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

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

```lean
theorem toLex_one [One α] : toLex (1 : α) = 1
```

Declaration: `Mathlib.Algebra.Order.Group.Synonym.toLex_one` (module Mathlib.Algebra.Order.Group.Synonym).

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

```lean
theorem toLex_one [One α] : toLex (1 : α) = 1
```

Declaration: `Mathlib.Algebra.Order.Group.Synonym.toLex_one` (module Mathlib.Algebra.Order.Group.Synonym).[/INST]


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

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

```lean
def mkDerivationEquiv : A ≃ₗ[R] Derivation R R[X] A
```

Declaration: `Mathlib.Algebra.Polynomial.Derivation.mkDerivationEquiv` (module Mathlib.Algebra.Polynomial.Derivation).

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

```lean
def mkDerivationEquiv : A ≃ₗ[R] Derivation R R[X] A
```

Declaration: `Mathlib.Algebra.Polynomial.Derivation.mkDerivationEquiv` (module Mathlib.Algebra.Polynomial.Derivation).[/INST]

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


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).

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

```lean
theorem Cofix.dest_corec' {α : TypeVec.{u} n} {β : Type u}
    (g : β → F (α.append1 (Cofix F α ⊕ β))) (x : β) :
    Cofix.dest (Cofix.corec' g x) =
      appendFun id (Sum.elim _root_.id (Cofix.corec' g)) <$$> g x
```

Declaration: `Mathlib.Data.QPF.Multivariate.Constructions.Cofix.Cofix` (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).

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

```lean
theorem Cofix.dest_corec' {α : TypeVec.{u} n} {β : Type u}
    (g : β → F (α.append1 (Cofix F α ⊕ β))) (x : β) :
    Cofix.dest (Cofix.corec' g x) =
      appendFun id (Sum.elim _root_.id (Cofix.corec' g)) <$$> g x
```

Declaration: `Mathlib.Data.QPF.Multivariate.Constructions.Cofix.Cofix` (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).[/INST]


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

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

```lean
lemma ModularForm.eq_const_of_weight_zero [𝒢.IsArithmetic] (f : ModularForm 𝒢 0) :
    ∃ c, (f : ℍ → ℂ) = Function.const ℍ c
```

Declaration: `Mathlib.NumberTheory.ModularForms.NormTrace.ModularForm` (module Mathlib.NumberTheory.ModularForms.NormTrace).

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

```lean
lemma ModularForm.eq_const_of_weight_zero [𝒢.IsArithmetic] (f : ModularForm 𝒢 0) :
    ∃ c, (f : ℍ → ℂ) = Function.const ℍ c
```

Declaration: `Mathlib.NumberTheory.ModularForms.NormTrace.ModularForm` (module Mathlib.NumberTheory.ModularForms.NormTrace).[/INST]


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

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

```lean
theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
    (ht₃ : IsOpen t) (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b
```

Declaration: `Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_open` (module Mathlib.Analysis.LocallyConvex.Separation).

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

```lean
theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
    (ht₃ : IsOpen t) (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b
```

Declaration: `Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_open` (module Mathlib.Analysis.LocallyConvex.Separation).[/INST]


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

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

```lean
theorem mem_ker {x : L} : x ∈ ker f ↔ f x = 0
```

Declaration: `Mathlib.Algebra.Lie.Ideal.mem_ker` (module Mathlib.Algebra.Lie.Ideal).

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

```lean
theorem mem_ker {x : L} : x ∈ ker f ↔ f x = 0
```

Declaration: `Mathlib.Algebra.Lie.Ideal.mem_ker` (module Mathlib.Algebra.Lie.Ideal).[/INST]


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

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

```lean
theorem comap_mulLeft_nhdsGT_zero {x : 𝕜} (hx : 0 < x) : comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0
```

Declaration: `Mathlib.Topology.Algebra.Order.Field.comap_mulLeft_nhdsGT_zero` (module Mathlib.Topology.Algebra.Order.Field).

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

```lean
theorem comap_mulLeft_nhdsGT_zero {x : 𝕜} (hx : 0 < x) : comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0
```

Declaration: `Mathlib.Topology.Algebra.Order.Field.comap_mulLeft_nhdsGT_zero` (module Mathlib.Topology.Algebra.Order.Field).[/INST]


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

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

```lean
theorem fixInduction'_fwd {C : α → Sort*} {f : α →. β ⊕ α} {b : β} {a a' : α} (h : b ∈ f.fix a)
    (h' : b ∈ f.fix a') (fa : Sum.inr a' ∈ f a)
    (hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
    (hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
    @fixInduction' _ _ C _ _ _ h hbase hind = hind a a' h' fa (fixInduction' h' hbase hind)
```

Declaration: `Mathlib.Data.PFun.fixInduction'_fwd` (module Mathlib.Data.PFun).

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

```lean
theorem fixInduction'_fwd {C : α → Sort*} {f : α →. β ⊕ α} {b : β} {a a' : α} (h : b ∈ f.fix a)
    (h' : b ∈ f.fix a') (fa : Sum.inr a' ∈ f a)
    (hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
    (hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
    @fixInduction' _ _ C _ _ _ h hbase hind = hind a a' h' fa (fixInduction' h' hbase hind)
```

Declaration: `Mathlib.Data.PFun.fixInduction'_fwd` (module Mathlib.Data.PFun).[/INST]


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

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

```lean
theorem ContDiffWithinAt.div {f g : E → 𝕜} {n} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
```

Declaration: `Mathlib.Analysis.Calculus.ContDiff.Operations.ContDiffWithinAt` (module Mathlib.Analysis.Calculus.ContDiff.Operations).

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

```lean
theorem ContDiffWithinAt.div {f g : E → 𝕜} {n} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
```

Declaration: `Mathlib.Analysis.Calculus.ContDiff.Operations.ContDiffWithinAt` (module Mathlib.Analysis.Calculus.ContDiff.Operations).[/INST]


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

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

```lean
lemma hasSumLocallyUniformly_aux (f : L.lattice → ℂ → ℂ)
    (u : ℝ → L.lattice → ℝ) (hu : ∀ r > 0, Summable (u r))
    (hf : ∀ r > 0, ∀ᶠ R in atTop, ∀ x, ‖x‖ < r → ∀ l : L.lattice, ‖l.1‖ = R → ‖f l x‖ ≤ u r l) :
    HasSumLocallyUniformly f (∑' j, f j ·)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.hasSumLocallyUniformly_aux` (module Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass).

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

```lean
lemma hasSumLocallyUniformly_aux (f : L.lattice → ℂ → ℂ)
    (u : ℝ → L.lattice → ℝ) (hu : ∀ r > 0, Summable (u r))
    (hf : ∀ r > 0, ∀ᶠ R in atTop, ∀ x, ‖x‖ < r → ∀ l : L.lattice, ‖l.1‖ = R → ‖f l x‖ ≤ u r l) :
    HasSumLocallyUniformly f (∑' j, f j ·)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.hasSumLocallyUniformly_aux` (module Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass).[/INST]


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

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

```lean
theorem IntermediateField.isCyclotomicExtension_lcm_sup [NeZero n₁] [NeZero n₂] :
    IsCyclotomicExtension {n₁.lcm n₂} K (F₁ ⊔ F₂ : IntermediateField K L)
```

Declaration: `Mathlib.NumberTheory.Cyclotomic.Basic.IntermediateField` (module Mathlib.NumberTheory.Cyclotomic.Basic).

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

```lean
theorem IntermediateField.isCyclotomicExtension_lcm_sup [NeZero n₁] [NeZero n₂] :
    IsCyclotomicExtension {n₁.lcm n₂} K (F₁ ⊔ F₂ : IntermediateField K L)
```

Declaration: `Mathlib.NumberTheory.Cyclotomic.Basic.IntermediateField` (module Mathlib.NumberTheory.Cyclotomic.Basic).[/INST]


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

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

```lean
def mapDomainRingHom [MulSemiringAction α β] [MulSemiringAction α₂ β]
    [MonoidHomClass F α α₂] {f : F} (hf : ∀ (a : α) (x : β), a • x = (f a) • x) :
    SkewMonoidAlgebra β α →+* SkewMonoidAlgebra β α₂ where
  __
:= (mapDomain f : SkewMonoidAlgebra β α →+ SkewMonoidAlgebra β α₂) .someMissingNamespaceXYZ
map_one' := mapDomain_one f
  map_mul' x y := mapDomain_mul x y hf

end MapDomain

section of

variable (k G)

variable [Monoid G] [MulSemiringAction G k]

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The embedding of a monoid into its skew monoid algebra. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=342, messages=[Message(data='failed to synthesize instance of type class\n  Monoid 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=22), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=342, i_end=425, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=0), pos_end=Position(line=6, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=425, i_end=437, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=437, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=453, i_end=498, messages=[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=15, column=21), pos_end=Position(line=15, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=498, i_end=642, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=62), 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.SkewMonoidAlgebra.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def mapDomainRingHom [MulSemiringAction α β] [MulSemiringAction α₂ β]
    [MonoidHomClass F α α₂] {f : F} (hf : ∀ (a : α) (x : β), a • x = (f a) • x) :
    SkewMonoidAlgebra β α →+* SkewMonoidAlgebra β α₂ where
  __
:= (mapDomain f : SkewMonoidAlgebra β α →+ SkewMonoidAlgebra β α₂) .someMissingNamespaceXYZ
map_one' := mapDomain_one f
  map_mul' x y := mapDomain_mul x y hf

end MapDomain

section of

variable (k G)

variable [Monoid G] [MulSemiringAction G k]

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The embedding of a monoid into its skew monoid algebra. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=342, messages=[Message(data='failed to synthesize instance of type class\n  Monoid 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=22), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=342, i_end=425, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=0), pos_end=Position(line=6, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=425, i_end=437, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=437, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=453, i_end=498, messages=[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=15, column=21), pos_end=Position(line=15, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=498, i_end=642, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=62), 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.Sum.Order).

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

```lean
def sumLexCongr (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ₗ β₁ ≃o α₂ ⊕ₗ β₂ where
  toEquiv
  frobnicateTacticThatDoesNotExist := ofLex.trans ((Equiv.sumCongr ea eb).trans toLex)
map_rel_iff' := by simp_rw [Lex.forall]; rintro (a | a) (b | b) <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=212, messages=[Message(data='failed to synthesize instance of type class\n  LE α₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE β₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Lex (α₁ ⊕ β₁))\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=50), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=212, i_end=284, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=12), 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.Sum.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def sumLexCongr (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ₗ β₁ ≃o α₂ ⊕ₗ β₂ where
  toEquiv
  frobnicateTacticThatDoesNotExist := ofLex.trans ((Equiv.sumCongr ea eb).trans toLex)
map_rel_iff' := by simp_rw [Lex.forall]; rintro (a | a) (b | b) <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=212, messages=[Message(data='failed to synthesize instance of type class\n  LE α₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE β₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Lex (α₁ ⊕ β₁))\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=50), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=212, i_end=284, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=12), 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.SuccPred.Basic).

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

```lean
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b
:= by .someMissingNamespaceXYZ
rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ioo\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (pred a)\n\nHint: The identifier `Ioo` 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=53), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ico\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Ico` 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=70), pos_end=Position(line=1, column=77), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα✝ : Type u_1\na : α✝\nx✝¹ : Sort u_2\nIoo : x✝¹\nx✝ : Sort u_3\nIco : x✝\nha : ¬IsMin a\n⊢ sorry = sorry', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=173, 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.Order.SuccPred.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b
:= by .someMissingNamespaceXYZ
rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ioo\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (pred a)\n\nHint: The identifier `Ioo` 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=53), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ico\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Ico` 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=70), pos_end=Position(line=1, column=77), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα✝ : Type u_1\na : α✝\nx✝¹ : Sort u_2\nIoo : x✝¹\nx✝ : Sort u_3\nIco : x✝\nha : ¬IsMin a\n⊢ sorry = sorry', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=173, 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.Topology.Algebra.Valued.LocallyCompact).

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

```lean
lemma norm_coe_unit (u : 𝒪[K]ˣ) : ‖((u : 𝒪[K]) : K)‖ = 1
:= by thisIsWrongLemmaNameXYZ
simpa [← NNReal.coe_inj] using
    (Valuation.integer.integers (NormedField.valuation (K := K))).valuation_unit u

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Norm 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=34), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Monoid 𝒪[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=25), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=271, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma norm_coe_unit (u : 𝒪[K]ˣ) : ‖((u : 𝒪[K]) : K)‖ = 1
:= by thisIsWrongLemmaNameXYZ
simpa [← NNReal.coe_inj] using
    (Valuation.integer.integers (NormedField.valuation (K := K))).valuation_unit u

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Norm 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=34), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Monoid 𝒪[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=25), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=271, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma ofPreGNS_toPreGNS (a : A) : f.ofPreGNS (f.toPreGNS a) = a
:= rfl .someMissingNamespaceXYZ
variable [StarOrderedRing A]

/--
The (semi-)inner product space whose elements are the elements of `A`, but which has an
inner product-induced norm that is different from the norm on `A` and which is induced by `f`.
-/
noncomputable abbrev preGNSpreInnerProdSpace : PreInnerProductSpace.Core ℂ f.PreGNS where
  inner a b := f (star (f.ofPreGNS a) * f.ofPreGNS b)
  conj_inner_symm := by simp [← Complex.star_def, ← map_star f]
  re_inner_nonneg _ := RCLike.nonneg_iff.mp (f.map_nonneg (star_mul_self_nonneg _)) |>.1
  add_left _ _ _ := by rw [map_add, star_add, add_mul, map_add]
  smul_left := by simp [smul_mul_assoc]

noncomputable instance : SeminormedAddCommGroup f.PreGNS :=
  InnerProductSpace.Core.toSeminormedAddCommGroup (c := f.preGNSpreInnerProdSpace)
noncomputable instance : InnerProductSpace ℂ f.PreGNS :=
  InnerProductSpace.ofCore f.preGNSpreInnerProdSpace
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=96, messages=[Message(data='Unknown identifier `f.ofPreGNS`', pos=Position(line=1, column=34), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  rfl\nbut this term has type\n  ?m.5 = ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  .someMissingNamespaceXYZ', pos=Position(line=2, column=3), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=96, i_end=126, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalSemiring A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=10), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=724, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=9, column=75), pos_end=Position(line=9, column=83), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=724, i_end=867, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=16, column=48), pos_end=Position(line=16, column=56), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=17, column=56), pos_end=Position(line=17, column=81), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=867, i_end=978, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=18, column=45), pos_end=Position(line=18, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=19, column=27), pos_end=Position(line=19, column=52), 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.CStarAlgebra.GelfandNaimarkSegal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma ofPreGNS_toPreGNS (a : A) : f.ofPreGNS (f.toPreGNS a) = a
:= rfl .someMissingNamespaceXYZ
variable [StarOrderedRing A]

/--
The (semi-)inner product space whose elements are the elements of `A`, but which has an
inner product-induced norm that is different from the norm on `A` and which is induced by `f`.
-/
noncomputable abbrev preGNSpreInnerProdSpace : PreInnerProductSpace.Core ℂ f.PreGNS where
  inner a b := f (star (f.ofPreGNS a) * f.ofPreGNS b)
  conj_inner_symm := by simp [← Complex.star_def, ← map_star f]
  re_inner_nonneg _ := RCLike.nonneg_iff.mp (f.map_nonneg (star_mul_self_nonneg _)) |>.1
  add_left _ _ _ := by rw [map_add, star_add, add_mul, map_add]
  smul_left := by simp [smul_mul_assoc]

noncomputable instance : SeminormedAddCommGroup f.PreGNS :=
  InnerProductSpace.Core.toSeminormedAddCommGroup (c := f.preGNSpreInnerProdSpace)
noncomputable instance : InnerProductSpace ℂ f.PreGNS :=
  InnerProductSpace.ofCore f.preGNSpreInnerProdSpace
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=96, messages=[Message(data='Unknown identifier `f.ofPreGNS`', pos=Position(line=1, column=34), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  rfl\nbut this term has type\n  ?m.5 = ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  .someMissingNamespaceXYZ', pos=Position(line=2, column=3), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=96, i_end=126, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalSemiring A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=10), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=724, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=9, column=75), pos_end=Position(line=9, column=83), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=724, i_end=867, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=16, column=48), pos_end=Position(line=16, column=56), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=17, column=56), pos_end=Position(line=17, column=81), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=867, i_end=978, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=18, column=45), pos_end=Position(line=18, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=19, column=27), pos_end=Position(line=19, column=52), 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.SimpleGraph.StronglyRegular).

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

```lean
theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
    Gᶜ.IsRegularOfDegree (n - k - 1)
:= by thisIsWrongLemmaNameXYZ
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
  exact h.regular.compl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  G\nis not known; cannot resolve field `IsSRGWith`\n\nHint: Consider replacing the field projection `.IsSRGWith` with a call to the function `SimpleGraph.IsSRGWith`.', pos=Position(line=1, column=40), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  Gᶜ\nis not known; cannot resolve field `IsRegularOfDegree`\n\nHint: Consider replacing the field projection `.IsRegularOfDegree` with a call to the function `SimpleGraph.IsRegularOfDegree`.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝ : Type u_1\nG : x✝\nh : sorry\n⊢ 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=112, i_end=214, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
    Gᶜ.IsRegularOfDegree (n - k - 1)
:= by thisIsWrongLemmaNameXYZ
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
  exact h.regular.compl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  G\nis not known; cannot resolve field `IsSRGWith`\n\nHint: Consider replacing the field projection `.IsSRGWith` with a call to the function `SimpleGraph.IsSRGWith`.', pos=Position(line=1, column=40), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  Gᶜ\nis not known; cannot resolve field `IsRegularOfDegree`\n\nHint: Consider replacing the field projection `.IsRegularOfDegree` with a call to the function `SimpleGraph.IsRegularOfDegree`.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝ : Type u_1\nG : x✝\nh : sorry\n⊢ 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=112, i_end=214, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem isCycle_cycleOf (f : Perm α) [DecidableRel f.SameCycle] (hx : f x ≠ x) :
    IsCycle (cycleOf f x)
:= thisIsWrongLemmaNameXYZ
have : cycleOf f x x ≠ x := by rwa [SameCycle.rfl.cycleOf_apply]
  (isCycle_iff_sameCycle this).2 @fun y =>
    ⟨fun h => mt h.apply_eq_self_iff.2 this, fun h =>
      if hxy : SameCycle f x y then
        let ⟨i, hi⟩ := hxy
        ⟨i, by rw [cycleOf_zpow_apply_self, hi]⟩
      else by
        rw [cycleOf_apply_of_not_sameCycle hxy] at h
        exact (h rfl).elim⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[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=29), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsCycle\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (cycleOf f x)\n\nHint: The identifier `IsCycle` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=519, messages=[Message(data="unexpected token 'have'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=4), 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.Factors).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem isCycle_cycleOf (f : Perm α) [DecidableRel f.SameCycle] (hx : f x ≠ x) :
    IsCycle (cycleOf f x)
:= thisIsWrongLemmaNameXYZ
have : cycleOf f x x ≠ x := by rwa [SameCycle.rfl.cycleOf_apply]
  (isCycle_iff_sameCycle this).2 @fun y =>
    ⟨fun h => mt h.apply_eq_self_iff.2 this, fun h =>
      if hxy : SameCycle f x y then
        let ⟨i, hi⟩ := hxy
        ⟨i, by rw [cycleOf_zpow_apply_self, hi]⟩
      else by
        rw [cycleOf_apply_of_not_sameCycle hxy] at h
        exact (h rfl).elim⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[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=29), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsCycle\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (cycleOf f x)\n\nHint: The identifier `IsCycle` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=519, messages=[Message(data="unexpected token 'have'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=4), 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.TwoSidedIdeal.Operations).

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

```lean
lemma mem_fromIdeal {I : Ideal R} {x : R} :
    x ∈ fromIdeal I ↔ x ∈ span I
:= by simp [fromIdeal] _unbalancedParenXYZ
/-- Every two-sided ideal is also a left ideal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  fromIdeal\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `fromIdeal` 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="Function expected at\n  span\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `span` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid argument: Variable `fromIdeal` is not a proposition or let-declaration', pos=Position(line=3, column=12), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nx✝¹ : Sort u_2\nfromIdeal : x✝¹\nx✝ : Sort u_3\nspan : x✝\nI : sorry\nx : R\n⊢ x ∈ sorry () ↔ x ∈ sorry ()', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=106, i_end=176, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=23), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=50), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma mem_fromIdeal {I : Ideal R} {x : R} :
    x ∈ fromIdeal I ↔ x ∈ span I
:= by simp [fromIdeal] _unbalancedParenXYZ
/-- Every two-sided ideal is also a left ideal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  fromIdeal\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `fromIdeal` 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="Function expected at\n  span\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `span` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid argument: Variable `fromIdeal` is not a proposition or let-declaration', pos=Position(line=3, column=12), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nx✝¹ : Sort u_2\nfromIdeal : x✝¹\nx✝ : Sort u_3\nspan : x✝\nI : sorry\nx : R\n⊢ x ∈ sorry () ↔ x ∈ sorry ()', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=106, i_end=176, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=23), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=50), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem coe_units_inv (A : (Matrix n n α)ˣ) : ↑A⁻¹ = (A⁻¹ : Matrix n n α)
:= by thisIsWrongLemmaNameXYZ
letI := A.invertible
  rw [← invOf_eq_nonsing_inv, invOf_units]

/-- The nonsingular inverse is the same as the general `Ring.inverse`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=92, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Monoid (Matrix n n 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=27), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Inv (Matrix n n 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=54), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nn : Type u_1\nα : Type u_2\nA : sorry\n⊢ sorry = 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=92, i_end=255, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=73), 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.Matrix.NonsingularInverse).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem coe_units_inv (A : (Matrix n n α)ˣ) : ↑A⁻¹ = (A⁻¹ : Matrix n n α)
:= by thisIsWrongLemmaNameXYZ
letI := A.invertible
  rw [← invOf_eq_nonsing_inv, invOf_units]

/-- The nonsingular inverse is the same as the general `Ring.inverse`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=92, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Monoid (Matrix n n 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=27), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Inv (Matrix n n 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=54), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nn : Type u_1\nα : Type u_2\nA : sorry\n⊢ sorry = 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=92, i_end=255, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=73), 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.LSeries.Linearity).

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

```lean
lemma LSeries_neg (f : ℕ → ℂ) (s : ℂ) : LSeries (-f) s = -LSeries f s
:= by .someMissingNamespaceXYZ
simp [LSeries, term_neg_apply, tsum_neg]

/-!
### Subtraction
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=86, messages=[Message(data='unknown tactic', pos=Position(line=2, 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=86, i_end=173, 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.LSeries.Linearity).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma LSeries_neg (f : ℕ → ℂ) (s : ℂ) : LSeries (-f) s = -LSeries f s
:= by .someMissingNamespaceXYZ
simp [LSeries, term_neg_apply, tsum_neg]

/-!
### Subtraction
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=86, messages=[Message(data='unknown tactic', pos=Position(line=2, 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=86, i_end=173, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma zpow_eq_one_iff_cases₀ : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
  frobnicateTacticThatDoesNotExist := by
simp [← zpow_eq_zpow_iff_cases₀]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='`zpow_eq_one_iff_cases₀` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=28), 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.Order.Field.Power).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma zpow_eq_one_iff_cases₀ : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
  frobnicateTacticThatDoesNotExist := by
simp [← zpow_eq_zpow_iff_cases₀]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='`zpow_eq_one_iff_cases₀` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=28), 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.Stream.Init).

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

```lean
theorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :
    get (iterate f a) (succ n) = get (iterate f (f a)) n
:= by rw [get_succ, tail_iterate] thisIsWrongLemmaNameXYZ
section Bisim

variable (R : Stream' α → Stream' α → Prop)

/-- equivalence relation -/
local infixl:50 " ~ " => R

/-- Streams `s₁` and `s₂` are defined to be bisimulations if
their heads are equal and tails are bisimulations. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=153, messages=[Message(data='Function expected at\n  get\nbut this term has type\n  ?m.3 ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f a)', pos=Position(line=2, column=4), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  get\nbut this term has type\n  ?m.8 ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f (f a))', pos=Position(line=2, column=33), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `get_succ`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Sort u_1\nn : ℕ\nf : α → α\na : α\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=153, i_end=192, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=34), pos_end=Position(line=3, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=243, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=299, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=299, i_end=417, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=53), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :
    get (iterate f a) (succ n) = get (iterate f (f a)) n
:= by rw [get_succ, tail_iterate] thisIsWrongLemmaNameXYZ
section Bisim

variable (R : Stream' α → Stream' α → Prop)

/-- equivalence relation -/
local infixl:50 " ~ " => R

/-- Streams `s₁` and `s₂` are defined to be bisimulations if
their heads are equal and tails are bisimulations. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=153, messages=[Message(data='Function expected at\n  get\nbut this term has type\n  ?m.3 ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f a)', pos=Position(line=2, column=4), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  get\nbut this term has type\n  ?m.8 ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f (f a))', pos=Position(line=2, column=33), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `get_succ`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Sort u_1\nn : ℕ\nf : α → α\na : α\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=153, i_end=192, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=34), pos_end=Position(line=3, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=243, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=299, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=299, i_end=417, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=53), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem GameAdd.induction {C : α → β → Prop} :
    WellFounded rα →
      WellFounded rβ →
        (∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) → ∀ a b, C a b
:= thisIsWrongLemmaNameXYZ
GameAdd.recursion

end Prod

/-! ### `Sym2.GameAdd` -/

namespace Sym2

/-- `Sym2.GameAdd rα x y` means that `x` can be reached from `y` by decreasing either entry with
  respect to the relation `rα`.

  See `Prod.GameAdd` for the ordered pair analog. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=284, messages=[Message(data="Function expected at\n  GameAdd\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  rα\n\nHint: The identifier `GameAdd` 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=4, column=28), pos_end=Position(line=4, column=59), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=5, column=3), pos_end=Position(line=5, 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=284, i_end=294, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=321, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=321, i_end=337, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=337, i_end=521, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=17, column=52), 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.GameAdd).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem GameAdd.induction {C : α → β → Prop} :
    WellFounded rα →
      WellFounded rβ →
        (∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) → ∀ a b, C a b
:= thisIsWrongLemmaNameXYZ
GameAdd.recursion

end Prod

/-! ### `Sym2.GameAdd` -/

namespace Sym2

/-- `Sym2.GameAdd rα x y` means that `x` can be reached from `y` by decreasing either entry with
  respect to the relation `rα`.

  See `Prod.GameAdd` for the ordered pair analog. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=284, messages=[Message(data="Function expected at\n  GameAdd\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  rα\n\nHint: The identifier `GameAdd` 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=4, column=28), pos_end=Position(line=4, column=59), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=5, column=3), pos_end=Position(line=5, 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=284, i_end=294, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=321, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=321, i_end=337, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=337, i_end=521, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=17, column=52), 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.Module.LinearMap).

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

```lean
theorem toSpanSingleton_comp_toSpanSingleton [ContinuousMul R₁] {x : M₂} {c : R₁} :
    (toSpanSingleton R₁ x).comp (toSpanSingleton R₁ c) =
      toSpanSingleton R₁ (c • x)
:= smulRight_comp_smulRight 1 1 ⟨⟩⟨⟩
end ToSpanSingleton

end Semiring

section Ring

variable {R : Type*} [Ring R] {R₂ : Type*} [Ring R₂] {R₃ : Type*} [Ring R₃] {M : Type*}
  [TopologicalSpace M] [AddCommGroup M] {M₂ : Type*} [TopologicalSpace M₂] [AddCommGroup M₂]
  {M₃ : Type*} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type*} [TopologicalSpace M₄]
  [AddCommGroup M₄] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃}
  {σ₁₃ : R →+* R₃}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=226, messages=[Message(data='failed to synthesize instance of type class\n  TopologicalSpace R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=62), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=226, i_end=254, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=4, column=34), pos_end=Position(line=4, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=5, column=0), pos_end=Position(line=5, column=19), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=254, i_end=268, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=268, i_end=282, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=282, i_end=732, 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.Topology.Algebra.Module.LinearMap).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem toSpanSingleton_comp_toSpanSingleton [ContinuousMul R₁] {x : M₂} {c : R₁} :
    (toSpanSingleton R₁ x).comp (toSpanSingleton R₁ c) =
      toSpanSingleton R₁ (c • x)
:= smulRight_comp_smulRight 1 1 ⟨⟩⟨⟩
end ToSpanSingleton

end Semiring

section Ring

variable {R : Type*} [Ring R] {R₂ : Type*} [Ring R₂] {R₃ : Type*} [Ring R₃] {M : Type*}
  [TopologicalSpace M] [AddCommGroup M] {M₂ : Type*} [TopologicalSpace M₂] [AddCommGroup M₂]
  {M₃ : Type*} [TopologicalSpace M₃] [AddCommGroup M₃] {M₄ : Type*} [TopologicalSpace M₄]
  [AddCommGroup M₄] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃}
  {σ₁₃ : R →+* R₃}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=226, messages=[Message(data='failed to synthesize instance of type class\n  TopologicalSpace R₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=62), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=226, i_end=254, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=4, column=34), pos_end=Position(line=4, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=5, column=0), pos_end=Position(line=5, column=19), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=254, i_end=268, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=268, i_end=282, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=282, i_end=732, 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.Analysis.Calculus.FDeriv.Mul).

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

```lean
theorem HasFDerivWithinAt.const_mul (ha : HasFDerivWithinAt a a' s x) (b : 𝔸) :
    HasFDerivWithinAt (fun y => b * a y) (b • a') s x
:= ⟨⟩⟨⟩
((ContinuousLinearMap.mul 𝕜 𝔸) b).hasFDerivAt.comp_hasFDerivWithinAt x ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=148, messages=[Message(data='`HasFDerivWithinAt.const_mul` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=148, i_end=234, 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.Analysis.Calculus.FDeriv.Mul).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem HasFDerivWithinAt.const_mul (ha : HasFDerivWithinAt a a' s x) (b : 𝔸) :
    HasFDerivWithinAt (fun y => b * a y) (b • a') s x
:= ⟨⟩⟨⟩
((ContinuousLinearMap.mul 𝕜 𝔸) b).hasFDerivAt.comp_hasFDerivWithinAt x ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=148, messages=[Message(data='`HasFDerivWithinAt.const_mul` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=148, i_end=234, 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.Combinatorics.SimpleGraph.UniversalVerts).

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

```lean
def deleteUniversalVerts (G : SimpleGraph V) : Subgraph G
:= ⟨⟩⟨⟩
(⊤ : Subgraph G).deleteVerts G.universalVerts
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=67, messages=[Message(data="Function expected at\n  Subgraph\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  G\n\nHint: The identifier `Subgraph` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=47), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=67, i_end=121, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, 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.Combinatorics.SimpleGraph.UniversalVerts).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def deleteUniversalVerts (G : SimpleGraph V) : Subgraph G
:= ⟨⟩⟨⟩
(⊤ : Subgraph G).deleteVerts G.universalVerts
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=67, messages=[Message(data="Function expected at\n  Subgraph\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  G\n\nHint: The identifier `Subgraph` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=47), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=67, i_end=121, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, 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.GroupTheory.Subgroup.Center).

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

```lean
instance instNormalCenter : (center G).Normal
:= .someMissingNamespaceXYZ
⟨fun a ha b ↦ by simp [mul_assoc, mem_center_iff.mp ha b, ha]⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=142, messages=[Message(data="Function expected at\n  center\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  G\n\nHint: The identifier `center` 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=29), 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)]
```

Return a corrected, compiling version of the code.

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

```lean
instance instNormalCenter : (center G).Normal
:= .someMissingNamespaceXYZ
⟨fun a ha b ↦ by simp [mul_assoc, mem_center_iff.mp ha b, ha]⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=142, messages=[Message(data="Function expected at\n  center\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  G\n\nHint: The identifier `center` 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=29), 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)]
```

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.MvPolynomial.Homogeneous).

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

```lean
lemma aeval [Algebra R S] (hφ : φ.IsHomogeneous m)
    (g : σ → MvPolynomial τ S) (hg : ∀ i, (g i).IsHomogeneous n) :
    (aeval g φ).IsHomogeneous (n * m)
:= .someMissingNamespaceXYZ
hφ.eval₂ _ _ (fun _ ↦ isHomogeneous_C _ _) hg

section CommRing

-- In this section we shadow the semiring `R` with a ring `R`.
variable {R σ : Type*} [CommRing R] {φ ψ : MvPolynomial σ R} {n : ℕ}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=245, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=13), pos_end=Position(line=1, column=24), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=245, i_end=326, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=400, 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.MvPolynomial.Homogeneous).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma aeval [Algebra R S] (hφ : φ.IsHomogeneous m)
    (g : σ → MvPolynomial τ S) (hg : ∀ i, (g i).IsHomogeneous n) :
    (aeval g φ).IsHomogeneous (n * m)
:= .someMissingNamespaceXYZ
hφ.eval₂ _ _ (fun _ ↦ isHomogeneous_C _ _) hg

section CommRing

-- In this section we shadow the semiring `R` with a ring `R`.
variable {R σ : Type*} [CommRing R] {φ ψ : MvPolynomial σ R} {n : ℕ}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=245, messages=[Message(data='failed to synthesize instance of type class\n  CommSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=13), pos_end=Position(line=1, column=24), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=245, i_end=326, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=400, 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.Dual.Defs).

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

```lean
theorem dualCoannihilator_sup_eq (U V : Submodule R (Module.Dual R M)) :
    (U ⊔ V).dualCoannihilator = U.dualCoannihilator ⊓ V.dualCoannihilator
:= thisIsWrongLemmaNameXYZ
(dualAnnihilator_gc R M).u_inf
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=208, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=53), pos_end=Position(line=1, column=68), 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=40), pos_end=Position(line=1, column=69), 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=53), pos_end=Position(line=1, column=68), 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=40), pos_end=Position(line=1, column=69), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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

```lean
theorem dualCoannihilator_sup_eq (U V : Submodule R (Module.Dual R M)) :
    (U ⊔ V).dualCoannihilator = U.dualCoannihilator ⊓ V.dualCoannihilator
:= thisIsWrongLemmaNameXYZ
(dualAnnihilator_gc R M).u_inf
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=208, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=53), pos_end=Position(line=1, column=68), 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=40), pos_end=Position(line=1, column=69), 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=53), pos_end=Position(line=1, column=68), 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=40), pos_end=Position(line=1, column=69), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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