mem_iff]
  refine ⟨fun ⟨h, h'⟩ ↦ ?_, fun ⟨hx, h⟩ ↦ ?_⟩
  · have : IsStarNormal x := ⟨h.trans h'.symm⟩
    exact ⟨this, by simp [sq, ← star_mul_self_eq_realPart_sq_add_imaginaryPart_sq x, h]⟩
  · simp [← hx.star_comm_self.eq, star_mul_self_eq_realPart_sq_add_imaginaryPart_sq, ← sq, h]

end RealImaginaryPart
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=278, 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)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=278, i_end=642, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=11, column=0), pos_end=Position(line=11, 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.LinearAlgebra.Complex.Module).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma mem_unitary_iff_isStarNormal_and_realPart_sq_add_imaginaryPart_sq_eq_one [Ring A]
    [StarRing A] [Module ℂ A] [SMulCommClass ℂ A A] [IsScalarTower ℂ A A] [StarModule ℂ A] {x : A} :
    x ∈ unitary A ↔ IsStarNormal x ∧ ℜ x ^ 2 + ℑ x ^ 2 = (1 : A)
:= by ⟨⟩⟨⟩
rw [Unitary.mem_iff]
  refine ⟨fun ⟨h, h'⟩ ↦ ?_, fun ⟨hx, h⟩ ↦ ?_⟩
  · have : IsStarNormal x := ⟨h.trans h'.symm⟩
    exact ⟨this, by simp [sq, ← star_mul_self_eq_realPart_sq_add_imaginaryPart_sq x, h]⟩
  · simp [← hx.star_comm_self.eq, star_mul_self_eq_realPart_sq_add_imaginaryPart_sq, ← sq, h]

end RealImaginaryPart
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=278, 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)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=278, i_end=642, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=11, column=0), pos_end=Position(line=11, 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]


[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.PythagoreanTriples).

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

```lean
theorem classified : h.IsClassified
:= by thisIsWrongLemmaNameXYZ
by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    use 0, 1, 0
    simp [hx, hy]
  apply h.isClassified_of_normalize_isPrimitiveClassified
  apply h.normalize.isPrimitiveClassified_of_coprime
  apply Int.gcd_div_gcd_div_gcd (Nat.pos_of_ne_zero h0)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=43, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `h.IsClassified`', pos=Position(line=1, column=21), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\n⊢ sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=43, i_end=512, 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.PythagoreanTriples).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem classified : h.IsClassified
:= by thisIsWrongLemmaNameXYZ
by_cases h0 : Int.gcd x y = 0
  · have hx : x = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_left h0
    have hy : y = 0 := by
      apply Int.natAbs_eq_zero.mp
      apply Nat.eq_zero_of_gcd_eq_zero_right h0
    use 0, 1, 0
    simp [hx, hy]
  apply h.isClassified_of_normalize_isPrimitiveClassified
  apply h.normalize.isPrimitiveClassified_of_coprime
  apply Int.gcd_div_gcd_div_gcd (Nat.pos_of_ne_zero h0)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=43, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `h.IsClassified`', pos=Position(line=1, column=21), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\n⊢ sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=43, i_end=512, 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.Field.Subfield.Basic).

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

```lean
theorem mem_fieldRange_self (x : K) : f x ∈ f.fieldRange
:= .someMissingNamespaceXYZ
exists_apply_eq_apply _ _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='Invalid field notation: Type of\n  f\nis not known; cannot resolve field `fieldRange`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AlgHom.fieldRange`\n  • `RingHom.fieldRange`', pos=Position(line=1, column=44), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=38), pos_end=Position(line=1, column=41), 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.Field.Subfield.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem mem_fieldRange_self (x : K) : f x ∈ f.fieldRange
:= .someMissingNamespaceXYZ
exists_apply_eq_apply _ _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='Invalid field notation: Type of\n  f\nis not known; cannot resolve field `fieldRange`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AlgHom.fieldRange`\n  • `RingHom.fieldRange`', pos=Position(line=1, column=44), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=38), pos_end=Position(line=1, column=41), 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.Rat.BigOperators).

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

```lean
theorem cast_list_sum (s : List ℚ) : (↑s.sum : α) = (s.map (↑)).sum
:= .someMissingNamespaceXYZ
map_list_sum (Rat.castHom α) _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=134, messages=[Message(data='Type mismatch\n  s.sum\nhas type\n  ℚ\nbut is expected to have type\n  α', pos=Position(line=1, column=38), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind=None), Message(data='cannot coerce\n  x\nto type\n  α', pos=Position(line=1, column=59), pos_end=Position(line=1, column=62), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Add 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=52), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Zero 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=52), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._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.Rat.BigOperators).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem cast_list_sum (s : List ℚ) : (↑s.sum : α) = (s.map (↑)).sum
:= .someMissingNamespaceXYZ
map_list_sum (Rat.castHom α) _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=134, messages=[Message(data='Type mismatch\n  s.sum\nhas type\n  ℚ\nbut is expected to have type\n  α', pos=Position(line=1, column=38), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind=None), Message(data='cannot coerce\n  x\nto type\n  α', pos=Position(line=1, column=59), pos_end=Position(line=1, column=62), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Add 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=52), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Zero 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=52), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._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.Filter.CountableInter).

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

```lean
theorem EventuallyLE.countable_bUnion {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
    ⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_›
  frobnicateTacticThatDoesNotExist := by
simp only [biUnion_eq_iUnion]
  haveI := hS.toEncodable
  exact EventuallyLE.countable_iUnion fun i => h i i.2
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=294, messages=[Message(data='`EventuallyLE.countable_bUnion` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=37), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=372, messages=[Message(data="unexpected token 'haveI'; expected command", pos=Position(line=6, column=2), pos_end=Position(line=6, column=7), 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.Filter.CountableInter).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem EventuallyLE.countable_bUnion {ι : Type*} {S : Set ι} (hS : S.Countable)
    {s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
    ⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_›
  frobnicateTacticThatDoesNotExist := by
simp only [biUnion_eq_iUnion]
  haveI := hS.toEncodable
  exact EventuallyLE.countable_iUnion fun i => h i i.2
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=294, messages=[Message(data='`EventuallyLE.countable_bUnion` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=37), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=372, messages=[Message(data="unexpected token 'haveI'; expected command", pos=Position(line=6, column=2), pos_end=Position(line=6, column=7), 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.MetricSpace.Basic).

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

```lean
theorem isClosedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
    {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
    IsClosedEmbedding f
:= thisIsWrongLemmaNameXYZ
isClosedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf

/-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
`f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=333, messages=[Message(data='failed to synthesize instance of type class\n  Dist γ\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=67), pos_end=Position(line=2, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  IsClosedEmbedding\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 `IsClosedEmbedding` 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=4), pos_end=Position(line=3, column=23), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=4, column=3), pos_end=Position(line=4, 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=333, i_end=510, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=77), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem isClosedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
    {ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
    IsClosedEmbedding f
:= thisIsWrongLemmaNameXYZ
isClosedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf

/-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
`f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=333, messages=[Message(data='failed to synthesize instance of type class\n  Dist γ\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=67), pos_end=Position(line=2, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  IsClosedEmbedding\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 `IsClosedEmbedding` 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=4), pos_end=Position(line=3, column=23), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=4, column=3), pos_end=Position(line=4, 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=333, i_end=510, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=77), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x < y ^ z⁻¹ ↔ y < x ^ z
:= .someMissingNamespaceXYZ
Real.lt_rpow_inv_iff_of_neg hx hy hz
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=163, messages=[Message(data='failed to synthesize instance of type class\n  Inv ℕ\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=78), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown constant `Iff.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  x < y ^ z⁻¹ ↔ y < x ^ z', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), 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.Analysis.SpecialFunctions.Pow.NNReal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x < y ^ z⁻¹ ↔ y < x ^ z
:= .someMissingNamespaceXYZ
Real.lt_rpow_inv_iff_of_neg hx hy hz
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=163, messages=[Message(data='failed to synthesize instance of type class\n  Inv ℕ\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=78), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown constant `Iff.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  x < y ^ z⁻¹ ↔ y < x ^ z', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), 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.Combinatorics.Enumerative.DyckWord).

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

```lean
lemma insidePart_add : (p + q).insidePart = p.insidePart
:= by _unbalancedParenXYZ
simp_rw [insidePart, firstReturn_add, add_eq_zero', h, false_and, dite_false, ite_false,
    DyckWord.ext_iff, take]
  congr 3
  exact take_append_of_le_length (firstReturn_lt_length h)

include h in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=64, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  p + q\nis not known; cannot resolve field `insidePart`\n\nHint: Consider replacing the field projection `.insidePart` with a call to the function `DyckWord.insidePart`.', pos=Position(line=1, column=23), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  p\nis not known; cannot resolve field `insidePart`\n\nHint: Consider replacing the field projection `.insidePart` with a call to the function `DyckWord.insidePart`.', pos=Position(line=1, column=44), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝¹ : Type u_1\np : x✝¹\nx✝ : Type u_2\nq : x✝\n⊢ sorry = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=64, i_end=282, messages=[Message(data='unexpected end of input', pos=Position(line=8, column=12), 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.Enumerative.DyckWord).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma insidePart_add : (p + q).insidePart = p.insidePart
:= by _unbalancedParenXYZ
simp_rw [insidePart, firstReturn_add, add_eq_zero', h, false_and, dite_false, ite_false,
    DyckWord.ext_iff, take]
  congr 3
  exact take_append_of_le_length (firstReturn_lt_length h)

include h in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=64, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  p + q\nis not known; cannot resolve field `insidePart`\n\nHint: Consider replacing the field projection `.insidePart` with a call to the function `DyckWord.insidePart`.', pos=Position(line=1, column=23), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  p\nis not known; cannot resolve field `insidePart`\n\nHint: Consider replacing the field projection `.insidePart` with a call to the function `DyckWord.insidePart`.', pos=Position(line=1, column=44), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝¹ : Type u_1\np : x✝¹\nx✝ : Type u_2\nq : x✝\n⊢ sorry = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=64, i_end=282, messages=[Message(data='unexpected end of input', pos=Position(line=8, column=12), 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]