[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.

Open namespaces: IsLocalization, Pointwise, nonZeroDivisors, Set, Submodule

## Surrounding context in Mathlib/RingTheory/FractionalIdeal/Basic.lean
```lean
-- Mathlib.RingTheory.FractionalIdeal.Basic.IsFractional (L71-L81)
def IsFractional (I : Submodule R P)

-- Mathlib.RingTheory.FractionalIdeal.Basic.FractionalIdeal (L82-L97)
def FractionalIdeal

-- Mathlib.RingTheory.FractionalIdeal.Basic.coeToSubmodule (L99-L108)
def coeToSubmodule (I : FractionalIdeal S P) : Submodule R P
```

```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).

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

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

Open namespaces: IsLocalization, Pointwise, nonZeroDivisors, Set, Submodule

## Surrounding context in Mathlib/RingTheory/FractionalIdeal/Basic.lean
```lean
-- Mathlib.RingTheory.FractionalIdeal.Basic.IsFractional (L71-L81)
def IsFractional (I : Submodule R P)

-- Mathlib.RingTheory.FractionalIdeal.Basic.FractionalIdeal (L82-L97)
def FractionalIdeal

-- Mathlib.RingTheory.FractionalIdeal.Basic.coeToSubmodule (L99-L108)
def coeToSubmodule (I : FractionalIdeal S P) : Submodule R P
```

```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).


[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.

Open namespaces: Nat

## Surrounding context in Mathlib/Data/List/Chain.lean
```lean
-- Mathlib.Data.List.Chain.isChain_nil (L38-L38)
theorem isChain_nil : IsChain R []

-- Mathlib.Data.List.Chain.isChain_singleton (L39-L40)
theorem isChain_singleton (a : α) : IsChain R [a]

-- Mathlib.Data.List.Chain.isChain_cons_iff (L41-L48)
theorem isChain_cons_iff (R : α → α → Prop) (a : α) (l : List α) :
    IsChain R (a :: l) ↔ l = [] ∨
      ∃ (b : α) (l' : List α), R a b ∧ IsChain R (b :: l') ∧ l = b :: l'
```

```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).

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

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

Open namespaces: Nat

## Surrounding context in Mathlib/Data/List/Chain.lean
```lean
-- Mathlib.Data.List.Chain.isChain_nil (L38-L38)
theorem isChain_nil : IsChain R []

-- Mathlib.Data.List.Chain.isChain_singleton (L39-L40)
theorem isChain_singleton (a : α) : IsChain R [a]

-- Mathlib.Data.List.Chain.isChain_cons_iff (L41-L48)
theorem isChain_cons_iff (R : α → α → Prop) (a : α) (l : List α) :
    IsChain R (a :: l) ↔ l = [] ∨
      ∃ (b : α) (l' : List α), R a b ∧ IsChain R (b :: l') ∧ l = b :: l'
```

```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).

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


[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.

Open namespaces: Function

## Surrounding context in Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
```lean
-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.one (L32-L34)
instance one : One (WithTop α)

-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.coe_one (L36-L38)
theorem coe_one : ((1 : α) : WithTop α) = 1

-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.coe_eq_one (L40-L41)
lemma coe_eq_one : (a : WithTop α) = 1 ↔ a = 1
```

```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).

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

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

Open namespaces: Function

## Surrounding context in Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
```lean
-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.one (L32-L34)
instance one : One (WithTop α)

-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.coe_one (L36-L38)
theorem coe_one : ((1 : α) : WithTop α) = 1

-- Mathlib.Algebra.Order.Monoid.Unbundled.WithTop.coe_eq_one (L40-L41)
lemma coe_eq_one : (a : WithTop α) = 1 ↔ a = 1
```

```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).


[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.

Open namespaces: CategoryTheory, Filter, Ultrafilter, TopologicalSpace, CategoryTheory.Limits, FiniteInter, scoped, Topology

## Surrounding context in Mathlib/Topology/Category/Compactum.lean
```lean
-- Mathlib.Topology.Category.Compactum.Compactum (L85-L90)
def Compactum

-- Mathlib.Topology.Category.Compactum.forget (L91-L93)
def forget : Compactum ⥤ Type*

-- Mathlib.Topology.Category.Compactum.free (L101-L104)
def free : Type* ⥤ Compactum
```

```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).

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

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

Open namespaces: CategoryTheory, Filter, Ultrafilter, TopologicalSpace, CategoryTheory.Limits, FiniteInter, scoped, Topology

## Surrounding context in Mathlib/Topology/Category/Compactum.lean
```lean
-- Mathlib.Topology.Category.Compactum.Compactum (L85-L90)
def Compactum

-- Mathlib.Topology.Category.Compactum.forget (L91-L93)
def forget : Compactum ⥤ Type*

-- Mathlib.Topology.Category.Compactum.free (L101-L104)
def free : Type* ⥤ Compactum
```

```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).


[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.

Open namespaces: Function, Set, Filter, Topology, TopologicalSpace

## Surrounding context in Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean
```lean
-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const (L52-L58)
def const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y where
  toPartialEquiv

-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const_apply (L60-L61)
lemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b

-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const_source (L63-L64)
lemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a}
```

```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).

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

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

Open namespaces: Function, Set, Filter, Topology, TopologicalSpace

## Surrounding context in Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean
```lean
-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const (L52-L58)
def const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y where
  toPartialEquiv

-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const_apply (L60-L61)
lemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b

-- Mathlib.Topology.OpenPartialHomeomorph.Constructions.const_source (L63-L64)
lemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a}
```

```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).


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

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

Open namespaces: OrderDual

## Surrounding context in Mathlib/Algebra/Order/Group/Synonym.lean
```lean
-- Mathlib.Algebra.Order.Group.Synonym.toDual_one (L128-L129)
theorem toDual_one [One α] : toDual (1 : α) = 1

-- Mathlib.Algebra.Order.Group.Synonym.ofDual_one (L131-L132)
theorem ofDual_one [One α] : (ofDual 1 : α) = 1

-- Mathlib.Algebra.Order.Group.Synonym.toDual_mul (L137-L138)
theorem toDual_mul [Mul α] (a b : α) : toDual (a * b) = toDual a * toDual b
```

```lean
theorem toLex_one [One α] : toLex (1 : α) = 1
```

Declaration: `Mathlib.Algebra.Order.Group.Synonym.toLex_one` (module Mathlib.Algebra.Order.Group.Synonym).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Group.Synonym).

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

Open namespaces: OrderDual

## Surrounding context in Mathlib/Algebra/Order/Group/Synonym.lean
```lean
-- Mathlib.Algebra.Order.Group.Synonym.toDual_one (L128-L129)
theorem toDual_one [One α] : toDual (1 : α) = 1

-- Mathlib.Algebra.Order.Group.Synonym.ofDual_one (L131-L132)
theorem ofDual_one [One α] : (ofDual 1 : α) = 1

-- Mathlib.Algebra.Order.Group.Synonym.toDual_mul (L137-L138)
theorem toDual_mul [Mul α] (a b : α) : toDual (a * b) = toDual a * toDual b
```

```lean
theorem toLex_one [One α] : toLex (1 : α) = 1
```

Declaration: `Mathlib.Algebra.Order.Group.Synonym.toLex_one` (module Mathlib.Algebra.Order.Group.Synonym).


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

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

```lean
def mkDerivationEquiv : A ≃ₗ[R] Derivation R R[X] A
```

Declaration: `Mathlib.Algebra.Polynomial.Derivation.mkDerivationEquiv` (module Mathlib.Algebra.Polynomial.Derivation).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Polynomial.Derivation).[/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 mkDerivationEquiv : A ≃ₗ[R] Derivation R R[X] A
```

Declaration: `Mathlib.Algebra.Polynomial.Derivation.mkDerivationEquiv` (module Mathlib.Algebra.Polynomial.Derivation).[/INST]


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

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

Open namespaces: MvFunctor, TypeVec, MvPFunctor, (LiftP, LiftR)

## Surrounding context in Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
```lean
-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.corecF (L61-L63)
def corecF {α : TypeVec n} {β : Type u} (g : β → F (α.append1 β)) : β → q.P.M α

-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.corecF_eq (L64-L68)
theorem corecF_eq {α : TypeVec n} {β : Type u} (g : β → F (α.append1 β)) (x : β) :
    M.dest q.P (corecF g x) = appendFun id (corecF g) <$$> repr (g x)

-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.IsPrecongr (L69-L75)
def IsPrecongr {α : TypeVec n} (r : q.P.M α → q.P.M α → Prop) : Prop
```

```lean
theorem Cofix.dest_corec' {α : TypeVec.{u} n} {β : Type u}
    (g : β → F (α.append1 (Cofix F α ⊕ β))) (x : β) :
    Cofix.dest (Cofix.corec' g x) =
      appendFun id (Sum.elim _root_.id (Cofix.corec' g)) <$$> g x
```

Declaration: `Mathlib.Data.QPF.Multivariate.Constructions.Cofix.Cofix` (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).

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

Open namespaces: MvFunctor, TypeVec, MvPFunctor, (LiftP, LiftR)

## Surrounding context in Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
```lean
-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.corecF (L61-L63)
def corecF {α : TypeVec n} {β : Type u} (g : β → F (α.append1 β)) : β → q.P.M α

-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.corecF_eq (L64-L68)
theorem corecF_eq {α : TypeVec n} {β : Type u} (g : β → F (α.append1 β)) (x : β) :
    M.dest q.P (corecF g x) = appendFun id (corecF g) <$$> repr (g x)

-- Mathlib.Data.QPF.Multivariate.Constructions.Cofix.IsPrecongr (L69-L75)
def IsPrecongr {α : TypeVec n} (r : q.P.M α → q.P.M α → Prop) : Prop
```

```lean
theorem Cofix.dest_corec' {α : TypeVec.{u} n} {β : Type u}
    (g : β → F (α.append1 (Cofix F α ⊕ β))) (x : β) :
    Cofix.dest (Cofix.corec' g x) =
      appendFun id (Sum.elim _root_.id (Cofix.corec' g)) <$$> g x
```

Declaration: `Mathlib.Data.QPF.Multivariate.Constructions.Cofix.Cofix` (module Mathlib.Data.QPF.Multivariate.Constructions.Cofix).


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

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

Open namespaces: UpperHalfPlane, scoped, ModularForm, Topology, Filter, Manifold, SlashInvariantForm

## Surrounding context in Mathlib/NumberTheory/ModularForms/NormTrace.lean
```lean
-- Mathlib.NumberTheory.ModularForms.NormTrace.quotientFunc (L35-L41)
def quotientFunc (q : 𝒬) (τ : ℍ) : ℂ

-- Mathlib.NumberTheory.ModularForms.NormTrace.quotientFunc_smul (L45-L52)
lemma quotientFunc_smul {h} (hh : h ∈ ℋ) (q : 𝒬) :
    quotientFunc f q ∣[k] h = quotientFunc f ((⟨h, hh⟩ : ℋ)⁻¹ • q)
```

Relevant lemmas available in scope: `Quotient.forall.mpr`

```lean
lemma ModularForm.eq_const_of_weight_zero [𝒢.IsArithmetic] (f : ModularForm 𝒢 0) :
    ∃ c, (f : ℍ → ℂ) = Function.const ℍ c
```

Declaration: `Mathlib.NumberTheory.ModularForms.NormTrace.ModularForm` (module Mathlib.NumberTheory.ModularForms.NormTrace).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.ModularForms.NormTrace).

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

Open namespaces: UpperHalfPlane, scoped, ModularForm, Topology, Filter, Manifold, SlashInvariantForm

## Surrounding context in Mathlib/NumberTheory/ModularForms/NormTrace.lean
```lean
-- Mathlib.NumberTheory.ModularForms.NormTrace.quotientFunc (L35-L41)
def quotientFunc (q : 𝒬) (τ : ℍ) : ℂ

-- Mathlib.NumberTheory.ModularForms.NormTrace.quotientFunc_smul (L45-L52)
lemma quotientFunc_smul {h} (hh : h ∈ ℋ) (q : 𝒬) :
    quotientFunc f q ∣[k] h = quotientFunc f ((⟨h, hh⟩ : ℋ)⁻¹ • q)
```

Relevant lemmas available in scope: `Quotient.forall.mpr`

```lean
lemma ModularForm.eq_const_of_weight_zero [𝒢.IsArithmetic] (f : ModularForm 𝒢 0) :
    ∃ c, (f : ℍ → ℂ) = Function.const ℍ c
```

Declaration: `Mathlib.NumberTheory.ModularForms.NormTrace.ModularForm` (module Mathlib.NumberTheory.ModularForms.NormTrace).


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

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

Open namespaces: Set, Pointwise

## Surrounding context in Mathlib/Analysis/LocallyConvex/Separation.lean
```lean
-- Mathlib.Analysis.LocallyConvex.Separation.separate_convex_open_set (L54-L87)
theorem separate_convex_open_set [TopologicalSpace E] [AddCommGroup E] [IsTopologicalAddGroup E]
    [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
    (hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s) :
    ∃ f : StrongDual ℝ E, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1

-- Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open (L94-L123)
theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
    (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b

-- Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_point (L124-L129)
theorem geometric_hahn_banach_open_point (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (disj : x ∉ s) :
    ∃ f : StrongDual ℝ E, ∀ a ∈ s, f a < f x
```

```lean
theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
    (ht₃ : IsOpen t) (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b
```

Declaration: `Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_open` (module Mathlib.Analysis.LocallyConvex.Separation).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.LocallyConvex.Separation).

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

Open namespaces: Set, Pointwise

## Surrounding context in Mathlib/Analysis/LocallyConvex/Separation.lean
```lean
-- Mathlib.Analysis.LocallyConvex.Separation.separate_convex_open_set (L54-L87)
theorem separate_convex_open_set [TopologicalSpace E] [AddCommGroup E] [IsTopologicalAddGroup E]
    [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
    (hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s) :
    ∃ f : StrongDual ℝ E, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1

-- Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open (L94-L123)
theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
    (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b

-- Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_point (L124-L129)
theorem geometric_hahn_banach_open_point (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (disj : x ∉ s) :
    ∃ f : StrongDual ℝ E, ∀ a ∈ s, f a < f x
```

```lean
theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
    (ht₃ : IsOpen t) (disj : Disjoint s t) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b
```

Declaration: `Mathlib.Analysis.LocallyConvex.Separation.geometric_hahn_banach_open_open` (module Mathlib.Analysis.LocallyConvex.Separation).


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

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

```lean
theorem mem_ker {x : L} : x ∈ ker f ↔ f x = 0
```

Declaration: `Mathlib.Algebra.Lie.Ideal.mem_ker` (module Mathlib.Algebra.Lie.Ideal).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Lie.Ideal).[/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_ker {x : L} : x ∈ ker f ↔ f x = 0
```

Declaration: `Mathlib.Algebra.Lie.Ideal.mem_ker` (module Mathlib.Algebra.Lie.Ideal).[/INST]


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

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

Open namespaces: Set, Filter, TopologicalSpace, Function, scoped, Pointwise, Topology, OrderDual

## Surrounding context in Mathlib/Topology/Algebra/Order/Field.lean
```lean
-- Mathlib.Topology.Algebra.Order.Field.Filter (L37-L47)
theorem Filter.Tendsto.atTop_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop

-- Mathlib.Topology.Algebra.Order.Field.Filter (L48-L51)
theorem Filter.Tendsto.pos_mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop

-- Mathlib.Topology.Algebra.Order.Field.inv_atTop₀ (L53-L56)
lemma inv_atTop₀ : (atTop : Filter 𝕜)⁻¹ = 𝓝[>] 0
```

```lean
theorem comap_mulLeft_nhdsGT_zero {x : 𝕜} (hx : 0 < x) : comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0
```

Declaration: `Mathlib.Topology.Algebra.Order.Field.comap_mulLeft_nhdsGT_zero` (module Mathlib.Topology.Algebra.Order.Field).

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

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

Open namespaces: Set, Filter, TopologicalSpace, Function, scoped, Pointwise, Topology, OrderDual

## Surrounding context in Mathlib/Topology/Algebra/Order/Field.lean
```lean
-- Mathlib.Topology.Algebra.Order.Field.Filter (L37-L47)
theorem Filter.Tendsto.atTop_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop

-- Mathlib.Topology.Algebra.Order.Field.Filter (L48-L51)
theorem Filter.Tendsto.pos_mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop

-- Mathlib.Topology.Algebra.Order.Field.inv_atTop₀ (L53-L56)
lemma inv_atTop₀ : (atTop : Filter 𝕜)⁻¹ = 𝓝[>] 0
```

```lean
theorem comap_mulLeft_nhdsGT_zero {x : 𝕜} (hx : 0 < x) : comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0
```

Declaration: `Mathlib.Topology.Algebra.Order.Field.comap_mulLeft_nhdsGT_zero` (module Mathlib.Topology.Algebra.Order.Field).


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

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

Open namespaces: Function

## Surrounding context in Mathlib/Data/PFun.lean
```lean
-- Mathlib.Data.PFun.PFun (L61-L70)
def PFun (α β : Type*)

-- Mathlib.Data.PFun.inhabited (L71-L74)
instance inhabited : Inhabited (α →. β)

-- Mathlib.Data.PFun.Dom (L75-L77)
def Dom (f : α →. β) : Set α
```

```lean
theorem fixInduction'_fwd {C : α → Sort*} {f : α →. β ⊕ α} {b : β} {a a' : α} (h : b ∈ f.fix a)
    (h' : b ∈ f.fix a') (fa : Sum.inr a' ∈ f a)
    (hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
    (hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
    @fixInduction' _ _ C _ _ _ h hbase hind = hind a a' h' fa (fixInduction' h' hbase hind)
```

Declaration: `Mathlib.Data.PFun.fixInduction'_fwd` (module Mathlib.Data.PFun).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.PFun).

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

Open namespaces: Function

## Surrounding context in Mathlib/Data/PFun.lean
```lean
-- Mathlib.Data.PFun.PFun (L61-L70)
def PFun (α β : Type*)

-- Mathlib.Data.PFun.inhabited (L71-L74)
instance inhabited : Inhabited (α →. β)

-- Mathlib.Data.PFun.Dom (L75-L77)
def Dom (f : α →. β) : Set α
```

```lean
theorem fixInduction'_fwd {C : α → Sort*} {f : α →. β ⊕ α} {b : β} {a a' : α} (h : b ∈ f.fix a)
    (h' : b ∈ f.fix a') (fa : Sum.inr a' ∈ f a)
    (hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
    (hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
    @fixInduction' _ _ C _ _ _ h hbase hind = hind a a' h' fa (fixInduction' h' hbase hind)
```

Declaration: `Mathlib.Data.PFun.fixInduction'_fwd` (module Mathlib.Data.PFun).


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

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

Open namespaces: scoped, NNReal, Nat, ContDiff, Set, Fin, Filter, Function

## Surrounding context in Mathlib/Analysis/Calculus/ContDiff/Operations.lean
```lean
-- Mathlib.Analysis.Calculus.ContDiff.Operations.hasFTaylorSeriesUpToOn_pi (L59-L75)
theorem hasFTaylorSeriesUpToOn_pi {n : WithTop ℕ∞} :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s

-- Mathlib.Analysis.Calculus.ContDiff.Operations.hasFTaylorSeriesUpToOn_pi' (L77-L83)
theorem hasFTaylorSeriesUpToOn_pi' {n : WithTop ℕ∞} :
    HasFTaylorSeriesUpToOn n Φ P' s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (fun x => Φ x i)
        (fun x m => (@ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ i).compContinuousMultilinearMap
          (P' x m)) s

-- Mathlib.Analysis.Calculus.ContDiff.Operations.contDiffWithinAt_pi (L84-L103)
theorem contDiffWithinAt_pi :
    ContDiffWithinAt 𝕜 n Φ s x ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => Φ x i) s x
```

```lean
theorem ContDiffWithinAt.div {f g : E → 𝕜} {n} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
```

Declaration: `Mathlib.Analysis.Calculus.ContDiff.Operations.ContDiffWithinAt` (module Mathlib.Analysis.Calculus.ContDiff.Operations).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.ContDiff.Operations).

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

Open namespaces: scoped, NNReal, Nat, ContDiff, Set, Fin, Filter, Function

## Surrounding context in Mathlib/Analysis/Calculus/ContDiff/Operations.lean
```lean
-- Mathlib.Analysis.Calculus.ContDiff.Operations.hasFTaylorSeriesUpToOn_pi (L59-L75)
theorem hasFTaylorSeriesUpToOn_pi {n : WithTop ℕ∞} :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s

-- Mathlib.Analysis.Calculus.ContDiff.Operations.hasFTaylorSeriesUpToOn_pi' (L77-L83)
theorem hasFTaylorSeriesUpToOn_pi' {n : WithTop ℕ∞} :
    HasFTaylorSeriesUpToOn n Φ P' s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (fun x => Φ x i)
        (fun x m => (@ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ i).compContinuousMultilinearMap
          (P' x m)) s

-- Mathlib.Analysis.Calculus.ContDiff.Operations.contDiffWithinAt_pi (L84-L103)
theorem contDiffWithinAt_pi :
    ContDiffWithinAt 𝕜 n Φ s x ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => Φ x i) s x
```

```lean
theorem ContDiffWithinAt.div {f g : E → 𝕜} {n} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
```

Declaration: `Mathlib.Analysis.Calculus.ContDiff.Operations.ContDiffWithinAt` (module Mathlib.Analysis.Calculus.ContDiff.Operations).


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

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

Open namespaces: Module, Filter, scoped, Topology, Nat, in

## Surrounding context in Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean
```lean
-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.lattice (L79-L80)
def lattice : Submodule ℤ ℂ

-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.mem_lattice (L81-L84)
lemma mem_lattice {L : PeriodPair} {x : ℂ} :
    x ∈ L.lattice ↔ ∃ m n : ℤ, m * L.ω₁ + n * L.ω₂ = x

-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.ω₁_mem_lattice (L85-L85)
lemma ω₁_mem_lattice : L.ω₁ ∈ L.lattice
```

Relevant lemmas available in scope: `x`, `hR`

```lean
lemma hasSumLocallyUniformly_aux (f : L.lattice → ℂ → ℂ)
    (u : ℝ → L.lattice → ℝ) (hu : ∀ r > 0, Summable (u r))
    (hf : ∀ r > 0, ∀ᶠ R in atTop, ∀ x, ‖x‖ < r → ∀ l : L.lattice, ‖l.1‖ = R → ‖f l x‖ ≤ u r l) :
    HasSumLocallyUniformly f (∑' j, f j ·)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.hasSumLocallyUniformly_aux` (module Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass).

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

Open namespaces: Module, Filter, scoped, Topology, Nat, in

## Surrounding context in Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean
```lean
-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.lattice (L79-L80)
def lattice : Submodule ℤ ℂ

-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.mem_lattice (L81-L84)
lemma mem_lattice {L : PeriodPair} {x : ℂ} :
    x ∈ L.lattice ↔ ∃ m n : ℤ, m * L.ω₁ + n * L.ω₂ = x

-- Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.ω₁_mem_lattice (L85-L85)
lemma ω₁_mem_lattice : L.ω₁ ∈ L.lattice
```

Relevant lemmas available in scope: `x`, `hR`

```lean
lemma hasSumLocallyUniformly_aux (f : L.lattice → ℂ → ℂ)
    (u : ℝ → L.lattice → ℝ) (hu : ∀ r > 0, Summable (u r))
    (hf : ∀ r > 0, ∀ᶠ R in atTop, ∀ x, ‖x‖ < r → ∀ l : L.lattice, ‖l.1‖ = R → ‖f l x‖ ≤ u r l) :
    HasSumLocallyUniformly f (∑' j, f j ·)
```

Declaration: `Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.hasSumLocallyUniformly_aux` (module Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass).


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

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

Open namespaces: Polynomial, Algebra, Module, Set

## Surrounding context in Mathlib/NumberTheory/Cyclotomic/Basic.lean
```lean
-- Mathlib.NumberTheory.Cyclotomic.Basic.iff_adjoin_eq_top (L94-L101)
theorem iff_adjoin_eq_top :
    IsCyclotomicExtension S A B ↔
      (∀ n : ℕ, n ∈ S → n ≠ 0 → ∃ r : B, IsPrimitiveRoot r n) ∧
        adjoin A {b : B | ∃ n : ℕ, n ∈ S ∧ n ≠ 0 ∧ b ^ n = 1} = ⊤

-- Mathlib.NumberTheory.Cyclotomic.Basic.iff_singleton (L102-L106)
theorem iff_singleton :
    IsCyclotomicExtension {n} A B ↔
      (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ n = 1}

-- Mathlib.NumberTheory.Cyclotomic.Basic.instSubsingleton (L107-L109)
instance instSubsingleton [h : IsCyclotomicExtension ∅ A B] : Subsingleton (Subalgebra A B)
```

Relevant lemmas available in scope: `_root_.isCyclotomicExtension_singleton_iff_eq_adjoin`

```lean
theorem IntermediateField.isCyclotomicExtension_lcm_sup [NeZero n₁] [NeZero n₂] :
    IsCyclotomicExtension {n₁.lcm n₂} K (F₁ ⊔ F₂ : IntermediateField K L)
```

Declaration: `Mathlib.NumberTheory.Cyclotomic.Basic.IntermediateField` (module Mathlib.NumberTheory.Cyclotomic.Basic).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Cyclotomic.Basic).

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

Open namespaces: Polynomial, Algebra, Module, Set

## Surrounding context in Mathlib/NumberTheory/Cyclotomic/Basic.lean
```lean
-- Mathlib.NumberTheory.Cyclotomic.Basic.iff_adjoin_eq_top (L94-L101)
theorem iff_adjoin_eq_top :
    IsCyclotomicExtension S A B ↔
      (∀ n : ℕ, n ∈ S → n ≠ 0 → ∃ r : B, IsPrimitiveRoot r n) ∧
        adjoin A {b : B | ∃ n : ℕ, n ∈ S ∧ n ≠ 0 ∧ b ^ n = 1} = ⊤

-- Mathlib.NumberTheory.Cyclotomic.Basic.iff_singleton (L102-L106)
theorem iff_singleton :
    IsCyclotomicExtension {n} A B ↔
      (∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ n = 1}

-- Mathlib.NumberTheory.Cyclotomic.Basic.instSubsingleton (L107-L109)
instance instSubsingleton [h : IsCyclotomicExtension ∅ A B] : Subsingleton (Subalgebra A B)
```

Relevant lemmas available in scope: `_root_.isCyclotomicExtension_singleton_iff_eq_adjoin`

```lean
theorem IntermediateField.isCyclotomicExtension_lcm_sup [NeZero n₁] [NeZero n₂] :
    IsCyclotomicExtension {n₁.lcm n₂} K (F₁ ⊔ F₂ : IntermediateField K L)
```

Declaration: `Mathlib.NumberTheory.Cyclotomic.Basic.IntermediateField` (module Mathlib.NumberTheory.Cyclotomic.Basic).


[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.SkewMonoidAlgebra.Basic).

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

```lean
def mapDomainRingHom [MulSemiringAction α β] [MulSemiringAction α₂ β]
    [MonoidHomClass F α α₂] {f : F} (hf : ∀ (a : α) (x : β), a • x = (f a) • x) :
    SkewMonoidAlgebra β α →+* SkewMonoidAlgebra β α₂ where
  __
:= (mapDomain f : SkewMonoidAlgebra β α →+ SkewMonoidAlgebra β α₂) .someMissingNamespaceXYZ
map_one' := mapDomain_one f
  map_mul' x y := mapDomain_mul x y hf

end MapDomain

section of

variable (k G)

variable [Monoid G] [MulSemiringAction G k]

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The embedding of a monoid into its skew monoid algebra. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=342, messages=[Message(data='failed to synthesize instance of type class\n  Monoid α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=342, i_end=425, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=0), pos_end=Position(line=6, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=425, i_end=437, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=437, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=453, i_end=498, messages=[Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=15, column=21), pos_end=Position(line=15, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=498, i_end=642, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=62), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def mapDomainRingHom [MulSemiringAction α β] [MulSemiringAction α₂ β]
    [MonoidHomClass F α α₂] {f : F} (hf : ∀ (a : α) (x : β), a • x = (f a) • x) :
    SkewMonoidAlgebra β α →+* SkewMonoidAlgebra β α₂ where
  __
:= (mapDomain f : SkewMonoidAlgebra β α →+ SkewMonoidAlgebra β α₂) .someMissingNamespaceXYZ
map_one' := mapDomain_one f
  map_mul' x y := mapDomain_mul x y hf

end MapDomain

section of

variable (k G)

variable [Monoid G] [MulSemiringAction G k]

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The embedding of a monoid into its skew monoid algebra. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=342, messages=[Message(data='failed to synthesize instance of type class\n  Monoid α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=342, i_end=425, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=6, column=0), pos_end=Position(line=6, column=8), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=9, column=0), pos_end=Position(line=9, column=13), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=425, i_end=437, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=437, i_end=453, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=453, i_end=498, messages=[Message(data='failed to synthesize instance of type class\n  Semiring k\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=15, column=21), pos_end=Position(line=15, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=498, i_end=642, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=18, column=62), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
def sumLexCongr (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ₗ β₁ ≃o α₂ ⊕ₗ β₂ where
  toEquiv
  frobnicateTacticThatDoesNotExist := ofLex.trans ((Equiv.sumCongr ea eb).trans toLex)
map_rel_iff' := by simp_rw [Lex.forall]; rintro (a | a) (b | b) <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=212, messages=[Message(data='failed to synthesize instance of type class\n  LE α₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE β₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Lex (α₁ ⊕ β₁))\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=50), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=212, i_end=284, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def sumLexCongr (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ₗ β₁ ≃o α₂ ⊕ₗ β₂ where
  toEquiv
  frobnicateTacticThatDoesNotExist := ofLex.trans ((Equiv.sumCongr ea eb).trans toLex)
map_rel_iff' := by simp_rw [Lex.forall]; rintro (a | a) (b | b) <;> simp
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=212, messages=[Message(data='failed to synthesize instance of type class\n  LE α₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=22), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE β₁\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=38), pos_end=Position(line=1, column=46), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  LE (Lex (α₁ ⊕ β₁))\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=50), pos_end=Position(line=1, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=212, i_end=284, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=12), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b
:= by .someMissingNamespaceXYZ
rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ioo\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (pred a)\n\nHint: The identifier `Ioo` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=53), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ico\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Ico` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=70), pos_end=Position(line=1, column=77), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα✝ : Type u_1\na : α✝\nx✝¹ : Sort u_2\nIoo : x✝¹\nx✝ : Sort u_3\nIco : x✝\nha : ¬IsMin a\n⊢ sorry = sorry', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=173, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b
:= by .someMissingNamespaceXYZ
rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ioo\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (pred a)\n\nHint: The identifier `Ioo` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=53), pos_end=Position(line=1, column=67), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  Ico\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `Ico` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=70), pos_end=Position(line=1, column=77), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nα✝ : Type u_1\na : α✝\nx✝¹ : Sort u_2\nIoo : x✝¹\nx✝ : Sort u_3\nIco : x✝\nha : ¬IsMin a\n⊢ sorry = sorry', pos=Position(line=2, column=6), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=173, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma norm_coe_unit (u : 𝒪[K]ˣ) : ‖((u : 𝒪[K]) : K)‖ = 1
:= by thisIsWrongLemmaNameXYZ
simpa [← NNReal.coe_inj] using
    (Valuation.integer.integers (NormedField.valuation (K := K))).valuation_unit u

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Norm K\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=34), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Monoid 𝒪[K]\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=271, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma norm_coe_unit (u : 𝒪[K]ˣ) : ‖((u : 𝒪[K]) : K)‖ = 1
:= by thisIsWrongLemmaNameXYZ
simpa [← NNReal.coe_inj] using
    (Valuation.integer.integers (NormedField.valuation (K := K))).valuation_unit u

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=75, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Norm K\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=34), pos_end=Position(line=1, column=52), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Monoid 𝒪[K]\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=30), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=75, i_end=271, messages=[Message(data='unexpected end of input', pos=Position(line=6, column=56), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma ofPreGNS_toPreGNS (a : A) : f.ofPreGNS (f.toPreGNS a) = a
:= rfl .someMissingNamespaceXYZ
variable [StarOrderedRing A]

/--
The (semi-)inner product space whose elements are the elements of `A`, but which has an
inner product-induced norm that is different from the norm on `A` and which is induced by `f`.
-/
noncomputable abbrev preGNSpreInnerProdSpace : PreInnerProductSpace.Core ℂ f.PreGNS where
  inner a b := f (star (f.ofPreGNS a) * f.ofPreGNS b)
  conj_inner_symm := by simp [← Complex.star_def, ← map_star f]
  re_inner_nonneg _ := RCLike.nonneg_iff.mp (f.map_nonneg (star_mul_self_nonneg _)) |>.1
  add_left _ _ _ := by rw [map_add, star_add, add_mul, map_add]
  smul_left := by simp [smul_mul_assoc]

noncomputable instance : SeminormedAddCommGroup f.PreGNS :=
  InnerProductSpace.Core.toSeminormedAddCommGroup (c := f.preGNSpreInnerProdSpace)
noncomputable instance : InnerProductSpace ℂ f.PreGNS :=
  InnerProductSpace.ofCore f.preGNSpreInnerProdSpace
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=96, messages=[Message(data='Unknown identifier `f.ofPreGNS`', pos=Position(line=1, column=34), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  rfl\nbut this term has type\n  ?m.5 = ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  .someMissingNamespaceXYZ', pos=Position(line=2, column=3), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=96, i_end=126, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalSemiring A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=10), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=724, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=9, column=75), pos_end=Position(line=9, column=83), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=724, i_end=867, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=16, column=48), pos_end=Position(line=16, column=56), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=17, column=56), pos_end=Position(line=17, column=81), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=867, i_end=978, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=18, column=45), pos_end=Position(line=18, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=19, column=27), pos_end=Position(line=19, column=52), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma ofPreGNS_toPreGNS (a : A) : f.ofPreGNS (f.toPreGNS a) = a
:= rfl .someMissingNamespaceXYZ
variable [StarOrderedRing A]

/--
The (semi-)inner product space whose elements are the elements of `A`, but which has an
inner product-induced norm that is different from the norm on `A` and which is induced by `f`.
-/
noncomputable abbrev preGNSpreInnerProdSpace : PreInnerProductSpace.Core ℂ f.PreGNS where
  inner a b := f (star (f.ofPreGNS a) * f.ofPreGNS b)
  conj_inner_symm := by simp [← Complex.star_def, ← map_star f]
  re_inner_nonneg _ := RCLike.nonneg_iff.mp (f.map_nonneg (star_mul_self_nonneg _)) |>.1
  add_left _ _ _ := by rw [map_add, star_add, add_mul, map_add]
  smul_left := by simp [smul_mul_assoc]

noncomputable instance : SeminormedAddCommGroup f.PreGNS :=
  InnerProductSpace.Core.toSeminormedAddCommGroup (c := f.preGNSpreInnerProdSpace)
noncomputable instance : InnerProductSpace ℂ f.PreGNS :=
  InnerProductSpace.ofCore f.preGNSpreInnerProdSpace
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=96, messages=[Message(data='Unknown identifier `f.ofPreGNS`', pos=Position(line=1, column=34), pos_end=Position(line=1, column=44), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Function expected at\n  rfl\nbut this term has type\n  ?m.5 = ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  .someMissingNamespaceXYZ', pos=Position(line=2, column=3), pos_end=Position(line=2, column=31), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=96, i_end=126, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalSemiring A\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=10), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=126, i_end=724, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=9, column=75), pos_end=Position(line=9, column=83), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=724, i_end=867, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=16, column=48), pos_end=Position(line=16, column=56), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=17, column=56), pos_end=Position(line=17, column=81), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=867, i_end=978, messages=[Message(data='Unknown identifier `f.PreGNS`', pos=Position(line=18, column=45), pos_end=Position(line=18, column=53), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown identifier `f.preGNSpreInnerProdSpace`', pos=Position(line=19, column=27), pos_end=Position(line=19, column=52), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
    Gᶜ.IsRegularOfDegree (n - k - 1)
:= by thisIsWrongLemmaNameXYZ
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
  exact h.regular.compl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  G\nis not known; cannot resolve field `IsSRGWith`\n\nHint: Consider replacing the field projection `.IsSRGWith` with a call to the function `SimpleGraph.IsSRGWith`.', pos=Position(line=1, column=40), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  Gᶜ\nis not known; cannot resolve field `IsRegularOfDegree`\n\nHint: Consider replacing the field projection `.IsRegularOfDegree` with a call to the function `SimpleGraph.IsRegularOfDegree`.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝ : Type u_1\nG : x✝\nh : sorry\n⊢ sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=112, i_end=214, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
    Gᶜ.IsRegularOfDegree (n - k - 1)
:= by thisIsWrongLemmaNameXYZ
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
  exact h.regular.compl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=112, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid field notation: Type of\n  G\nis not known; cannot resolve field `IsSRGWith`\n\nHint: Consider replacing the field projection `.IsSRGWith` with a call to the function `SimpleGraph.IsSRGWith`.', pos=Position(line=1, column=40), pos_end=Position(line=1, column=59), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  Gᶜ\nis not known; cannot resolve field `IsRegularOfDegree`\n\nHint: Consider replacing the field projection `.IsRegularOfDegree` with a call to the function `SimpleGraph.IsRegularOfDegree`.', pos=Position(line=2, column=4), pos_end=Position(line=2, column=36), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='unsolved goals\nx✝ : Type u_1\nG : x✝\nh : sorry\n⊢ sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=112, i_end=214, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem isCycle_cycleOf (f : Perm α) [DecidableRel f.SameCycle] (hx : f x ≠ x) :
    IsCycle (cycleOf f x)
:= thisIsWrongLemmaNameXYZ
have : cycleOf f x x ≠ x := by rwa [SameCycle.rfl.cycleOf_apply]
  (isCycle_iff_sameCycle this).2 @fun y =>
    ⟨fun h => mt h.apply_eq_self_iff.2 this, fun h =>
      if hxy : SameCycle f x y then
        let ⟨i, hi⟩ := hxy
        ⟨i, by rw [cycleOf_zpow_apply_self, hi]⟩
      else by
        rw [cycleOf_apply_of_not_sameCycle hxy] at h
        exact (h rfl).elim⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=29), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsCycle\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (cycleOf f x)\n\nHint: The identifier `IsCycle` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=519, messages=[Message(data="unexpected token 'have'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=4), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem isCycle_cycleOf (f : Perm α) [DecidableRel f.SameCycle] (hx : f x ≠ x) :
    IsCycle (cycleOf f x)
:= thisIsWrongLemmaNameXYZ
have : cycleOf f x x ≠ x := by rwa [SameCycle.rfl.cycleOf_apply]
  (isCycle_iff_sameCycle this).2 @fun y =>
    ⟨fun h => mt h.apply_eq_self_iff.2 this, fun h =>
      if hxy : SameCycle f x y then
        let ⟨i, hi⟩ := hxy
        ⟨i, by rw [cycleOf_zpow_apply_self, hi]⟩
      else by
        rw [cycleOf_apply_of_not_sameCycle hxy] at h
        exact (h rfl).elim⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=137, messages=[Message(data="Function expected at\n  Perm\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `Perm` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=29), pos_end=Position(line=1, column=35), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  IsCycle\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (cycleOf f x)\n\nHint: The identifier `IsCycle` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=4), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=137, i_end=519, messages=[Message(data="unexpected token 'have'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=4), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma mem_fromIdeal {I : Ideal R} {x : R} :
    x ∈ fromIdeal I ↔ x ∈ span I
:= by simp [fromIdeal] _unbalancedParenXYZ
/-- Every two-sided ideal is also a left ideal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  fromIdeal\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `fromIdeal` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  span\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `span` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid argument: Variable `fromIdeal` is not a proposition or let-declaration', pos=Position(line=3, column=12), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nx✝¹ : Sort u_2\nfromIdeal : x✝¹\nx✝ : Sort u_3\nspan : x✝\nI : sorry\nx : R\n⊢ x ∈ sorry () ↔ x ∈ sorry ()', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=106, i_end=176, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=23), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=50), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma mem_fromIdeal {I : Ideal R} {x : R} :
    x ∈ fromIdeal I ↔ x ∈ span I
:= by simp [fromIdeal] _unbalancedParenXYZ
/-- Every two-sided ideal is also a left ideal. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=106, messages=[Message(data='failed to synthesize instance of type class\n  Semiring R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=32), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  fromIdeal\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `fromIdeal` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=8), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  span\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  I\n\nHint: The identifier `span` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=2, column=26), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Invalid argument: Variable `fromIdeal` is not a proposition or let-declaration', pos=Position(line=3, column=12), pos_end=Position(line=3, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data='unsolved goals\nR : Type u_1\nx✝¹ : Sort u_2\nfromIdeal : x✝¹\nx✝ : Sort u_3\nspan : x✝\nI : sorry\nx : R\n⊢ x ∈ sorry () ↔ x ∈ sorry ()', pos=Position(line=3, column=3), pos_end=Position(line=3, column=22), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=106, i_end=176, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=23), pos_end=Position(line=3, column=42), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=4, column=50), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem coe_units_inv (A : (Matrix n n α)ˣ) : ↑A⁻¹ = (A⁻¹ : Matrix n n α)
:= by thisIsWrongLemmaNameXYZ
letI := A.invertible
  rw [← invOf_eq_nonsing_inv, invOf_units]

/-- The nonsingular inverse is the same as the general `Ring.inverse`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=92, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Monoid (Matrix n n α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=27), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Inv (Matrix n n α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=54), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nn : Type u_1\nα : Type u_2\nA : sorry\n⊢ sorry = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=92, i_end=255, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=73), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem coe_units_inv (A : (Matrix n n α)ˣ) : ↑A⁻¹ = (A⁻¹ : Matrix n n α)
:= by thisIsWrongLemmaNameXYZ
letI := A.invertible
  rw [← invOf_eq_nonsing_inv, invOf_units]

/-- The nonsingular inverse is the same as the general `Ring.inverse`. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=92, messages=[Message(data='unknown tactic', pos=Position(line=2, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Monoid (Matrix n n α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=27), pos_end=Position(line=1, column=42), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Inv (Matrix n n α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=54), pos_end=Position(line=1, column=57), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='unsolved goals\nn : Type u_1\nα : Type u_2\nA : sorry\n⊢ sorry = sorry', pos=Position(line=2, column=3), pos_end=Position(line=2, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=92, i_end=255, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=73), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma LSeries_neg (f : ℕ → ℂ) (s : ℂ) : LSeries (-f) s = -LSeries f s
:= by .someMissingNamespaceXYZ
simp [LSeries, term_neg_apply, tsum_neg]

/-!
### Subtraction
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=86, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=86, i_end=173, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma LSeries_neg (f : ℕ → ℂ) (s : ℂ) : LSeries (-f) s = -LSeries f s
:= by .someMissingNamespaceXYZ
simp [LSeries, term_neg_apply, tsum_neg]

/-!
### Subtraction
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=86, messages=[Message(data='unknown tactic', pos=Position(line=2, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=86, i_end=173, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
lemma zpow_eq_one_iff_cases₀ : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
  frobnicateTacticThatDoesNotExist := by
simp [← zpow_eq_zpow_iff_cases₀]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='`zpow_eq_one_iff_cases₀` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=28), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma zpow_eq_one_iff_cases₀ : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
  frobnicateTacticThatDoesNotExist := by
simp [← zpow_eq_zpow_iff_cases₀]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=162, messages=[Message(data='`zpow_eq_one_iff_cases₀` has already been declared', pos=Position(line=1, column=6), pos_end=Position(line=1, column=28), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :
    get (iterate f a) (succ n) = get (iterate f (f a)) n
:= by rw [get_succ, tail_iterate] thisIsWrongLemmaNameXYZ
section Bisim

variable (R : Stream' α → Stream' α → Prop)

/-- equivalence relation -/
local infixl:50 " ~ " => R

/-- Streams `s₁` and `s₂` are defined to be bisimulations if
their heads are equal and tails are bisimulations. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=153, messages=[Message(data='Function expected at\n  get\nbut this term has type\n  ?m.3 ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f a)', pos=Position(line=2, column=4), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  get\nbut this term has type\n  ?m.8 ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f (f a))', pos=Position(line=2, column=33), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `get_succ`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Sort u_1\nn : ℕ\nf : α → α\na : α\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=153, i_end=192, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=34), pos_end=Position(line=3, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=243, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=299, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=299, i_end=417, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=53), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :
    get (iterate f a) (succ n) = get (iterate f (f a)) n
:= by rw [get_succ, tail_iterate] thisIsWrongLemmaNameXYZ
section Bisim

variable (R : Stream' α → Stream' α → Prop)

/-- equivalence relation -/
local infixl:50 " ~ " => R

/-- Streams `s₁` and `s₂` are defined to be bisimulations if
their heads are equal and tails are bisimulations. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=153, messages=[Message(data='Function expected at\n  get\nbut this term has type\n  ?m.3 ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f a)', pos=Position(line=2, column=4), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data='Function expected at\n  get\nbut this term has type\n  ?m.8 ?m.7\n\nNote: Expected a function because this term is being applied to the argument\n  (iterate f (f a))', pos=Position(line=2, column=33), pos_end=Position(line=2, column=56), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `get_succ`', pos=Position(line=3, column=10), pos_end=Position(line=3, column=18), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nα : Sort u_1\nn : ℕ\nf : α → α\na : α\n⊢ sorry = sorry', pos=Position(line=3, column=3), pos_end=Position(line=3, column=33), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=153, i_end=192, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=34), pos_end=Position(line=3, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=243, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=299, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=299, i_end=417, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=53), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

```lean
theorem GameAdd.induction {C : α → β → Prop} :
    WellFounded rα →
      WellFounded rβ →
        (∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) → ∀ a b, C a b
:= thisIsWrongLemmaNameXYZ
GameAdd.recursion

end Prod

/-! ### `Sym2.GameAdd` -/

namespace Sym2

/-- `Sym2.GameAdd rα x y` means that `x` can be reached from `y` by decreasing either entry with
  respect to the relation `rα`.

  See `Prod.GameAdd` for the ordered pair analog. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=284, messages=[Message(data="Function expected at\n  GameAdd\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  rα\n\nHint: The identifier `GameAdd` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=28), pos_end=Position(line=4, column=59), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=5, column=3), pos_end=Position(line=5, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=284, i_end=294, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=321, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=321, i_end=337, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=337, i_end=521, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=17, column=52), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
theorem GameAdd.induction {C : α → β → Prop} :
    WellFounded rα →
      WellFounded rβ →
        (∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) → ∀ a b, C a b
:= thisIsWrongLemmaNameXYZ
GameAdd.recursion

end Prod

/-! ### `Sym2.GameAdd` -/

namespace Sym2

/-- `Sym2.GameAdd rα x y` means that `x` can be reached from `y` by decreasing either entry with
  respect to the relation `rα`.

  See `Prod.GameAdd` for the ordered pair analog. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=284, messages=[Message(data="Function expected at\n  GameAdd\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  rα\n\nHint: The identifier `GameAdd` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=4, column=28), pos_end=Position(line=4, column=59), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=5, column=3), pos_end=Position(line=5, column=26), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=284, i_end=294, messages=[Message(data='Invalid `end`: There is no current scope to end\n\nNote: Scopes are introduced using `namespace` and `section`', pos=Position(line=8, column=0), pos_end=Position(line=8, column=8), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=294, i_end=321, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=321, i_end=337, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=337, i_end=521, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=17, column=52), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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