[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.InformationTheory.KullbackLeibler.Basic).

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

```lean
lemma mul_log_le_klDiv (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ)
        + ν.real univ - μ.real univ)
      ≤ klDiv μ ν
:= by .someMissingNamespaceXYZ
by_cases hμν : μ ≪ ν
  swap; · simp [hμν]
  by_cases h_int : Integrable (llr μ ν) μ
  swap; · simp [h_int]
  rw [← ENNReal.ofReal_toReal (a := klDiv μ ν)]
  · exact ENNReal.ofReal_le_ofReal (mul_log_le_toReal_klDiv hμν h_int)
  · rw [klDiv_ne_top_iff]
    exact ⟨hμν, h_int⟩

end Inequalities

/-- **Converse Gibbs' inequality**: the Kullback-Leibler divergence between two finite measures is
zero if and only if the two measures are equal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=226, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=226, i_end=570, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=15, column=0), pos_end=Position(line=15, column=16), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=570, i_end=719, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, 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.InformationTheory.KullbackLeibler.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma mul_log_le_klDiv (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ)
        + ν.real univ - μ.real univ)
      ≤ klDiv μ ν
:= by .someMissingNamespaceXYZ
by_cases hμν : μ ≪ ν
  swap; · simp [hμν]
  by_cases h_int : Integrable (llr μ ν) μ
  swap; · simp [h_int]
  rw [← ENNReal.ofReal_toReal (a := klDiv μ ν)]
  · exact ENNReal.ofReal_le_ofReal (mul_log_le_toReal_klDiv hμν h_int)
  · rw [klDiv_ne_top_iff]
    exact ⟨hμν, h_int⟩

end Inequalities

/-- **Converse Gibbs' inequality**: the Kullback-Leibler divergence between two finite measures is
zero if and only if the two measures are equal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=226, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=226, i_end=570, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=15, column=0), pos_end=Position(line=15, column=16), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=570, i_end=719, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, 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.ModelTheory.Satisfiability).

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

```lean
theorem IsSatisfiable.mono (h : T'.IsSatisfiable) (hs : T ⊆ T') : T.IsSatisfiable
:= .someMissingNamespaceXYZ
⟨(Theory.Model.mono (ModelType.is_model h.some) hs).bundled⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=176, messages=[Message(data="Invalid field notation: Type of\n  T'\nis not known; cannot resolve field `IsSatisfiable`\n\nHint: Consider replacing the field projection `.IsSatisfiable` with a call to the function `FirstOrder.Language.Theory.IsSatisfiable`.", pos=Position(line=1, column=32), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  T\nis not known; cannot resolve field `IsSatisfiable`\n\nHint: Consider replacing the field projection `.IsSatisfiable` with a call to the function `FirstOrder.Language.Theory.IsSatisfiable`.', pos=Position(line=1, column=66), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.invalidField._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.ModelTheory.Satisfiability).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsSatisfiable.mono (h : T'.IsSatisfiable) (hs : T ⊆ T') : T.IsSatisfiable
:= .someMissingNamespaceXYZ
⟨(Theory.Model.mono (ModelType.is_model h.some) hs).bundled⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=176, messages=[Message(data="Invalid field notation: Type of\n  T'\nis not known; cannot resolve field `IsSatisfiable`\n\nHint: Consider replacing the field projection `.IsSatisfiable` with a call to the function `FirstOrder.Language.Theory.IsSatisfiable`.", pos=Position(line=1, column=32), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  T\nis not known; cannot resolve field `IsSatisfiable`\n\nHint: Consider replacing the field projection `.IsSatisfiable` with a call to the function `FirstOrder.Language.Theory.IsSatisfiable`.', pos=Position(line=1, column=66), pos_end=Position(line=1, column=81), severity=<Severity.ERROR: 3>, kind='lean.invalidField._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.RepresentationTheory.Rep.Basic).

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

```lean
abbrev ofHom (f : ρ.IntertwiningMap σ) : of ρ ⟶ of σ
:= thisIsWrongLemmaNameXYZ
ConcreteCategory.ofHom (C := Rep.{w} k G) f

/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=131, messages=[Message(data='Unknown identifier `ρ.IntertwiningMap`', pos=Position(line=1, column=18), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  of\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  ρ\n\nHint: The identifier `of` 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=41), pos_end=Position(line=1, column=45), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  of\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  σ\n\nHint: The identifier `of` 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=48), pos_end=Position(line=1, column=52), 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=131, i_end=202, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=71), 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.RepresentationTheory.Rep.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
abbrev ofHom (f : ρ.IntertwiningMap σ) : of ρ ⟶ of σ
:= thisIsWrongLemmaNameXYZ
ConcreteCategory.ofHom (C := Rep.{w} k G) f

/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=131, messages=[Message(data='Unknown identifier `ρ.IntertwiningMap`', pos=Position(line=1, column=18), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  of\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  ρ\n\nHint: The identifier `of` 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=41), pos_end=Position(line=1, column=45), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  of\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  σ\n\nHint: The identifier `of` 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=48), pos_end=Position(line=1, column=52), 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=131, i_end=202, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=71), 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.AlgebraicGeometry.PullbackCarrier).

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

```lean
lemma range_snd_comp :
    Set.range (pullback.snd f g ≫ g) = Set.range f ∩ Set.range g
:= by .someMissingNamespaceXYZ
rw [← pullback.condition, range_fst_comp]

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=55, messages=[Message(data='expected token', pos=Position(line=2, column=32), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=55, i_end=224, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma range_snd_comp :
    Set.range (pullback.snd f g ≫ g) = Set.range f ∩ Set.range g
:= by .someMissingNamespaceXYZ
rw [← pullback.condition, range_fst_comp]

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=55, messages=[Message(data='expected token', pos=Position(line=2, column=32), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=55, i_end=224, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem normalizedTrace_algebraMap_apply_eq_self (a : F) :
    normalizedTrace F K (algebraMap F K a) = a
:= by simp .someMissingNamespaceXYZ
/-- The normalized trace map is a left inverse of the algebra map. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=117, messages=[Message(data="Function expected at\n  normalizedTrace\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  F\n\nHint: The identifier `normalizedTrace` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nF : Sort u_1\nx✝ : Sort u_2\nnormalizedTrace : x✝\na : F\n⊢ sorry () = a', pos=Position(line=3, column=3), pos_end=Position(line=3, column=10), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=117, i_end=211, messages=[Message(data="unexpected token '.'; expected command", pos=Position(line=3, column=11), pos_end=Position(line=3, column=12), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=69), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem normalizedTrace_algebraMap_apply_eq_self (a : F) :
    normalizedTrace F K (algebraMap F K a) = a
:= by simp .someMissingNamespaceXYZ
/-- The normalized trace map is a left inverse of the algebra map. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=117, messages=[Message(data="Function expected at\n  normalizedTrace\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  F\n\nHint: The identifier `normalizedTrace` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nF : Sort u_1\nx✝ : Sort u_2\nnormalizedTrace : x✝\na : F\n⊢ sorry () = a', pos=Position(line=3, column=3), pos_end=Position(line=3, column=10), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=117, i_end=211, messages=[Message(data="unexpected token '.'; expected command", pos=Position(line=3, column=11), pos_end=Position(line=3, column=12), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=69), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

[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.FieldTheory.RatFunc.Luroth.swap_Φ_natDegree_eq_θ_natDegree`:
State 0: ⊢ lemma swap_Φ_natDegree_eq_θ_natDegree (h : E ≠ ⊥) :
    (Bivariate.swap (Φ E)).natDegree = (θ E).natDegree
Tactic: have := congr((Bivariate.swap $(Q₃_mul_Φ h)).natDegree)  rwa [map_mul, Polynomial.map_C, Bivariate.swap_C_C,  natDegree_mul (C_ne_zero.mpr (Q₃_map h ▸ Q₂_ne_zero h))  ((map_ne_zero_iff _ Bivariate.swap.injective).mpr (Φ_ne_zero h)),  natDegree_C, zero_add, swap_θ, natDegree_neg] at this  /-- Lüroth's theorem. Any intermediate field between `K` and `K⟮X⟯` is  generated by a single element `generator E`. See also `transcendental_generator`  for the statement that the generator is transcendental if `E ≠ ⊥`. -/  public theorem eq_adjoin_generator : E = K⟮generator E⟯ := by  by_cases h : E = ⊥  · rwa [generator_eq_zero h, adjoin_zero]
State 1: ⊢ <goal after have := congr((Bivariate.swap $(Q₃_mul_Φ h)).natDegree)>
Tactic: refine le_antisymm (relfinrank_eq_one_iff.mp ?_) adjoin_generator_le  suffices (φ E).natDegree = m E by
State 2: ⊢ <goal after refine le_antisymm (relfinrank_eq_one_iff.mp ?_) adjoin_generator_le>
Tactic: refine (mul_eq_right₀ ?_).mp <| this ▸ (generator E).finrank_eq_max_natDegree ▸  φ_natDegree h ▸ relfinrank_mul_finrank_top (adjoin_generator_le (E := E))
State 3: ⊢ <goal after refine (mul_eq_right₀ ?_).mp <| this ▸ (generator E).finrank_eq_max_natDegree ▸>
Tactic: intro H
State 4: ⊢ <goal after intro H>
Tactic: exact generator_ne_C h ((eq_C_iff _).mpr (Nat.max_eq_zero_iff.mp H))
State 5: ⊢ <goal after exact generator_ne_C h ((eq_C_iff _).mpr (Nat.max_eq_zero_iff.mp H))>
Tactic: rw [← Φ_natDegree_eq_φ_natDegree h, Φ_natDegree_eq_θ_natDegree h]
State 6: ⊢ <goal after rw [← Φ_natDegree_eq_φ_natDegree h, Φ_natDegree_eq_θ_natDegree h]>
Tactic: exact le_antisymm (θ_natDegree_le h) (swap_Φ_natDegree_eq_θ_natDegree h ▸ m_le_swap_Φ_natDegree h)  /-- The `K`-algebra equivalence between `K⟮X⟯` and an intermediate field `E` given  by sending `X` to `generator E`. See also `Luroth.eq_adjoin_generator`. -/  public def algEquiv (h : E ≠ ⊥) : K⟮X⟯ ≃ₐ[K] E :=  (algEquivOfTranscendental (generator E) (transcendental_of_ne_C _ (generator_ne_C h))).trans <|  IntermediateField.equivOfEq eq_adjoin_generator.symm
```

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.FieldTheory.RatFunc.Luroth).[/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.FieldTheory.RatFunc.Luroth.swap_Φ_natDegree_eq_θ_natDegree`:
State 0: ⊢ lemma swap_Φ_natDegree_eq_θ_natDegree (h : E ≠ ⊥) :
    (Bivariate.swap (Φ E)).natDegree = (θ E).natDegree
Tactic: have := congr((Bivariate.swap $(Q₃_mul_Φ h)).natDegree)  rwa [map_mul, Polynomial.map_C, Bivariate.swap_C_C,  natDegree_mul (C_ne_zero.mpr (Q₃_map h ▸ Q₂_ne_zero h))  ((map_ne_zero_iff _ Bivariate.swap.injective).mpr (Φ_ne_zero h)),  natDegree_C, zero_add, swap_θ, natDegree_neg] at this  /-- Lüroth's theorem. Any intermediate field between `K` and `K⟮X⟯` is  generated by a single element `generator E`. See also `transcendental_generator`  for the statement that the generator is transcendental if `E ≠ ⊥`. -/  public theorem eq_adjoin_generator : E = K⟮generator E⟯ := by  by_cases h : E = ⊥  · rwa [generator_eq_zero h, adjoin_zero]
State 1: ⊢ <goal after have := congr((Bivariate.swap $(Q₃_mul_Φ h)).natDegree)>
Tactic: refine le_antisymm (relfinrank_eq_one_iff.mp ?_) adjoin_generator_le  suffices (φ E).natDegree = m E by
State 2: ⊢ <goal after refine le_antisymm (relfinrank_eq_one_iff.mp ?_) adjoin_generator_le>
Tactic: refine (mul_eq_right₀ ?_).mp <| this ▸ (generator E).finrank_eq_max_natDegree ▸  φ_natDegree h ▸ relfinrank_mul_finrank_top (adjoin_generator_le (E := E))
State 3: ⊢ <goal after refine (mul_eq_right₀ ?_).mp <| this ▸ (generator E).finrank_eq_max_natDegree ▸>
Tactic: intro H
State 4: ⊢ <goal after intro H>
Tactic: exact generator_ne_C h ((eq_C_iff _).mpr (Nat.max_eq_zero_iff.mp H))
State 5: ⊢ <goal after exact generator_ne_C h ((eq_C_iff _).mpr (Nat.max_eq_zero_iff.mp H))>
Tactic: rw [← Φ_natDegree_eq_φ_natDegree h, Φ_natDegree_eq_θ_natDegree h]
State 6: ⊢ <goal after rw [← Φ_natDegree_eq_φ_natDegree h, Φ_natDegree_eq_θ_natDegree h]>
Tactic: exact le_antisymm (θ_natDegree_le h) (swap_Φ_natDegree_eq_θ_natDegree h ▸ m_le_swap_Φ_natDegree h)  /-- The `K`-algebra equivalence between `K⟮X⟯` and an intermediate field `E` given  by sending `X` to `generator E`. See also `Luroth.eq_adjoin_generator`. -/  public def algEquiv (h : E ≠ ⊥) : K⟮X⟯ ≃ₐ[K] E :=  (algEquivOfTranscendental (generator E) (transcendental_of_ne_C _ (generator_ne_C h))).trans <|  IntermediateField.equivOfEq eq_adjoin_generator.symm
```

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.MeasureTheory.Function.EssSup).

[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.MeasureTheory.Function.EssSup.essSup_map_measure_of_measurable`:
State 0: ⊢ theorem essSup_map_measure_of_measurable (hg : Measurable g) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ
Tactic: refine le_antisymm ?_ (essSup_comp_le_essSup_map_measure hf)
State 1: ⊢ <goal after refine le_antisymm ?_ (essSup_comp_le_essSup_map_measure hf)>
Tactic: refine limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => ?_)
State 2: ⊢ <goal after refine limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => ?_)>
Tactic: rw [eventually_map] at h_le ⊢
State 3: ⊢ <goal after rw [eventually_map] at h_le ⊢>
Tactic: rw [ae_map_iff hf (measurableSet_le hg measurable_const)]
State 4: ⊢ <goal after rw [ae_map_iff hf (measurableSet_le hg measurable_const)]>
Tactic: exact h_le
```

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.MeasureTheory.Function.EssSup).[/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.MeasureTheory.Function.EssSup.essSup_map_measure_of_measurable`:
State 0: ⊢ theorem essSup_map_measure_of_measurable (hg : Measurable g) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ
Tactic: refine le_antisymm ?_ (essSup_comp_le_essSup_map_measure hf)
State 1: ⊢ <goal after refine le_antisymm ?_ (essSup_comp_le_essSup_map_measure hf)>
Tactic: refine limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => ?_)
State 2: ⊢ <goal after refine limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => ?_)>
Tactic: rw [eventually_map] at h_le ⊢
State 3: ⊢ <goal after rw [eventually_map] at h_le ⊢>
Tactic: rw [ae_map_iff hf (measurableSet_le hg measurable_const)]
State 4: ⊢ <goal after rw [ae_map_iff hf (measurableSet_le hg measurable_const)]>
Tactic: exact h_le
```

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.Probability.Martingale.BorelCantelli).

[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.Probability.Martingale.BorelCantelli.Submartingale`:
State 0: ⊢ theorem Submartingale.exists_tendsto_of_abs_bddAbove_aux [IsFiniteMeasure μ]
    (hf : Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
Tactic: have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedAbove f i n ω) atTop (𝓝 c) := by
State 1: ⊢ <goal after have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedAbove f i n ω) atTop (𝓝 c) := by>
Tactic: rw [ae_all_iff]
State 2: ⊢ <goal after rw [ae_all_iff]>
Tactic: exact fun i ↦ Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedAbove i)  (hf.eLpNorm_stoppedAbove_le' i.cast_nonneg hf0 hbdd)  filter_upwards [ht] with ω hω hωb
State 3: ⊢ <goal after exact fun i ↦ Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedAbove i)>
Tactic: rw [BddAbove] at hωb
State 4: ⊢ <goal after rw [BddAbove] at hωb>
Tactic: obtain ⟨i, hi⟩ := exists_nat_gt hωb.some
State 5: ⊢ <goal after obtain ⟨i, hi⟩ := exists_nat_gt hωb.some>
Tactic: have hib : ∀ n, f n ω < i := by
State 6: ⊢ <goal after have hib : ∀ n, f n ω < i := by>
Tactic: intro n
State 7: ⊢ <goal after intro n>
Tactic: exact lt_of_le_of_lt ((mem_upperBounds.1 hωb.some_mem) _ ⟨n, rfl⟩) hi
State 8: ⊢ <goal after exact lt_of_le_of_lt ((mem_upperBounds.1 hωb.some_mem) _ ⟨n, rfl⟩) hi>
Tactic: have heq : ∀ n, stoppedAbove f i n ω = f n ω := by
State 9: ⊢ <goal after have heq : ∀ n, stoppedAbove f i n ω = f n ω := by>
Tactic: intro n
State 10: ⊢ <goal after intro n>
Tactic: rw [stoppedAbove, stoppedProcess, leastGE, hittingAfter_eq_top_iff.mpr]  · simp only [le_top, inf_of_le_left]  congr  · simp [hib]
State 11: ⊢ <goal after rw [stoppedAbove, stoppedProcess, leastGE, hittingAfter_eq_top_iff.mpr]>
Tactic: simp only [← heq, hω i]
```

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.Probability.Martingale.BorelCantelli).[/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.Probability.Martingale.BorelCantelli.Submartingale`:
State 0: ⊢ theorem Submartingale.exists_tendsto_of_abs_bddAbove_aux [IsFiniteMeasure μ]
    (hf : Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
Tactic: have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedAbove f i n ω) atTop (𝓝 c) := by
State 1: ⊢ <goal after have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedAbove f i n ω) atTop (𝓝 c) := by>
Tactic: rw [ae_all_iff]
State 2: ⊢ <goal after rw [ae_all_iff]>
Tactic: exact fun i ↦ Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedAbove i)  (hf.eLpNorm_stoppedAbove_le' i.cast_nonneg hf0 hbdd)  filter_upwards [ht] with ω hω hωb
State 3: ⊢ <goal after exact fun i ↦ Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedAbove i)>
Tactic: rw [BddAbove] at hωb
State 4: ⊢ <goal after rw [BddAbove] at hωb>
Tactic: obtain ⟨i, hi⟩ := exists_nat_gt hωb.some
State 5: ⊢ <goal after obtain ⟨i, hi⟩ := exists_nat_gt hωb.some>
Tactic: have hib : ∀ n, f n ω < i := by
State 6: ⊢ <goal after have hib : ∀ n, f n ω < i := by>
Tactic: intro n
State 7: ⊢ <goal after intro n>
Tactic: exact lt_of_le_of_lt ((mem_upperBounds.1 hωb.some_mem) _ ⟨n, rfl⟩) hi
State 8: ⊢ <goal after exact lt_of_le_of_lt ((mem_upperBounds.1 hωb.some_mem) _ ⟨n, rfl⟩) hi>
Tactic: have heq : ∀ n, stoppedAbove f i n ω = f n ω := by
State 9: ⊢ <goal after have heq : ∀ n, stoppedAbove f i n ω = f n ω := by>
Tactic: intro n
State 10: ⊢ <goal after intro n>
Tactic: rw [stoppedAbove, stoppedProcess, leastGE, hittingAfter_eq_top_iff.mpr]  · simp only [le_top, inf_of_le_left]  congr  · simp [hib]
State 11: ⊢ <goal after rw [stoppedAbove, stoppedProcess, leastGE, hittingAfter_eq_top_iff.mpr]>
Tactic: simp only [← heq, hω i]
```

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.InformationTheory.KullbackLeibler.KLFun).

[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.InformationTheory.KullbackLeibler.KLFun.deriv_klFun`:
State 0: ⊢ lemma deriv_klFun : deriv klFun = log
Tactic: ext x  by_cases h0 : x = 0  · simp only [h0, log_zero]
State 1: ⊢ <goal after ext x>
Tactic: exact deriv_zero_of_not_differentiableAt not_differentiableAt_klFun_zero  · exact (hasDerivAt_klFun h0).deriv
```

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.InformationTheory.KullbackLeibler.KLFun).[/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.InformationTheory.KullbackLeibler.KLFun.deriv_klFun`:
State 0: ⊢ lemma deriv_klFun : deriv klFun = log
Tactic: ext x  by_cases h0 : x = 0  · simp only [h0, log_zero]
State 1: ⊢ <goal after ext x>
Tactic: exact deriv_zero_of_not_differentiableAt not_differentiableAt_klFun_zero  · exact (hasDerivAt_klFun h0).deriv
```

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.AlgebraicGeometry.Morphisms.UniversallyClosed).

[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.AlgebraicGeometry.Morphisms.UniversallyClosed.universallyClosed_eq_universallySpecializing`:
State 0: ⊢ lemma universallyClosed_eq_universallySpecializing :
    @UniversallyClosed = (topologically @SpecializingMap).universally ⊓ @QuasiCompact
Tactic: rw [← universally_eq_iff (P := @QuasiCompact).mpr inferInstance, ← universally_inf]
State 1: ⊢ <goal after rw [← universally_eq_iff (P := @QuasiCompact).mpr inferInstance, ← universally_inf]>
Tactic: apply le_antisymm  · rw [← universally_eq_iff (P := @UniversallyClosed).mpr inferInstance]
State 2: ⊢ <goal after apply le_antisymm>
Tactic: exact universally_mono fun X Y f H ↦ ⟨f.isClosedMap.specializingMap, inferInstance⟩  · rw [universallyClosed_eq]
State 3: ⊢ <goal after exact universally_mono fun X Y f H ↦ ⟨f.isClosedMap.specializingMap, inferInstance⟩>
Tactic: exact universally_mono fun X Y f ⟨h₁, h₂⟩ ↦ (isClosedMap_iff_specializingMap _).mpr h₁
```

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.AlgebraicGeometry.Morphisms.UniversallyClosed).[/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.AlgebraicGeometry.Morphisms.UniversallyClosed.universallyClosed_eq_universallySpecializing`:
State 0: ⊢ lemma universallyClosed_eq_universallySpecializing :
    @UniversallyClosed = (topologically @SpecializingMap).universally ⊓ @QuasiCompact
Tactic: rw [← universally_eq_iff (P := @QuasiCompact).mpr inferInstance, ← universally_inf]
State 1: ⊢ <goal after rw [← universally_eq_iff (P := @QuasiCompact).mpr inferInstance, ← universally_inf]>
Tactic: apply le_antisymm  · rw [← universally_eq_iff (P := @UniversallyClosed).mpr inferInstance]
State 2: ⊢ <goal after apply le_antisymm>
Tactic: exact universally_mono fun X Y f H ↦ ⟨f.isClosedMap.specializingMap, inferInstance⟩  · rw [universallyClosed_eq]
State 3: ⊢ <goal after exact universally_mono fun X Y f H ↦ ⟨f.isClosedMap.specializingMap, inferInstance⟩>
Tactic: exact universally_mono fun X Y f ⟨h₁, h₂⟩ ↦ (isClosedMap_iff_specializingMap _).mpr h₁
```

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.FieldTheory.SeparableClosure).

[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.FieldTheory.SeparableClosure.IsSepClosed`:
State 0: ⊢ theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
    separableClosure F E = ⊥ ↔ IsSepClosed F
Tactic: refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,  fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩
State 1: ⊢ <goal after refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,>
Tactic: obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
State 2: ⊢ <goal after obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep>
Tactic: obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd _ x hx)
State 3: ⊢ <goal after obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd _ x hx)>
Tactic: exact ⟨x, by simpa [Algebra.ofId_apply] using hx⟩  /-- If `E` is separably closed, then the separable closure of `F` in `E` is an absolute  separable closure of `F`. -/
```

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.FieldTheory.SeparableClosure).[/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.FieldTheory.SeparableClosure.IsSepClosed`:
State 0: ⊢ theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
    separableClosure F E = ⊥ ↔ IsSepClosed F
Tactic: refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,  fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩
State 1: ⊢ <goal after refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,>
Tactic: obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
State 2: ⊢ <goal after obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep>
Tactic: obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd _ x hx)
State 3: ⊢ <goal after obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd _ x hx)>
Tactic: exact ⟨x, by simpa [Algebra.ofId_apply] using hx⟩  /-- If `E` is separably closed, then the separable closure of `F` in `E` is an absolute  separable closure of `F`. -/
```

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.MeasureTheory.OuterMeasure.OfFunction).

[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.MeasureTheory.OuterMeasure.OfFunction.comap_ofFunction`:
State 0: ⊢ theorem comap_ofFunction {β} (f : β → α) (h : Monotone m ∨ Surjective f) :
    comap f (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun s => m (f '' s)) (by simp; simp [m_empty])
Tactic: refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_  · rw [comap_apply]
State 1: ⊢ <goal after refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_>
Tactic: apply ofFunction_le  · rw [comap_apply, ofFunction_apply, ofFunction_apply]
State 2: ⊢ <goal after apply ofFunction_le>
Tactic: refine iInf_mono' fun t => ⟨fun k => f ⁻¹' t k, ?_⟩
State 3: ⊢ <goal after refine iInf_mono' fun t => ⟨fun k => f ⁻¹' t k, ?_⟩>
Tactic: refine iInf_mono' fun ht => ?_
State 4: ⊢ <goal after refine iInf_mono' fun ht => ?_>
Tactic: rw [Set.image_subset_iff, preimage_iUnion] at ht
State 5: ⊢ <goal after rw [Set.image_subset_iff, preimage_iUnion] at ht>
Tactic: refine ⟨ht, ENNReal.tsum_le_tsum fun n => ?_⟩  rcases h with hl | hr  exacts [hl (image_preimage_subset _ _), (congr_arg m (hr.image_preimage (t n))).le]
```

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.MeasureTheory.OuterMeasure.OfFunction).[/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.MeasureTheory.OuterMeasure.OfFunction.comap_ofFunction`:
State 0: ⊢ theorem comap_ofFunction {β} (f : β → α) (h : Monotone m ∨ Surjective f) :
    comap f (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun s => m (f '' s)) (by simp; simp [m_empty])
Tactic: refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_  · rw [comap_apply]
State 1: ⊢ <goal after refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_>
Tactic: apply ofFunction_le  · rw [comap_apply, ofFunction_apply, ofFunction_apply]
State 2: ⊢ <goal after apply ofFunction_le>
Tactic: refine iInf_mono' fun t => ⟨fun k => f ⁻¹' t k, ?_⟩
State 3: ⊢ <goal after refine iInf_mono' fun t => ⟨fun k => f ⁻¹' t k, ?_⟩>
Tactic: refine iInf_mono' fun ht => ?_
State 4: ⊢ <goal after refine iInf_mono' fun ht => ?_>
Tactic: rw [Set.image_subset_iff, preimage_iUnion] at ht
State 5: ⊢ <goal after rw [Set.image_subset_iff, preimage_iUnion] at ht>
Tactic: refine ⟨ht, ENNReal.tsum_le_tsum fun n => ?_⟩  rcases h with hl | hr  exacts [hl (image_preimage_subset _ _), (congr_arg m (hr.image_preimage (t n))).le]
```

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.InformationTheory.KullbackLeibler.Basic).

[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.InformationTheory.KullbackLeibler.Basic.klDiv_smul_same`:
State 0: ⊢ lemma klDiv_smul_same [IsFiniteMeasure μ] [IsFiniteMeasure ν] (c : ℝ≥0) :
    klDiv (c • μ) (c • ν) = c * klDiv μ ν
Tactic: have hc' : (c : ℝ≥0∞) ≠ 0 := by simpa
State 1: ⊢ <goal after have hc' : (c : ℝ≥0∞) ≠ 0 := by simpa>
Tactic: have hμ_smul (μ : Measure α) : μ = c⁻¹ • (c • μ) := by
State 2: ⊢ <goal after have hμ_smul (μ : Measure α) : μ = c⁻¹ • (c • μ) := by>
Tactic: rw [smul_smul, inv_mul_cancel₀ hc, one_smul]  by_cases hμν : μ ≪ ν  swap  · rw [klDiv_of_not_ac hμν, klDiv_of_not_ac, ENNReal.mul_top hc']
State 3: ⊢ <goal after rw [smul_smul, inv_mul_cancel₀ hc, one_smul]>
Tactic: refine fun h_contra ↦ hμν ?_
State 4: ⊢ <goal after refine fun h_contra ↦ hμν ?_>
Tactic: rw [hμ_smul μ, hμ_smul ν]
State 5: ⊢ <goal after rw [hμ_smul μ, hμ_smul ν]>
Tactic: exact h_contra.smul _  by_cases h_int : Integrable (llr μ ν) μ  swap  · rw [klDiv_of_not_integrable h_int, klDiv_of_not_integrable, ENNReal.mul_top hc']
State 6: ⊢ <goal after exact h_contra.smul _>
Tactic: refine fun h_contra ↦ h_int ?_
State 7: ⊢ <goal after refine fun h_contra ↦ h_int ?_>
Tactic: rw [hμ_smul μ, hμ_smul ν]
State 8: ⊢ <goal after rw [hμ_smul μ, hμ_smul ν]>
Tactic: refine Integrable.smul_measure_nnreal ?_
State 9: ⊢ <goal after refine Integrable.smul_measure_nnreal ?_>
Tactic: rw [integrable_congr (llr_smul_nnreal_same (hμν.smul c) c⁻¹ (by simpa))]  fun_prop
State 10: ⊢ <goal after rw [integrable_congr (llr_smul_nnreal_same (hμν.smul c) c⁻¹ (by simpa))]>
Tactic: rw [← ENNReal.ofReal_toReal (klDiv_ne_top (hμν.smul c) _),  ← ENNReal.ofReal_toReal (klDiv_ne_top hμν h_int)]  swap  · refine Integrable.smul_measure_nnreal ?_
State 11: ⊢ <goal after rw [← ENNReal.ofReal_toReal (klDiv_ne_top (hμν.smul c) _),>
Tactic: rw [integrable_congr (llr_smul_nnreal_same hμν c hc)]  fun_prop
State 12: ⊢ <goal after rw [integrable_congr (llr_smul_nnreal_same hμν c hc)]>
Tactic: simp [toReal_klDiv_smul_same hμν h_int]  section Inequalities  variable [IsFiniteMeasure μ] [IsFiniteMeasure ν]
```

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.InformationTheory.KullbackLeibler.Basic).[/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.InformationTheory.KullbackLeibler.Basic.klDiv_smul_same`:
State 0: ⊢ lemma klDiv_smul_same [IsFiniteMeasure μ] [IsFiniteMeasure ν] (c : ℝ≥0) :
    klDiv (c • μ) (c • ν) = c * klDiv μ ν
Tactic: have hc' : (c : ℝ≥0∞) ≠ 0 := by simpa
State 1: ⊢ <goal after have hc' : (c : ℝ≥0∞) ≠ 0 := by simpa>
Tactic: have hμ_smul (μ : Measure α) : μ = c⁻¹ • (c • μ) := by
State 2: ⊢ <goal after have hμ_smul (μ : Measure α) : μ = c⁻¹ • (c • μ) := by>
Tactic: rw [smul_smul, inv_mul_cancel₀ hc, one_smul]  by_cases hμν : μ ≪ ν  swap  · rw [klDiv_of_not_ac hμν, klDiv_of_not_ac, ENNReal.mul_top hc']
State 3: ⊢ <goal after rw [smul_smul, inv_mul_cancel₀ hc, one_smul]>
Tactic: refine fun h_contra ↦ hμν ?_
State 4: ⊢ <goal after refine fun h_contra ↦ hμν ?_>
Tactic: rw [hμ_smul μ, hμ_smul ν]
State 5: ⊢ <goal after rw [hμ_smul μ, hμ_smul ν]>
Tactic: exact h_contra.smul _  by_cases h_int : Integrable (llr μ ν) μ  swap  · rw [klDiv_of_not_integrable h_int, klDiv_of_not_integrable, ENNReal.mul_top hc']
State 6: ⊢ <goal after exact h_contra.smul _>
Tactic: refine fun h_contra ↦ h_int ?_
State 7: ⊢ <goal after refine fun h_contra ↦ h_int ?_>
Tactic: rw [hμ_smul μ, hμ_smul ν]
State 8: ⊢ <goal after rw [hμ_smul μ, hμ_smul ν]>
Tactic: refine Integrable.smul_measure_nnreal ?_
State 9: ⊢ <goal after refine Integrable.smul_measure_nnreal ?_>
Tactic: rw [integrable_congr (llr_smul_nnreal_same (hμν.smul c) c⁻¹ (by simpa))]  fun_prop
State 10: ⊢ <goal after rw [integrable_congr (llr_smul_nnreal_same (hμν.smul c) c⁻¹ (by simpa))]>
Tactic: rw [← ENNReal.ofReal_toReal (klDiv_ne_top (hμν.smul c) _),  ← ENNReal.ofReal_toReal (klDiv_ne_top hμν h_int)]  swap  · refine Integrable.smul_measure_nnreal ?_
State 11: ⊢ <goal after rw [← ENNReal.ofReal_toReal (klDiv_ne_top (hμν.smul c) _),>
Tactic: rw [integrable_congr (llr_smul_nnreal_same hμν c hc)]  fun_prop
State 12: ⊢ <goal after rw [integrable_congr (llr_smul_nnreal_same hμν c hc)]>
Tactic: simp [toReal_klDiv_smul_same hμν h_int]  section Inequalities  variable [IsFiniteMeasure μ] [IsFiniteMeasure ν]
```

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.ModelTheory.Substructures).

[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.ModelTheory.Substructures.codRestrict`:
State 0: ⊢ def codRestrict (p : L.Substructure N) (f : M ↪[L] N) (h : ∀ c, f c ∈ p) : M ↪[L] p where
  toFun
Tactic: rw [← p.subtype.map_rel]  change RelMap r (Hom.comp p.subtype.toHom (f.toHom.codRestrict p h) ∘ x) ↔ _
State 1: ⊢ <goal after rw [← p.subtype.map_rel]>
Tactic: rw [Hom.subtype_comp_codRestrict, ← f.map_rel]
State 2: ⊢ <goal after rw [Hom.subtype_comp_codRestrict, ← f.map_rel]>
Tactic: rfl
```

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.ModelTheory.Substructures).[/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.ModelTheory.Substructures.codRestrict`:
State 0: ⊢ def codRestrict (p : L.Substructure N) (f : M ↪[L] N) (h : ∀ c, f c ∈ p) : M ↪[L] p where
  toFun
Tactic: rw [← p.subtype.map_rel]  change RelMap r (Hom.comp p.subtype.toHom (f.toHom.codRestrict p h) ∘ x) ↔ _
State 1: ⊢ <goal after rw [← p.subtype.map_rel]>
Tactic: rw [Hom.subtype_comp_codRestrict, ← f.map_rel]
State 2: ⊢ <goal after rw [Hom.subtype_comp_codRestrict, ← f.map_rel]>
Tactic: rfl
```

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.FieldTheory.IsAlgClosed.Classification).

[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.FieldTheory.IsAlgClosed.Classification.ringEquiv_of_equiv_of_char_eq`:
State 0: ⊢ theorem ringEquiv_of_equiv_of_char_eq (p : ℕ) [CharP K p] [CharP L p] (hK : ℵ₀ < #K)
    (hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L)
Tactic: exact ringEquiv_of_Cardinal_eq_of_charP p hK hKL  · simp only [hp] at *  letI : CharZero K := CharP.charP_to_charZero K  letI : CharZero L := CharP.charP_to_charZero L
State 1: ⊢ <goal after exact ringEquiv_of_Cardinal_eq_of_charP p hK hKL>
Tactic: exact ringEquiv_of_equiv_of_charZero hK hKL  end IsAlgClosed
```

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.FieldTheory.IsAlgClosed.Classification).[/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.FieldTheory.IsAlgClosed.Classification.ringEquiv_of_equiv_of_char_eq`:
State 0: ⊢ theorem ringEquiv_of_equiv_of_char_eq (p : ℕ) [CharP K p] [CharP L p] (hK : ℵ₀ < #K)
    (hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L)
Tactic: exact ringEquiv_of_Cardinal_eq_of_charP p hK hKL  · simp only [hp] at *  letI : CharZero K := CharP.charP_to_charZero K  letI : CharZero L := CharP.charP_to_charZero L
State 1: ⊢ <goal after exact ringEquiv_of_Cardinal_eq_of_charP p hK hKL>
Tactic: exact ringEquiv_of_equiv_of_charZero hK hKL  end IsAlgClosed
```

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.MeasureTheory.Integral.BoundedContinuousFunction).

[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.MeasureTheory.Integral.BoundedContinuousFunction.norm_integral_le_mul_norm`:
State 0: ⊢ lemma norm_integral_le_mul_norm [IsFiniteMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ μ.real Set.univ * ‖f‖
Tactic: calc  ‖∫ x, f x ∂μ‖  _ ≤ ∫ x, ‖f x‖ ∂μ := norm_integral_le_integral_norm _  _ ≤ ∫ _, ‖f‖ ∂μ := ?_  _ = μ.real Set.univ • ‖f‖ := by rw [integral_const]
State 1: ⊢ <goal after calc  ‖∫ x, f x ∂μ‖>
Tactic: apply integral_mono _ (integrable_const ‖f‖) (fun x ↦ f.norm_coe_le_norm x) -- NOTE: `gcongr`?
State 2: ⊢ <goal after apply integral_mono _ (integrable_const ‖f‖) (fun x ↦ f.norm_coe_le_norm x) -- NOTE: `gcongr`?>
Tactic: exact (integrable_norm_iff f.continuous.measurable.aestronglyMeasurable).mpr (f.integrable μ)
```

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.MeasureTheory.Integral.BoundedContinuousFunction).[/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.MeasureTheory.Integral.BoundedContinuousFunction.norm_integral_le_mul_norm`:
State 0: ⊢ lemma norm_integral_le_mul_norm [IsFiniteMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ μ.real Set.univ * ‖f‖
Tactic: calc  ‖∫ x, f x ∂μ‖  _ ≤ ∫ x, ‖f x‖ ∂μ := norm_integral_le_integral_norm _  _ ≤ ∫ _, ‖f‖ ∂μ := ?_  _ = μ.real Set.univ • ‖f‖ := by rw [integral_const]
State 1: ⊢ <goal after calc  ‖∫ x, f x ∂μ‖>
Tactic: apply integral_mono _ (integrable_const ‖f‖) (fun x ↦ f.norm_coe_le_norm x) -- NOTE: `gcongr`?
State 2: ⊢ <goal after apply integral_mono _ (integrable_const ‖f‖) (fun x ↦ f.norm_coe_le_norm x) -- NOTE: `gcongr`?>
Tactic: exact (integrable_norm_iff f.continuous.measurable.aestronglyMeasurable).mpr (f.integrable μ)
```

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.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic).

[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.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic.map_Φ`:
State 0: ⊢ lemma map_Φ (n : ℤ) : (W.map f).Φ n = (W.Φ n).map f
Tactic: rw [← coe_mapRingHom]
State 1: ⊢ <goal after rw [← coe_mapRingHom]>
Tactic: simp [Φ, map_sub, apply_ite <| mapRingHom f]
```

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.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic).[/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.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic.map_Φ`:
State 0: ⊢ lemma map_Φ (n : ℤ) : (W.map f).Φ n = (W.Φ n).map f
Tactic: rw [← coe_mapRingHom]
State 1: ⊢ <goal after rw [← coe_mapRingHom]>
Tactic: simp [Φ, map_sub, apply_ite <| mapRingHom f]
```

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