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

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

```lean
theorem Monic.ne_zero_of_ne (h : (0 : R) ≠ 1) {p : R[X]} (hp : p.Monic) : p ≠ 0
:= by thisIsWrongLemmaNameXYZ
nontriviality R
  exact hp.ne_zero
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=91, 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  OfNat R 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  R\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=34), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat R 1\nnumerals are polymorphic in Lean, but the numeral `1` cannot be used in a context where the expected type is\n  R\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Invalid field notation: Field projection operates on types of the form `C ...` where C is a constant. The expression\n  p\nhas type `?m.12.1 R X ?m.13` which does not have the necessary form.', pos=Position(line=1, column=63), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat R[X] 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  R[X]\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=78), pos_end=Position(line=1, column=79), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nR : Type u_1\nidx✝ : Type u_2\nX : idx✝\nh : sorry ≠ 1\np : R[X]\nhp : sorry\n⊢ p ≠ 0', 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=91, i_end=148, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem Monic.ne_zero_of_ne (h : (0 : R) ≠ 1) {p : R[X]} (hp : p.Monic) : p ≠ 0
:= by thisIsWrongLemmaNameXYZ
nontriviality R
  exact hp.ne_zero
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=91, 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  OfNat R 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  R\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=34), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat R 1\nnumerals are polymorphic in Lean, but the numeral `1` cannot be used in a context where the expected type is\n  R\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Invalid field notation: Field projection operates on types of the form `C ...` where C is a constant. The expression\n  p\nhas type `?m.12.1 R X ?m.13` which does not have the necessary form.', pos=Position(line=1, column=63), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat R[X] 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  R[X]\ndue to the absence of the instance above\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=78), pos_end=Position(line=1, column=79), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nR : Type u_1\nidx✝ : Type u_2\nX : idx✝\nh : sorry ≠ 1\np : R[X]\nhp : sorry\n⊢ p ≠ 0', 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=91, i_end=148, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma inv_inv_add_inv : (p⁻¹ + q⁻¹)⁻¹ = r
  frobnicateTacticThatDoesNotExist := by exact_mod_cast h.coe.inv_inv_add_inv
protected lemma inv_lt_inv : p⁻¹ < r⁻¹ := h.coe.inv_lt_inv
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=129, messages=[Message(data="Function expected at\n  r\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist\n\nHint: The identifier `r` 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=40), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `h.coe.inv_inv_add_inv`', pos=Position(line=2, column=56), pos_end=Position(line=2, column=77), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=129, i_end=193, messages=[Message(data='protected declarations must be in a namespace', pos=Position(line=3, column=16), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma inv_inv_add_inv : (p⁻¹ + q⁻¹)⁻¹ = r
  frobnicateTacticThatDoesNotExist := by exact_mod_cast h.coe.inv_inv_add_inv
protected lemma inv_lt_inv : p⁻¹ < r⁻¹ := h.coe.inv_lt_inv
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=129, messages=[Message(data="Function expected at\n  r\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist\n\nHint: The identifier `r` 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=40), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `h.coe.inv_inv_add_inv`', pos=Position(line=2, column=56), pos_end=Position(line=2, column=77), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=129, i_end=193, messages=[Message(data='protected declarations must be in a namespace', pos=Position(line=3, column=16), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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

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


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

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

```lean
def Hom.Simps.hom (X Y : BddDistLat.{u}) (f : Hom X Y)
:= thisIsWrongLemmaNameXYZ
f.hom

initialize_simps_projections Hom (hom' → hom)

/-!
The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=89, messages=[Message(data="Function expected at\n  Hom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  X\n\nHint: The identifier `Hom` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=46), pos_end=Position(line=1, column=53), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=89, i_end=138, messages=[Message(data='Unknown constant `Hom`', pos=Position(line=5, column=29), pos_end=Position(line=5, column=32), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=138, i_end=242, 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.Category.BddDistLat).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def Hom.Simps.hom (X Y : BddDistLat.{u}) (f : Hom X Y)
:= thisIsWrongLemmaNameXYZ
f.hom

initialize_simps_projections Hom (hom' → hom)

/-!
The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=89, messages=[Message(data="Function expected at\n  Hom\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  X\n\nHint: The identifier `Hom` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=46), pos_end=Position(line=1, column=53), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=89, i_end=138, messages=[Message(data='Unknown constant `Hom`', pos=Position(line=5, column=29), pos_end=Position(line=5, column=32), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=138, i_end=242, 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.QuasiSeparated).

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

```lean
lemma QuasiSeparatedSpace.isCompact_sInter_of_nonempty {s : Set (Set α)} (hf : s.Finite)
    (hne : s.Nonempty) (ho : ∀ t ∈ s, IsOpen t ∨ IsClosed t) (hc : ∀ t ∈ s, IsCompact t) :
    IsCompact (⋂₀ s)
:= by _unbalancedParenXYZ
wlog h : ∀ t ∈ s, IsOpen t
  · let a := { t ∈ s | IsOpen t }
    let b := { t ∈ s | IsClosed t }
    have heq : s = a ∪ b := subset_antisymm (by grind) (by grind)
    rw [heq, Set.sInter_union]
    simp only [not_forall] at h
    obtain ⟨t, ht, hno⟩ := h
    obtain (ha | ha) := a.eq_empty_or_nonempty
    · simp only [ha, Set.sInter_empty, Set.univ_inter]
      exact IsCompact.of_isClosed_subset (hc _ ht) (isClosed_sInter (by grind)) (by grind)
    · apply IsCompact.inter_right
      · apply this (hf.subset (by grind)) ha <;> grind
      · exact isClosed_sInter (by grind)
  revert hne
  induction s, hf using Set.Finite.induction_on with
  | empty => simp
  | insert ha hs ih =>
    rename_i s
    obtain (rfl | hne) := s.eq_empty_or_nonempty
    · grind
    · grind [IsCompact.inter_of_isOpen, hs.isOpen_sInter, Set.sInter_insert]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=223, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=223, i_end=1100, 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.QuasiSeparated).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma QuasiSeparatedSpace.isCompact_sInter_of_nonempty {s : Set (Set α)} (hf : s.Finite)
    (hne : s.Nonempty) (ho : ∀ t ∈ s, IsOpen t ∨ IsClosed t) (hc : ∀ t ∈ s, IsCompact t) :
    IsCompact (⋂₀ s)
:= by _unbalancedParenXYZ
wlog h : ∀ t ∈ s, IsOpen t
  · let a := { t ∈ s | IsOpen t }
    let b := { t ∈ s | IsClosed t }
    have heq : s = a ∪ b := subset_antisymm (by grind) (by grind)
    rw [heq, Set.sInter_union]
    simp only [not_forall] at h
    obtain ⟨t, ht, hno⟩ := h
    obtain (ha | ha) := a.eq_empty_or_nonempty
    · simp only [ha, Set.sInter_empty, Set.univ_inter]
      exact IsCompact.of_isClosed_subset (hc _ ht) (isClosed_sInter (by grind)) (by grind)
    · apply IsCompact.inter_right
      · apply this (hf.subset (by grind)) ha <;> grind
      · exact isClosed_sInter (by grind)
  revert hne
  induction s, hf using Set.Finite.induction_on with
  | empty => simp
  | insert ha hs ih =>
    rename_i s
    obtain (rfl | hne) := s.eq_empty_or_nonempty
    · grind
    · grind [IsCompact.inter_of_isOpen, hs.isOpen_sInter, Set.sInter_insert]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=223, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=223, i_end=1100, 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.Equiv).

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

```lean
theorem UniqueDiffOn.image {f' : E → E →L[𝕜] F} (hs : UniqueDiffOn 𝕜 s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hd : ∀ x ∈ s, DenseRange (f' x)) :
    UniqueDiffOn 𝕜 (f '' s)
:= .someMissingNamespaceXYZ
forall_mem_image.2 fun x hx => (hf' x hx).uniqueDiffWithinAt (hs x hx) (hd x hx)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=317, messages=[Message(data='`UniqueDiffOn.image` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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

```lean
theorem UniqueDiffOn.image {f' : E → E →L[𝕜] F} (hs : UniqueDiffOn 𝕜 s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hd : ∀ x ∈ s, DenseRange (f' x)) :
    UniqueDiffOn 𝕜 (f '' s)
:= .someMissingNamespaceXYZ
forall_mem_image.2 fun x hx => (hf' x hx).uniqueDiffWithinAt (hs x hx) (hd x hx)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=317, messages=[Message(data='`UniqueDiffOn.image` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=26), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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.Finsubgraph).

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

```lean
def FinsubgraphHom.restrict {G' G'' : G.Finsubgraph} (h : G'' ≤ G') (f : G' →fg F) : G'' →fg F
  frobnicateTacticThatDoesNotExist := by
refine ⟨fun ⟨v, hv⟩ => f.toFun ⟨v, h.1 hv⟩, ?_⟩
  rintro ⟨u, hu⟩ ⟨v, hv⟩ huv
  exact f.map_rel' (h.2 huv)

/-- The inverse system of finite homomorphisms. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=269, messages=[Message(data='Unknown identifier `G.Finsubgraph`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `G.Finsubgraph`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Invalid `⟨...⟩` notation: The expected type of this term could not be determined', pos=Position(line=3, column=7), pos_end=Position(line=3, column=47), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=269, i_end=319, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, 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.Combinatorics.SimpleGraph.Finsubgraph).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def FinsubgraphHom.restrict {G' G'' : G.Finsubgraph} (h : G'' ≤ G') (f : G' →fg F) : G'' →fg F
  frobnicateTacticThatDoesNotExist := by
refine ⟨fun ⟨v, hv⟩ => f.toFun ⟨v, h.1 hv⟩, ?_⟩
  rintro ⟨u, hu⟩ ⟨v, hv⟩ huv
  exact f.map_rel' (h.2 huv)

/-- The inverse system of finite homomorphisms. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=269, messages=[Message(data='Unknown identifier `G.Finsubgraph`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `G.Finsubgraph`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Invalid `⟨...⟩` notation: The expected type of this term could not be determined', pos=Position(line=3, column=7), pos_end=Position(line=3, column=47), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=269, i_end=319, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, 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.GroupTheory.Sylow).

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

```lean
theorem normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
    [Finite (Sylow p N)] (P : Sylow p G) (hP : P ≤ N) : normalizer (P : Subgroup G) ⊔ N = ⊤
:= by .someMissingNamespaceXYZ
rw [← normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
    inf_of_le_left hP]

end Sylow

end InfiniteSylow

open Equiv Equiv.Perm Finset Function List QuotientGroup

universe u

variable {G : Type u} [Group G]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=190, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Group G\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=59), pos_end=Position(line=1, column=69), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=190, i_end=333, 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=9), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=333, i_end=352, messages=[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=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=352, i_end=410, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=410, i_end=422, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=422, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
    [Finite (Sylow p N)] (P : Sylow p G) (hP : P ≤ N) : normalizer (P : Subgroup G) ⊔ N = ⊤
:= by .someMissingNamespaceXYZ
rw [← normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
    inf_of_le_left hP]

end Sylow

end InfiniteSylow

open Equiv Equiv.Perm Finset Function List QuotientGroup

universe u

variable {G : Type u} [Group G]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=190, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Group G\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=59), pos_end=Position(line=1, column=69), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=190, i_end=333, 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=9), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=333, i_end=352, messages=[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=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=352, i_end=410, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=410, i_end=422, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=422, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem truncLT_smul [DecidableLT Γ] (c : Γ) (r : R) (x : V⟦Γ⟧) :
    truncLT c (r • x) = r • truncLT c x
  frobnicateTacticThatDoesNotExist := by ext; simp
end SMulZeroClass

section Addition

variable [PartialOrder Γ]

section AddMonoid

variable [AddMonoid R]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=168, messages=[Message(data='failed to synthesize instance of type class\n  LT Γ\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=35), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=168, i_end=187, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=4, column=0), pos_end=Position(line=4, column=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=187, i_end=205, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=205, i_end=233, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=233, i_end=252, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=252, i_end=274, 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.HahnSeries.Addition).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem truncLT_smul [DecidableLT Γ] (c : Γ) (r : R) (x : V⟦Γ⟧) :
    truncLT c (r • x) = r • truncLT c x
  frobnicateTacticThatDoesNotExist := by ext; simp
end SMulZeroClass

section Addition

variable [PartialOrder Γ]

section AddMonoid

variable [AddMonoid R]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=168, messages=[Message(data='failed to synthesize instance of type class\n  LT Γ\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=35), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=168, i_end=187, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=4, column=0), pos_end=Position(line=4, column=17), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=187, i_end=205, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=205, i_end=233, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=233, i_end=252, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=252, i_end=274, 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.Finsupp.LSum).

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

```lean
theorem domLCongr_apply {α₁ : Type*} {α₂ : Type*} (e : α₁ ≃ α₂) (v : α₁ →₀ M) :
    (Finsupp.domLCongr e : _ ≃ₗ[R] _) v = Finsupp.domCongr e v
  frobnicateTacticThatDoesNotExist :=
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=209, messages=[Message(data='failed to synthesize instance of type class\n  Zero M\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=69), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  NonAssocSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=27), pos_end=Position(line=2, column=36), 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=2, column=27), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Application type mismatch: The argument\n  frobnicateTacticThatDoesNotExist\nhas type\n  ?m.3\nbut is expected to have type\n  α₂\nin the application\n  ((Finsupp.domCongr e) ?m.34) frobnicateTacticThatDoesNotExist', pos=Position(line=3, column=2), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem domLCongr_apply {α₁ : Type*} {α₂ : Type*} (e : α₁ ≃ α₂) (v : α₁ →₀ M) :
    (Finsupp.domLCongr e : _ ≃ₗ[R] _) v = Finsupp.domCongr e v
  frobnicateTacticThatDoesNotExist :=
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=209, messages=[Message(data='failed to synthesize instance of type class\n  Zero M\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=69), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  NonAssocSemiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=27), pos_end=Position(line=2, column=36), 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=2, column=27), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Application type mismatch: The argument\n  frobnicateTacticThatDoesNotExist\nhas type\n  ?m.3\nbut is expected to have type\n  α₂\nin the application\n  ((Finsupp.domCongr e) ?m.34) frobnicateTacticThatDoesNotExist', pos=Position(line=3, column=2), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem prodPrimes_ne_zero : s.prodPrimes ≠ 0
  frobnicateTacticThatDoesNotExist :=
Squarefree.ne_zero s.prodPrimes_squarefree
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data='Unknown identifier `s.prodPrimes`', pos=Position(line=1, column=29), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  0\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=44), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `s.prodPrimes_squarefree`', pos=Position(line=3, column=19), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem prodPrimes_ne_zero : s.prodPrimes ≠ 0
  frobnicateTacticThatDoesNotExist :=
Squarefree.ne_zero s.prodPrimes_squarefree
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=128, messages=[Message(data='Unknown identifier `s.prodPrimes`', pos=Position(line=1, column=29), pos_end=Position(line=1, column=41), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  0\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=44), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `s.prodPrimes_squarefree`', pos=Position(line=3, column=19), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem algebraRat.charP_zero [Semiring R] [Algebra ℚ R] : CharP R 0
:= .someMissingNamespaceXYZ
charP_of_injective_algebraMap (algebraMap ℚ R).injective 0

/-- A nontrivial `ℚ`-algebra has characteristic zero.

This cannot be a (local) instance because it would immediately form a loop with the
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=161, messages=[Message(data='`algebraRat.charP_zero` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=29), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=161, i_end=301, messages=[Message(data='unterminated comment', pos=Position(line=7, column=83), 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.CharP.Algebra).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem algebraRat.charP_zero [Semiring R] [Algebra ℚ R] : CharP R 0
:= .someMissingNamespaceXYZ
charP_of_injective_algebraMap (algebraMap ℚ R).injective 0

/-- A nontrivial `ℚ`-algebra has characteristic zero.

This cannot be a (local) instance because it would immediately form a loop with the
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=161, messages=[Message(data='`algebraRat.charP_zero` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=29), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=161, i_end=301, messages=[Message(data='unterminated comment', pos=Position(line=7, column=83), 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.Nat.Log).

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

```lean
lemma log_lt_self (b : ℕ) {x : ℕ} (hx : x ≠ 0) : log b x < x
:= thisIsWrongLemmaNameXYZ
match le_or_gt b 1 with
  | .inl h => log_of_left_le_one h x ▸ Nat.pos_iff_ne_zero.2 hx
  | .inr h => log_lt_of_lt_pow hx <| Nat.lt_pow_self h
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data="unexpected token '|'; expected '.'", pos=Position(line=3, column=23), pos_end=Position(line=4, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  log\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  b\n\nHint: The identifier `log` 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=49), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=238, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma log_lt_self (b : ℕ) {x : ℕ} (hx : x ≠ 0) : log b x < x
:= thisIsWrongLemmaNameXYZ
match le_or_gt b 1 with
  | .inl h => log_of_left_le_one h x ▸ Nat.pos_iff_ne_zero.2 hx
  | .inr h => log_lt_of_lt_pow hx <| Nat.lt_pow_self h
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data="unexpected token '|'; expected '.'", pos=Position(line=3, column=23), pos_end=Position(line=4, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  log\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  b\n\nHint: The identifier `log` 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=49), pos_end=Position(line=1, column=56), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=238, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem csInf_image_le {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddBelow s) :
    sInf (f '' s) ≤ f c
  frobnicateTacticThatDoesNotExist := by
let f' : αᵒᵈ → βᵒᵈ := f
  exact le_csSup_image (α := αᵒᵈ) (β := βᵒᵈ)
    (show Monotone f' from fun x y hxy => h_mono hxy) hcs h_bdd

-- Porting note: in mathlib3 `f'` is not needed
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=356, messages=[Message(data='failed to synthesize instance of type class\n  LE 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=66), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Function expected at\n  f c\nbut this term has type\n  ?m.13\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=2, column=20), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `β`', pos=Position(line=4, column=15), pos_end=Position(line=4, column=16), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `exact`', pos=Position(line=5, column=2), pos_end=Position(line=5, column=7), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Invalid argument name `α` for function', pos=Position(line=5, column=23), pos_end=Position(line=5, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nα : Type u_1\nα✝ : Type u_2\nf : α → α✝\ns : Set α\nc : α\nhcs : c ∈ s\nh_bdd : sorry\nf' : αᵒᵈ → sorryᵒᵈ := sorry\n⊢ sInf (f '' s) ≤ sorry", pos=Position(line=3, column=38), pos_end=Position(line=6, column=63), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem csInf_image_le {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddBelow s) :
    sInf (f '' s) ≤ f c
  frobnicateTacticThatDoesNotExist := by
let f' : αᵒᵈ → βᵒᵈ := f
  exact le_csSup_image (α := αᵒᵈ) (β := βᵒᵈ)
    (show Monotone f' from fun x y hxy => h_mono hxy) hcs h_bdd

-- Porting note: in mathlib3 `f'` is not needed
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=356, messages=[Message(data='failed to synthesize instance of type class\n  LE 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=66), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Function expected at\n  f c\nbut this term has type\n  ?m.13\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=2, column=20), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `β`', pos=Position(line=4, column=15), pos_end=Position(line=4, column=16), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `exact`', pos=Position(line=5, column=2), pos_end=Position(line=5, column=7), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Invalid argument name `α` for function', pos=Position(line=5, column=23), pos_end=Position(line=5, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nα : Type u_1\nα✝ : Type u_2\nf : α → α✝\ns : Set α\nc : α\nhcs : c ∈ s\nh_bdd : sorry\nf' : αᵒᵈ → sorryᵒᵈ := sorry\n⊢ sInf (f '' s) ≤ sorry", pos=Position(line=3, column=38), pos_end=Position(line=6, column=63), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], 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.WithVal).

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

```lean
def withValEquiv (R : Type*) [CommRing R] [Algebra R K] [IsIntegralClosure R ℤ K] :
    𝓞 (WithVal v) ≃+* R
:= NumberField.RingOfIntegers.equiv R ⟨⟩⟨⟩
end NumberField.RingOfIntegers

open scoped NumberField in
/-- The ring of integers of `WithVal v`, when `v` is a valuation on `ℚ`, is
equivalent to `ℤ`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=159, 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=1, column=43), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=159, i_end=198, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=3, column=40), pos_end=Position(line=3, column=41), 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=4, column=0), pos_end=Position(line=4, column=30), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=198, i_end=326, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=21), 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.WithVal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def withValEquiv (R : Type*) [CommRing R] [Algebra R K] [IsIntegralClosure R ℤ K] :
    𝓞 (WithVal v) ≃+* R
:= NumberField.RingOfIntegers.equiv R ⟨⟩⟨⟩
end NumberField.RingOfIntegers

open scoped NumberField in
/-- The ring of integers of `WithVal v`, when `v` is a valuation on `ℚ`, is
equivalent to `ℤ`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=159, 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=1, column=43), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=159, i_end=198, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=3, column=40), pos_end=Position(line=3, column=41), 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=4, column=0), pos_end=Position(line=4, column=30), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=198, i_end=326, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=21), 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.Fourier.AddCircleMulti).

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

```lean
theorem hasSum_mFourier_series_apply_of_summable (h : Summable (mFourierCoeff f))
    (x : UnitAddTorus d) : HasSum (fun i ↦ mFourierCoeff f i • mFourier i x) (f x)
:= by .someMissingNamespaceXYZ
simpa only [map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
    (hasSum_mFourier_series_of_summable h)

end Convergence

end UnitAddTorus
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=177, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourierCoeff\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `mFourierCoeff` 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=64), pos_end=Position(line=1, column=79), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourierCoeff\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `mFourierCoeff` 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=43), pos_end=Position(line=2, column=60), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourier\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 `mFourier` 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=63), pos_end=Position(line=2, column=75), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=78), pos_end=Position(line=2, column=81), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝² : Sort u_1\nmFourierCoeff : x✝²\nd : Type u_2\nx✝¹ : Sort u_3\nmFourier : x✝¹\nx✝ : Sort u_4\nf : x✝\nh : Summable sorry\nx : UnitAddTorus d\n⊢ HasSum (fun i => ?m.16 • ?m.18) sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=177, i_end=326, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=15), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=342, messages=[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=16), 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.Fourier.AddCircleMulti).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem hasSum_mFourier_series_apply_of_summable (h : Summable (mFourierCoeff f))
    (x : UnitAddTorus d) : HasSum (fun i ↦ mFourierCoeff f i • mFourier i x) (f x)
:= by .someMissingNamespaceXYZ
simpa only [map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
    (hasSum_mFourier_series_of_summable h)

end Convergence

end UnitAddTorus
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=177, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourierCoeff\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `mFourierCoeff` 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=64), pos_end=Position(line=1, column=79), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourierCoeff\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `mFourierCoeff` 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=43), pos_end=Position(line=2, column=60), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  mFourier\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 `mFourier` 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=63), pos_end=Position(line=2, column=75), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=78), pos_end=Position(line=2, column=81), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝² : Sort u_1\nmFourierCoeff : x✝²\nd : Type u_2\nx✝¹ : Sort u_3\nmFourier : x✝¹\nx✝ : Sort u_4\nf : x✝\nh : Summable sorry\nx : UnitAddTorus d\n⊢ HasSum (fun i => ?m.16 • ?m.18) sorry', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=177, i_end=326, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=7, column=0), pos_end=Position(line=7, column=15), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=326, i_end=342, messages=[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=16), 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.Matroid.Loop).

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

```lean
lemma closure_inter_eq_of_subset_coloops (X : Set α) (hK : K ⊆ M.coloops) :
     M.closure X ∩ K = X ∩ K
:= by .someMissingNamespaceXYZ
nth_rw 1 [← inter_eq_self_of_subset_right hK]
  rw [← inter_assoc, closure_inter_coloops_eq, inter_assoc, inter_eq_self_of_subset_right hK]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.coloops`', pos=Position(line=1, column=63), pos_end=Position(line=1, column=72), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.closure`', pos=Position(line=2, column=5), pos_end=Position(line=2, column=14), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Type u_1\nK X : Set α\nhK : K ⊆ sorry\n⊢ sorry ∩ K = X ∩ K', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=286, 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.Matroid.Loop).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma closure_inter_eq_of_subset_coloops (X : Set α) (hK : K ⊆ M.coloops) :
     M.closure X ∩ K = X ∩ K
:= by .someMissingNamespaceXYZ
nth_rw 1 [← inter_eq_self_of_subset_right hK]
  rw [← inter_assoc, closure_inter_coloops_eq, inter_assoc, inter_eq_self_of_subset_right hK]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=120, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.coloops`', pos=Position(line=1, column=63), pos_end=Position(line=1, column=72), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.closure`', pos=Position(line=2, column=5), pos_end=Position(line=2, column=14), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Type u_1\nK X : Set α\nhK : K ⊆ sorry\n⊢ sorry ∩ K = X ∩ K', pos=Position(line=3, column=6), pos_end=Position(line=3, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=120, i_end=286, 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.OrderOfElement).

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

```lean
lemma pow_mod_card (a : G) (n : ℕ) : a ^ (n % card G) = a ^ n
  frobnicateTacticThatDoesNotExist := by
rw [eq_comm, ← pow_mod_orderOf, ← Nat.mod_mod_of_dvd n orderOf_dvd_card, pow_mod_orderOf]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=198, messages=[Message(data='`pow_mod_card` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=18), 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.OrderOfElement).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma pow_mod_card (a : G) (n : ℕ) : a ^ (n % card G) = a ^ n
  frobnicateTacticThatDoesNotExist := by
rw [eq_comm, ← pow_mod_orderOf, ← Nat.mod_mod_of_dvd n orderOf_dvd_card, pow_mod_orderOf]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=198, messages=[Message(data='`pow_mod_card` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=18), 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.Polynomial.Basic).

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

```lean
theorem isNoetherianRing_fin_0 [IsNoetherianRing R] :
    IsNoetherianRing (MvPolynomial (Fin 0) R)
:= by .someMissingNamespaceXYZ
apply isNoetherianRing_of_ringEquiv R
  symm; apply MvPolynomial.isEmptyRingEquiv R (Fin 0)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=108, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  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=32), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=108, i_end=222, 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.Polynomial.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem isNoetherianRing_fin_0 [IsNoetherianRing R] :
    IsNoetherianRing (MvPolynomial (Fin 0) R)
:= by .someMissingNamespaceXYZ
apply isNoetherianRing_of_ringEquiv R
  symm; apply MvPolynomial.isEmptyRingEquiv R (Fin 0)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=108, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  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=32), pos_end=Position(line=1, column=50), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=108, i_end=222, 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.Matrix.Charpoly.Basic).

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

```lean
lemma BlockTriangular.charpoly {α : Type*} {b : n → α} [LinearOrder α] (h : M.BlockTriangular b) :
    M.charpoly = ∏ a ∈ image b univ, (M.toSquareBlock b a).charpoly
:= by thisIsWrongLemmaNameXYZ
simp only [Matrix.charpoly, h.charmatrix.det, charmatrix_toSquareBlock]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.BlockTriangular`', pos=Position(line=1, column=76), pos_end=Position(line=1, column=93), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.charpoly`', pos=Position(line=2, column=4), pos_end=Position(line=2, column=14), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  image\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  b\n\nHint: The identifier `image` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=23), pos_end=Position(line=2, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.toSquareBlock`', pos=Position(line=2, column=38), pos_end=Position(line=2, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nn : Sort u_2\nx✝ : Sort u_3\nimage : x✝\nα : Type u_1\nb : n → α\ninst✝ : LinearOrder α\nh : sorry\n⊢ sorry = ∏ a ∈ sorry, ?m.10', 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=183, i_end=277, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma BlockTriangular.charpoly {α : Type*} {b : n → α} [LinearOrder α] (h : M.BlockTriangular b) :
    M.charpoly = ∏ a ∈ image b univ, (M.toSquareBlock b a).charpoly
:= by thisIsWrongLemmaNameXYZ
simp only [Matrix.charpoly, h.charmatrix.det, charmatrix_toSquareBlock]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=183, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.BlockTriangular`', pos=Position(line=1, column=76), pos_end=Position(line=1, column=93), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `M.charpoly`', pos=Position(line=2, column=4), pos_end=Position(line=2, column=14), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  image\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  b\n\nHint: The identifier `image` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=23), pos_end=Position(line=2, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `M.toSquareBlock`', pos=Position(line=2, column=38), pos_end=Position(line=2, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nn : Sort u_2\nx✝ : Sort u_3\nimage : x✝\nα : Type u_1\nb : n → α\ninst✝ : LinearOrder α\nh : sorry\n⊢ sorry = ∏ a ∈ sorry, ?m.10', 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=183, i_end=277, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem mem_realUnits_iff (u : (𝓞 K)ˣ) :
    u ∈ realUnits K ↔ ∃ v : (𝓞 K⁺)ˣ, algebraMap (𝓞 K⁺) (𝓞 K) v = u
:= by .someMissingNamespaceXYZ
simp [realUnits, MonoidHom.mem_range, RingHom.toMonoidHom_eq_coe, Units.ext_iff]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=140, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `𝓞` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=32), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  realUnits\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `realUnits` 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  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K⁺\n\nHint: The identifier `𝓞` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=29), pos_end=Position(line=2, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K⁺\n\nHint: The identifier `𝓞` 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=49), pos_end=Position(line=2, column=53), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `𝓞` 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=56), pos_end=Position(line=2, column=59), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=140, i_end=243, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem mem_realUnits_iff (u : (𝓞 K)ˣ) :
    u ∈ realUnits K ↔ ∃ v : (𝓞 K⁺)ˣ, algebraMap (𝓞 K⁺) (𝓞 K) v = u
:= by .someMissingNamespaceXYZ
simp [realUnits, MonoidHom.mem_range, RingHom.toMonoidHom_eq_coe, Units.ext_iff]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=140, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `𝓞` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=32), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  realUnits\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `realUnits` 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  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K⁺\n\nHint: The identifier `𝓞` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=29), pos_end=Position(line=2, column=33), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K⁺\n\nHint: The identifier `𝓞` 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=49), pos_end=Position(line=2, column=53), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  𝓞\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  K\n\nHint: The identifier `𝓞` 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=56), pos_end=Position(line=2, column=59), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=140, i_end=243, 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.Category.TopCat.Limits.Pullbacks).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Category.TopCat.Limits.Pullbacks.fst_iso_of_right_embedding_range_subset`:
State 0: ⊢ theorem fst_iso_of_right_embedding_range_subset {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S}
    (hg : IsEmbedding g) (H : Set.range f ⊆ Set.range g) :
    IsIso (pullback.fst f g)
Tactic: rw [pullback_fst_range]
State 1: ⊢ <goal after rw [pullback_fst_range]>
Tactic: exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ }
State 2: ⊢ <goal after exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ }>
Tactic: convert (isoOfHomeo esto).isIso_hom
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Category.TopCat.Limits.Pullbacks).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Category.TopCat.Limits.Pullbacks.fst_iso_of_right_embedding_range_subset`:
State 0: ⊢ theorem fst_iso_of_right_embedding_range_subset {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S}
    (hg : IsEmbedding g) (H : Set.range f ⊆ Set.range g) :
    IsIso (pullback.fst f g)
Tactic: rw [pullback_fst_range]
State 1: ⊢ <goal after rw [pullback_fst_range]>
Tactic: exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ }
State 2: ⊢ <goal after exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ }>
Tactic: convert (isoOfHomeo esto).isIso_hom
```

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


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

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.Graph.Subgraph.le_of_le_subset`:
State 0: ⊢ lemma le_of_le_subset (h' : K ≤ G) (hsu : V(K) ⊆ V(H)) (h : H ≤i G) : K ≤ H
Tactic: refine (Compatible.of_le_le h' h.le).le_iff.mpr ⟨hsu, fun e he ↦ ?_⟩
State 1: ⊢ <goal after refine (Compatible.of_le_le h' h.le).le_iff.mpr ⟨hsu, fun e he ↦ ?_⟩>
Tactic: obtain ⟨u, v, huv⟩ := K.exists_isLink_of_mem_edgeSet he
State 2: ⊢ <goal after obtain ⟨u, v, huv⟩ := K.exists_isLink_of_mem_edgeSet he>
Tactic: exact h.2 (huv.mono h') (hsu huv.left_mem) (hsu huv.right_mem) |>.edge_mem
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.Graph.Subgraph).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.Graph.Subgraph.le_of_le_subset`:
State 0: ⊢ lemma le_of_le_subset (h' : K ≤ G) (hsu : V(K) ⊆ V(H)) (h : H ≤i G) : K ≤ H
Tactic: refine (Compatible.of_le_le h' h.le).le_iff.mpr ⟨hsu, fun e he ↦ ?_⟩
State 1: ⊢ <goal after refine (Compatible.of_le_le h' h.le).le_iff.mpr ⟨hsu, fun e he ↦ ?_⟩>
Tactic: obtain ⟨u, v, huv⟩ := K.exists_isLink_of_mem_edgeSet he
State 2: ⊢ <goal after obtain ⟨u, v, huv⟩ := K.exists_isLink_of_mem_edgeSet he>
Tactic: exact h.2 (huv.mono h') (hsu huv.left_mem) (hsu huv.right_mem) |>.edge_mem
```

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


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

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.PGroup.nonempty_fixed_point_of_prime_not_dvd_card`:
State 0: ⊢ theorem nonempty_fixed_point_of_prime_not_dvd_card (α) [MulAction G α] (hpα : ¬p ∣ Nat.card α) :
    (fixedPoints G α).Nonempty
Tactic: have : Finite α := Nat.finite_of_card_ne_zero (fun h ↦ (h ▸ hpα) (dvd_zero p))  @Set.Nonempty.of_subtype _ _  (by
State 1: ⊢ <goal after have : Finite α := Nat.finite_of_card_ne_zero (fun h ↦ (h ▸ hpα) (dvd_zero p))>
Tactic: rw [← Finite.card_pos_iff, pos_iff_ne_zero]
State 2: ⊢ <goal after rw [← Finite.card_pos_iff, pos_iff_ne_zero]>
Tactic: contrapose! hpα
State 3: ⊢ <goal after contrapose! hpα>
Tactic: rw [← Nat.modEq_zero_iff_dvd, ← hpα]
State 4: ⊢ <goal after rw [← Nat.modEq_zero_iff_dvd, ← hpα]>
Tactic: exact hG.card_modEq_card_fixedPoints α)  /-- If a p-group acts on `α` and the cardinality of `α` is a multiple  of `p`, and the action has one fixed point, then it has another fixed point. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.PGroup).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.PGroup.nonempty_fixed_point_of_prime_not_dvd_card`:
State 0: ⊢ theorem nonempty_fixed_point_of_prime_not_dvd_card (α) [MulAction G α] (hpα : ¬p ∣ Nat.card α) :
    (fixedPoints G α).Nonempty
Tactic: have : Finite α := Nat.finite_of_card_ne_zero (fun h ↦ (h ▸ hpα) (dvd_zero p))  @Set.Nonempty.of_subtype _ _  (by
State 1: ⊢ <goal after have : Finite α := Nat.finite_of_card_ne_zero (fun h ↦ (h ▸ hpα) (dvd_zero p))>
Tactic: rw [← Finite.card_pos_iff, pos_iff_ne_zero]
State 2: ⊢ <goal after rw [← Finite.card_pos_iff, pos_iff_ne_zero]>
Tactic: contrapose! hpα
State 3: ⊢ <goal after contrapose! hpα>
Tactic: rw [← Nat.modEq_zero_iff_dvd, ← hpα]
State 4: ⊢ <goal after rw [← Nat.modEq_zero_iff_dvd, ← hpα]>
Tactic: exact hG.card_modEq_card_fixedPoints α)  /-- If a p-group acts on `α` and the cardinality of `α` is a multiple  of `p`, and the action has one fixed point, then it has another fixed point. -/
```

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


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

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Localization.AsSubring.isLocalization_range_mapToFractionRing`:
State 0: ⊢ instance isLocalization_range_mapToFractionRing (B : Type*) [CommRing B] [Algebra A B]
    [IsLocalization S B] (hS : S ≤ A⁰) : IsLocalization S (mapToFractionRing K S B hS).range
Tactic: refine ⟨fun a b h => ?_, Set.rangeFactorization_surjective⟩
State 1: ⊢ <goal after refine ⟨fun a b h => ?_, Set.rangeFactorization_surjective⟩>
Tactic: refine (IsLocalization.lift_injective_iff _).2 (fun a b => ?_) (Subtype.ext_iff.1 h)
State 2: ⊢ <goal after refine (IsLocalization.lift_injective_iff _).2 (fun a b => ?_) (Subtype.ext_iff.1 h)>
Tactic: exact ⟨fun h => congr_arg _ (IsLocalization.injective _ hS h),  fun h => congr_arg _ (IsFractionRing.injective A K h)⟩)
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Localization.AsSubring).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Localization.AsSubring.isLocalization_range_mapToFractionRing`:
State 0: ⊢ instance isLocalization_range_mapToFractionRing (B : Type*) [CommRing B] [Algebra A B]
    [IsLocalization S B] (hS : S ≤ A⁰) : IsLocalization S (mapToFractionRing K S B hS).range
Tactic: refine ⟨fun a b h => ?_, Set.rangeFactorization_surjective⟩
State 1: ⊢ <goal after refine ⟨fun a b h => ?_, Set.rangeFactorization_surjective⟩>
Tactic: refine (IsLocalization.lift_injective_iff _).2 (fun a b => ?_) (Subtype.ext_iff.1 h)
State 2: ⊢ <goal after refine (IsLocalization.lift_injective_iff _).2 (fun a b => ?_) (Subtype.ext_iff.1 h)>
Tactic: exact ⟨fun h => congr_arg _ (IsLocalization.injective _ hS h),  fun h => congr_arg _ (IsFractionRing.injective A K h)⟩)
```

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


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

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Algebra.Polynomial.EraseLead.mono_map_natDegree_eq`:
State 0: ⊢ theorem mono_map_natDegree_eq {S F : Type*} [Semiring S]
    [FunLike F R[X] S[X]] [AddMonoidHomClass F R[X] S[X]] {φ : F}
    {p : R[X]} (k : ℕ) (fu : ℕ → ℕ) (fu0 : ∀ {n}, n ≤ k → fu n = 0)
    (fc : ∀ {n m}, k ≤ n → n < m → fu n < fu m) (φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0)
    (φ_mon_nat : ∀ n c, c ≠ 0 → (φ (monomial n c)).natDegree = fu n) :
    (φ p).natDegree = fu p.natDegree
Tactic: refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)  p.natDegree (by simp [fu0]) ?_ ?_ _ rfl.le  · intro n r r0 _
State 1: ⊢ <goal after refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)>
Tactic: rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]  · intro f g fg _ fk gk
State 2: ⊢ <goal after rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]>
Tactic: rw [natDegree_add_eq_right_of_natDegree_lt fg, map_add]  by_cases! FG : k ≤ f.natDegree  · rw [natDegree_add_eq_right_of_natDegree_lt, gk]
State 3: ⊢ <goal after rw [natDegree_add_eq_right_of_natDegree_lt fg, map_add]>
Tactic: rw [fk, gk]
State 4: ⊢ <goal after rw [fk, gk]>
Tactic: exact fc FG fg  · cases k  · nomatch FG  · rwa [φ_k FG, zero_add]
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Polynomial.EraseLead).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Algebra.Polynomial.EraseLead.mono_map_natDegree_eq`:
State 0: ⊢ theorem mono_map_natDegree_eq {S F : Type*} [Semiring S]
    [FunLike F R[X] S[X]] [AddMonoidHomClass F R[X] S[X]] {φ : F}
    {p : R[X]} (k : ℕ) (fu : ℕ → ℕ) (fu0 : ∀ {n}, n ≤ k → fu n = 0)
    (fc : ∀ {n m}, k ≤ n → n < m → fu n < fu m) (φ_k : ∀ {f : R[X]}, f.natDegree < k → φ f = 0)
    (φ_mon_nat : ∀ n c, c ≠ 0 → (φ (monomial n c)).natDegree = fu n) :
    (φ p).natDegree = fu p.natDegree
Tactic: refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)  p.natDegree (by simp [fu0]) ?_ ?_ _ rfl.le  · intro n r r0 _
State 1: ⊢ <goal after refine induction_with_natDegree_le (fun p => (φ p).natDegree = fu p.natDegree)>
Tactic: rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]  · intro f g fg _ fk gk
State 2: ⊢ <goal after rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, φ_mon_nat _ _ r0]>
Tactic: rw [natDegree_add_eq_right_of_natDegree_lt fg, map_add]  by_cases! FG : k ≤ f.natDegree  · rw [natDegree_add_eq_right_of_natDegree_lt, gk]
State 3: ⊢ <goal after rw [natDegree_add_eq_right_of_natDegree_lt fg, map_add]>
Tactic: rw [fk, gk]
State 4: ⊢ <goal after rw [fk, gk]>
Tactic: exact fc FG fg  · cases k  · nomatch FG  · rwa [φ_k FG, zero_add]
```

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