[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.SetTheory.Descriptive.Tree).

[User]
You are editing a file in `Mathlib/SetTheory/Descriptive/Tree.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.SetTheory.Descriptive.Tree.mem_of_prefix`, L46-L48)
```lean
lemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by
obtain ⟨_, rfl⟩ := h'; exact mem_of_append h
```

## Edit instruction
Edit `mem_of_prefix` so it still compiles and the proof is simplified where possible. Return the full updated `mem_of_prefix` block.

<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.SetTheory.Descriptive.Tree).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/SetTheory/Descriptive/Tree.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.SetTheory.Descriptive.Tree.mem_of_prefix`, L46-L48)
```lean
lemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by
obtain ⟨_, rfl⟩ := h'; exact mem_of_append h
```

## Edit instruction
Edit `mem_of_prefix` so it still compiles and the proof is simplified where possible. Return the full updated `mem_of_prefix` block.
[/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.CategoryTheory.Functor.Derived.PointwiseRightDerived).

[User]
You are editing a file in `Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived.hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor`, L78-L84)
```lean
lemma hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor [L.IsLocalization W] :
    HasPointwiseLeftKanExtension L F := fun Y ↦ by
have := Localization.essSurj L W
  rw [← hasPointwiseLeftKanExtensionAt_iff_of_iso _ F (L.objObjPreimageIso Y),
    ← F.hasPointwiseRightDerivedFunctorAt_iff L W]
  infer_instance
```

## Edit instruction
Edit `hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor` so it still compiles and the proof is simplified where possible. Return the full updated `hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor` block.

<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.CategoryTheory.Functor.Derived.PointwiseRightDerived).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived.hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor`, L78-L84)
```lean
lemma hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor [L.IsLocalization W] :
    HasPointwiseLeftKanExtension L F := fun Y ↦ by
have := Localization.essSurj L W
  rw [← hasPointwiseLeftKanExtensionAt_iff_of_iso _ F (L.objObjPreimageIso Y),
    ← F.hasPointwiseRightDerivedFunctorAt_iff L W]
  infer_instance
```

## Edit instruction
Edit `hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor` so it still compiles and the proof is simplified where possible. Return the full updated `hasPointwiseLeftKanExtension_of_hasPointwiseRightDerivedFunctor` block.
[/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.CategoryTheory.Abelian.Exact).

[User]
You are editing a file in `Mathlib/CategoryTheory/Abelian/Exact.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Abelian.Exact.exact_cokernel`, L159-L164)
```lean
theorem exact_cokernel {X Y : C} (f : X ⟶ Y) :
    (ShortComplex.mk f (cokernel.π f) (by simp)).Exact := exact_of_g_is_cokernel _ (cokernelIsCokernel f)

variable (S)
```

## Edit instruction
Edit `exact_cokernel` so it still compiles and the proof is simplified where possible. Return the full updated `exact_cokernel` block.

<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.CategoryTheory.Abelian.Exact).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Abelian/Exact.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Abelian.Exact.exact_cokernel`, L159-L164)
```lean
theorem exact_cokernel {X Y : C} (f : X ⟶ Y) :
    (ShortComplex.mk f (cokernel.π f) (by simp)).Exact := exact_of_g_is_cokernel _ (cokernelIsCokernel f)

variable (S)
```

## Edit instruction
Edit `exact_cokernel` so it still compiles and the proof is simplified where possible. Return the full updated `exact_cokernel` block.
[/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.CategoryTheory.Functor.KanExtension.Basic).

[User]
You are editing a file in `Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Functor.KanExtension.Basic.rightExtensionEquivalenceOfIso₁`, L511-L514)
```lean
def rightExtensionEquivalenceOfIso₁ : RightExtension L F ≌ RightExtension L' F := CostructuredArrow.mapNatIso ((whiskeringLeft C D H).mapIso iso₁)

include iso₁ in
```

## Edit instruction
Edit `rightExtensionEquivalenceOfIso₁` so it still compiles and the proof is simplified where possible. Return the full updated `rightExtensionEquivalenceOfIso₁` block.

<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.CategoryTheory.Functor.KanExtension.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Functor.KanExtension.Basic.rightExtensionEquivalenceOfIso₁`, L511-L514)
```lean
def rightExtensionEquivalenceOfIso₁ : RightExtension L F ≌ RightExtension L' F := CostructuredArrow.mapNatIso ((whiskeringLeft C D H).mapIso iso₁)

include iso₁ in
```

## Edit instruction
Edit `rightExtensionEquivalenceOfIso₁` so it still compiles and the proof is simplified where possible. Return the full updated `rightExtensionEquivalenceOfIso₁` block.
[/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.CategoryTheory.Endomorphism).

[User]
You are editing a file in `Mathlib/CategoryTheory/Endomorphism.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Endomorphism.asHom`, L55-L58)
```lean
abbrev asHom (f : End X) : X ⟶ X := f
-- TODO: to fix defeq abuse, this should be `(1 : End x) = of (𝟙 X)`.
-- But that would require many more extra simp lemmas to get rid of the `of`.
```

## Edit instruction
Edit `asHom` so it still compiles and the proof is simplified where possible. Return the full updated `asHom` block.

<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.CategoryTheory.Endomorphism).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Endomorphism.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Endomorphism.asHom`, L55-L58)
```lean
abbrev asHom (f : End X) : X ⟶ X := f
-- TODO: to fix defeq abuse, this should be `(1 : End x) = of (𝟙 X)`.
-- But that would require many more extra simp lemmas to get rid of the `of`.
```

## Edit instruction
Edit `asHom` so it still compiles and the proof is simplified where possible. Return the full updated `asHom` block.
[/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.LocallyIntegrable).

[User]
You are editing a file in `Mathlib/MeasureTheory/Function/LocallyIntegrable.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.MeasureTheory.Function.LocallyIntegrable.integrable_iff_integrableAtFilter_atBot`, L473-L479)
```lean
theorem integrable_iff_integrableAtFilter_atBot [LinearOrder X] [OrderTop X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrable f μ := by
constructor
  · exact fun hf ↦ ⟨hf.integrableAtFilter _, hf.locallyIntegrable⟩
  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
    exact h.1.filter_mono cocompact_le_atBot
```

## Edit instruction
Edit `integrable_iff_integrableAtFilter_atBot` so it still compiles and the proof is simplified where possible. Return the full updated `integrable_iff_integrableAtFilter_atBot` block.

<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.LocallyIntegrable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/MeasureTheory/Function/LocallyIntegrable.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.MeasureTheory.Function.LocallyIntegrable.integrable_iff_integrableAtFilter_atBot`, L473-L479)
```lean
theorem integrable_iff_integrableAtFilter_atBot [LinearOrder X] [OrderTop X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrable f μ := by
constructor
  · exact fun hf ↦ ⟨hf.integrableAtFilter _, hf.locallyIntegrable⟩
  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
    exact h.1.filter_mono cocompact_le_atBot
```

## Edit instruction
Edit `integrable_iff_integrableAtFilter_atBot` so it still compiles and the proof is simplified where possible. Return the full updated `integrable_iff_integrableAtFilter_atBot` block.
[/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.Geometry.Manifold.Diffeomorph).

[User]
You are editing a file in `Mathlib/Geometry/Manifold/Diffeomorph.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Geometry.Manifold.Diffeomorph.coe_toDiffeomorph_symm`, L387-L396)
```lean
theorem coe_toDiffeomorph_symm : ⇑e.toDiffeomorph.symm = e.symm := rfl

end ContinuousLinearEquiv

namespace ModelWithCorners

variable (I) (e : E ≃L[𝕜] E')

/-- Apply a continuous linear equivalence to the model vector space. -/
```

## Edit instruction
Edit `coe_toDiffeomorph_symm` so it still compiles and the proof is simplified where possible. Return the full updated `coe_toDiffeomorph_symm` block.

<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.Geometry.Manifold.Diffeomorph).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Geometry/Manifold/Diffeomorph.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Geometry.Manifold.Diffeomorph.coe_toDiffeomorph_symm`, L387-L396)
```lean
theorem coe_toDiffeomorph_symm : ⇑e.toDiffeomorph.symm = e.symm := rfl

end ContinuousLinearEquiv

namespace ModelWithCorners

variable (I) (e : E ≃L[𝕜] E')

/-- Apply a continuous linear equivalence to the model vector space. -/
```

## Edit instruction
Edit `coe_toDiffeomorph_symm` so it still compiles and the proof is simplified where possible. Return the full updated `coe_toDiffeomorph_symm` block.
[/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.CategoryTheory.GradedObject).

[User]
You are editing a file in `Mathlib/CategoryTheory/GradedObject.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.GradedObject.eval`, L74-L77)
```lean
def eval {β : Type w} (b : β) : GradedObject β C ⥤ C where
  obj X := X b
map f := f b
```

## Edit instruction
Edit `eval` so it still compiles and the proof is simplified where possible. Return the full updated `eval` block.

<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.CategoryTheory.GradedObject).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/GradedObject.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.GradedObject.eval`, L74-L77)
```lean
def eval {β : Type w} (b : β) : GradedObject β C ⥤ C where
  obj X := X b
map f := f b
```

## Edit instruction
Edit `eval` so it still compiles and the proof is simplified where possible. Return the full updated `eval` block.
[/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.CategoryTheory.Bicategory.Free).

[User]
You are editing a file in `Mathlib/CategoryTheory/Bicategory/Free.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Bicategory.Free.of`, L273-L276)
```lean
def of : Prefunctor B (FreeBicategory B) where
  obj := id
map := @fun _ _ => Hom.of
```

## Edit instruction
Edit `of` so it still compiles and the proof is simplified where possible. Return the full updated `of` block.

<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.CategoryTheory.Bicategory.Free).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Bicategory/Free.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Bicategory.Free.of`, L273-L276)
```lean
def of : Prefunctor B (FreeBicategory B) where
  obj := id
map := @fun _ _ => Hom.of
```

## Edit instruction
Edit `of` so it still compiles and the proof is simplified where possible. Return the full updated `of` block.
[/INST]


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

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

```lean
theorem aemeasurable_indicator_iff₀ {s} (hs : NullMeasurableSet s μ) :
    AEMeasurable (indicator s f) μ ↔ AEMeasurable f (μ.restrict s)
```

Declaration: `Mathlib.MeasureTheory.Measure.AEMeasurable.aemeasurable_indicator_iff₀` (module Mathlib.MeasureTheory.Measure.AEMeasurable).

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

```lean
theorem aemeasurable_indicator_iff₀ {s} (hs : NullMeasurableSet s μ) :
    AEMeasurable (indicator s f) μ ↔ AEMeasurable f (μ.restrict s)
```

Declaration: `Mathlib.MeasureTheory.Measure.AEMeasurable.aemeasurable_indicator_iff₀` (module Mathlib.MeasureTheory.Measure.AEMeasurable).[/INST]


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

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

```lean
lemma isSmall : IsSmall.{w} I
```

Declaration: `Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument.isSmall` (module Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument).

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

```lean
lemma isSmall : IsSmall.{w} I
```

Declaration: `Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument.isSmall` (module Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument).[/INST]


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

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

```lean
lemma mass_map_le (f : Ω → Ω') (μ : FiniteMeasure Ω) : (μ.map f).mass ≤ μ.mass
```

Declaration: `Mathlib.MeasureTheory.Measure.FiniteMeasure.mass_map_le` (module Mathlib.MeasureTheory.Measure.FiniteMeasure).

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

```lean
lemma mass_map_le (f : Ω → Ω') (μ : FiniteMeasure Ω) : (μ.map f).mass ≤ μ.mass
```

Declaration: `Mathlib.MeasureTheory.Measure.FiniteMeasure.mass_map_le` (module Mathlib.MeasureTheory.Measure.FiniteMeasure).[/INST]


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

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

```lean
instance prod.instNoAtoms_fst [NoAtoms μ] :
    NoAtoms (Measure.prod μ ν) where
  measure_singleton
  | (x, y) => nonpos_iff_eq_zero.mp <| calc
    μ.prod ν {(x, y)} = μ.prod ν ({x} ×ˢ {y})
```

Declaration: `Mathlib.MeasureTheory.Measure.Prod.prod` (module Mathlib.MeasureTheory.Measure.Prod).

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

```lean
instance prod.instNoAtoms_fst [NoAtoms μ] :
    NoAtoms (Measure.prod μ ν) where
  measure_singleton
  | (x, y) => nonpos_iff_eq_zero.mp <| calc
    μ.prod ν {(x, y)} = μ.prod ν ({x} ×ˢ {y})
```

Declaration: `Mathlib.MeasureTheory.Measure.Prod.prod` (module Mathlib.MeasureTheory.Measure.Prod).[/INST]


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

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

```lean
theorem natSepDegree_X_sub_C (x : F) : (X - C x).natSepDegree = 1
```

Declaration: `Mathlib.FieldTheory.SeparableDegree.natSepDegree_X_sub_C` (module Mathlib.FieldTheory.SeparableDegree).

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

```lean
theorem natSepDegree_X_sub_C (x : F) : (X - C x).natSepDegree = 1
```

Declaration: `Mathlib.FieldTheory.SeparableDegree.natSepDegree_X_sub_C` (module Mathlib.FieldTheory.SeparableDegree).[/INST]


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

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

```lean
theorem condExp_ae_eq_restrict_of_measurableSpace_eq_on {m m₂ m0 : MeasurableSpace α}
    {μ : Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm)]
    [SigmaFinite (μ.trim hm₂)] (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) ↔ MeasurableSet[m₂] (s ∩ t)) :
    μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂]
```

Declaration: `Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator.condExp_ae_eq_restrict_of_measurableSpace_eq_on` (module Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator).

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

```lean
theorem condExp_ae_eq_restrict_of_measurableSpace_eq_on {m m₂ m0 : MeasurableSpace α}
    {μ : Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm)]
    [SigmaFinite (μ.trim hm₂)] (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) ↔ MeasurableSet[m₂] (s ∩ t)) :
    μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂]
```

Declaration: `Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator.condExp_ae_eq_restrict_of_measurableSpace_eq_on` (module Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator).[/INST]


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

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

```lean
theorem one_toNat : toNat 1 = 1
```

Declaration: `Mathlib.SetTheory.Cardinal.ToNat.one_toNat` (module Mathlib.SetTheory.Cardinal.ToNat).

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

```lean
theorem one_toNat : toNat 1 = 1
```

Declaration: `Mathlib.SetTheory.Cardinal.ToNat.one_toNat` (module Mathlib.SetTheory.Cardinal.ToNat).[/INST]


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

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

```lean
lemma mpullback_zero : mpullback I I' f 0 = 0
```

Declaration: `Mathlib.Geometry.Manifold.VectorField.Pullback.mpullback_zero` (module Mathlib.Geometry.Manifold.VectorField.Pullback).

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

```lean
lemma mpullback_zero : mpullback I I' f 0 = 0
```

Declaration: `Mathlib.Geometry.Manifold.VectorField.Pullback.mpullback_zero` (module Mathlib.Geometry.Manifold.VectorField.Pullback).[/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.Galois.Basic).

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

```lean
def galoisInsertionIntermediateFieldSubgroup [FiniteDimensional F E] :
    GaloisInsertion (OrderDual.toDual ∘
      (IntermediateField.fixingSubgroup : IntermediateField F E → Subgroup Gal(E/F)))
      ((IntermediateField.fixedField : Subgroup Gal(E/F) → IntermediateField F E) ∘
        OrderDual.toDual) where
  choice K _
:= IntermediateField.fixingSubgroup K .someMissingNamespaceXYZ
gc K H := (IntermediateField.le_iff_le H K).symm
  le_l_u H := le_of_eq (IntermediateField.fixingSubgroup_fixedField H).symm
  choice_eq _ _ := rfl

/-- The Galois correspondence as a `GaloisCoinsertion`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=397, messages=[Message(data='failed to synthesize instance of type class\n  DivisionRing F\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=397, i_end=604, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=8, column=0), pos_end=Position(line=8, column=2), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=58), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def galoisInsertionIntermediateFieldSubgroup [FiniteDimensional F E] :
    GaloisInsertion (OrderDual.toDual ∘
      (IntermediateField.fixingSubgroup : IntermediateField F E → Subgroup Gal(E/F)))
      ((IntermediateField.fixedField : Subgroup Gal(E/F) → IntermediateField F E) ∘
        OrderDual.toDual) where
  choice K _
:= IntermediateField.fixingSubgroup K .someMissingNamespaceXYZ
gc K H := (IntermediateField.le_iff_le H K).symm
  le_l_u H := le_of_eq (IntermediateField.fixingSubgroup_fixedField H).symm
  choice_eq _ _ := rfl

/-- The Galois correspondence as a `GaloisCoinsertion`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=397, messages=[Message(data='failed to synthesize instance of type class\n  DivisionRing F\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=46), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=397, i_end=604, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=8, column=0), pos_end=Position(line=8, column=2), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=58), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem measure_preimage_smul_of_nullMeasurableSet (hs : NullMeasurableSet s μ) (c : G) :
    μ ((c • ·) ⁻¹' s) = μ s
  frobnicateTacticThatDoesNotExist := by
rw [← measure_toMeasurable s,
    ← SMulInvariantMeasure.measure_preimage_smul c (measurableSet_toMeasurable μ s)]
  exact measure_congr (tendsto_smul_ae μ c hs.toMeasurable_ae_eq) |>.symm

end AE_smul

section AE

variable {m : MeasurableSpace α} [Group G] [MulAction G α]
  (μ : Measure α) [SMulInvariantMeasure G α μ]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=290, messages=[Message(data="Function expected at\n  NullMeasurableSet\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `NullMeasurableSet` 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=57), pos_end=Position(line=1, column=78), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  μ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  ((c • ·) ⁻¹' s)\n\nHint: The identifier `μ` 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=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  μ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `μ` 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=24), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `measure_toMeasurable`', pos=Position(line=4, column=6), pos_end=Position(line=4, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nx✝¹ : Sort u_1\nNullMeasurableSet : x✝¹\nG : Sort u_2\nx✝ : Sort u_3\nμ : x✝\nhs : sorry\nc : G\n⊢ sorry = sorry', pos=Position(line=3, column=38), pos_end=Position(line=5, column=84), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=290, i_end=377, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=2), pos_end=Position(line=6, column=7), 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=8, column=0), pos_end=Position(line=8, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=377, i_end=389, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=389, i_end=500, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.7\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=13, column=19), pos_end=Position(line=13, column=45), 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.MeasureTheory.Group.Action).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem measure_preimage_smul_of_nullMeasurableSet (hs : NullMeasurableSet s μ) (c : G) :
    μ ((c • ·) ⁻¹' s) = μ s
  frobnicateTacticThatDoesNotExist := by
rw [← measure_toMeasurable s,
    ← SMulInvariantMeasure.measure_preimage_smul c (measurableSet_toMeasurable μ s)]
  exact measure_congr (tendsto_smul_ae μ c hs.toMeasurable_ae_eq) |>.symm

end AE_smul

section AE

variable {m : MeasurableSpace α} [Group G] [MulAction G α]
  (μ : Measure α) [SMulInvariantMeasure G α μ]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=290, messages=[Message(data="Function expected at\n  NullMeasurableSet\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `NullMeasurableSet` 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=57), pos_end=Position(line=1, column=78), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  μ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  ((c • ·) ⁻¹' s)\n\nHint: The identifier `μ` 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=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  μ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  s\n\nHint: The identifier `μ` 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=24), pos_end=Position(line=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `measure_toMeasurable`', pos=Position(line=4, column=6), pos_end=Position(line=4, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nx✝¹ : Sort u_1\nNullMeasurableSet : x✝¹\nG : Sort u_2\nx✝ : Sort u_3\nμ : x✝\nhs : sorry\nc : G\n⊢ sorry = sorry', pos=Position(line=3, column=38), pos_end=Position(line=5, column=84), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=290, i_end=377, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=2), pos_end=Position(line=6, column=7), 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=8, column=0), pos_end=Position(line=8, column=11), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=377, i_end=389, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=389, i_end=500, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.7\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=13, column=19), pos_end=Position(line=13, column=45), 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.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq).

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

```lean
theorem inr_fst' {b : BinaryBicone X Y} (h : b.IsBilimit) :
    IsPullback b.inr (0 : Y ⟶ 0) b.fst (0 : 0 ⟶ X)
:= by .someMissingNamespaceXYZ
apply flip
  refine of_bot ?_ (by simp) (of_isBilimit h)
  simp

/-- The square
```
  Y --inr--> X ⊞ Y
  |            |
  0           fst
  |            |
  v            v
  0 ---0-----> X
```
is a pullback square.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=123, 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  BinaryBicone\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 `BinaryBicone` 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=22), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsPullback\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  b.inr\n\nHint: The identifier `IsPullback` 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=50), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝¹ : Sort u_1\nBinaryBicone : x✝¹\nx✝ : Sort u_2\nIsPullback : x✝\nb : sorry\nh : sorry\n⊢ 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=123, i_end=365, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=2), 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.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem inr_fst' {b : BinaryBicone X Y} (h : b.IsBilimit) :
    IsPullback b.inr (0 : Y ⟶ 0) b.fst (0 : 0 ⟶ X)
:= by .someMissingNamespaceXYZ
apply flip
  refine of_bot ?_ (by simp) (of_isBilimit h)
  simp

/-- The square
```
  Y --inr--> X ⊞ Y
  |            |
  0           fst
  |            |
  v            v
  0 ---0-----> X
```
is a pullback square.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=123, 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  BinaryBicone\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 `BinaryBicone` 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=22), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsPullback\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  b.inr\n\nHint: The identifier `IsPullback` 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=50), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝¹ : Sort u_1\nBinaryBicone : x✝¹\nx✝ : Sort u_2\nIsPullback : x✝\nb : sorry\nh : sorry\n⊢ 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=123, i_end=365, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=2), 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.Probability.Distributions.Gaussian.Real).

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

```lean
def gaussianPDFReal (μ : ℝ) (v : ℝ≥0) (x : ℝ) : ℝ
:= thisIsWrongLemmaNameXYZ
(√(2 * π * v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * v))
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=143, messages=[Message(data='failed to synthesize instance of type class\n  LE Type\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=33), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat Type 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  Type\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=35), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `rexp`', pos=Position(line=3, column=19), pos_end=Position(line=3, column=23), 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.Probability.Distributions.Gaussian.Real).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def gaussianPDFReal (μ : ℝ) (v : ℝ≥0) (x : ℝ) : ℝ
:= thisIsWrongLemmaNameXYZ
(√(2 * π * v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * v))
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=143, messages=[Message(data='failed to synthesize instance of type class\n  LE Type\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=33), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat Type 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  Type\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=35), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `rexp`', pos=Position(line=3, column=19), pos_end=Position(line=3, column=23), 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.Geometry.Euclidean.Sphere.Basic).

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

```lean
theorem sbtw_of_collinear_of_dist_center_lt_radius {s : Sphere P} {p₁ p₂ p₃ : P}
    (h : Collinear ℝ ({p₁, p₂, p₃} : Set P)) (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center < s.radius)
    (hp₃ : p₃ ∈ s) (hp₁p₃ : p₁ ≠ p₃) : Sbtw ℝ p₁ p₂ p₃
:= .someMissingNamespaceXYZ
h.sbtw_of_dist_eq_of_dist_lt hp₁ hp₂ hp₃ hp₁p₃

namespace Sphere

variable {s : Sphere P} {p₁ p₂ : P}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=366, messages=[Message(data='failed to synthesize instance of type class\n  AddTorsor ?m.4 P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=9), pos_end=Position(line=2, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Dist P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=67), pos_end=Position(line=2, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  AddTorsor ?m.34 P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=39), pos_end=Position(line=3, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  Sphere\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  P\n\nHint: The identifier `Sphere` 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=56), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=366, i_end=384, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=384, i_end=423, messages=[Message(data='Function expected at\n  Sphere\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  P', pos=Position(line=9, column=14), pos_end=Position(line=9, column=22), 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.Geometry.Euclidean.Sphere.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem sbtw_of_collinear_of_dist_center_lt_radius {s : Sphere P} {p₁ p₂ p₃ : P}
    (h : Collinear ℝ ({p₁, p₂, p₃} : Set P)) (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center < s.radius)
    (hp₃ : p₃ ∈ s) (hp₁p₃ : p₁ ≠ p₃) : Sbtw ℝ p₁ p₂ p₃
:= .someMissingNamespaceXYZ
h.sbtw_of_dist_eq_of_dist_lt hp₁ hp₂ hp₃ hp₁p₃

namespace Sphere

variable {s : Sphere P} {p₁ p₂ : P}
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=366, messages=[Message(data='failed to synthesize instance of type class\n  AddTorsor ?m.4 P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=9), pos_end=Position(line=2, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Dist P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=67), pos_end=Position(line=2, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  AddTorsor ?m.34 P\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=39), pos_end=Position(line=3, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  Sphere\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  P\n\nHint: The identifier `Sphere` 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=56), pos_end=Position(line=1, column=64), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=366, i_end=384, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=384, i_end=423, messages=[Message(data='Function expected at\n  Sphere\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  P', pos=Position(line=9, column=14), pos_end=Position(line=9, column=22), 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.SetTheory.Ordinal.Veblen).

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

```lean
theorem veblenWith_eq_self_of_le (h : o₁ ≤ o₂) (h' : veblenWith f o₂ a = a) :
    veblenWith f o₁ a = a
:= by thisIsWrongLemmaNameXYZ
obtain rfl | h := h.eq_or_lt
  · assumption
  · rw [← h', veblenWith_veblenWith_of_lt hf h]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=121, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  veblenWith\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `veblenWith` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=53), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  veblenWith\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `veblenWith` 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=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nα✝¹ : Type u_1\no₁ o₂ : α✝¹\nx✝ : Sort u_2\nveblenWith : x✝\nα✝ : Sort u_3\na : α✝\nh : o₁ ≤ o₂\nh' : sorry = a\n⊢ sorry = a", 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=121, i_end=239, 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.SetTheory.Ordinal.Veblen).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem veblenWith_eq_self_of_le (h : o₁ ≤ o₂) (h' : veblenWith f o₂ a = a) :
    veblenWith f o₁ a = a
:= by thisIsWrongLemmaNameXYZ
obtain rfl | h := h.eq_or_lt
  · assumption
  · rw [← h', veblenWith_veblenWith_of_lt hf h]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=121, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  veblenWith\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `veblenWith` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=53), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  veblenWith\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `veblenWith` 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=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="unsolved goals\nα✝¹ : Type u_1\no₁ o₂ : α✝¹\nx✝ : Sort u_2\nveblenWith : x✝\nα✝ : Sort u_3\na : α✝\nh : o₁ ≤ o₂\nh' : sorry = a\n⊢ sorry = a", 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=121, i_end=239, 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.InformationTheory.Hamming).

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

```lean
theorem nndist_eq_hammingDist (x y : Hamming β) :
    nndist x y = hammingDist (ofHamming x) (ofHamming y)
:= thisIsWrongLemmaNameXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=138, messages=[Message(data="Function expected at\n  ofHamming\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `ofHamming` 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=30), pos_end=Position(line=2, column=41), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ofHamming\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  y\n\nHint: The identifier `ofHamming` 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=44), pos_end=Position(line=2, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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

```lean
theorem nndist_eq_hammingDist (x y : Hamming β) :
    nndist x y = hammingDist (ofHamming x) (ofHamming y)
:= thisIsWrongLemmaNameXYZ
rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=138, messages=[Message(data="Function expected at\n  ofHamming\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `ofHamming` 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=30), pos_end=Position(line=2, column=41), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ofHamming\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  y\n\nHint: The identifier `ofHamming` 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=44), pos_end=Position(line=2, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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.FinitelyGenerated).

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

```lean
theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S
:= by _unbalancedParenXYZ
rw [Structure.cg_def]
  refine ⟨fun h => CG.of_map_embedding S.subtype ?_, fun h => ?_⟩
  · rw [← Hom.range_eq_map, range_subtype]
    exact h
  · have h := h.map S.subtype.toHom
    rw [← Hom.range_eq_map, range_subtype] at h
    exact h
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=99, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `L.Substructure`', pos=Position(line=1, column=46), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `Structure.CG`', pos=Position(line=1, column=73), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nS : sorry\n⊢ sorry ↔ sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=99, i_end=366, 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.ModelTheory.FinitelyGenerated).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Substructure.cg_iff_structure_cg (S : L.Substructure M) : S.CG ↔ Structure.CG L S
:= by _unbalancedParenXYZ
rw [Structure.cg_def]
  refine ⟨fun h => CG.of_map_embedding S.subtype ?_, fun h => ?_⟩
  · rw [← Hom.range_eq_map, range_subtype]
    exact h
  · have h := h.map S.subtype.toHom
    rw [← Hom.range_eq_map, range_subtype] at h
    exact h
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=99, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `L.Substructure`', pos=Position(line=1, column=46), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `Structure.CG`', pos=Position(line=1, column=73), pos_end=Position(line=1, column=85), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nS : sorry\n⊢ sorry ↔ sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=99, i_end=366, 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.RepresentationTheory.Invariants).

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

```lean
theorem invariants_eq_inter : (invariants ρ).carrier = ⋂ g : G, Function.fixedPoints (ρ g)
:= by thisIsWrongLemmaNameXYZ
ext; simp [Function.IsFixedPt]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=102, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invariants\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 `invariants` 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=31), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ρ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  g\n\nHint: The identifier `ρ` 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=86), pos_end=Position(line=1, column=89), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝¹ : Sort u_1\ninvariants : x✝¹\nG : Sort u_2\nx✝ : Sort u_3\nρ : x✝\n⊢ sorry = ⋂ g, Function.fixedPoints ?m.10', 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=102, i_end=155, 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.RepresentationTheory.Invariants).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem invariants_eq_inter : (invariants ρ).carrier = ⋂ g : G, Function.fixedPoints (ρ g)
:= by thisIsWrongLemmaNameXYZ
ext; simp [Function.IsFixedPt]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=102, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  invariants\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 `invariants` 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=31), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ρ\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  g\n\nHint: The identifier `ρ` 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=86), pos_end=Position(line=1, column=89), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝¹ : Sort u_1\ninvariants : x✝¹\nG : Sort u_2\nx✝ : Sort u_3\nρ : x✝\n⊢ sorry = ⋂ g, Function.fixedPoints ?m.10', 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=102, i_end=155, 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.AlgebraicGeometry.AffineScheme).

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

```lean
theorem isAffineOpen_opensRange {X Y : Scheme} [IsAffine X] (f : X ⟶ Y)
    [H : IsOpenImmersion f] : IsAffineOpen f.opensRange
:= by .someMissingNamespaceXYZ
refine .of_isIso (IsOpenImmersion.isoOfRangeEq f (Y.ofRestrict _) ?_).inv
  exact Subtype.range_val.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=138, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=138, i_end=265, 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.AlgebraicGeometry.AffineScheme).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem isAffineOpen_opensRange {X Y : Scheme} [IsAffine X] (f : X ⟶ Y)
    [H : IsOpenImmersion f] : IsAffineOpen f.opensRange
:= by .someMissingNamespaceXYZ
refine .of_isIso (IsOpenImmersion.isoOfRangeEq f (Y.ofRestrict _) ?_).inv
  exact Subtype.range_val.symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=138, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=138, i_end=265, 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.FieldTheory.PurelyInseparable.PerfectClosure).

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

```lean
theorem LinearIndependent.map_pow_expChar_pow_of_isSeparable'
    (hsep : ∀ i : ι, IsSeparable F (v i))
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n)
  frobnicateTacticThatDoesNotExist := by
let E' := adjoin F (Set.range v)
  haveI : Algebra.IsSeparable F E' := (isSeparable_adjoin_iff_isSeparable F _).2 <| by
    rintro _ ⟨y, rfl⟩; exact hsep y
  let v' (i : ι) : E' := ⟨v i, subset_adjoin F _ ⟨i, rfl⟩⟩
  have h' : LinearIndependent F v' := h.of_comp E'.val.toLinearMap
  exact (h'.map_pow_expChar_pow_of_isSeparable q n).map'
    E'.val.toLinearMap (LinearMap.ker_eq_bot_of_injective E'.val.injective)

/-- If `E / F` is a separable extension of exponential characteristic `q`, if `{ u_i }` is an
`F`-basis of `E`, then `{ u_i ^ (q ^ n) }` is also an `F`-basis of `E`
for any natural number `n`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=646, messages=[Message(data="`LinearIndependent.map_pow_expChar_pow_of_isSeparable'` has already been declared", pos=Position(line=1, column=8), pos_end=Position(line=1, column=61), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=646, i_end=841, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=15, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem LinearIndependent.map_pow_expChar_pow_of_isSeparable'
    (hsep : ∀ i : ι, IsSeparable F (v i))
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n)
  frobnicateTacticThatDoesNotExist := by
let E' := adjoin F (Set.range v)
  haveI : Algebra.IsSeparable F E' := (isSeparable_adjoin_iff_isSeparable F _).2 <| by
    rintro _ ⟨y, rfl⟩; exact hsep y
  let v' (i : ι) : E' := ⟨v i, subset_adjoin F _ ⟨i, rfl⟩⟩
  have h' : LinearIndependent F v' := h.of_comp E'.val.toLinearMap
  exact (h'.map_pow_expChar_pow_of_isSeparable q n).map'
    E'.val.toLinearMap (LinearMap.ker_eq_bot_of_injective E'.val.injective)

/-- If `E / F` is a separable extension of exponential characteristic `q`, if `{ u_i }` is an
`F`-basis of `E`, then `{ u_i ^ (q ^ n) }` is also an `F`-basis of `E`
for any natural number `n`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=646, messages=[Message(data="`LinearIndependent.map_pow_expChar_pow_of_isSeparable'` has already been declared", pos=Position(line=1, column=8), pos_end=Position(line=1, column=61), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=646, i_end=841, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=15, column=30), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem Continuous.map_eq_borel {X Y : Type*} [TopologicalSpace X] [PolishSpace X]
    [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y]
    {f : X → Y} (hf : Continuous f) (hsurj : Surjective f) :
    MeasurableSpace.map f ‹MeasurableSpace X› = borel Y
:= by thisIsWrongLemmaNameXYZ
borelize Y
  exact hf.measurable.map_measurableSpace_eq hsurj
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=313, messages=[Message(data='unknown tactic', pos=Position(line=5, 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=313, i_end=397, 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.MeasureTheory.Constructions.Polish.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem Continuous.map_eq_borel {X Y : Type*} [TopologicalSpace X] [PolishSpace X]
    [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y]
    {f : X → Y} (hf : Continuous f) (hsurj : Surjective f) :
    MeasurableSpace.map f ‹MeasurableSpace X› = borel Y
:= by thisIsWrongLemmaNameXYZ
borelize Y
  exact hf.measurable.map_measurableSpace_eq hsurj
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=313, messages=[Message(data='unknown tactic', pos=Position(line=5, 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=313, i_end=397, 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.CategoryTheory.Limits.Shapes.Pullback.Cospan).

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

```lean
abbrev Hom.snd : zero ⟶ right
:= thisIsWrongLemmaNameXYZ
WidePushoutShape.Hom.init _

/-- The identity arrows of the walking span. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=88, messages=[Message(data='typeclass instance problem is stuck\n  Quiver ?m.5\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `Quiver` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.', pos=Position(line=1, column=17), 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=88, i_end=135, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=47), 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.CategoryTheory.Limits.Shapes.Pullback.Cospan).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
abbrev Hom.snd : zero ⟶ right
:= thisIsWrongLemmaNameXYZ
WidePushoutShape.Hom.init _

/-- The identity arrows of the walking span. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=88, messages=[Message(data='typeclass instance problem is stuck\n  Quiver ?m.5\n\nNote: Lean will not try to resolve this typeclass instance problem because the type argument to `Quiver` is a metavariable. This argument must be fully determined before Lean will try to resolve the typeclass.\n\nHint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.', pos=Position(line=1, column=17), 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=88, i_end=135, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=47), 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.Probability.ProbabilityMassFunction.Binomial).

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

```lean
theorem binomial_apply_self (p : ℝ≥0) (h : p ≤ 1) (n : ℕ) :
    binomial p h n (.last n) = p ^ n
:= by simp ⟨⟩⟨⟩
/-- The binomial distribution on one coin is the Bernoulli distribution. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=116, messages=[Message(data='failed to synthesize instance of type class\n  LE Type\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=33), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat Type 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  Type\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=35), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  binomial\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  p\n\nHint: The identifier `binomial` 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=28), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nbinomial : x✝\np : sorry\nh : sorry\nn : ℕ\n⊢ sorry ()', 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=116, i_end=204, 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=75), 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.Probability.ProbabilityMassFunction.Binomial).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem binomial_apply_self (p : ℝ≥0) (h : p ≤ 1) (n : ℕ) :
    binomial p h n (.last n) = p ^ n
:= by simp ⟨⟩⟨⟩
/-- The binomial distribution on one coin is the Bernoulli distribution. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=116, messages=[Message(data='failed to synthesize instance of type class\n  LE Type\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=33), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  OfNat Type 0\nnumerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is\n  Type\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=35), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  binomial\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  p\n\nHint: The identifier `binomial` 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=28), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nbinomial : x✝\np : sorry\nh : sorry\nn : ℕ\n⊢ sorry ()', 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=116, i_end=204, 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=75), 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.Geometry.Manifold.LocalDiffeomorph).

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

```lean
lemma IsLocalDiffeomorph.mfderivToContinuousLinearEquiv_coe
    (hf : IsLocalDiffeomorph I J n f) (hn : n ≠ 0) (x : M) :
    hf.mfderivToContinuousLinearEquiv hn x = mfderiv% f x
:= ⟨⟩⟨⟩
(hf x).mfderivToContinuousLinearEquiv_coe hn

end Differential
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=190, messages=[Message(data='`IsLocalDiffeomorph.mfderivToContinuousLinearEquiv_coe` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=190, i_end=259, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=4, column=5), pos_end=Position(line=4, column=6), 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=7, column=0), pos_end=Position(line=7, 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.Geometry.Manifold.LocalDiffeomorph).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma IsLocalDiffeomorph.mfderivToContinuousLinearEquiv_coe
    (hf : IsLocalDiffeomorph I J n f) (hn : n ≠ 0) (x : M) :
    hf.mfderivToContinuousLinearEquiv hn x = mfderiv% f x
:= ⟨⟩⟨⟩
(hf x).mfderivToContinuousLinearEquiv_coe hn

end Differential
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=190, messages=[Message(data='`IsLocalDiffeomorph.mfderivToContinuousLinearEquiv_coe` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=190, i_end=259, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=4, column=5), pos_end=Position(line=4, column=6), 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=7, column=0), pos_end=Position(line=7, 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.SetTheory.Ordinal.Basic).

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

```lean
instance small_Iio (o : Ordinal.{u}) : Small.{u} (Iio o)
:= ⟨⟩⟨⟩
⟨_, ⟨ToType.mk.toEquiv⟩⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=66, messages=[Message(data="Function expected at\n  Iio\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  o\n\nHint: The identifier `Iio` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=50), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data='Insufficient number of fields for `⟨...⟩` constructor: Constructor `Small.mk` has 1 explicit fields, but none were provided', pos=Position(line=2, column=3), pos_end=Position(line=2, column=5), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=66, i_end=105, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, column=6), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
instance small_Iio (o : Ordinal.{u}) : Small.{u} (Iio o)
:= ⟨⟩⟨⟩
⟨_, ⟨ToType.mk.toEquiv⟩⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=66, messages=[Message(data="Function expected at\n  Iio\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  o\n\nHint: The identifier `Iio` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=50), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data='Insufficient number of fields for `⟨...⟩` constructor: Constructor `Small.mk` has 1 explicit fields, but none were provided', pos=Position(line=2, column=3), pos_end=Position(line=2, column=5), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=66, i_end=105, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, column=6), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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