[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.Independence.Kernel.Indep).

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

```lean
lemma iIndep.precomp (hg : Function.Injective g) (h : iIndep m κ μ) :
    iIndep (m ∘ g) κ μ
  frobnicateTacticThatDoesNotExist :=
(iIndepSets.precomp hg h :)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=164, messages=[Message(data="Function expected at\n  iIndep\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  m\n\nHint: The identifier `iIndep` 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=54), pos_end=Position(line=1, column=66), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  iIndep\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (m ∘ g)\n\nHint: The identifier `iIndep` 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=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `iIndepSets.precomp`', pos=Position(line=4, column=1), pos_end=Position(line=4, column=19), 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.Independence.Kernel.Indep).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma iIndep.precomp (hg : Function.Injective g) (h : iIndep m κ μ) :
    iIndep (m ∘ g) κ μ
  frobnicateTacticThatDoesNotExist :=
(iIndepSets.precomp hg h :)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=164, messages=[Message(data="Function expected at\n  iIndep\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  m\n\nHint: The identifier `iIndep` 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=54), pos_end=Position(line=1, column=66), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  iIndep\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (m ∘ g)\n\nHint: The identifier `iIndep` 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=3, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `iIndepSets.precomp`', pos=Position(line=4, column=1), pos_end=Position(line=4, column=19), 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]


Instruction: Edit `Periodic` so it still compiles and the proof is simplified where possible. Return the full updated `Periodic` block.

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

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

## Current excerpt (around `Mathlib.Algebra.Ring.Periodic.Periodic`, L187-L189)
```lean
theorem Periodic.nat_mul_eq [NonAssocSemiring α] (h : Periodic f c) (n : ℕ) : f (n * c) = f 0 := (h.nat_mul n).eq
```

## Edit instruction
Edit `Periodic` so it still compiles and the proof is simplified where possible. Return the full updated `Periodic` 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.Algebra.Ring.Periodic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Algebra/Ring/Periodic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Algebra.Ring.Periodic.Periodic`, L187-L189)
```lean
theorem Periodic.nat_mul_eq [NonAssocSemiring α] (h : Periodic f c) (n : ℕ) : f (n * c) = f 0 := (h.nat_mul n).eq
```

## Edit instruction
Edit `Periodic` so it still compiles and the proof is simplified where possible. Return the full updated `Periodic` 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.RepresentationTheory.Basic).

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

```lean
lemma apply_eq_of_coe_eq [IsTrivial (ρ.comp S.subtype)] (g h : G) (hgh : (g : G ⧸ S) = h) :
    ρ g = ρ h
:= by _unbalancedParenXYZ
ext x
  apply (ρ.apply_bijective g⁻¹).1
  simpa [← Module.End.mul_apply, ← map_mul, -isTrivial_def] using
    (congr($(isTrivial_def (ρ.comp S.subtype) ⟨g⁻¹ * h, QuotientGroup.eq.1 hgh⟩) x)).symm

variable [S.Normal]

/-- Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` which is trivial on `S` factors
through `G ⧸ S`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=118, messages=[Message(data='unknown tactic', pos=Position(line=3, 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=118, i_end=371, messages=[Message(data='Unknown identifier `S.Normal`', pos=Position(line=9, column=10), pos_end=Position(line=9, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=371, i_end=489, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=19), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma apply_eq_of_coe_eq [IsTrivial (ρ.comp S.subtype)] (g h : G) (hgh : (g : G ⧸ S) = h) :
    ρ g = ρ h
:= by _unbalancedParenXYZ
ext x
  apply (ρ.apply_bijective g⁻¹).1
  simpa [← Module.End.mul_apply, ← map_mul, -isTrivial_def] using
    (congr($(isTrivial_def (ρ.comp S.subtype) ⟨g⁻¹ * h, QuotientGroup.eq.1 hgh⟩) x)).symm

variable [S.Normal]

/-- Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` which is trivial on `S` factors
through `G ⧸ S`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=118, messages=[Message(data='unknown tactic', pos=Position(line=3, 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=118, i_end=371, messages=[Message(data='Unknown identifier `S.Normal`', pos=Position(line=9, column=10), pos_end=Position(line=9, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=371, i_end=489, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=19), 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]


Instruction: Edit `map_add_right` so it still compiles and the proof is simplified where possible. Return the full updated `map_add_right` block.

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

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

## Current excerpt (around `Mathlib.LinearAlgebra.TensorProduct.Map.map_add_right`, L161-L165)
```lean
theorem map_add_right (f : M →ₛₗ[σ₁₂] M₂) (g₁ g₂ : N →ₛₗ[σ₁₂] N₂) :
    map f (g₁ + g₂) = map f g₁ + map f g₂ := by
ext
  simp only [tmul_add, compr₂ₛₗ_apply, mk_apply, map_tmul, add_apply]
```

## Edit instruction
Edit `map_add_right` so it still compiles and the proof is simplified where possible. Return the full updated `map_add_right` 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.LinearAlgebra.TensorProduct.Map).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/LinearAlgebra/TensorProduct/Map.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.LinearAlgebra.TensorProduct.Map.map_add_right`, L161-L165)
```lean
theorem map_add_right (f : M →ₛₗ[σ₁₂] M₂) (g₁ g₂ : N →ₛₗ[σ₁₂] N₂) :
    map f (g₁ + g₂) = map f g₁ + map f g₂ := by
ext
  simp only [tmul_add, compr₂ₛₗ_apply, mk_apply, map_tmul, add_apply]
```

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


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

[User]
Complete the following proof.

```lean
lemma isUnit_basicOpen (f : R) :
    IsUnit ((algebraMap R Γ(R, basicOpen f)) f)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicGeometry.StructureSheaf).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma isUnit_basicOpen (f : R) :
    IsUnit ((algebraMap R Γ(R, basicOpen f)) f)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
instance smooth_comp {Z : Scheme.{u}} (g : Y ⟶ Z) [Smooth f] [Smooth g] :
    Smooth (f ≫ g)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicGeometry.Morphisms.Smooth).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
instance smooth_comp {Z : Scheme.{u}} (g : Y ⟶ Z) [Smooth f] [Smooth g] :
    Smooth (f ≫ g)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


Instruction: Edit `unsym` so it still compiles and the proof is simplified where possible. Return the full updated `unsym` block.

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

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

## Current excerpt (around `Mathlib.Algebra.Symmetrized.unsym`, L60-L62)
```lean
def unsym : αˢʸᵐ ≃ α := Equiv.refl _
```

## Edit instruction
Edit `unsym` so it still compiles and the proof is simplified where possible. Return the full updated `unsym` 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.Algebra.Symmetrized).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Algebra/Symmetrized.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Algebra.Symmetrized.unsym`, L60-L62)
```lean
def unsym : αˢʸᵐ ≃ α := Equiv.refl _
```

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


Instruction: Edit `forget₂Mon_preservesLimits` so it still compiles and the proof is simplified where possible. Return the full updated `forget₂Mon_preservesLimits` block.

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

[User]
You are editing a file in `Mathlib/Algebra/Category/Ring/Limits.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Algebra.Category.Ring.Limits.forget₂Mon_preservesLimits`, L172-L176)
```lean
instance forget₂Mon_preservesLimits : PreservesLimits (forget₂ SemiRingCat MonCat.{u}) := SemiRingCat.forget₂Mon_preservesLimitsOfSize.{u, u}

/-- The forgetful functor from semirings to types preserves all limits.
-/
```

## Edit instruction
Edit `forget₂Mon_preservesLimits` so it still compiles and the proof is simplified where possible. Return the full updated `forget₂Mon_preservesLimits` 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.Algebra.Category.Ring.Limits).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Algebra/Category/Ring/Limits.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Algebra.Category.Ring.Limits.forget₂Mon_preservesLimits`, L172-L176)
```lean
instance forget₂Mon_preservesLimits : PreservesLimits (forget₂ SemiRingCat MonCat.{u}) := SemiRingCat.forget₂Mon_preservesLimitsOfSize.{u, u}

/-- The forgetful functor from semirings to types preserves all limits.
-/
```

## Edit instruction
Edit `forget₂Mon_preservesLimits` so it still compiles and the proof is simplified where possible. Return the full updated `forget₂Mon_preservesLimits` block.
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral).

[User]
Complete the following proof.

```lean
theorem tendsto_lintegral_norm_of_dominated_convergence
    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem tendsto_lintegral_norm_of_dominated_convergence
    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/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.Condensed.Epi).

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

```lean
lemma epi_iff_surjective_on_stonean : Epi f ↔
    ∀ (S : Stonean), Function.Surjective (f.hom.app (op S.compHaus))
:= by thisIsWrongLemmaNameXYZ
rw [← (StoneanCompHaus.equivalence A).inverse.epi_map_iff_epi,
    ← Presheaf.coherentExtensiveEquivalence.functor.epi_map_iff_epi,
    ← isLocallySurjective_iff_epi']
  exact extensiveTopology.isLocallySurjective_iff (D := A) _

end Condensed

namespace CondensedSet

variable {X Y : CondensedSet.{u}} (f : X ⟶ Y)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=126, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `f.hom.app`', pos=Position(line=2, column=42), pos_end=Position(line=2, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  Epi\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `Epi` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=38), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nEpi : x✝\n⊢ sorry ↔ ∀ (S : Stonean), Function.Surjective sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=400, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=400, i_end=424, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=424, i_end=471, 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.Condensed.Epi).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma epi_iff_surjective_on_stonean : Epi f ↔
    ∀ (S : Stonean), Function.Surjective (f.hom.app (op S.compHaus))
:= by thisIsWrongLemmaNameXYZ
rw [← (StoneanCompHaus.equivalence A).inverse.epi_map_iff_epi,
    ← Presheaf.coherentExtensiveEquivalence.functor.epi_map_iff_epi,
    ← isLocallySurjective_iff_epi']
  exact extensiveTopology.isLocallySurjective_iff (D := A) _

end Condensed

namespace CondensedSet

variable {X Y : CondensedSet.{u}} (f : X ⟶ Y)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=126, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `f.hom.app`', pos=Position(line=2, column=42), pos_end=Position(line=2, column=51), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data="Function expected at\n  Epi\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `Epi` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=38), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nEpi : x✝\n⊢ sorry ↔ ∀ (S : Stonean), Function.Surjective sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=400, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=400, i_end=424, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=424, i_end=471, 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.MeasureTheory.Group.Prod).

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

```lean
theorem inv_ae : (ae μ)⁻¹ = ae μ
:= by thisIsWrongLemmaNameXYZ
refine le_antisymm (quasiMeasurePreserving_inv μ).tendsto_ae ?_
  nth_rewrite 1 [← inv_inv (ae μ)]
  exact Filter.map_mono (quasiMeasurePreserving_inv μ).tendsto_ae
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=45, 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  ae\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  μ\n\nHint: The identifier `ae` 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=18), pos_end=Position(line=1, column=22), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ae\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  μ\n\nHint: The identifier `ae` 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=28), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nae : x✝\n⊢ sorry⁻¹ = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=45, i_end=237, 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.Group.Prod).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem inv_ae : (ae μ)⁻¹ = ae μ
:= by thisIsWrongLemmaNameXYZ
refine le_antisymm (quasiMeasurePreserving_inv μ).tendsto_ae ?_
  nth_rewrite 1 [← inv_inv (ae μ)]
  exact Filter.map_mono (quasiMeasurePreserving_inv μ).tendsto_ae
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=45, 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  ae\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  μ\n\nHint: The identifier `ae` 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=18), pos_end=Position(line=1, column=22), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  ae\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  μ\n\nHint: The identifier `ae` 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=28), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝ : Sort u_1\nae : x✝\n⊢ sorry⁻¹ = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=45, i_end=237, 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.MeasureTheory.Integral.IntegrableOn).

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

```lean
theorem integrableOn_Icc_iff_integrableOn_Ico (hb : ‖f b‖ₑ ≠ ∞
:= by finiteness) : .someMissingNamespaceXYZ
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ico' (by rw [measure_singleton]; exact ENNReal.zero_ne_top) hb
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=69, messages=[Message(data='expected token', pos=Position(line=1, column=61), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=69, i_end=279, 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.Integral.IntegrableOn).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem integrableOn_Icc_iff_integrableOn_Ico (hb : ‖f b‖ₑ ≠ ∞
:= by finiteness) : .someMissingNamespaceXYZ
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ico' (by rw [measure_singleton]; exact ENNReal.zero_ne_top) hb
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=69, messages=[Message(data='expected token', pos=Position(line=1, column=61), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=69, i_end=279, 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.Homological.GroupHomology.LowDegree).

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

```lean
theorem mem_cycles₂_iff (x : G × G →₀ A) :
    x ∈ cycles₂ A ↔ x.sum (fun g a => single g.2 (A.ρ g.1⁻¹ a) + single g.1 a) =
      x.sum (fun g a => single (g.1 * g.2) a)
:= by thisIsWrongLemmaNameXYZ
change x.sum (fun g a => _) = 0 ↔ _
  simp [sub_add_eq_add_sub, sub_eq_zero]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=194, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Zero A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=29), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  cycles₂\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  A\n\nHint: The identifier `cycles₂` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=8), pos_end=Position(line=2, column=17), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nG : Type u_1\nA : Type u_2\nx✝ : Sort u_3\ncycles₂ : x✝\nx : sorry\n⊢ x ∈ sorry ↔ sorry = sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=194, i_end=295, 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.Homological.GroupHomology.LowDegree).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem mem_cycles₂_iff (x : G × G →₀ A) :
    x ∈ cycles₂ A ↔ x.sum (fun g a => single g.2 (A.ρ g.1⁻¹ a) + single g.1 a) =
      x.sum (fun g a => single (g.1 * g.2) a)
:= by thisIsWrongLemmaNameXYZ
change x.sum (fun g a => _) = 0 ↔ _
  simp [sub_add_eq_add_sub, sub_eq_zero]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=194, messages=[Message(data='unknown tactic', pos=Position(line=4, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Zero A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=29), pos_end=Position(line=1, column=39), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  cycles₂\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  A\n\nHint: The identifier `cycles₂` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=8), pos_end=Position(line=2, column=17), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nG : Type u_1\nA : Type u_2\nx✝ : Sort u_3\ncycles₂ : x✝\nx : sorry\n⊢ x ∈ sorry ↔ sorry = sorry', pos=Position(line=4, column=3), pos_end=Position(line=4, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=194, i_end=295, 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.AlgebraicTopology.SimplicialSet.Finite).

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

```lean
lemma finite_iff_of_iso {Y : SSet.{u}} (e : X ≅ Y) : X.Finite ↔ Y.Finite
:= .someMissingNamespaceXYZ
⟨fun _ ↦ finite_of_iso e, fun _ ↦ finite_of_iso e.symm⟩

variable (X) in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=170, messages=[Message(data='Unknown constant `Iff.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  X.Finite ↔ Y.Finite', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=170, i_end=185, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=15), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma finite_iff_of_iso {Y : SSet.{u}} (e : X ≅ Y) : X.Finite ↔ Y.Finite
:= .someMissingNamespaceXYZ
⟨fun _ ↦ finite_of_iso e, fun _ ↦ finite_of_iso e.symm⟩

variable (X) in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=170, messages=[Message(data='Unknown constant `Iff.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  X.Finite ↔ Y.Finite', pos=Position(line=2, column=3), pos_end=Position(line=2, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=170, i_end=185, messages=[Message(data='unexpected end of input', pos=Position(line=5, column=15), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
def NormNums.erase {m : Type → Type} [Monad m] [MonadError m] (d : NormNums) (declName : Name) :
    m NormNums
:= do thisIsWrongLemmaNameXYZ
unless d.tree.values.any (·.name == declName) && ! d.erased.contains declName
  do
    throwError "'{declName}' does not have [norm_num] attribute"
  return d.eraseCore declName

initialize registerBuiltinAttribute {
  name := `norm_num
  descr := "adds a norm_num extension"
  applicationTime := .afterCompilation
  add := fun declName stx kind ↦ match stx with
    | `(attr| norm_num $es,*) => do
      let env ← getEnv
      ensureAttrDeclIsMeta `norm_num declName kind
      unless (env.getModuleIdxFor? declName).isNone do
        throwError "invalid attribute 'norm_num', declaration is in an imported module"
      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions
      let ext ← mkNormNumExt declName
      let keys ← MetaM.run' <| es.getElems.mapM fun stx ↦ do
        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|
          withReader ({ · with ignoreTCFailures := true }) do
            let e ← elabTerm stx none
            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)
            return e
        DiscrTree.mkPath e
      normNumExt.add ((keys, declName), ext) kind
      -- TODO: track what `[norm_num]` decls are actually used at use sites
      recordExtraRevUseOfCurrentModule
    | _ => throwUnsupportedSyntax
  erase := fun declName => do
    let s := normNumExt.getState (← getEnv)
    let s ← s.erase declName
    modifyEnv fun env => normNumExt.modifyState env fun _ => s
}

/-- A simp plugin which calls `NormNum.eval`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=144, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.2\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=1, column=48), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=144, i_end=1664, messages=[Message(data="unexpected token 'unless'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `registerBuiltinAttribute`', pos=Position(line=9, column=11), pos_end=Position(line=9, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1664, i_end=1712, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=38, column=48), 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.Tactic.NormNum.Core).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def NormNums.erase {m : Type → Type} [Monad m] [MonadError m] (d : NormNums) (declName : Name) :
    m NormNums
:= do thisIsWrongLemmaNameXYZ
unless d.tree.values.any (·.name == declName) && ! d.erased.contains declName
  do
    throwError "'{declName}' does not have [norm_num] attribute"
  return d.eraseCore declName

initialize registerBuiltinAttribute {
  name := `norm_num
  descr := "adds a norm_num extension"
  applicationTime := .afterCompilation
  add := fun declName stx kind ↦ match stx with
    | `(attr| norm_num $es,*) => do
      let env ← getEnv
      ensureAttrDeclIsMeta `norm_num declName kind
      unless (env.getModuleIdxFor? declName).isNone do
        throwError "invalid attribute 'norm_num', declaration is in an imported module"
      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions
      let ext ← mkNormNumExt declName
      let keys ← MetaM.run' <| es.getElems.mapM fun stx ↦ do
        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|
          withReader ({ · with ignoreTCFailures := true }) do
            let e ← elabTerm stx none
            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)
            return e
        DiscrTree.mkPath e
      normNumExt.add ((keys, declName), ext) kind
      -- TODO: track what `[norm_num]` decls are actually used at use sites
      recordExtraRevUseOfCurrentModule
    | _ => throwUnsupportedSyntax
  erase := fun declName => do
    let s := normNumExt.getState (← getEnv)
    let s ← s.erase declName
    modifyEnv fun env => normNumExt.modifyState env fun _ => s
}

/-- A simp plugin which calls `NormNum.eval`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=144, messages=[Message(data='invalid binder annotation, type is not a class instance\n  ?m.2\n\nNote: Use the command `set_option checkBinderAnnotations false` to disable the check', pos=Position(line=1, column=48), pos_end=Position(line=1, column=60), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=144, i_end=1664, messages=[Message(data="unexpected token 'unless'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `registerBuiltinAttribute`', pos=Position(line=9, column=11), pos_end=Position(line=9, column=35), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=1664, i_end=1712, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=38, column=48), 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.RepresentationTheory.FiniteIndex).

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

```lean
lemma resIndAdjunction_homEquiv_apply (A : Rep.{u} k S)
    {B : Rep.{u} k G} (f : res S.subtype B ⟶ A) :
    (resIndAdjunction k S).homEquiv _ _ f =
      resCoindHomEquiv S.subtype B A f ≫ (indCoindIso A).inv
  frobnicateTacticThatDoesNotExist := by
rw [resIndAdjunction, Adjunction.homEquiv_ofNatIsoRight_apply]
  simp [resCoindHomEquiv]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=191, messages=[Message(data='expected token', pos=Position(line=4, column=39), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=9), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  res\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  S.subtype\n\nHint: The identifier `res` 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=27), pos_end=Position(line=2, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  resIndAdjunction\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  k\n\nHint: The identifier `resIndAdjunction` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=5), pos_end=Position(line=3, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  resCoindHomEquiv\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  S.subtype\n\nHint: The identifier `resCoindHomEquiv` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=6), pos_end=Position(line=4, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=191, i_end=344, 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.FiniteIndex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma resIndAdjunction_homEquiv_apply (A : Rep.{u} k S)
    {B : Rep.{u} k G} (f : res S.subtype B ⟶ A) :
    (resIndAdjunction k S).homEquiv _ _ f =
      resCoindHomEquiv S.subtype B A f ≫ (indCoindIso A).inv
  frobnicateTacticThatDoesNotExist := by
rw [resIndAdjunction, Adjunction.homEquiv_ofNatIsoRight_apply]
  simp [resCoindHomEquiv]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=191, messages=[Message(data='expected token', pos=Position(line=4, column=39), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=43), pos_end=Position(line=1, column=54), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=2, column=9), pos_end=Position(line=2, column=20), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  res\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  S.subtype\n\nHint: The identifier `res` 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=27), pos_end=Position(line=2, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  resIndAdjunction\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  k\n\nHint: The identifier `resIndAdjunction` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=5), pos_end=Position(line=3, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  resCoindHomEquiv\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  S.subtype\n\nHint: The identifier `resCoindHomEquiv` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=6), pos_end=Position(line=4, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=191, i_end=344, 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.Tactic.IntervalCases).

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

```lean
theorem of_le_right [LE α] (h : (a : α) ≤ b) (eq : b = b') : a ≤ b'
:= eq ▸ h thisIsWrongLemmaNameXYZ
/--
Given a proof `pf`, attempts to parse it as an upper (`lb = false`) or lower (`lb = true`)
bound on `n`. If successful, it returns `(bound, n, pf')` where `n` is a numeral and
`pf'` proves `n ≤ e` or `n ≱ e` (as described by `bound`).
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=110, messages=[Message(data='Function expected at\n  h\nbut this term has type\n  a ≤ b\n\nNote: Expected a function because this term is being applied to the argument\n  thisIsWrongLemmaNameXYZ', pos=Position(line=2, column=8), pos_end=Position(line=2, column=33), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=110, i_end=355, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, 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.Tactic.IntervalCases).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem of_le_right [LE α] (h : (a : α) ≤ b) (eq : b = b') : a ≤ b'
:= eq ▸ h thisIsWrongLemmaNameXYZ
/--
Given a proof `pf`, attempts to parse it as an upper (`lb = false`) or lower (`lb = true`)
bound on `n`. If successful, it returns `(bound, n, pf')` where `n` is a numeral and
`pf'` proves `n ≤ e` or `n ≱ e` (as described by `bound`).
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=110, messages=[Message(data='Function expected at\n  h\nbut this term has type\n  a ≤ b\n\nNote: Expected a function because this term is being applied to the argument\n  thisIsWrongLemmaNameXYZ', pos=Position(line=2, column=8), pos_end=Position(line=2, column=33), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=110, i_end=355, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, 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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.VectorBundle.Basic).

[User]
Complete the following proof.

```lean
theorem contMDiffOn_zeroSection {t : Set B} :
    ContMDiffOn IB (IB.prod 𝓘(𝕜, F)) n (zeroSection F E) t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.VectorBundle.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem contMDiffOn_zeroSection {t : Set B} :
    ContMDiffOn IB (IB.prod 𝓘(𝕜, F)) n (zeroSection F E) t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
def dispatchLemma
    (twoH : Name × Name × (Name ⊕ Name) × List Bool) (debug : Bool
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.ComputeDegree).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def dispatchLemma
    (twoH : Name × Name × (Name ⊕ Name) × List Bool) (debug : Bool
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
instance IsRightCancelMul.mulRightStrictMono_of_mulRightMono
    [Mul N] [IsRightCancelMul N] [PartialOrder N] [MulRightMono N] :
    MulRightStrictMono N where
  elim a _ _ bc
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.Unbundled.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
instance IsRightCancelMul.mulRightStrictMono_of_mulRightMono
    [Mul N] [IsRightCancelMul N] [PartialOrder N] [MulRightMono N] :
    MulRightStrictMono N where
  elim a _ _ bc
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem measurable_of_Ioi {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Ioi x)) : Measurable f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Constructions.BorelSpace.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem measurable_of_Ioi {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Ioi x)) : Measurable f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem rightAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
    (rightAdjointUniq adj1 adj1).hom = 𝟙 _
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.CategoryTheory.Adjunction.Unique).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem rightAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
    (rightAdjointUniq adj1 adj1).hom = 𝟙 _
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem Measure.isComplete_iff : μ.IsComplete ↔ ∀ s, μ s = 0 → MeasurableSet s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Measure.NullMeasurable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Measure.isComplete_iff : μ.IsComplete ↔ ∀ s, μ s = 0 → MeasurableSet s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem parallelPair_functor_obj {F : WalkingParallelPair ⥤ C} (j : WalkingParallelPair) :
    (parallelPair (F.map left) (F.map right)).obj j = F.obj j
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.CategoryTheory.Limits.Shapes.Equalizers).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem parallelPair_functor_obj {F : WalkingParallelPair ⥤ C} (j : WalkingParallelPair) :
    (parallelPair (F.map left) (F.map right)).obj j = F.obj j
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
lemma Odd.pow_nonpos_iff (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Ring.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma Odd.pow_nonpos_iff (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
lemma descPochhammer_eval_eq_prod_range {R : Type*} [CommRing R] (n : ℕ) (r : R) :
    (descPochhammer R n).eval r = ∏ j ∈ Finset.range n, (r - j)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Polynomial.Pochhammer).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma descPochhammer_eval_eq_prod_range {R : Type*} [CommRing R] (n : ℕ) (r : R) :
    (descPochhammer R n).eval r = ∏ j ∈ Finset.range n, (r - j)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem image_succ_Ico (i j : Fin n) : succ '' Ico i j = Ico i.succ j.succ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Set.Fin).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem image_succ_Ico (i j : Fin n) : succ '' Ico i j = Ico i.succ j.succ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
theorem finprod_mem_pair (h : a ≠ b) : (∏ᶠ i ∈ ({a, b} : Set α), f i) = f a * f b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.BigOperators.Finprod).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem finprod_mem_pair (h : a ≠ b) : (∏ᶠ i ∈ ({a, b} : Set α), f i) = f a * f b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
lemma quasiIso_extendMap_iff [∀ j, K.HasHomology j] [∀ j, L.HasHomology j] :
    QuasiIso (extendMap φ e) ↔ QuasiIso φ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.Embedding.ExtendHomology).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma quasiIso_extendMap_iff [∀ j, K.HasHomology j] [∀ j, L.HasHomology j] :
    QuasiIso (extendMap φ e) ↔ QuasiIso φ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
lemma Integrable.prodLp_fst (h : Integrable f μ) : Integrable (fun x ↦ (f x).fst) μ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.SpecificCodomains.WithLp).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma Integrable.prodLp_fst (h : Integrable f μ) : Integrable (fun x ↦ (f x).fst) μ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

[User]
Complete the following proof.

```lean
lemma rescale_to_shell (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
    (hx : p x ≠ 0) :
    ∃ d : 𝕜, d ≠ 0 ∧ p (d • x) < ε ∧ (ε / ‖c‖ ≤ p (d • x)) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Seminorm).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma rescale_to_shell (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
    (hx : p x ≠ 0) :
    ∃ d : 𝕜, d ≠ 0 ∧ p (d • x) < ε ∧ (ε / ‖c‖ ≤ p (d • x)) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


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

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

```lean
theorem sumLift₂_nonempty :
    (sumLift₂ f g a b).Nonempty ↔
      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨
        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty
:= by .someMissingNamespaceXYZ
simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=240, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  sumLift₂\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `sumLift₂` 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=5), pos_end=Position(line=2, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inl\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a₁\n\nHint: The identifier `inl` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=20), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inl\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  b₁\n\nHint: The identifier `inl` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=33), pos_end=Position(line=3, column=39), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  a₁\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=43), pos_end=Position(line=3, column=50), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inr\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  a₂\n\nHint: The identifier `inr` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=21), pos_end=Position(line=4, column=27), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inr\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  b₂\n\nHint: The identifier `inr` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=34), pos_end=Position(line=4, column=40), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  g\nbut this term has type\n  ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  a₂\n\nHint: The identifier `g` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=44), pos_end=Position(line=4, column=51), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝⁴ : Sort u_1\nsumLift₂ : x✝⁴\nα✝¹ : Sort u_2\na : α✝¹\nx✝³ : Sort u_3\ninl : x✝³\nα✝ : Sort u_4\nb : α✝\nx✝² : Sort u_5\nf : x✝²\nx✝¹ : Sort u_6\ninr : x✝¹\nx✝ : Sort u_7\ng : x✝\n⊢ sorry ↔ (∃ a₁ b₁, a = ?m.14 ∧ b = ?m.18 ∧ sorry) ∨ ∃ a₂ b₂, a = ?m.34 ∧ b = ?m.38 ∧ sorry', pos=Position(line=5, column=6), pos_end=Position(line=5, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=240, i_end=358, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem sumLift₂_nonempty :
    (sumLift₂ f g a b).Nonempty ↔
      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨
        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty
:= by .someMissingNamespaceXYZ
simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=240, messages=[Message(data='unknown tactic', pos=Position(line=5, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  sumLift₂\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `sumLift₂` 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=5), pos_end=Position(line=2, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inl\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a₁\n\nHint: The identifier `inl` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=20), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inl\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  b₁\n\nHint: The identifier `inl` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=33), pos_end=Position(line=3, column=39), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  f\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  a₁\n\nHint: The identifier `f` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=3, column=43), pos_end=Position(line=3, column=50), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inr\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  a₂\n\nHint: The identifier `inr` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=21), pos_end=Position(line=4, column=27), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  inr\nbut this term has type\n  ?m.6\n\nNote: Expected a function because this term is being applied to the argument\n  b₂\n\nHint: The identifier `inr` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=34), pos_end=Position(line=4, column=40), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  g\nbut this term has type\n  ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  a₂\n\nHint: The identifier `g` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=44), pos_end=Position(line=4, column=51), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nx✝⁴ : Sort u_1\nsumLift₂ : x✝⁴\nα✝¹ : Sort u_2\na : α✝¹\nx✝³ : Sort u_3\ninl : x✝³\nα✝ : Sort u_4\nb : α✝\nx✝² : Sort u_5\nf : x✝²\nx✝¹ : Sort u_6\ninr : x✝¹\nx✝ : Sort u_7\ng : x✝\n⊢ sorry ↔ (∃ a₁ b₁, a = ?m.14 ∧ b = ?m.18 ∧ sorry) ∨ ∃ a₂ b₂, a = ?m.34 ∧ b = ?m.38 ∧ sorry', pos=Position(line=5, column=6), pos_end=Position(line=5, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=240, i_end=358, 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.ModelTheory.DirectLimit).

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

```text
Proof-state transcript for `Mathlib.ModelTheory.DirectLimit.exists_fg_substructure_in_Sigma`:
State 0: ⊢ theorem exists_fg_substructure_in_Sigma (S : L.Substructure (DirectLimit G f)) (S_fg : S.FG) :
    ∃ i, ∃ T : L.Substructure (G i), T.map (of L ι G f i).toHom = S
Tactic: use i
State 1: ⊢ <goal after use i>
Tactic: use Substructure.closure L (range y)
State 2: ⊢ <goal after use Substructure.closure L (range y)>
Tactic: rw [Substructure.map_closure]
State 3: ⊢ <goal after rw [Substructure.map_closure]>
Tactic: simp only [Embedding.coe_toHom, of_apply]
State 4: ⊢ <goal after simp only [Embedding.coe_toHom, of_apply]>
Tactic: rw [← image_univ, image_image, image_univ, ← eq_y,  Subtype.range_coe_subtype, Finset.setOf_mem, A_closure]  variable {P : Type u₁} [L.Structure P]  set_option backward.isDefEq.respectTransparency false in  variable (L ι G f) in  /-- The universal property of the direct limit: maps from the components to another module  that respect the directed system structure (i.e. make some diagram commute) give rise  to a unique map out of the direct limit. -/  noncomputable def lift (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i x) :  DirectLimit G f ↪[L] P where  toFun :=  Quotient.lift (fun x : Σˣ f => (g x.1) x.2) fun x y xy => by
State 5: ⊢ <goal after rw [← image_univ, image_image, image_univ, ← eq_y,>
Tactic: simp only
State 6: ⊢ <goal after simp only>
Tactic: obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.1 y.1
State 7: ⊢ <goal after obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.1 y.1>
Tactic: rw [← Hg x.1 i hx, ← Hg y.1 i hy]
State 8: ⊢ <goal after rw [← Hg x.1 i hx, ← Hg y.1 i hy]>
Tactic: exact congr_arg _ ((equiv_iff ..).1 xy)  inj' x y xy := by
State 9: ⊢ <goal after exact congr_arg _ ((equiv_iff ..).1 xy)>
Tactic: rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.lift_mk, Quotient.lift_mk] at xy
State 10: ⊢ <goal after rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.lift_mk, Quotient.lift_mk] at xy>
Tactic: obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.out.1 y.out.1
State 11: ⊢ <goal after obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.out.1 y.out.1>
Tactic: rw [← Hg x.out.1 i hx, ← Hg y.out.1 i hy] at xy
State 12: ⊢ <goal after rw [← Hg x.out.1 i hx, ← Hg y.out.1 i hy] at xy>
Tactic: rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.eq_iff_equiv, equiv_iff G f hx hy]
State 13: ⊢ <goal after rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.eq_iff_equiv, equiv_iff G f hx hy]>
Tactic: exact (g i).injective xy  map_fun' F x := by
State 14: ⊢ <goal after exact (g i).injective xy>
Tactic: obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x  change _ = funMap F (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y)
State 15: ⊢ <goal after obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x>
Tactic: rw [funMap_quotient_mk'_sigma_mk', ← Function.comp_assoc, Quotient.lift_comp_mk]
State 16: ⊢ <goal after rw [funMap_quotient_mk'_sigma_mk', ← Function.comp_assoc, Quotient.lift_comp_mk]>
Tactic: simp only [Quotient.lift_mk, Embedding.map_fun]
State 17: ⊢ <goal after simp only [Quotient.lift_mk, Embedding.map_fun]>
Tactic: rfl  map_rel' R x := by
State 18: ⊢ <goal after rfl>
Tactic: obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x  change RelMap R (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y) ↔ _
State 19: ⊢ <goal after obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x>
Tactic: rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp_assoc,  Quotient.lift_comp_mk]
State 20: ⊢ <goal after rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp_assoc,>
Tactic: rfl  variable (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i x)
```

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

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

```text
Proof-state transcript for `Mathlib.ModelTheory.DirectLimit.exists_fg_substructure_in_Sigma`:
State 0: ⊢ theorem exists_fg_substructure_in_Sigma (S : L.Substructure (DirectLimit G f)) (S_fg : S.FG) :
    ∃ i, ∃ T : L.Substructure (G i), T.map (of L ι G f i).toHom = S
Tactic: use i
State 1: ⊢ <goal after use i>
Tactic: use Substructure.closure L (range y)
State 2: ⊢ <goal after use Substructure.closure L (range y)>
Tactic: rw [Substructure.map_closure]
State 3: ⊢ <goal after rw [Substructure.map_closure]>
Tactic: simp only [Embedding.coe_toHom, of_apply]
State 4: ⊢ <goal after simp only [Embedding.coe_toHom, of_apply]>
Tactic: rw [← image_univ, image_image, image_univ, ← eq_y,  Subtype.range_coe_subtype, Finset.setOf_mem, A_closure]  variable {P : Type u₁} [L.Structure P]  set_option backward.isDefEq.respectTransparency false in  variable (L ι G f) in  /-- The universal property of the direct limit: maps from the components to another module  that respect the directed system structure (i.e. make some diagram commute) give rise  to a unique map out of the direct limit. -/  noncomputable def lift (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i x) :  DirectLimit G f ↪[L] P where  toFun :=  Quotient.lift (fun x : Σˣ f => (g x.1) x.2) fun x y xy => by
State 5: ⊢ <goal after rw [← image_univ, image_image, image_univ, ← eq_y,>
Tactic: simp only
State 6: ⊢ <goal after simp only>
Tactic: obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.1 y.1
State 7: ⊢ <goal after obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.1 y.1>
Tactic: rw [← Hg x.1 i hx, ← Hg y.1 i hy]
State 8: ⊢ <goal after rw [← Hg x.1 i hx, ← Hg y.1 i hy]>
Tactic: exact congr_arg _ ((equiv_iff ..).1 xy)  inj' x y xy := by
State 9: ⊢ <goal after exact congr_arg _ ((equiv_iff ..).1 xy)>
Tactic: rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.lift_mk, Quotient.lift_mk] at xy
State 10: ⊢ <goal after rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.lift_mk, Quotient.lift_mk] at xy>
Tactic: obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.out.1 y.out.1
State 11: ⊢ <goal after obtain ⟨i, hx, hy⟩ := directed_of (· ≤ ·) x.out.1 y.out.1>
Tactic: rw [← Hg x.out.1 i hx, ← Hg y.out.1 i hy] at xy
State 12: ⊢ <goal after rw [← Hg x.out.1 i hx, ← Hg y.out.1 i hy] at xy>
Tactic: rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.eq_iff_equiv, equiv_iff G f hx hy]
State 13: ⊢ <goal after rw [← Quotient.out_eq x, ← Quotient.out_eq y, Quotient.eq_iff_equiv, equiv_iff G f hx hy]>
Tactic: exact (g i).injective xy  map_fun' F x := by
State 14: ⊢ <goal after exact (g i).injective xy>
Tactic: obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x  change _ = funMap F (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y)
State 15: ⊢ <goal after obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x>
Tactic: rw [funMap_quotient_mk'_sigma_mk', ← Function.comp_assoc, Quotient.lift_comp_mk]
State 16: ⊢ <goal after rw [funMap_quotient_mk'_sigma_mk', ← Function.comp_assoc, Quotient.lift_comp_mk]>
Tactic: simp only [Quotient.lift_mk, Embedding.map_fun]
State 17: ⊢ <goal after simp only [Quotient.lift_mk, Embedding.map_fun]>
Tactic: rfl  map_rel' R x := by
State 18: ⊢ <goal after rfl>
Tactic: obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x  change RelMap R (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y) ↔ _
State 19: ⊢ <goal after obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x>
Tactic: rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp_assoc,  Quotient.lift_comp_mk]
State 20: ⊢ <goal after rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp_assoc,>
Tactic: rfl  variable (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i x)
```

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