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

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

## Current excerpt (around `Mathlib.Analysis.Calculus.VectorField.pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt`, L619-L654)
```lean
lemma pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt
    {f : E → F} {V W : F → F} {x : E} {t : Set F}
    (hf : IsSymmSndFDerivWithinAt 𝕜 f s x) (h'f : ContDiffWithinAt 𝕜 2 f s x)
    (hV : DifferentiableWithinAt 𝕜 V t (f x)) (hW : DifferentiableWithinAt 𝕜 W t (f x))
    (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hst : MapsTo f s t) :
    pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x
      = lieBracketWithin 𝕜 (pullbackWithin 𝕜 f V s) (pullbackWithin 𝕜 f W s) s x := by
by_cases h : (fderivWithin 𝕜 f s x).IsInvertible; swap
  · simp [pullbackWithin_eq_of_not_isInvertible h, lieBracketWithin_eq]
  rcases exists_continuousLinearEquiv_fderivWithin_symm_eq h'f h hu hx
    with ⟨M, -, M_symm_smooth, hM, M_diff⟩
  have hMx : M x = fderivWithin 𝕜 f s x := (mem_of_mem_nhdsWithin hx hM :)
  have AV : fderivWithin 𝕜 (pullbackWithin 𝕜 f V s) s x =
      fderivWithin 𝕜 (fun y ↦ ((M y).symm : F →L[𝕜] E) (V (f y))) s x := by
    apply Filter.EventuallyEq.fderivWithin_eq_of_mem _ hx
    filter_upwards [hM] with y hy using pullbackWithin_eq_of_fderivWithin_eq hy _
  have AW : fderivWithin 𝕜 (pullbackWithin 𝕜 f W s) s x =
      fderivWithin 𝕜 (fun y ↦ ((M y).symm : F →L[𝕜] E) (W (f y))) s x := by
    apply Filter.EventuallyEq.fderivWithin_eq_of_mem _ hx
    filter_upwards [hM] with y hy using pullbackWithin_eq_of_fderivWithin_eq hy _
  have Af : DifferentiableWithinAt 𝕜 f s x := h'f.differentiableWithinAt two_ne_zero
  simp only [lieBracketWithin_eq, pullbackWithin_eq_of_fderivWithin_eq hMx, map_sub, AV, AW]
  rw [fderivWithin_clm_apply, fderivWithin_clm_apply]
  · simp [fderivWithin_comp' x hW Af hst (hu x hx), ← hMx,
      fderivWithin_comp' x hV Af hst (hu x hx), M_diff, hf.eq]
  · exact hu x hx
  · exact M_symm_smooth.differentiableWithinAt one_ne_zero
  · exact hV.comp x Af hst
  · exact hu x hx
  · exact M_symm_smooth.differentiableWithinAt one_ne_zero
  · exact hW.comp x Af hst

/-- The Lie bracket commutes with taking pullbacks. This requires the function to have symmetric
second derivative. Version in a complete space. One could also give a version avoiding
completeness but requiring that `f` is a local diffeomorphism. Variant where unique
differentiability and the invariance property are only required in a smaller set `u`. -/
```

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

## Current excerpt (around `Mathlib.Analysis.Calculus.VectorField.pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt`, L619-L654)
```lean
lemma pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt
    {f : E → F} {V W : F → F} {x : E} {t : Set F}
    (hf : IsSymmSndFDerivWithinAt 𝕜 f s x) (h'f : ContDiffWithinAt 𝕜 2 f s x)
    (hV : DifferentiableWithinAt 𝕜 V t (f x)) (hW : DifferentiableWithinAt 𝕜 W t (f x))
    (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hst : MapsTo f s t) :
    pullbackWithin 𝕜 f (lieBracketWithin 𝕜 V W t) s x
      = lieBracketWithin 𝕜 (pullbackWithin 𝕜 f V s) (pullbackWithin 𝕜 f W s) s x := by
by_cases h : (fderivWithin 𝕜 f s x).IsInvertible; swap
  · simp [pullbackWithin_eq_of_not_isInvertible h, lieBracketWithin_eq]
  rcases exists_continuousLinearEquiv_fderivWithin_symm_eq h'f h hu hx
    with ⟨M, -, M_symm_smooth, hM, M_diff⟩
  have hMx : M x = fderivWithin 𝕜 f s x := (mem_of_mem_nhdsWithin hx hM :)
  have AV : fderivWithin 𝕜 (pullbackWithin 𝕜 f V s) s x =
      fderivWithin 𝕜 (fun y ↦ ((M y).symm : F →L[𝕜] E) (V (f y))) s x := by
    apply Filter.EventuallyEq.fderivWithin_eq_of_mem _ hx
    filter_upwards [hM] with y hy using pullbackWithin_eq_of_fderivWithin_eq hy _
  have AW : fderivWithin 𝕜 (pullbackWithin 𝕜 f W s) s x =
      fderivWithin 𝕜 (fun y ↦ ((M y).symm : F →L[𝕜] E) (W (f y))) s x := by
    apply Filter.EventuallyEq.fderivWithin_eq_of_mem _ hx
    filter_upwards [hM] with y hy using pullbackWithin_eq_of_fderivWithin_eq hy _
  have Af : DifferentiableWithinAt 𝕜 f s x := h'f.differentiableWithinAt two_ne_zero
  simp only [lieBracketWithin_eq, pullbackWithin_eq_of_fderivWithin_eq hMx, map_sub, AV, AW]
  rw [fderivWithin_clm_apply, fderivWithin_clm_apply]
  · simp [fderivWithin_comp' x hW Af hst (hu x hx), ← hMx,
      fderivWithin_comp' x hV Af hst (hu x hx), M_diff, hf.eq]
  · exact hu x hx
  · exact M_symm_smooth.differentiableWithinAt one_ne_zero
  · exact hV.comp x Af hst
  · exact hu x hx
  · exact M_symm_smooth.differentiableWithinAt one_ne_zero
  · exact hW.comp x Af hst

/-- The Lie bracket commutes with taking pullbacks. This requires the function to have symmetric
second derivative. Version in a complete space. One could also give a version avoiding
completeness but requiring that `f` is a local diffeomorphism. Variant where unique
differentiability and the invariance property are only required in a smaller set `u`. -/
```

## Edit instruction
Edit `pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt` so it still compiles and the proof is simplified where possible. Return the full updated `pullbackWithin_lieBracketWithin_of_isSymmSndFDerivWithinAt` 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.NumberTheory.ModularForms.Cusps).

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

## Current excerpt (around `Mathlib.NumberTheory.ModularForms.Cusps.IsRegularAtInfty`, L246-L248)
```lean
def IsRegularAtInfty : Prop := 𝒢.strictPeriods = 𝒢.periods
```

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

## Current excerpt (around `Mathlib.NumberTheory.ModularForms.Cusps.IsRegularAtInfty`, L246-L248)
```lean
def IsRegularAtInfty : Prop := 𝒢.strictPeriods = 𝒢.periods
```

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

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


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

[User]
You are editing a file in `Mathlib/NumberTheory/Padics/PadicVal/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.NumberTheory.Padics.PadicVal.Basic.pow_succ_padicValNat_not_dvd`, L435-L439)
```lean
theorem pow_succ_padicValNat_not_dvd {n : ℕ} [hp : Fact p.Prime] (hn : n ≠ 0) :
    ¬p ^ (padicValNat p n + 1) ∣ n := by
rw [padicValNat_dvd_iff_le hn, not_le]
  exact Nat.lt_succ_self _
```

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

## Current excerpt (around `Mathlib.NumberTheory.Padics.PadicVal.Basic.pow_succ_padicValNat_not_dvd`, L435-L439)
```lean
theorem pow_succ_padicValNat_not_dvd {n : ℕ} [hp : Fact p.Prime] (hn : n ≠ 0) :
    ¬p ^ (padicValNat p n + 1) ∣ n := by
rw [padicValNat_dvd_iff_le hn, not_le]
  exact Nat.lt_succ_self _
```

## Edit instruction
Edit `pow_succ_padicValNat_not_dvd` so it still compiles and the proof is simplified where possible. Return the full updated `pow_succ_padicValNat_not_dvd` 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.Algebra.Category.Ring.FilteredColimits).

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

## Current excerpt (around `Mathlib.Algebra.Category.Ring.FilteredColimits.forget₂Ring_preservesFilteredColimits`, L352-L359)
```lean
instance forget₂Ring_preservesFilteredColimits :
    PreservesFilteredColimits (forget₂ CommRingCat RingCat.{u}) where
  preserves_filtered_colimits {J hJ1 _} := letI : Category J := hJ1
    { preservesColimit := fun {F} =>
        preservesColimit_of_preserves_colimit_cocone (colimitCoconeIsColimit.{u, u} F)
          (RingCat.FilteredColimits.colimitCoconeIsColimit (F ⋙ forget₂ CommRingCat RingCat.{u})) }
```

## Edit instruction
Edit `forget₂Ring_preservesFilteredColimits` so it still compiles and the proof is simplified where possible. Return the full updated `forget₂Ring_preservesFilteredColimits` 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.FilteredColimits).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Algebra/Category/Ring/FilteredColimits.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Algebra.Category.Ring.FilteredColimits.forget₂Ring_preservesFilteredColimits`, L352-L359)
```lean
instance forget₂Ring_preservesFilteredColimits :
    PreservesFilteredColimits (forget₂ CommRingCat RingCat.{u}) where
  preserves_filtered_colimits {J hJ1 _} := letI : Category J := hJ1
    { preservesColimit := fun {F} =>
        preservesColimit_of_preserves_colimit_cocone (colimitCoconeIsColimit.{u, u} F)
          (RingCat.FilteredColimits.colimitCoconeIsColimit (F ⋙ forget₂ CommRingCat RingCat.{u})) }
```

## Edit instruction
Edit `forget₂Ring_preservesFilteredColimits` so it still compiles and the proof is simplified where possible. Return the full updated `forget₂Ring_preservesFilteredColimits` 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.RingTheory.Valuation.Basic).

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

## Current excerpt (around `Mathlib.RingTheory.Valuation.Basic.ne_zero_of_unit`, L218-L220)
```lean
theorem ne_zero_of_unit [Nontrivial Γ₀] (v : Valuation K Γ₀) (x : Kˣ) : v x ≠ (0 : Γ₀) := by
simp only [ne_eq, Valuation.zero_iff, Units.ne_zero x, not_false_iff]
```

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

## Current excerpt (around `Mathlib.RingTheory.Valuation.Basic.ne_zero_of_unit`, L218-L220)
```lean
theorem ne_zero_of_unit [Nontrivial Γ₀] (v : Valuation K Γ₀) (x : Kˣ) : v x ≠ (0 : Γ₀) := by
simp only [ne_eq, Valuation.zero_iff, Units.ne_zero x, not_false_iff]
```

## Edit instruction
Edit `ne_zero_of_unit` so it still compiles and the proof is simplified where possible. Return the full updated `ne_zero_of_unit` 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.Combinatorics.SimpleGraph.Walks.Decomp).

[User]
You are editing a file in `Mathlib/Combinatorics/SimpleGraph/Walks/Decomp.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Combinatorics.SimpleGraph.Walks.Decomp.rotate`, L300-L302)
```lean
def rotate (c : G.Walk v v) (u : V) (h : u ∈ c.support) : G.Walk u u := (c.dropUntil u h).append (c.takeUntil u h)
```

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

## Current excerpt (around `Mathlib.Combinatorics.SimpleGraph.Walks.Decomp.rotate`, L300-L302)
```lean
def rotate (c : G.Walk v v) (u : V) (h : u ∈ c.support) : G.Walk u u := (c.dropUntil u h).append (c.takeUntil u h)
```

## Edit instruction
Edit `rotate` so it still compiles and the proof is simplified where possible. Return the full updated `rotate` 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.NumberTheory.NumberField.CMField).

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

## Current excerpt (around `Mathlib.NumberTheory.NumberField.CMField.regulator_div_regulator_eq_two_pow_mul_indexRealUnits_inv`, L443-L460)
```lean
theorem regulator_div_regulator_eq_two_pow_mul_indexRealUnits_inv :
    regulator K / regulator K⁺ = 2 ^ rank K * (indexRealUnits K : ℝ)⁻¹ := by
rw [indexRealUnits, ← closure_realFundSystem_sup_torsion, ← regOfFamily_div_regulator
    (realFundSystem K), regOfFamily_realFunSystem, inv_div, ← mul_div_assoc, mul_div_mul_comm,
    div_self (by positivity), one_mul]

end units

end IsCMField

end maximalRealSubfield

namespace CMExtension

variable (F K : Type*) [Field F] [IsTotallyReal F] [Field K] [CharZero K] [Algebra.IsIntegral ℚ K]
  [IsTotallyComplex K] [Algebra F K] [IsQuadraticExtension F K]

set_option backward.isDefEq.respectTransparency false in
```

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

## Current excerpt (around `Mathlib.NumberTheory.NumberField.CMField.regulator_div_regulator_eq_two_pow_mul_indexRealUnits_inv`, L443-L460)
```lean
theorem regulator_div_regulator_eq_two_pow_mul_indexRealUnits_inv :
    regulator K / regulator K⁺ = 2 ^ rank K * (indexRealUnits K : ℝ)⁻¹ := by
rw [indexRealUnits, ← closure_realFundSystem_sup_torsion, ← regOfFamily_div_regulator
    (realFundSystem K), regOfFamily_realFunSystem, inv_div, ← mul_div_assoc, mul_div_mul_comm,
    div_self (by positivity), one_mul]

end units

end IsCMField

end maximalRealSubfield

namespace CMExtension

variable (F K : Type*) [Field F] [IsTotallyReal F] [Field K] [CharZero K] [Algebra.IsIntegral ℚ K]
  [IsTotallyComplex K] [Algebra F K] [IsQuadraticExtension F K]

set_option backward.isDefEq.respectTransparency false in
```

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

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

## Current excerpt (around `Mathlib.LinearAlgebra.Multilinear.Basic.map_nonempty`, L1432-L1436)
```lean
theorem map_nonempty [Nonempty ι] (f : MultilinearMap R M₁ M₂) (p : ∀ i, Submodule R (M₁ i)) :
    (map f p : Set M₂).Nonempty := ⟨f 0, 0, fun i => (p i).zero_mem, rfl⟩

/-- The range of a multilinear map, closed under scalar multiplication. -/
```

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

## Current excerpt (around `Mathlib.LinearAlgebra.Multilinear.Basic.map_nonempty`, L1432-L1436)
```lean
theorem map_nonempty [Nonempty ι] (f : MultilinearMap R M₁ M₂) (p : ∀ i, Submodule R (M₁ i)) :
    (map f p : Set M₂).Nonempty := ⟨f 0, 0, fun i => (p i).zero_mem, rfl⟩

/-- The range of a multilinear map, closed under scalar multiplication. -/
```

## Edit instruction
Edit `map_nonempty` so it still compiles and the proof is simplified where possible. Return the full updated `map_nonempty` 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.RingTheory.MvPowerSeries.PiTopology).

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

## Current excerpt (around `Mathlib.RingTheory.MvPowerSeries.PiTopology.tendsto_iff_coeff_tendsto`, L131-L137)
```lean
theorem tendsto_iff_coeff_tendsto [Semiring R] {ι : Type*}
    (f : ι → MvPowerSeries σ R) (u : Filter ι) (g : MvPowerSeries σ R) :
    Tendsto f u (nhds g) ↔
    ∀ d : σ →₀ ℕ, Tendsto (fun i => coeff d (f i)) u (nhds (coeff d g)) := by
rw [nhds_pi, tendsto_pi]
  exact forall_congr' (fun d => Iff.rfl)
```

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

## Current excerpt (around `Mathlib.RingTheory.MvPowerSeries.PiTopology.tendsto_iff_coeff_tendsto`, L131-L137)
```lean
theorem tendsto_iff_coeff_tendsto [Semiring R] {ι : Type*}
    (f : ι → MvPowerSeries σ R) (u : Filter ι) (g : MvPowerSeries σ R) :
    Tendsto f u (nhds g) ↔
    ∀ d : σ →₀ ℕ, Tendsto (fun i => coeff d (f i)) u (nhds (coeff d g)) := by
rw [nhds_pi, tendsto_pi]
  exact forall_congr' (fun d => Iff.rfl)
```

## Edit instruction
Edit `tendsto_iff_coeff_tendsto` so it still compiles and the proof is simplified where possible. Return the full updated `tendsto_iff_coeff_tendsto` 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.Topology.Sheaves.Stalks).

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

## Current excerpt (around `Mathlib.Topology.Sheaves.Stalks.stalkFunctor_map_germ`, L145-L149)
```lean
theorem stalkFunctor_map_germ {F G : X.Presheaf C} (U : Opens X) (x : X) (hx : x ∈ U) (f : F ⟶ G) :
    F.germ U x hx ≫ (stalkFunctor C x).map f = f.app (op U) ≫ G.germ U x hx := colimit.ι_map (whiskerLeft (OpenNhds.inclusion x).op f) (op ⟨U, hx⟩)

set_option backward.isDefEq.respectTransparency false in
```

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

## Current excerpt (around `Mathlib.Topology.Sheaves.Stalks.stalkFunctor_map_germ`, L145-L149)
```lean
theorem stalkFunctor_map_germ {F G : X.Presheaf C} (U : Opens X) (x : X) (hx : x ∈ U) (f : F ⟶ G) :
    F.germ U x hx ≫ (stalkFunctor C x).map f = f.app (op U) ≫ G.germ U x hx := colimit.ι_map (whiskerLeft (OpenNhds.inclusion x).op f) (op ⟨U, hx⟩)

set_option backward.isDefEq.respectTransparency false in
```

## Edit instruction
Edit `stalkFunctor_map_germ` so it still compiles and the proof is simplified where possible. Return the full updated `stalkFunctor_map_germ` 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.Order.Preorder.Chain).

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

## Current excerpt (around `Mathlib.Order.Preorder.Chain.IsChain`, L77-L79)
```lean
theorem IsChain.symm (h : IsChain r s) : IsChain (flip r) s := h.mono' fun _ _ => Or.symm
```

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

## Current excerpt (around `Mathlib.Order.Preorder.Chain.IsChain`, L77-L79)
```lean
theorem IsChain.symm (h : IsChain r s) : IsChain (flip r) s := h.mono' fun _ _ => Or.symm
```

## Edit instruction
Edit `IsChain` so it still compiles and the proof is simplified where possible. Return the full updated `IsChain` 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.Algebra.Ring.Parity).

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

## Current excerpt (around `Mathlib.Algebra.Ring.Parity.Even`, L84-L86)
```lean
lemma Even.trans_dvd (ha : Even a) (hab : a ∣ b) : Even b := even_iff_two_dvd.2 <| ha.two_dvd.trans hab
```

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

## Current excerpt (around `Mathlib.Algebra.Ring.Parity.Even`, L84-L86)
```lean
lemma Even.trans_dvd (ha : Even a) (hab : a ∣ b) : Even b := even_iff_two_dvd.2 <| ha.two_dvd.trans hab
```

## Edit instruction
Edit `Even` so it still compiles and the proof is simplified where possible. Return the full updated `Even` 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.LinearAlgebra.Matrix.Basis).

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

## Current excerpt (around `Mathlib.LinearAlgebra.Matrix.Basis.toMatrix_transpose_apply`, L58-L60)
```lean
theorem toMatrix_transpose_apply : (e.toMatrix v)ᵀ j = e.repr (v j) := funext fun _ => rfl
```

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

## Current excerpt (around `Mathlib.LinearAlgebra.Matrix.Basis.toMatrix_transpose_apply`, L58-L60)
```lean
theorem toMatrix_transpose_apply : (e.toMatrix v)ᵀ j = e.repr (v j) := funext fun _ => rfl
```

## Edit instruction
Edit `toMatrix_transpose_apply` so it still compiles and the proof is simplified where possible. Return the full updated `toMatrix_transpose_apply` 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.Combinatorics.SimpleGraph.Matching).

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

## Current excerpt (around `Mathlib.Combinatorics.SimpleGraph.Matching.IsMatching`, L79-L82)
```lean
theorem IsMatching.toEdge.surjective (h : M.IsMatching) : Surjective h.toEdge := by
rintro ⟨⟨x, y⟩, he⟩
  exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
```

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

## Current excerpt (around `Mathlib.Combinatorics.SimpleGraph.Matching.IsMatching`, L79-L82)
```lean
theorem IsMatching.toEdge.surjective (h : M.IsMatching) : Surjective h.toEdge := by
rintro ⟨⟨x, y⟩, he⟩
  exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
```

## Edit instruction
Edit `IsMatching` so it still compiles and the proof is simplified where possible. Return the full updated `IsMatching` 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.RingTheory.Spectrum.Prime.Topology).

[User]
You are editing a file in `Mathlib/RingTheory/Spectrum/Prime/Topology.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.RingTheory.Spectrum.Prime.Topology.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical`, L308-L322)
```lean
theorem discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical :
    letI s := {I : Ideal R | I.IsMaximal}
DiscreteTopology (PrimeSpectrum R) ↔ Finite s ∧ sInf s ≤ nilradical R := by
  rw [discreteTopology_iff_finite_and_krullDimLE_zero, Ring.krullDimLE_zero_iff,
    (equivSubtype R).finite_iff, ← Set.coe_setOf, Set.finite_coe_iff, Set.finite_coe_iff]
  refine ⟨fun h ↦ ⟨h.1.subset fun _ h ↦ h.isPrime, nilradical_eq_sInf R ▸ sInf_le_sInf h.2⟩,
    fun ⟨fin, le⟩ ↦ ?_⟩
  have hpm (I : Ideal R) (hI : I.IsPrime) : I.IsMaximal := by
    replace le := le.trans (nilradical_le_prime I)
    rw [← fin.coe_toFinset, ← Finset.inf_id_eq_sInf, hI.inf_le'] at le
    have ⟨M, hM, hMI⟩ := le
    rw [fin.mem_toFinset] at hM
    rwa [← hM.eq_of_le hI.1 hMI]
  exact ⟨fin.subset hpm, hpm⟩
```

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

## Current excerpt (around `Mathlib.RingTheory.Spectrum.Prime.Topology.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical`, L308-L322)
```lean
theorem discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical :
    letI s := {I : Ideal R | I.IsMaximal}
DiscreteTopology (PrimeSpectrum R) ↔ Finite s ∧ sInf s ≤ nilradical R := by
  rw [discreteTopology_iff_finite_and_krullDimLE_zero, Ring.krullDimLE_zero_iff,
    (equivSubtype R).finite_iff, ← Set.coe_setOf, Set.finite_coe_iff, Set.finite_coe_iff]
  refine ⟨fun h ↦ ⟨h.1.subset fun _ h ↦ h.isPrime, nilradical_eq_sInf R ▸ sInf_le_sInf h.2⟩,
    fun ⟨fin, le⟩ ↦ ?_⟩
  have hpm (I : Ideal R) (hI : I.IsPrime) : I.IsMaximal := by
    replace le := le.trans (nilradical_le_prime I)
    rw [← fin.coe_toFinset, ← Finset.inf_id_eq_sInf, hI.inf_le'] at le
    have ⟨M, hM, hMI⟩ := le
    rw [fin.mem_toFinset] at hM
    rwa [← hM.eq_of_le hI.1 hMI]
  exact ⟨fin.subset hpm, hpm⟩
```

## Edit instruction
Edit `discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical` so it still compiles and the proof is simplified where possible. Return the full updated `discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical` 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.Analysis.SpecialFunctions.Trigonometric.Angle).

[User]
You are editing a file in `Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.abs_toReal_coe_eq_self_iff`, L542-L546)
```lean
theorem abs_toReal_coe_eq_self_iff {θ : ℝ} : |(θ : Angle).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π := ⟨fun h => h ▸ ⟨abs_nonneg _, abs_toReal_le_pi _⟩, fun h =>
    (toReal_coe_eq_self_iff.2 ⟨(Left.neg_neg_iff.2 Real.pi_pos).trans_le h.1, h.2⟩).symm ▸
      abs_eq_self.2 h.1⟩
```

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

## Current excerpt (around `Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.abs_toReal_coe_eq_self_iff`, L542-L546)
```lean
theorem abs_toReal_coe_eq_self_iff {θ : ℝ} : |(θ : Angle).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ π := ⟨fun h => h ▸ ⟨abs_nonneg _, abs_toReal_le_pi _⟩, fun h =>
    (toReal_coe_eq_self_iff.2 ⟨(Left.neg_neg_iff.2 Real.pi_pos).trans_le h.1, h.2⟩).symm ▸
      abs_eq_self.2 h.1⟩
```

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

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

## Current excerpt (around `Mathlib.Data.Option.Basic.elim_apply`, L225-L228)
```lean
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y := by rw [elim_comp fun f : α → β => f y]
open Function in
```

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

## Current excerpt (around `Mathlib.Data.Option.Basic.elim_apply`, L225-L228)
```lean
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y := by rw [elim_comp fun f : α → β => f y]
open Function in
```

## Edit instruction
Edit `elim_apply` so it still compiles and the proof is simplified where possible. Return the full updated `elim_apply` 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.Algebra.Homology.DerivedCategory.Ext.ExactSequences).

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

## Current excerpt (around `Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences.covariantSequence_exact`, L131-L137)
```lean
lemma covariantSequence_exact :
    (covariantSequence X hS n₀ n₁ h).Exact := exact_of_δ₀ (covariant_sequence_exact₂' X hS n₀).exact_toComposableArrows
    (exact_of_δ₀ (covariant_sequence_exact₃' X hS n₀ n₁ h).exact_toComposableArrows
      (exact_of_δ₀ (covariant_sequence_exact₁' X hS n₀ n₁ h).exact_toComposableArrows
        (covariant_sequence_exact₂' X hS n₁).exact_toComposableArrows))
```

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

## Current excerpt (around `Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences.covariantSequence_exact`, L131-L137)
```lean
lemma covariantSequence_exact :
    (covariantSequence X hS n₀ n₁ h).Exact := exact_of_δ₀ (covariant_sequence_exact₂' X hS n₀).exact_toComposableArrows
    (exact_of_δ₀ (covariant_sequence_exact₃' X hS n₀ n₁ h).exact_toComposableArrows
      (exact_of_δ₀ (covariant_sequence_exact₁' X hS n₀ n₁ h).exact_toComposableArrows
        (covariant_sequence_exact₂' X hS n₁).exact_toComposableArrows))
```

## Edit instruction
Edit `covariantSequence_exact` so it still compiles and the proof is simplified where possible. Return the full updated `covariantSequence_exact` 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.RingTheory.Polynomial.GaussNorm).

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

## Current excerpt (around `Mathlib.RingTheory.Polynomial.GaussNorm.le_gaussNorm`, L137-L141)
```lean
lemma le_gaussNorm (hc : 0 ≤ c) (i : ℕ) : v (p.coeff i) * c ^ i ≤ p.gaussNorm v c := by
rw [← gaussNorm_coe_powerSeries _ _ hc, ← coeff_coe]
  apply PowerSeries.le_gaussNorm
  simpa using aux_bdd v p
```

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

## Current excerpt (around `Mathlib.RingTheory.Polynomial.GaussNorm.le_gaussNorm`, L137-L141)
```lean
lemma le_gaussNorm (hc : 0 ≤ c) (i : ℕ) : v (p.coeff i) * c ^ i ≤ p.gaussNorm v c := by
rw [← gaussNorm_coe_powerSeries _ _ hc, ← coeff_coe]
  apply PowerSeries.le_gaussNorm
  simpa using aux_bdd v p
```

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

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

## Current excerpt (around `Mathlib.Topology.Separation.Basic.t0Space_iff_not_inseparable`, L85-L88)
```lean
theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by
simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise]
```

## Edit instruction
Edit `t0Space_iff_not_inseparable` so it still compiles and the proof is simplified where possible. Return the full updated `t0Space_iff_not_inseparable` 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.Topology.Separation.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Topology/Separation/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Topology.Separation.Basic.t0Space_iff_not_inseparable`, L85-L88)
```lean
theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
    T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by
simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise]
```

## Edit instruction
Edit `t0Space_iff_not_inseparable` so it still compiles and the proof is simplified where possible. Return the full updated `t0Space_iff_not_inseparable` 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.Combinatorics.Configuration).

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

## Current excerpt (around `Mathlib.Combinatorics.Configuration.eq_or_eq_of_orthogonal`, L500-L509)
```lean
lemma eq_or_eq_of_orthogonal {a b c d : ℙ K (Fin 3 → K)} (hac : a.orthogonal c)
    (hbc : b.orthogonal c) (had : a.orthogonal d) (hbd : b.orthogonal d) :
    a = b ∨ c = d := by
induction a with | h a ha =>
  induction b with | h b hb =>
  induction c with | h c hc =>
  induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
```

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

## Current excerpt (around `Mathlib.Combinatorics.Configuration.eq_or_eq_of_orthogonal`, L500-L509)
```lean
lemma eq_or_eq_of_orthogonal {a b c d : ℙ K (Fin 3 → K)} (hac : a.orthogonal c)
    (hbc : b.orthogonal c) (had : a.orthogonal d) (hbd : b.orthogonal d) :
    a = b ∨ c = d := by
induction a with | h a ha =>
  induction b with | h b hb =>
  induction c with | h c hc =>
  induction d with | h d hd =>
  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
```

## Edit instruction
Edit `eq_or_eq_of_orthogonal` so it still compiles and the proof is simplified where possible. Return the full updated `eq_or_eq_of_orthogonal` 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.Data.Finset.Prod).

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

## Current excerpt (around `Mathlib.Data.Finset.Prod.disjoint_product`, L222-L224)
```lean
theorem disjoint_product : Disjoint (s ×ˢ t) (s' ×ˢ t') ↔ Disjoint s s' ∨ Disjoint t t' := by
simp_rw [← disjoint_coe, coe_product, Set.disjoint_prod]
```

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

## Current excerpt (around `Mathlib.Data.Finset.Prod.disjoint_product`, L222-L224)
```lean
theorem disjoint_product : Disjoint (s ×ˢ t) (s' ×ˢ t') ↔ Disjoint s s' ∨ Disjoint t t' := by
simp_rw [← disjoint_coe, coe_product, Set.disjoint_prod]
```

## Edit instruction
Edit `disjoint_product` so it still compiles and the proof is simplified where possible. Return the full updated `disjoint_product` 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.Algebra.Lie.SkewAdjoint).

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

## Current excerpt (around `Mathlib.Algebra.Lie.SkewAdjoint.mem_skewAdjointMatricesLieSubalgebra_unit_smul`, L153-L161)
```lean
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> intro h
  · simpa using congr_arg (fun B => u⁻¹ • B) h
  · simp [h]

end SkewAdjointMatrices
```

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

## Current excerpt (around `Mathlib.Algebra.Lie.SkewAdjoint.mem_skewAdjointMatricesLieSubalgebra_unit_smul`, L153-L161)
```lean
theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J := by
change A ∈ skewAdjointMatricesSubmodule (u • J) ↔ A ∈ skewAdjointMatricesSubmodule J
  simp only [mem_skewAdjointMatricesSubmodule, Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
  constructor <;> intro h
  · simpa using congr_arg (fun B => u⁻¹ • B) h
  · simp [h]

end SkewAdjointMatrices
```

## Edit instruction
Edit `mem_skewAdjointMatricesLieSubalgebra_unit_smul` so it still compiles and the proof is simplified where possible. Return the full updated `mem_skewAdjointMatricesLieSubalgebra_unit_smul` 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.Algebra.GroupWithZero.Range).

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

## Current excerpt (around `Mathlib.Algebra.GroupWithZero.Range.valueGroup_def`, L99-L102)
```lean
lemma valueGroup_def : valueGroup f = Subgroup.closure (valueMonoid f) := rfl
/-- For a morphism of monoids with zero `f`, this is the smallest submonoid with zero of the
codomain containing the range of `f`. -/
```

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

## Current excerpt (around `Mathlib.Algebra.GroupWithZero.Range.valueGroup_def`, L99-L102)
```lean
lemma valueGroup_def : valueGroup f = Subgroup.closure (valueMonoid f) := rfl
/-- For a morphism of monoids with zero `f`, this is the smallest submonoid with zero of the
codomain containing the range of `f`. -/
```

## Edit instruction
Edit `valueGroup_def` so it still compiles and the proof is simplified where possible. Return the full updated `valueGroup_def` 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.Data.Set.MemPartition).

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

## Current excerpt (around `Mathlib.Data.Set.MemPartition.mem_memPartitionSet`, L136-L143)
```lean
lemma mem_memPartitionSet (f : ℕ → Set α) (n : ℕ) (a : α) : a ∈ memPartitionSet f n a := by
induction n with
  | zero => simp [memPartitionSet]
  | succ n ih =>
    classical
    rw [memPartitionSet_succ]
    split_ifs with h <;> exact ⟨ih, h⟩
```

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

## Current excerpt (around `Mathlib.Data.Set.MemPartition.mem_memPartitionSet`, L136-L143)
```lean
lemma mem_memPartitionSet (f : ℕ → Set α) (n : ℕ) (a : α) : a ∈ memPartitionSet f n a := by
induction n with
  | zero => simp [memPartitionSet]
  | succ n ih =>
    classical
    rw [memPartitionSet_succ]
    split_ifs with h <;> exact ⟨ih, h⟩
```

## Edit instruction
Edit `mem_memPartitionSet` so it still compiles and the proof is simplified where possible. Return the full updated `mem_memPartitionSet` 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.Algebra.Algebra.Subalgebra.MulOpposite).

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

## Current excerpt (around `Mathlib.Algebra.Algebra.Subalgebra.MulOpposite.op_sSup`, L104-L106)
```lean
theorem op_sSup (S : Set (Subalgebra R A)) : (sSup S).op = sSup (.unop ⁻¹' S) := opEquiv.map_sSup_eq_sSup_symm_preimage _
```

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

## Current excerpt (around `Mathlib.Algebra.Algebra.Subalgebra.MulOpposite.op_sSup`, L104-L106)
```lean
theorem op_sSup (S : Set (Subalgebra R A)) : (sSup S).op = sSup (.unop ⁻¹' S) := opEquiv.map_sSup_eq_sSup_symm_preimage _
```

## Edit instruction
Edit `op_sSup` so it still compiles and the proof is simplified where possible. Return the full updated `op_sSup` 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.Analysis.Normed.Group.Hom).

[User]
You are editing a file in `Mathlib/Analysis/Normed/Group/Hom.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.Normed.Group.Hom.coe_nsmul`, L471-L473)
```lean
theorem coe_nsmul (r : ℕ) (f : NormedAddGroupHom V₁ V₂) : ⇑(r • f) = r • ⇑f := rfl
```

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

## Current excerpt (around `Mathlib.Analysis.Normed.Group.Hom.coe_nsmul`, L471-L473)
```lean
theorem coe_nsmul (r : ℕ) (f : NormedAddGroupHom V₁ V₂) : ⇑(r • f) = r • ⇑f := rfl
```

## Edit instruction
Edit `coe_nsmul` so it still compiles and the proof is simplified where possible. Return the full updated `coe_nsmul` 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.Order.CompleteBooleanAlgebra).

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

## Current excerpt (around `Mathlib.Order.CompleteBooleanAlgebra.Pi`, L626-L630)
```lean
instance Pi.instCompleteBooleanAlgebra {ι : Type*} {π : ι → Type*}
    [∀ i, CompleteBooleanAlgebra (π i)] : CompleteBooleanAlgebra (∀ i, π i) where
  __ := instBooleanAlgebra
__ := instCompleteDistribLattice
```

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

## Current excerpt (around `Mathlib.Order.CompleteBooleanAlgebra.Pi`, L626-L630)
```lean
instance Pi.instCompleteBooleanAlgebra {ι : Type*} {π : ι → Type*}
    [∀ i, CompleteBooleanAlgebra (π i)] : CompleteBooleanAlgebra (∀ i, π i) where
  __ := instBooleanAlgebra
__ := instCompleteDistribLattice
```

## Edit instruction
Edit `Pi` so it still compiles and the proof is simplified where possible. Return the full updated `Pi` 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.Analysis.Distribution.SchwartzSpace.Basic).

[User]
You are editing a file in `Mathlib/Analysis/Distribution/SchwartzSpace/Basic.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.Distribution.SchwartzSpace.Basic.smulLeftCLM_const`, L754-L759)
```lean
theorem smulLeftCLM_const (c : 𝕜) :
    smulLeftCLM F (fun (_ : E) ↦ c) = c • ContinuousLinearMap.id 𝕜 _ := by
ext f x
  have : (fun (_ : E) ↦ c).HasTemperateGrowth := by fun_prop
  simp [this]
```

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

## Current excerpt (around `Mathlib.Analysis.Distribution.SchwartzSpace.Basic.smulLeftCLM_const`, L754-L759)
```lean
theorem smulLeftCLM_const (c : 𝕜) :
    smulLeftCLM F (fun (_ : E) ↦ c) = c • ContinuousLinearMap.id 𝕜 _ := by
ext f x
  have : (fun (_ : E) ↦ c).HasTemperateGrowth := by fun_prop
  simp [this]
```

## Edit instruction
Edit `smulLeftCLM_const` so it still compiles and the proof is simplified where possible. Return the full updated `smulLeftCLM_const` 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.Data.Int.GCD).

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

## Current excerpt (around `Mathlib.Data.Int.GCD.gcd_eq_gcd_ab`, L174-L184)
```lean
theorem gcd_eq_gcd_ab : ∀ x y : ℤ, (gcd x y : ℤ) = x * gcdA x y + y * gcdB x y
  | (m : ℕ), (n : ℕ) => Nat.gcd_eq_gcd_ab _ _
  | (m : ℕ), -[n+1] =>
    show (_ : ℤ) = _ + -(n + 1) * -_ by rw [Int.neg_mul_neg]; apply Nat.gcd_eq_gcd_ab
  | -[m+1], (n : ℕ) =>
    show (_ : ℤ) = -(m + 1) * -_ + _ by rw [Int.neg_mul_neg]; apply Nat.gcd_eq_gcd_ab
  | -[m+1], -[n+1] =>
    show (_ : ℤ) = -(m + 1) * -_ + -(n + 1) * -_ by
      rw [Int.neg_mul_neg, Int.neg_mul_neg]
      apply Nat.gcd_eq_gcd_ab
```

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

## Current excerpt (around `Mathlib.Data.Int.GCD.gcd_eq_gcd_ab`, L174-L184)
```lean
theorem gcd_eq_gcd_ab : ∀ x y : ℤ, (gcd x y : ℤ) = x * gcdA x y + y * gcdB x y
  | (m : ℕ), (n : ℕ) => Nat.gcd_eq_gcd_ab _ _
  | (m : ℕ), -[n+1] =>
    show (_ : ℤ) = _ + -(n + 1) * -_ by rw [Int.neg_mul_neg]; apply Nat.gcd_eq_gcd_ab
  | -[m+1], (n : ℕ) =>
    show (_ : ℤ) = -(m + 1) * -_ + _ by rw [Int.neg_mul_neg]; apply Nat.gcd_eq_gcd_ab
  | -[m+1], -[n+1] =>
    show (_ : ℤ) = -(m + 1) * -_ + -(n + 1) * -_ by
      rw [Int.neg_mul_neg, Int.neg_mul_neg]
      apply Nat.gcd_eq_gcd_ab
```

## Edit instruction
Edit `gcd_eq_gcd_ab` so it still compiles and the proof is simplified where possible. Return the full updated `gcd_eq_gcd_ab` 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.Algebra.Algebra.Spectrum.Quasispectrum).

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

## Current excerpt (around `Mathlib.Algebra.Algebra.Spectrum.Quasispectrum.apply_mem`, L506-L509)
```lean
theorem apply_mem (h : QuasispectrumRestricts a f) {s : S} (hs : s ∈ quasispectrum S a) :
    f s ∈ quasispectrum R a := h.image ▸ ⟨s, hs, rfl⟩
```

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

## Current excerpt (around `Mathlib.Algebra.Algebra.Spectrum.Quasispectrum.apply_mem`, L506-L509)
```lean
theorem apply_mem (h : QuasispectrumRestricts a f) {s : S} (hs : s ∈ quasispectrum S a) :
    f s ∈ quasispectrum R a := h.image ▸ ⟨s, hs, rfl⟩
```

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


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

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

```lean
theorem antitone_const_tsub : Antitone fun x => c - x
```

Declaration: `Mathlib.Algebra.Order.Sub.Defs.antitone_const_tsub` (module Mathlib.Algebra.Order.Sub.Defs).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Sub.Defs).[/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 antitone_const_tsub : Antitone fun x => c - x
```

Declaration: `Mathlib.Algebra.Order.Sub.Defs.antitone_const_tsub` (module Mathlib.Algebra.Order.Sub.Defs).[/INST]


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

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

```lean
def subtypeₐᵢ (s : AffineSubspace 𝕜 P) [Nonempty s] : s →ᵃⁱ[𝕜] P
```

Declaration: `Mathlib.Analysis.Normed.Affine.Isometry.subtypeₐᵢ` (module Mathlib.Analysis.Normed.Affine.Isometry).

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

```lean
def subtypeₐᵢ (s : AffineSubspace 𝕜 P) [Nonempty s] : s →ᵃⁱ[𝕜] P
```

Declaration: `Mathlib.Analysis.Normed.Affine.Isometry.subtypeₐᵢ` (module Mathlib.Analysis.Normed.Affine.Isometry).[/INST]


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

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

```lean
theorem intrinsicStar_mul' [SMulCommClass R' E E] [IsScalarTower R' E E] :
    star (toConv (mul' R' E)) = toConv (mul' R' E ∘ₗ TensorProduct.comm R' E E)
```

Declaration: `Mathlib.Algebra.Star.LinearMap.intrinsicStar_mul'` (module Mathlib.Algebra.Star.LinearMap).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Star.LinearMap).[/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 intrinsicStar_mul' [SMulCommClass R' E E] [IsScalarTower R' E E] :
    star (toConv (mul' R' E)) = toConv (mul' R' E ∘ₗ TensorProduct.comm R' E E)
```

Declaration: `Mathlib.Algebra.Star.LinearMap.intrinsicStar_mul'` (module Mathlib.Algebra.Star.LinearMap).[/INST]


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

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

```lean
theorem Monotone.pow_const {f : β → M} (hf : Monotone f) : ∀ n : ℕ, Monotone fun a => f a ^ n
  | 0 => by simpa using monotone_const
  | n + 1 => by
    simp_rw [pow_succ]
    exact (Monotone.pow_const hf _).mul' hf
```

Declaration: `Mathlib.Algebra.Order.Monoid.Unbundled.Pow.Monotone` (module Mathlib.Algebra.Order.Monoid.Unbundled.Pow).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.Unbundled.Pow).[/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 Monotone.pow_const {f : β → M} (hf : Monotone f) : ∀ n : ℕ, Monotone fun a => f a ^ n
  | 0 => by simpa using monotone_const
  | n + 1 => by
    simp_rw [pow_succ]
    exact (Monotone.pow_const hf _).mul' hf
```

Declaration: `Mathlib.Algebra.Order.Monoid.Unbundled.Pow.Monotone` (module Mathlib.Algebra.Order.Monoid.Unbundled.Pow).[/INST]


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

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

```lean
theorem sublists'_cons (a : α) (l : List α) :
    sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l)
```

Declaration: `Mathlib.Data.List.Sublists.sublists'_cons` (module Mathlib.Data.List.Sublists).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.Sublists).[/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 sublists'_cons (a : α) (l : List α) :
    sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l)
```

Declaration: `Mathlib.Data.List.Sublists.sublists'_cons` (module Mathlib.Data.List.Sublists).[/INST]


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

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

```lean
theorem IsBigO.rpow_rpow_nhdsGE_zero_of_le {a b : ℝ} (h : a ≤ b) (hb : b ≠ 0) :
    (· ^ b : ℝ → ℝ) =O[𝓝[≥] 0] (· ^ a)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics.IsBigO` (module Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics).[/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 IsBigO.rpow_rpow_nhdsGE_zero_of_le {a b : ℝ} (h : a ≤ b) (hb : b ≠ 0) :
    (· ^ b : ℝ → ℝ) =O[𝓝[≥] 0] (· ^ a)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics.IsBigO` (module Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics).[/INST]


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

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

```lean
theorem Pi.mulSingle_inf [∀ i, SemilatticeInf (f i)] [∀ i, One (f i)] (i : I) (x y : f i) :
    Pi.mulSingle i (x ⊓ y) = Pi.mulSingle i x ⊓ Pi.mulSingle i y
```

Declaration: `Mathlib.Algebra.Group.Pi.Lemmas.Pi` (module Mathlib.Algebra.Group.Pi.Lemmas).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Group.Pi.Lemmas).[/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 Pi.mulSingle_inf [∀ i, SemilatticeInf (f i)] [∀ i, One (f i)] (i : I) (x y : f i) :
    Pi.mulSingle i (x ⊓ y) = Pi.mulSingle i x ⊓ Pi.mulSingle i y
```

Declaration: `Mathlib.Algebra.Group.Pi.Lemmas.Pi` (module Mathlib.Algebra.Group.Pi.Lemmas).[/INST]


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

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

```lean
theorem Antivary.sum_smul_le_sum_smul_comp_perm (hfg : Antivary f g) :
    ∑ i, f i • g i ≤ ∑ i, f i • g (σ i)
```

Declaration: `Mathlib.Algebra.Order.Rearrangement.Antivary` (module Mathlib.Algebra.Order.Rearrangement).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Rearrangement).[/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 Antivary.sum_smul_le_sum_smul_comp_perm (hfg : Antivary f g) :
    ∑ i, f i • g i ≤ ∑ i, f i • g (σ i)
```

Declaration: `Mathlib.Algebra.Order.Rearrangement.Antivary` (module Mathlib.Algebra.Order.Rearrangement).[/INST]


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

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

```lean
theorem le_hasProd_of_le_prod [ClosedIciTopology α] [L.NeBot]
    (hf : HasProd f a L) (h : ∀ s, c ≤ ∏ i ∈ s, f i) : c ≤ a
```

Declaration: `Mathlib.Topology.Algebra.InfiniteSum.Order.le_hasProd_of_le_prod` (module Mathlib.Topology.Algebra.InfiniteSum.Order).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.InfiniteSum.Order).[/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 le_hasProd_of_le_prod [ClosedIciTopology α] [L.NeBot]
    (hf : HasProd f a L) (h : ∀ s, c ≤ ∏ i ∈ s, f i) : c ≤ a
```

Declaration: `Mathlib.Topology.Algebra.InfiniteSum.Order.le_hasProd_of_le_prod` (module Mathlib.Topology.Algebra.InfiniteSum.Order).[/INST]


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

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

```lean
theorem attachFin_Iic : attachFin (Iic a) (fun _x hx ↦ (mem_Iic.mp hx).trans_lt a.2) = Iic a
```

Declaration: `Mathlib.Order.Interval.Finset.Fin.attachFin_Iic` (module Mathlib.Order.Interval.Finset.Fin).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.Fin).[/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 attachFin_Iic : attachFin (Iic a) (fun _x hx ↦ (mem_Iic.mp hx).trans_lt a.2) = Iic a
```

Declaration: `Mathlib.Order.Interval.Finset.Fin.attachFin_Iic` (module Mathlib.Order.Interval.Finset.Fin).[/INST]


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

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

```lean
theorem preimage_image (e : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : e ⁻¹' (e '' s) = s
```

Declaration: `Mathlib.Topology.Algebra.ContinuousAffineEquiv.preimage_image` (module Mathlib.Topology.Algebra.ContinuousAffineEquiv).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.ContinuousAffineEquiv).[/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 preimage_image (e : P₁ ≃ᴬ[k] P₂) (s : Set P₁) : e ⁻¹' (e '' s) = s
```

Declaration: `Mathlib.Topology.Algebra.ContinuousAffineEquiv.preimage_image` (module Mathlib.Topology.Algebra.ContinuousAffineEquiv).[/INST]


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

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

```lean
theorem coe_le_norm_of_mem {a : A} {r : ℝ≥0} (hr : r ∈ spectrum ℝ≥0 a) :
    r ≤ ‖a‖
```

Declaration: `Mathlib.Analysis.Normed.Algebra.Spectrum.coe_le_norm_of_mem` (module Mathlib.Analysis.Normed.Algebra.Spectrum).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Algebra.Spectrum).[/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 coe_le_norm_of_mem {a : A} {r : ℝ≥0} (hr : r ∈ spectrum ℝ≥0 a) :
    r ≤ ‖a‖
```

Declaration: `Mathlib.Analysis.Normed.Algebra.Spectrum.coe_le_norm_of_mem` (module Mathlib.Analysis.Normed.Algebra.Spectrum).[/INST]


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

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

```lean
theorem symm_bijective :
    Function.Bijective (LieModuleEquiv.symm : (M ≃ₗ⁅R,L⁆ N) → N ≃ₗ⁅R,L⁆ M)
```

Declaration: `Mathlib.Algebra.Lie.Basic.symm_bijective` (module Mathlib.Algebra.Lie.Basic).

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

```lean
theorem symm_bijective :
    Function.Bijective (LieModuleEquiv.symm : (M ≃ₗ⁅R,L⁆ N) → N ≃ₗ⁅R,L⁆ M)
```

Declaration: `Mathlib.Algebra.Lie.Basic.symm_bijective` (module Mathlib.Algebra.Lie.Basic).[/INST]


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

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

```lean
theorem coe_of (X : Type u) [CommRing X] [TopologicalSpace X] [IsTopologicalRing X] :
    (of X : Type u) = X
```

Declaration: `Mathlib.Topology.Category.TopCommRingCat.coe_of` (module Mathlib.Topology.Category.TopCommRingCat).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Category.TopCommRingCat).[/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 coe_of (X : Type u) [CommRing X] [TopologicalSpace X] [IsTopologicalRing X] :
    (of X : Type u) = X
```

Declaration: `Mathlib.Topology.Category.TopCommRingCat.coe_of` (module Mathlib.Topology.Category.TopCommRingCat).[/INST]


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

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

```lean
theorem l_ciSup_set (gc : GaloisConnection l u) {s : Set γ} {f : γ → α} (hf : BddAbove (f '' s))
    (hne : s.Nonempty) : l (⨆ i : s, f i) = ⨆ i : s, l (f i)
```

Declaration: `Mathlib.Order.ConditionallyCompleteLattice.Indexed.l_ciSup_set` (module Mathlib.Order.ConditionallyCompleteLattice.Indexed).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.ConditionallyCompleteLattice.Indexed).[/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 l_ciSup_set (gc : GaloisConnection l u) {s : Set γ} {f : γ → α} (hf : BddAbove (f '' s))
    (hne : s.Nonempty) : l (⨆ i : s, f i) = ⨆ i : s, l (f i)
```

Declaration: `Mathlib.Order.ConditionallyCompleteLattice.Indexed.l_ciSup_set` (module Mathlib.Order.ConditionallyCompleteLattice.Indexed).[/INST]


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

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

```lean
theorem finsetImage_castLE_Icc (h : n ≤ m) :
    (Icc a b).image (castLE h) = Icc (castLE h a) (castLE h b)
```

Declaration: `Mathlib.Order.Interval.Finset.Fin.finsetImage_castLE_Icc` (module Mathlib.Order.Interval.Finset.Fin).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.Fin).[/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 finsetImage_castLE_Icc (h : n ≤ m) :
    (Icc a b).image (castLE h) = Icc (castLE h a) (castLE h b)
```

Declaration: `Mathlib.Order.Interval.Finset.Fin.finsetImage_castLE_Icc` (module Mathlib.Order.Interval.Finset.Fin).[/INST]


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

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

```lean
theorem disjoint_iff_disjoint_support : Disjoint f g ↔ _root_.Disjoint f.support g.support
```

Declaration: `Mathlib.GroupTheory.Perm.Support.disjoint_iff_disjoint_support` (module Mathlib.GroupTheory.Perm.Support).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Perm.Support).[/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 disjoint_iff_disjoint_support : Disjoint f g ↔ _root_.Disjoint f.support g.support
```

Declaration: `Mathlib.GroupTheory.Perm.Support.disjoint_iff_disjoint_support` (module Mathlib.GroupTheory.Perm.Support).[/INST]


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

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

```lean
lemma coroot_mem_span_int (i : ι) :
    P.coroot i ∈ span ℤ (P.coroot '' b.support)
```

Declaration: `Mathlib.LinearAlgebra.RootSystem.Base.coroot_mem_span_int` (module Mathlib.LinearAlgebra.RootSystem.Base).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.RootSystem.Base).[/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 coroot_mem_span_int (i : ι) :
    P.coroot i ∈ span ℤ (P.coroot '' b.support)
```

Declaration: `Mathlib.LinearAlgebra.RootSystem.Base.coroot_mem_span_int` (module Mathlib.LinearAlgebra.RootSystem.Base).[/INST]


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

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

```lean
abbrev subDownRight {d u l r : Nat} (A : Matrix (Fin (u + d)) (Fin (l + r)) α) :
    Matrix (Fin d) (Fin r) α
```

Declaration: `Mathlib.LinearAlgebra.Matrix.Defs.subDownRight` (module Mathlib.LinearAlgebra.Matrix.Defs).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.Defs).[/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
abbrev subDownRight {d u l r : Nat} (A : Matrix (Fin (u + d)) (Fin (l + r)) α) :
    Matrix (Fin d) (Fin r) α
```

Declaration: `Mathlib.LinearAlgebra.Matrix.Defs.subDownRight` (module Mathlib.LinearAlgebra.Matrix.Defs).[/INST]


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

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

```lean
def lift (b : FreeGroupBasis ι G) : (ι → H) ≃ (G →* H)
```

Declaration: `Mathlib.GroupTheory.FreeGroup.IsFreeGroup.lift` (module Mathlib.GroupTheory.FreeGroup.IsFreeGroup).

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

```lean
def lift (b : FreeGroupBasis ι G) : (ι → H) ≃ (G →* H)
```

Declaration: `Mathlib.GroupTheory.FreeGroup.IsFreeGroup.lift` (module Mathlib.GroupTheory.FreeGroup.IsFreeGroup).[/INST]


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

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

```lean
theorem headI_mul_tail_prod_of_ne_nil [Inhabited M] (l : List M) (h : l ≠ []) :
    l.headI * l.tail.prod = l.prod
```

Declaration: `Mathlib.Algebra.BigOperators.Group.List.Basic.headI_mul_tail_prod_of_ne_nil` (module Mathlib.Algebra.BigOperators.Group.List.Basic).

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

```lean
theorem headI_mul_tail_prod_of_ne_nil [Inhabited M] (l : List M) (h : l ≠ []) :
    l.headI * l.tail.prod = l.prod
```

Declaration: `Mathlib.Algebra.BigOperators.Group.List.Basic.headI_mul_tail_prod_of_ne_nil` (module Mathlib.Algebra.BigOperators.Group.List.Basic).[/INST]


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

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

```lean
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] : ¬p ∣ Nat.card (Sylow p G)
```

Declaration: `Mathlib.GroupTheory.Sylow.not_dvd_card_sylow` (module Mathlib.GroupTheory.Sylow).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Sylow).[/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 not_dvd_card_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] : ¬p ∣ Nat.card (Sylow p G)
```

Declaration: `Mathlib.GroupTheory.Sylow.not_dvd_card_sylow` (module Mathlib.GroupTheory.Sylow).[/INST]


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

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

```lean
theorem mem_centralizer_iff {R} [Semiring R] {s : Set R} {z : R} :
    z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g
```

Declaration: `Mathlib.Algebra.Ring.Subsemiring.Basic.mem_centralizer_iff` (module Mathlib.Algebra.Ring.Subsemiring.Basic).

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

```lean
theorem mem_centralizer_iff {R} [Semiring R] {s : Set R} {z : R} :
    z ∈ centralizer s ↔ ∀ g ∈ s, g * z = z * g
```

Declaration: `Mathlib.Algebra.Ring.Subsemiring.Basic.mem_centralizer_iff` (module Mathlib.Algebra.Ring.Subsemiring.Basic).[/INST]


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

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

```lean
theorem pi_update_of_notMem [DecidableEq ι] (hi : i ∉ s) (f : ∀ j, α j) (a : α i)
    (t : ∀ j, α j → Set (β j)) : (s.pi fun j => t j (update f i a j)) = s.pi fun j => t j (f j)
```

Declaration: `Mathlib.Data.Set.Prod.pi_update_of_notMem` (module Mathlib.Data.Set.Prod).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.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
theorem pi_update_of_notMem [DecidableEq ι] (hi : i ∉ s) (f : ∀ j, α j) (a : α i)
    (t : ∀ j, α j → Set (β j)) : (s.pi fun j => t j (update f i a j)) = s.pi fun j => t j (f j)
```

Declaration: `Mathlib.Data.Set.Prod.pi_update_of_notMem` (module Mathlib.Data.Set.Prod).[/INST]


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

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

```lean
theorem lt_iff_ssubset : s < t ↔ s ⊂ t
```

Declaration: `Mathlib.Data.Set.Basic.lt_iff_ssubset` (module Mathlib.Data.Set.Basic).

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

```lean
theorem lt_iff_ssubset : s < t ↔ s ⊂ t
```

Declaration: `Mathlib.Data.Set.Basic.lt_iff_ssubset` (module Mathlib.Data.Set.Basic).[/INST]


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

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

```lean
theorem deriv_div_const (d : 𝕜') : deriv (fun x => c x / d) x = deriv c x / d
```

Declaration: `Mathlib.Analysis.Calculus.Deriv.Mul.deriv_div_const` (module Mathlib.Analysis.Calculus.Deriv.Mul).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.Deriv.Mul).[/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 deriv_div_const (d : 𝕜') : deriv (fun x => c x / d) x = deriv c x / d
```

Declaration: `Mathlib.Analysis.Calculus.Deriv.Mul.deriv_div_const` (module Mathlib.Analysis.Calculus.Deriv.Mul).[/INST]


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

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

```lean
theorem mul_eq_mul (I J : FractionalIdeal S P) : mul I J = I * J
```

Declaration: `Mathlib.RingTheory.FractionalIdeal.Basic.mul_eq_mul` (module Mathlib.RingTheory.FractionalIdeal.Basic).

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

```lean
theorem mul_eq_mul (I J : FractionalIdeal S P) : mul I J = I * J
```

Declaration: `Mathlib.RingTheory.FractionalIdeal.Basic.mul_eq_mul` (module Mathlib.RingTheory.FractionalIdeal.Basic).[/INST]


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

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

```lean
theorem IsChain.cons {x} : ∀ {l : List α}, IsChain R l → (∀ y ∈ l.head?, R x y) →
    IsChain R (x :: l)
  | [], _, _ => .singleton x
  | _ :: _, hl, H => hl.cons_cons <| H _ rfl
```

Declaration: `Mathlib.Data.List.Chain.IsChain` (module Mathlib.Data.List.Chain).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.Chain).[/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 IsChain.cons {x} : ∀ {l : List α}, IsChain R l → (∀ y ∈ l.head?, R x y) →
    IsChain R (x :: l)
  | [], _, _ => .singleton x
  | _ :: _, hl, H => hl.cons_cons <| H _ rfl
```

Declaration: `Mathlib.Data.List.Chain.IsChain` (module Mathlib.Data.List.Chain).[/INST]


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

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

```lean
lemma add_right_inj [IsRightCancelAdd α] (hz : z ≠ ⊤) : x + z = y + z ↔ x = y
```

Declaration: `Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.add_right_inj` (module Mathlib.Algebra.Order.Monoid.Unbundled.WithTop).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.Unbundled.WithTop).[/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 add_right_inj [IsRightCancelAdd α] (hz : z ≠ ⊤) : x + z = y + z ↔ x = y
```

Declaration: `Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.add_right_inj` (module Mathlib.Algebra.Order.Monoid.Unbundled.WithTop).[/INST]


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

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

```lean
instance isEquivalence : compactumToCompHaus.IsEquivalence where

end compactumToCompHaus

/-- The forgetful functors of `Compactum` and `CompHaus` are compatible via
`compactumToCompHaus`. -/
```

Declaration: `Mathlib.Topology.Category.Compactum.isEquivalence` (module Mathlib.Topology.Category.Compactum).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Category.Compactum).[/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 isEquivalence : compactumToCompHaus.IsEquivalence where

end compactumToCompHaus

/-- The forgetful functors of `Compactum` and `CompHaus` are compatible via
`compactumToCompHaus`. -/
```

Declaration: `Mathlib.Topology.Category.Compactum.isEquivalence` (module Mathlib.Topology.Category.Compactum).[/INST]


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

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

```lean
lemma lift_openEmbedding_trans_apply
    (e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) (z : Z) :
    (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) z = (e.symm.trans e') z
```

Declaration: `Mathlib.Topology.OpenPartialHomeomorph.Constructions.lift_openEmbedding_trans_apply` (module Mathlib.Topology.OpenPartialHomeomorph.Constructions).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.OpenPartialHomeomorph.Constructions).[/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 lift_openEmbedding_trans_apply
    (e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) (z : Z) :
    (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) z = (e.symm.trans e') z
```

Declaration: `Mathlib.Topology.OpenPartialHomeomorph.Constructions.lift_openEmbedding_trans_apply` (module Mathlib.Topology.OpenPartialHomeomorph.Constructions).[/INST]