[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.Finsupp.Pi).

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

```lean
theorem leftInverse_splittingOfFunOnFintypeSurjective [Finite α] (f : M →ₗ[R] α → R)
    (s : Surjective f) : LeftInverse f (splittingOfFunOnFintypeSurjective f s)
:= fun g => .someMissingNamespaceXYZ
LinearMap.congr_fun (splittingOfFunOnFintypeSurjective_splits f s) g
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=277, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=70), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), 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=70), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  Surjective\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `Surjective` 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=9), pos_end=Position(line=2, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  LeftInverse\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `LeftInverse` 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=25), pos_end=Position(line=2, column=78), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  max ?u.24856 (?u.25167+1) =?= ?u.24859+1\nwhile trying to unify\n  Sort (max ?u.24856 (?u.25167 + 1)) : Type (max ?u.24856 (?u.25167 + 1))\nwith\n  Type ?u.24859 : Type (?u.24859 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=78), 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.Finsupp.Pi).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem leftInverse_splittingOfFunOnFintypeSurjective [Finite α] (f : M →ₗ[R] α → R)
    (s : Surjective f) : LeftInverse f (splittingOfFunOnFintypeSurjective f s)
:= fun g => .someMissingNamespaceXYZ
LinearMap.congr_fun (splittingOfFunOnFintypeSurjective_splits f s) g
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=277, messages=[Message(data='failed to synthesize instance of type class\n  NonAssocSemiring 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=70), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), 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=70), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data="Function expected at\n  Surjective\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `Surjective` 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=9), pos_end=Position(line=2, column=21), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  LeftInverse\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `LeftInverse` 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=25), pos_end=Position(line=2, column=78), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  max ?u.24856 (?u.25167+1) =?= ?u.24859+1\nwhile trying to unify\n  Sort (max ?u.24856 (?u.25167 + 1)) : Type (max ?u.24856 (?u.25167 + 1))\nwith\n  Type ?u.24859 : Type (?u.24859 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=78), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem Zigzag.of_hom_hom {j₁ j₂ j₃ : J} (f₁₂ : j₁ ⟶ j₂) (f₂₃ : j₂ ⟶ j₃) : Zigzag j₁ j₃
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.CategoryTheory.IsConnected.isoConstant (L107-L115)
def isoConstant [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j : J) :
    F ≅ (Functor.const J).obj (F.obj j) :=
(IsPreconnected.IsoConstantAux.factorThroughDiscrete F).symm
    ≪≫ isoWhiskerRight (IsPreconnected.iso_constant _ j).some _
    ≪≫ NatIso.ofComponents (fun _ => eqToIso Function.apply_invFun_apply) (by simp)

/-- If `J` is connected, any functor to a discrete category is constant on objects.
The converse is given in `IsConnected.of_any_functor_const_on_obj`.
-/
```

-- snippet 2
```lean
-- Mathlib.CategoryTheory.IsConnected.any_functor_const_on_obj (L116-L122)
theorem any_functor_const_on_obj [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j j' : J) :
    F.obj j = F.obj j' := by
ext; exact ((isoConstant F j').hom.app j).down.1

/-- If any functor to a discrete category is constant on objects, J is connected.
The converse of `any_functor_const_on_obj`.
-/
```

-- snippet 3
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L123-L127)
theorem IsPreconnected.of_any_functor_const_on_obj
    (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') :
    IsPreconnected J where
  iso_constant := fun F j' => ⟨NatIso.ofComponents fun j => eqToIso (h F j j')⟩
```

-- snippet 4
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L128-L132)
instance IsPreconnected.prod [IsPreconnected J] [IsPreconnected K] : IsPreconnected (J × K) := by
refine .of_any_functor_const_on_obj (fun {a} F ⟨j, k⟩ ⟨j', k'⟩ => ?_)
  exact (any_functor_const_on_obj (Prod.sectL J k ⋙ F) j j').trans
    (any_functor_const_on_obj (Prod.sectR j' K ⋙ F) k k')
```

-- snippet 5
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L133-L137)
instance IsConnected.prod [IsConnected J] [IsConnected K] : IsConnected (J × K) where

/-- If any functor to a discrete category is constant on objects, J is connected.
The converse of `any_functor_const_on_obj`.
-/
```

-- snippet 6
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L138-L147)
theorem IsConnected.of_any_functor_const_on_obj [Nonempty J]
    (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') : IsConnected J :=
{ IsPreconnected.of_any_functor_const_on_obj h with }

/-- If `J` is connected, then given any function `F` such that the presence of a
morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, we have that `F` is constant.
This can be thought of as a local-to-global property.

The converse is shown in `IsConnected.of_constant_of_preserves_morphisms`
-/
```

-- snippet 7
```lean
-- Mathlib.CategoryTheory.IsConnected.constant_of_preserves_morphisms (L148-L163)
theorem constant_of_preserves_morphisms [IsPreconnected J] {α : Type u₂} (F : J → α)
    (h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) (j j' : J) : F j = F j' := by
simpa using
    any_functor_const_on_obj
      { obj := Discrete.mk ∘ F
        map := fun f => eqToHom (by ext; exact h _ _ f) }
      j j'

/-- If `J` is connected, then given any function `F` such that the presence of a
morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, there exists `a` such that `F j = a`
holds for any `j`. See `constant_of_preserves_morphisms` for a different
formulation of the fact that `F` is constant.
This can be thought of as a local-to-global property.

The converse is shown in `IsConnected.of_constant_of_preserves_morphisms`
-/
```

-- snippet 8
```lean
-- Mathlib.CategoryTheory.IsConnected.constant_of_preserves_morphisms' (L164-L174)
theorem constant_of_preserves_morphisms' [IsConnected J] {α : Type u₂} (F : J → α)
    (h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) :
    ∃ (a : α), ∀ (j : J), F j = a :=
⟨F (Classical.arbitrary _), fun _ ↦ constant_of_preserves_morphisms _ h _ _⟩

/-- `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
This can be thought of as a local-to-global property.

The converse of `constant_of_preserves_morphisms`.
-/
```

-- snippet 9
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L175-L187)
theorem IsPreconnected.of_constant_of_preserves_morphisms
    (h : ∀ {α : Type u₁} (F : J → α),
      (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') :
    IsPreconnected J :=
IsPreconnected.of_any_functor_const_on_obj fun F =>
    h F.obj fun f => by ext; exact Discrete.eq_of_hom (F.map f)

/-- `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
This can be thought of as a local-to-global property.

The converse of `constant_of_preserves_morphisms`.
-/
```

-- snippet 10
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L188-L199)
theorem IsConnected.of_constant_of_preserves_morphisms [Nonempty J]
    (h : ∀ {α : Type u₁} (F : J → α),
      (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') :
    IsConnected J :=
{ IsPreconnected.of_constant_of_preserves_morphisms h with }

/-- An inductive-like property for the objects of a connected category.
If the set `p` is nonempty, and `p` is closed under morphisms of `J`,
then `p` contains all of `J`.

The converse is given in `IsConnected.of_induct`.
-/
```

-- snippet 11
```lean
-- Mathlib.CategoryTheory.IsConnected.induct_on_objects (L200-L210)
theorem induct_on_objects [IsPreconnected J] (p : Set J) {j₀ : J} (h0 : j₀ ∈ p)
    (h1 : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) (j : J) : j ∈ p := by
let aux (j₁ j₂ : J) (f : j₁ ⟶ j₂) := congrArg ULift.up <| (h1 f).eq
  injection constant_of_preserves_morphisms (fun k => ULift.up.{u₁} (k ∈ p)) aux j j₀ with i
  rwa [i]

/--
If any maximal connected component containing some element j₀ of J is all of J, then J is connected.

The converse of `induct_on_objects`.
-/
```

-- snippet 12
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L211-L221)
theorem IsConnected.of_induct {j₀ : J}
    (h : ∀ p : Set J, j₀ ∈ p → (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) → ∀ j : J, j ∈ p) :
    IsConnected J :=
have := Nonempty.intro j₀
  IsConnected.of_constant_of_preserves_morphisms fun {α} F a => by
    have w := h { j | F j = F j₀ } rfl (fun {j₁} {j₂} f => by
      change F j₁ = F j₀ ↔ F j₂ = F j₀
      simp [a f])
    intro j j'
    rw [w j, w j']
```

-- snippet 13
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_induction (L238-L245)
theorem isPreconnected_induction [IsPreconnected J] (Z : J → Sort*)
    (h₁ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₁ → Z j₂) (h₂ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₂ → Z j₁)
    {j₀ : J} (x : Z j₀) (j : J) : Nonempty (Z j) :=
(induct_on_objects { j | Nonempty (Z j) } ⟨x⟩
      (fun f => ⟨by rintro ⟨y⟩; exact ⟨h₁ f y⟩, by rintro ⟨y⟩; exact ⟨h₂ f y⟩⟩)
      j :)

/-- If `J` and `K` are equivalent, then if `J` is preconnected then `K` is as well. -/
```

-- snippet 14
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_of_equivalent (L246-L256)
theorem isPreconnected_of_equivalent {K : Type u₂} [Category.{v₂} K] [IsPreconnected J]
    (e : J ≌ K) : IsPreconnected K where
  iso_constant F k :=
⟨calc
        F ≅ e.inverse ⋙ e.functor ⋙ F := (e.invFunIdAssoc F).symm
        _ ≅ e.inverse ⋙ (Functor.const J).obj ((e.functor ⋙ F).obj (e.inverse.obj k)) :=
          isoWhiskerLeft e.inverse (isoConstant (e.functor ⋙ F) (e.inverse.obj k))
        _ ≅ e.inverse ⋙ (Functor.const J).obj (F.obj k) :=
          isoWhiskerLeft _ ((F ⋙ Functor.const J).mapIso (e.counitIso.app k))
        _ ≅ (Functor.const K).obj (F.obj k) := NatIso.ofComponents fun _ => Iso.refl _⟩
```

-- snippet 15
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_iff_of_equivalence (L257-L261)
lemma isPreconnected_iff_of_equivalence {K : Type u₂} [Category.{v₂} K] (e : J ≌ K) :
    IsPreconnected J ↔ IsPreconnected K :=
⟨fun _ => isPreconnected_of_equivalent e, fun _ => isPreconnected_of_equivalent e.symm⟩

/-- If `J` and `K` are equivalent, then if `J` is connected then `K` is as well. -/
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.CategoryTheory.IsConnected).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem Zigzag.of_hom_hom {j₁ j₂ j₃ : J} (f₁₂ : j₁ ⟶ j₂) (f₂₃ : j₂ ⟶ j₃) : Zigzag j₁ j₃
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.CategoryTheory.IsConnected.isoConstant (L107-L115)
def isoConstant [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j : J) :
    F ≅ (Functor.const J).obj (F.obj j) :=
(IsPreconnected.IsoConstantAux.factorThroughDiscrete F).symm
    ≪≫ isoWhiskerRight (IsPreconnected.iso_constant _ j).some _
    ≪≫ NatIso.ofComponents (fun _ => eqToIso Function.apply_invFun_apply) (by simp)

/-- If `J` is connected, any functor to a discrete category is constant on objects.
The converse is given in `IsConnected.of_any_functor_const_on_obj`.
-/
```

-- snippet 2
```lean
-- Mathlib.CategoryTheory.IsConnected.any_functor_const_on_obj (L116-L122)
theorem any_functor_const_on_obj [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j j' : J) :
    F.obj j = F.obj j' := by
ext; exact ((isoConstant F j').hom.app j).down.1

/-- If any functor to a discrete category is constant on objects, J is connected.
The converse of `any_functor_const_on_obj`.
-/
```

-- snippet 3
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L123-L127)
theorem IsPreconnected.of_any_functor_const_on_obj
    (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') :
    IsPreconnected J where
  iso_constant := fun F j' => ⟨NatIso.ofComponents fun j => eqToIso (h F j j')⟩
```

-- snippet 4
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L128-L132)
instance IsPreconnected.prod [IsPreconnected J] [IsPreconnected K] : IsPreconnected (J × K) := by
refine .of_any_functor_const_on_obj (fun {a} F ⟨j, k⟩ ⟨j', k'⟩ => ?_)
  exact (any_functor_const_on_obj (Prod.sectL J k ⋙ F) j j').trans
    (any_functor_const_on_obj (Prod.sectR j' K ⋙ F) k k')
```

-- snippet 5
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L133-L137)
instance IsConnected.prod [IsConnected J] [IsConnected K] : IsConnected (J × K) where

/-- If any functor to a discrete category is constant on objects, J is connected.
The converse of `any_functor_const_on_obj`.
-/
```

-- snippet 6
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L138-L147)
theorem IsConnected.of_any_functor_const_on_obj [Nonempty J]
    (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') : IsConnected J :=
{ IsPreconnected.of_any_functor_const_on_obj h with }

/-- If `J` is connected, then given any function `F` such that the presence of a
morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, we have that `F` is constant.
This can be thought of as a local-to-global property.

The converse is shown in `IsConnected.of_constant_of_preserves_morphisms`
-/
```

-- snippet 7
```lean
-- Mathlib.CategoryTheory.IsConnected.constant_of_preserves_morphisms (L148-L163)
theorem constant_of_preserves_morphisms [IsPreconnected J] {α : Type u₂} (F : J → α)
    (h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) (j j' : J) : F j = F j' := by
simpa using
    any_functor_const_on_obj
      { obj := Discrete.mk ∘ F
        map := fun f => eqToHom (by ext; exact h _ _ f) }
      j j'

/-- If `J` is connected, then given any function `F` such that the presence of a
morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, there exists `a` such that `F j = a`
holds for any `j`. See `constant_of_preserves_morphisms` for a different
formulation of the fact that `F` is constant.
This can be thought of as a local-to-global property.

The converse is shown in `IsConnected.of_constant_of_preserves_morphisms`
-/
```

-- snippet 8
```lean
-- Mathlib.CategoryTheory.IsConnected.constant_of_preserves_morphisms' (L164-L174)
theorem constant_of_preserves_morphisms' [IsConnected J] {α : Type u₂} (F : J → α)
    (h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) :
    ∃ (a : α), ∀ (j : J), F j = a :=
⟨F (Classical.arbitrary _), fun _ ↦ constant_of_preserves_morphisms _ h _ _⟩

/-- `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
This can be thought of as a local-to-global property.

The converse of `constant_of_preserves_morphisms`.
-/
```

-- snippet 9
```lean
-- Mathlib.CategoryTheory.IsConnected.IsPreconnected (L175-L187)
theorem IsPreconnected.of_constant_of_preserves_morphisms
    (h : ∀ {α : Type u₁} (F : J → α),
      (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') :
    IsPreconnected J :=
IsPreconnected.of_any_functor_const_on_obj fun F =>
    h F.obj fun f => by ext; exact Discrete.eq_of_hom (F.map f)

/-- `J` is connected if: given any function `F : J → α` which is constant for any
`j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant.
This can be thought of as a local-to-global property.

The converse of `constant_of_preserves_morphisms`.
-/
```

-- snippet 10
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L188-L199)
theorem IsConnected.of_constant_of_preserves_morphisms [Nonempty J]
    (h : ∀ {α : Type u₁} (F : J → α),
      (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') :
    IsConnected J :=
{ IsPreconnected.of_constant_of_preserves_morphisms h with }

/-- An inductive-like property for the objects of a connected category.
If the set `p` is nonempty, and `p` is closed under morphisms of `J`,
then `p` contains all of `J`.

The converse is given in `IsConnected.of_induct`.
-/
```

-- snippet 11
```lean
-- Mathlib.CategoryTheory.IsConnected.induct_on_objects (L200-L210)
theorem induct_on_objects [IsPreconnected J] (p : Set J) {j₀ : J} (h0 : j₀ ∈ p)
    (h1 : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) (j : J) : j ∈ p := by
let aux (j₁ j₂ : J) (f : j₁ ⟶ j₂) := congrArg ULift.up <| (h1 f).eq
  injection constant_of_preserves_morphisms (fun k => ULift.up.{u₁} (k ∈ p)) aux j j₀ with i
  rwa [i]

/--
If any maximal connected component containing some element j₀ of J is all of J, then J is connected.

The converse of `induct_on_objects`.
-/
```

-- snippet 12
```lean
-- Mathlib.CategoryTheory.IsConnected.IsConnected (L211-L221)
theorem IsConnected.of_induct {j₀ : J}
    (h : ∀ p : Set J, j₀ ∈ p → (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) → ∀ j : J, j ∈ p) :
    IsConnected J :=
have := Nonempty.intro j₀
  IsConnected.of_constant_of_preserves_morphisms fun {α} F a => by
    have w := h { j | F j = F j₀ } rfl (fun {j₁} {j₂} f => by
      change F j₁ = F j₀ ↔ F j₂ = F j₀
      simp [a f])
    intro j j'
    rw [w j, w j']
```

-- snippet 13
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_induction (L238-L245)
theorem isPreconnected_induction [IsPreconnected J] (Z : J → Sort*)
    (h₁ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₁ → Z j₂) (h₂ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₂ → Z j₁)
    {j₀ : J} (x : Z j₀) (j : J) : Nonempty (Z j) :=
(induct_on_objects { j | Nonempty (Z j) } ⟨x⟩
      (fun f => ⟨by rintro ⟨y⟩; exact ⟨h₁ f y⟩, by rintro ⟨y⟩; exact ⟨h₂ f y⟩⟩)
      j :)

/-- If `J` and `K` are equivalent, then if `J` is preconnected then `K` is as well. -/
```

-- snippet 14
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_of_equivalent (L246-L256)
theorem isPreconnected_of_equivalent {K : Type u₂} [Category.{v₂} K] [IsPreconnected J]
    (e : J ≌ K) : IsPreconnected K where
  iso_constant F k :=
⟨calc
        F ≅ e.inverse ⋙ e.functor ⋙ F := (e.invFunIdAssoc F).symm
        _ ≅ e.inverse ⋙ (Functor.const J).obj ((e.functor ⋙ F).obj (e.inverse.obj k)) :=
          isoWhiskerLeft e.inverse (isoConstant (e.functor ⋙ F) (e.inverse.obj k))
        _ ≅ e.inverse ⋙ (Functor.const J).obj (F.obj k) :=
          isoWhiskerLeft _ ((F ⋙ Functor.const J).mapIso (e.counitIso.app k))
        _ ≅ (Functor.const K).obj (F.obj k) := NatIso.ofComponents fun _ => Iso.refl _⟩
```

-- snippet 15
```lean
-- Mathlib.CategoryTheory.IsConnected.isPreconnected_iff_of_equivalence (L257-L261)
lemma isPreconnected_iff_of_equivalence {K : Type u₂} [Category.{v₂} K] (e : J ≌ K) :
    IsPreconnected J ↔ IsPreconnected K :=
⟨fun _ => isPreconnected_of_equivalent e, fun _ => isPreconnected_of_equivalent e.symm⟩

/-- If `J` and `K` are equivalent, then if `J` is connected then `K` is as well. -/
```
[/INST]


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

[User]
Complete the following proof.

```lean
theorem coe_ideal_mul_inv (I : Ideal A) (hI0 : I ≠ ⊥) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1
```

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

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

```lean
theorem coe_ideal_mul_inv (I : Ideal A) (hI0 : I ≠ ⊥) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1
```

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


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

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

```text
Proof-state transcript for `Mathlib.LinearAlgebra.RootSystem.Chain.coe_chainBotCoeff_eq_sSup`:
State 0: ⊢ lemma coe_chainBotCoeff_eq_sSup :
    P.chainBotCoeff i j = sSup {k : ℤ | P.root j - k • P.root i ∈ range P.root}
Tactic: rw [setOf_root_sub_zsmul_mem_eq_Icc h]
State 1: ⊢ <goal after rw [setOf_root_sub_zsmul_mem_eq_Icc h]>
Tactic: simp  omit h  private lemma chainCoeff_reflectionPerm_left_aux :  letI := P.indexNeg  Icc (-P.chainTopCoeff i j : ℤ) (P.chainBotCoeff i j) =  Icc (-P.chainBotCoeff (-i) j : ℤ) (P.chainTopCoeff (-i) j) := by  letI := P.indexNeg  by_cases h : LinearIndependent R ![P.root i, P.root j]  · have h' : LinearIndependent R ![P.root (-i), P.root j] := by simpa
State 2: ⊢ <goal after simp>
Tactic: ext z
State 3: ⊢ <goal after ext z>
Tactic: rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,  reflection_apply_self, smul_neg, ← neg_smul, P.root_add_zsmul_mem_range_iff h, mem_Icc]  grind  · have h' : ¬ LinearIndependent R ![P.root (-i), P.root j] := by simpa
State 4: ⊢ <goal after rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,>
Tactic: simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',  chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h']  private lemma chainCoeff_reflectionPerm_right_aux :  letI := P.indexNeg  Icc (-P.chainTopCoeff i j : ℤ) (P.chainBotCoeff i j) =  Icc (-P.chainBotCoeff i (-j) : ℤ) (P.chainTopCoeff i (-j)) := by  letI := P.indexNeg  by_cases h : LinearIndependent R ![P.root i, P.root j]  · have h' : LinearIndependent R ![P.root i, P.root (-j)] := by simpa
State 5: ⊢ <goal after simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',>
Tactic: ext z
State 6: ⊢ <goal after ext z>
Tactic: rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,  reflection_apply_self, ← sub_neg_eq_add, ← neg_sub', neg_mem_range_root_iff,  P.root_sub_zsmul_mem_range_iff h, mem_Icc]  · have h' : ¬ LinearIndependent R ![P.root i, P.root (-j)] := by simpa
State 7: ⊢ <goal after rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,>
Tactic: simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',  chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h']
```

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

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

```text
Proof-state transcript for `Mathlib.LinearAlgebra.RootSystem.Chain.coe_chainBotCoeff_eq_sSup`:
State 0: ⊢ lemma coe_chainBotCoeff_eq_sSup :
    P.chainBotCoeff i j = sSup {k : ℤ | P.root j - k • P.root i ∈ range P.root}
Tactic: rw [setOf_root_sub_zsmul_mem_eq_Icc h]
State 1: ⊢ <goal after rw [setOf_root_sub_zsmul_mem_eq_Icc h]>
Tactic: simp  omit h  private lemma chainCoeff_reflectionPerm_left_aux :  letI := P.indexNeg  Icc (-P.chainTopCoeff i j : ℤ) (P.chainBotCoeff i j) =  Icc (-P.chainBotCoeff (-i) j : ℤ) (P.chainTopCoeff (-i) j) := by  letI := P.indexNeg  by_cases h : LinearIndependent R ![P.root i, P.root j]  · have h' : LinearIndependent R ![P.root (-i), P.root j] := by simpa
State 2: ⊢ <goal after simp>
Tactic: ext z
State 3: ⊢ <goal after ext z>
Tactic: rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,  reflection_apply_self, smul_neg, ← neg_smul, P.root_add_zsmul_mem_range_iff h, mem_Icc]  grind  · have h' : ¬ LinearIndependent R ![P.root (-i), P.root j] := by simpa
State 4: ⊢ <goal after rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,>
Tactic: simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',  chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h']  private lemma chainCoeff_reflectionPerm_right_aux :  letI := P.indexNeg  Icc (-P.chainTopCoeff i j : ℤ) (P.chainBotCoeff i j) =  Icc (-P.chainBotCoeff i (-j) : ℤ) (P.chainTopCoeff i (-j)) := by  letI := P.indexNeg  by_cases h : LinearIndependent R ![P.root i, P.root j]  · have h' : LinearIndependent R ![P.root i, P.root (-j)] := by simpa
State 5: ⊢ <goal after simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',>
Tactic: ext z
State 6: ⊢ <goal after ext z>
Tactic: rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,  reflection_apply_self, ← sub_neg_eq_add, ← neg_sub', neg_mem_range_root_iff,  P.root_sub_zsmul_mem_range_iff h, mem_Icc]  · have h' : ¬ LinearIndependent R ![P.root i, P.root (-j)] := by simpa
State 7: ⊢ <goal after rw [← P.root_add_zsmul_mem_range_iff h', indexNeg_neg, root_reflectionPerm, mem_Icc,>
Tactic: simp only [chainTopCoeff_of_not_linearIndependent h, chainTopCoeff_of_not_linearIndependent h',  chainBotCoeff_of_not_linearIndependent h, chainBotCoeff_of_not_linearIndependent h']
```

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


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

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

```text
Proof-state transcript for `Mathlib.FieldTheory.PrimitiveElement.FiniteDimensional`:
State 0: ⊢ theorem FiniteDimensional.of_exists_primitive_element [Algebra.IsAlgebraic F E]
    (h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E
Tactic: obtain ⟨α, hprim⟩ := h
State 1: ⊢ <goal after obtain ⟨α, hprim⟩ := h>
Tactic: have hfin := adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral (R := F) α)
State 2: ⊢ <goal after have hfin := adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral (R := F) α)>
Tactic: rw [hprim] at hfin
State 3: ⊢ <goal after rw [hprim] at hfin>
Tactic: exact topEquiv.toLinearEquiv.finiteDimensional  -- A finite simple extension has only finitely many intermediate fields
```

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

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

```text
Proof-state transcript for `Mathlib.FieldTheory.PrimitiveElement.FiniteDimensional`:
State 0: ⊢ theorem FiniteDimensional.of_exists_primitive_element [Algebra.IsAlgebraic F E]
    (h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E
Tactic: obtain ⟨α, hprim⟩ := h
State 1: ⊢ <goal after obtain ⟨α, hprim⟩ := h>
Tactic: have hfin := adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral (R := F) α)
State 2: ⊢ <goal after have hfin := adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral (R := F) α)>
Tactic: rw [hprim] at hfin
State 3: ⊢ <goal after rw [hprim] at hfin>
Tactic: exact topEquiv.toLinearEquiv.finiteDimensional  -- A finite simple extension has only finitely many intermediate fields
```

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


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

[User]
Complete the following proof.

```lean
lemma edist_le_of_mem_pairSet (ha : 1 < a) (hJ_card : #J ≤ a ^ n) {s t : T}
    (h : (s, t) ∈ pairSet J a c) : edist s t ≤ n * c
```

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

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

```lean
lemma edist_le_of_mem_pairSet (ha : 1 < a) (hJ_card : #J ≤ a ^ n) {s t : T}
    (h : (s, t) ∈ pairSet J a c) : edist s t ≤ n * c
```

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


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

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

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

```lean
instance isScalarTower_closedBall_ball_ball :
    IsScalarTower (closedBall (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1)
```

Declaration: `Mathlib.Analysis.Normed.Module.Ball.Action.isScalarTower_closedBall_ball_ball` (module Mathlib.Analysis.Normed.Module.Ball.Action).

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

```lean
instance isScalarTower_closedBall_ball_ball :
    IsScalarTower (closedBall (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1)
```

Declaration: `Mathlib.Analysis.Normed.Module.Ball.Action.isScalarTower_closedBall_ball_ball` (module Mathlib.Analysis.Normed.Module.Ball.Action).[/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.Dynamics.OmegaLimit).

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

```lean
theorem eventually_mapsTo_of_isOpen_of_omegaLimit_subset [CompactSpace β] {v : Set β}
    (hv₁ : IsOpen v) (hv₂ : ω f ϕ s ⊆ v) : ∀ᶠ t in f, MapsTo (ϕ t) s v
:= by .someMissingNamespaceXYZ
rcases eventually_closure_subset_of_isOpen_of_omegaLimit_subset f ϕ s hv₁ hv₂ with ⟨u, hu_mem, hu⟩
  refine mem_of_superset hu_mem fun t ht x hx ↦ ?_
  exact hu (subset_closure <| mem_image2_of_mem ht hx)

/-- The ω-limit of a nonempty set w.r.t. a nontrivial filter is nonempty. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=180, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=180, i_end=497, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=76), 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.Dynamics.OmegaLimit).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem eventually_mapsTo_of_isOpen_of_omegaLimit_subset [CompactSpace β] {v : Set β}
    (hv₁ : IsOpen v) (hv₂ : ω f ϕ s ⊆ v) : ∀ᶠ t in f, MapsTo (ϕ t) s v
:= by .someMissingNamespaceXYZ
rcases eventually_closure_subset_of_isOpen_of_omegaLimit_subset f ϕ s hv₁ hv₂ with ⟨u, hu_mem, hu⟩
  refine mem_of_superset hu_mem fun t ht x hx ↦ ?_
  exact hu (subset_closure <| mem_image2_of_mem ht hx)

/-- The ω-limit of a nonempty set w.r.t. a nontrivial filter is nonempty. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=180, messages=[Message(data='unknown tactic', pos=Position(line=3, column=8), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=180, i_end=497, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=76), 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.QuadraticForm.TensorProduct).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem tensorDistrib_tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) (m₁ : M₁) (m₂ : M₂) :
    tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₂ m₂ • Q₁ m₁
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.tensorDistrib (L48-L59)
def tensorDistrib :
    QuadraticMap A M₁ N₁ ⊗[R] QuadraticMap R M₂ N₂ →ₗ[A] QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  -- while `letI`s would produce a better term than `let`, they would make this already-slow
  -- definition even slower.
  let toQ := BilinMap.toQuadraticMapLinearMap A A (M₁ ⊗[R] M₂)
  let tmulB := BilinMap.tensorDistrib R A (M₁ := M₁) (M₂ := M₂)
  let toB := AlgebraTensorModule.map
      (QuadraticMap.associated : QuadraticMap A M₁ N₁ →ₗ[A] BilinMap A M₁ N₁)
      (QuadraticMap.associated : QuadraticMap R M₂ N₂ →ₗ[R] BilinMap R M₂ N₂)
  toQ ∘ₗ tmulB ∘ₗ toB
```

-- snippet 2
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_tmul (L72-L89)
theorem associated_tmul [Invertible (2 : A)]
    (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) :
    (Q₁.tmul Q₂).associated = Q₁.associated.tmul Q₂.associated := by
letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  rw [QuadraticMap.tmul, BilinMap.tmul]
  have : Subsingleton (Invertible (2 : A)) := inferInstance
  convert associated_left_inverse A (LinearMap.BilinMap.tmul_isSymm
    (QuadraticMap.associated_isSymm A Q₁) (QuadraticMap.associated_isSymm R Q₂))

end QuadraticMap

namespace QuadraticForm

variable (R A) in
/-- The tensor product of two quadratic forms injects into quadratic forms on tensor products.

Note this is heterobasic; the quadratic form on the left can take values in a larger ring than
the one on the right. -/
```

-- snippet 3
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.tensorDistrib (L90-L95)
def tensorDistrib :
    QuadraticForm A M₁ ⊗[R] QuadraticForm R M₂ →ₗ[A] QuadraticForm A (M₁ ⊗[R] M₂) :=
(AlgebraTensorModule.rid R A A).congrQuadraticMap.toLinearMap ∘ₗ QuadraticMap.tensorDistrib R A

-- TODO: make the RHS `MulOpposite.op (Q₂ m₂) • Q₁ m₁` so that this has a nicer defeq for
-- `R = A` of `Q₁ m₁ * Q₂ m₂`.
```

-- snippet 4
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_tmul (L108-L114)
theorem associated_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
    (Q₁.tmul Q₂).associated = BilinForm.tmul Q₁.associated Q₂.associated := by
rw [BilinForm.tmul, BilinForm.tensorDistrib, LinearMap.comp_apply, ← BilinMap.tmul,
    ← QuadraticMap.associated_tmul Q₁ Q₂]
  aesop

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 5
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.polarBilin_tmul (L115-L126)
theorem polarBilin_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
    polarBilin (Q₁.tmul Q₂) = ⅟(2 : A) • BilinForm.tmul (polarBilin Q₁) (polarBilin Q₂) := by
simp_rw [← two_nsmul_associated A, ← two_nsmul_associated R, BilinForm.tmul, tmul_smul,
    ← smul_tmul', map_nsmul, associated_tmul]
  rw [smul_comm (_ : A) (_ : ℕ), ← smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two,
    one_smul]

variable (A) in
/-- The base change of a quadratic form. -/
protected def baseChange (Q : QuadraticForm R M₂) : QuadraticForm A (A ⊗[R] M₂) :=
  QuadraticForm.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (QuadraticMap.sq (R := A)) Q
```

-- snippet 6
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.baseChange_tmul (L128-L131)
theorem baseChange_tmul (Q : QuadraticForm R M₂) (a : A) (m₂ : M₂) :
    Q.baseChange A (a ⊗ₜ m₂) = Q m₂ • (a * a) :=
tensorDistrib_tmul _ _ _ _
```

-- snippet 7
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_baseChange (L132-L137)
theorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :
    associated (R := A) (Q.baseChange A) = BilinForm.baseChange A (associated (R := R) Q) := by
dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]
  rw [associated_tmul (QuadraticMap.sq (R := A)) Q, associated_sq]
  exact rfl
```

-- snippet 8
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.polarBilin_baseChange (L138-L152)
theorem polarBilin_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :
    polarBilin (Q.baseChange A) = BilinForm.baseChange A (polarBilin Q) := by
rw [QuadraticForm.baseChange, BilinForm.baseChange, polarBilin_tmul, BilinForm.tmul,
    ← map_smul, smul_tmul', ← two_nsmul_associated R, coe_associatedHom, associated_sq,
    smul_comm, ← smul_assoc, two_smul, invOf_two_add_invOf_two, one_smul]

end QuadraticForm

end InvertibleTwo

/-- If two quadratic maps from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.

In other words, if a base change exists for a quadratic map, it is unique.

Note that unlike `QuadraticForm.baseChange`, this does not need `Invertible (2 : R)`. -/
```

-- snippet 9
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.baseChange_ext (L154-L172)
theorem baseChange_ext ⦃Q₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁⦄
    (h : ∀ m, Q₁ (1 ⊗ₜ m) = Q₂ (1 ⊗ₜ m)) :
    Q₁ = Q₂ := by
replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by
    rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]
  ext x
  induction x with
  | tmul => simp [h]
  | zero => simp
  | add x y hx hy =>
    have : Q₁.polarBilin = Q₂.polarBilin := by
      ext
      dsimp [polar]
      rw [← TensorProduct.tmul_add, h, h, h]
    replace := congr($this x y)
    dsimp [polar] at this
    linear_combination (norm := module) this + hx + hy

end CommRing
```

-- snippet 10
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.congr_fun (L198-L207)
theorem congr_fun (h : Q = Q') (x : M) : Q x = Q' x :=
DFunLike.congr_fun h _

/-- Copy of a `QuadraticMap` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : QuadraticMap R M N where
  toFun := Q'
  toFun_smul := h.symm ▸ Q.toFun_smul
  exists_companion' := h.symm ▸ Q.exists_companion'
```

-- snippet 11
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_smul_of_tower (L246-L251)
theorem map_smul_of_tower [CommSemiring S] [Algebra S R] [SMul S M] [IsScalarTower S R M]
    [Module S N] [IsScalarTower S R N] (a : S)
    (x : M) : Q (a • x) = (a * a) • Q x := by
rw [← IsScalarTower.algebraMap_smul R a x, Q.map_smul, ← map_mul, algebraMap_smul]

/-- Restrict the domain of a quadratic map -/
```

-- snippet 12
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_zero_left (L272-L274)
theorem polar_zero_left (y : M) : polar Q 0 y = 0 := by
simp only [polar, zero_add, QuadraticMap.map_zero, sub_zero, sub_self]
```

-- snippet 13
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.AlgClosed.equivalent_of_isAlgClosed (L59-L67)
theorem equivalent_of_isAlgClosed [Invertible (2 : K)] {M : Type*} [AddCommGroup M] [Module K M]
    [FiniteDimensional K M] (Q₁ Q₂ : QuadraticForm K M)
    (hQ₁ : (associated Q₁).SeparatingLeft)
    (hQ₂ : (associated Q₂).SeparatingLeft) : Equivalent Q₁ Q₂ :=
open Classical in
  (Q₁.equivalent_weightedSumSquares_of_isAlgClosed hQ₁).trans
  (Q₂.equivalent_weightedSumSquares_of_isAlgClosed hQ₂).symm

end QuadraticForm
```

-- snippet 14
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_add_self (L235-L242)
theorem map_add_self (x : M) : Q (x + x) = 4 • Q x := by
rw [← two_smul R x, Q.map_smul, ← Nat.cast_smul_eq_nsmul R]
  norm_num

-- not @[simp] because it is superseded by `ZeroHomClass.map_zero`
protected theorem map_zero : Q 0 = 0 := by
  rw [← @zero_smul R _ _ _ _ (0 : M), Q.map_smul, zero_mul, zero_smul]
```

-- snippet 15
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.coe_mk (L186-L193)
theorem coe_mk (toFun : M → N) (toFun_smul exists_companion') :
    ⇑({toFun, toFun_smul, exists_companion'} : QuadraticMap R M N) = toFun := rfl
-- this must come after the instFunLike definition
initialize_simps_projections QuadraticMap (toFun → apply)

variable {Q}
```

-- snippet 16
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_smul (L114-L116)
theorem polar_smul [Monoid S] [DistribMulAction S N] (f : M → N) (s : S) (x y : M) :
    polar (s • f) x y = s • polar f x y := by simp only [polar, Pi.smul_apply, smul_sub]
```

-- snippet 17
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.zeroHomClass (L243-L245)
instance zeroHomClass : ZeroHomClass (QuadraticMap R M N) M N :=
{ QuadraticMap.instFunLike (R := R) (M := M) (N := N) with map_zero := QuadraticMap.map_zero }
```

-- snippet 18
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.AlgClosed.equivalent_weightedSumSquares_of_isAlgClosed (L49-L58)
theorem equivalent_weightedSumSquares_of_isAlgClosed [Invertible (2 : K)] {M : Type*}
    [AddCommGroup M] [Module K M]
    [FiniteDimensional K M] (Q : QuadraticForm K M) (hQ : (associated Q).SeparatingLeft) :
    Equivalent Q (weightedSumSquares K (1 : Fin (Module.finrank K M) → K)) :=
open Classical in
  let ⟨w, ⟨hw₁⟩⟩ := Q.equivalent_weightedSumSquares_units_of_nondegenerate' hQ
  ⟨hw₁.trans (isometryEquivSumSquaresUnits w)⟩

/-- All nondegenerate quadratic forms on an algebraically closed field of characteristic not equal
to 2 are equivalent. -/
```

-- snippet 19
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.QuadraticForm (L163-L173)
abbrev QuadraticForm : Type _ := QuadraticMap R M R
end QuadraticForm

namespace QuadraticMap

section DFunLike

variable [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
variable {Q Q' : QuadraticMap R M N}
```

-- snippet 20
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_add_left_iff (L121-L129)
theorem polar_add_left_iff {f : M → N} {x x' y : M} :
    polar f (x + x') y = polar f x y + polar f x' y ↔
      f (x + x' + y) + (f x + f x' + f y) = f (x + x') + f (x' + y) + f (y + x) := by
simp only [← add_assoc]
  simp only [polar, sub_eq_iff_eq_add, eq_sub_iff_add_eq, sub_add_eq_add_sub, add_sub]
  simp only [add_right_comm _ (f y) _, add_right_comm _ (f x') (f x)]
  rw [add_comm y x, add_right_comm _ _ (f (x + y)), add_comm _ (f (x + y)),
    add_right_comm (f (x + y)), add_left_inj]
```

-- snippet 21
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.coe_copy (L209-L211)
theorem coe_copy (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : ⇑(Q.copy Q' h) = Q' :=
rfl
```

-- snippet 22
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polarSym2 (L136-L138)
def polarSym2 (f : M → N) : Sym2 M → N :=
Sym2.lift ⟨polar f, polar_comm _⟩
```

-- snippet 23
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar (L99-L106)
def polar (f : M → N) (x y : M) :=
f (x + y) - f x - f y

protected theorem map_add (f : M → N) (x y : M) :
    f (x + y) = f x + f y + polar f x y := by
  rw [polar]
  abel
```

-- snippet 24
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polarSym2_sym2Mk (L140-L162)
lemma polarSym2_sym2Mk (f : M → N) (x y : M) : polarSym2 f s(x, y) = polar f x y := rfl
end QuadraticMap

end Polar

/-- A quadratic map on a module.

For a more familiar constructor when `R` is a ring, see `QuadraticMap.ofPolar`. -/
structure QuadraticMap (R : Type u) (M : Type v) (N : Type w) [CommSemiring R] [AddCommMonoid M]
    [Module R M] [AddCommMonoid N] [Module R N] where
  /-- The underlying function.

  Do NOT use directly. Use the coercion instead. -/
  toFun : M → N
  toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x
  exists_companion' : ∃ B : BilinMap R M N, ∀ x y, toFun (x + y) = toFun x + toFun y + B x y

section QuadraticForm

variable (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M]

/-- A quadratic form on a module. -/
```

-- snippet 25
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.ext (L195-L197)
theorem ext (H : ∀ x : M, Q x = Q' x) : Q = Q' :=
DFunLike.ext _ _ H
```

-- snippet 26
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.instFunLike (L174-L180)
instance instFunLike : FunLike (QuadraticMap R M N) M N where
  coe := toFun
coe_injective' x y h := by cases x; cases y; congr

variable (Q)

/-- The `simp` normal form for a quadratic map is `DFunLike.coe`, not `toFun`. -/
```

-- snippet 27
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_add_add_add_map (L228-L234)
theorem map_add_add_add_map (x y z : M) :
    Q (x + y + z) + (Q x + Q y + Q z) = Q (x + y) + Q (y + z) + Q (z + x) := by
obtain ⟨B, h⟩ := Q.exists_companion
  rw [add_comm z x]
  simp only [h, LinearMap.map_add₂]
  abel
```

-- snippet 28
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.toFun_eq_coe (L182-L184)
theorem toFun_eq_coe : Q.toFun = ⇑Q :=
rfl
```

-- snippet 29
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_neg (L111-L113)
theorem polar_neg (f : M → N) (x y : M) : polar (-f) x y = -polar f x y := by
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
```

-- snippet 30
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_comm (L117-L120)
theorem polar_comm (f : M → N) (x y : M) : polar f x y = polar f y x := by
rw [polar, polar, add_comm, sub_sub, sub_sub, add_comm (f x) (f y)]

/-- Auxiliary lemma to express bilinearity of `QuadraticMap.polar` without subtraction. -/
```

-- snippet 31
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_add (L107-L110)
theorem polar_add (f g : M → N) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by
simp only [polar, Pi.add_apply]
  abel
```

-- snippet 32
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_comp (L130-L135)
theorem polar_comp {F : Type*} [AddCommGroup S] [FunLike F N S] [AddMonoidHomClass F N S]
    (f : M → N) (g : F) (x y : M) :
    polar (g ∘ f) x y = g (polar f x y) := by
simp only [polar, Function.comp_apply, map_sub]

/-- `QuadraticMap.polar` as a function from `Sym2`. -/
```

-- snippet 33
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.exists_companion (L225-L227)
theorem exists_companion : ∃ B : BilinMap R M N, ∀ x y, Q (x + y) = Q x + Q y + B x y :=
Q.exists_companion'
```

-- snippet 34
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.copy_eq (L212-L224)
theorem copy_eq (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : Q.copy Q' h = Q :=
DFunLike.ext' h

end DFunLike

section CommSemiring

variable [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
variable (Q : QuadraticMap R M N)

protected theorem map_smul (a : R) (x : M) : Q (a • x) = (a * a) • Q x :=
  Q.toFun_smul a x
```

<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.QuadraticForm.TensorProduct).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem tensorDistrib_tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) (m₁ : M₁) (m₂ : M₂) :
    tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₂ m₂ • Q₁ m₁
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.tensorDistrib (L48-L59)
def tensorDistrib :
    QuadraticMap A M₁ N₁ ⊗[R] QuadraticMap R M₂ N₂ →ₗ[A] QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  -- while `letI`s would produce a better term than `let`, they would make this already-slow
  -- definition even slower.
  let toQ := BilinMap.toQuadraticMapLinearMap A A (M₁ ⊗[R] M₂)
  let tmulB := BilinMap.tensorDistrib R A (M₁ := M₁) (M₂ := M₂)
  let toB := AlgebraTensorModule.map
      (QuadraticMap.associated : QuadraticMap A M₁ N₁ →ₗ[A] BilinMap A M₁ N₁)
      (QuadraticMap.associated : QuadraticMap R M₂ N₂ →ₗ[R] BilinMap R M₂ N₂)
  toQ ∘ₗ tmulB ∘ₗ toB
```

-- snippet 2
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_tmul (L72-L89)
theorem associated_tmul [Invertible (2 : A)]
    (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) :
    (Q₁.tmul Q₂).associated = Q₁.associated.tmul Q₂.associated := by
letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  rw [QuadraticMap.tmul, BilinMap.tmul]
  have : Subsingleton (Invertible (2 : A)) := inferInstance
  convert associated_left_inverse A (LinearMap.BilinMap.tmul_isSymm
    (QuadraticMap.associated_isSymm A Q₁) (QuadraticMap.associated_isSymm R Q₂))

end QuadraticMap

namespace QuadraticForm

variable (R A) in
/-- The tensor product of two quadratic forms injects into quadratic forms on tensor products.

Note this is heterobasic; the quadratic form on the left can take values in a larger ring than
the one on the right. -/
```

-- snippet 3
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.tensorDistrib (L90-L95)
def tensorDistrib :
    QuadraticForm A M₁ ⊗[R] QuadraticForm R M₂ →ₗ[A] QuadraticForm A (M₁ ⊗[R] M₂) :=
(AlgebraTensorModule.rid R A A).congrQuadraticMap.toLinearMap ∘ₗ QuadraticMap.tensorDistrib R A

-- TODO: make the RHS `MulOpposite.op (Q₂ m₂) • Q₁ m₁` so that this has a nicer defeq for
-- `R = A` of `Q₁ m₁ * Q₂ m₂`.
```

-- snippet 4
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_tmul (L108-L114)
theorem associated_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
    (Q₁.tmul Q₂).associated = BilinForm.tmul Q₁.associated Q₂.associated := by
rw [BilinForm.tmul, BilinForm.tensorDistrib, LinearMap.comp_apply, ← BilinMap.tmul,
    ← QuadraticMap.associated_tmul Q₁ Q₂]
  aesop

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 5
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.polarBilin_tmul (L115-L126)
theorem polarBilin_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
    polarBilin (Q₁.tmul Q₂) = ⅟(2 : A) • BilinForm.tmul (polarBilin Q₁) (polarBilin Q₂) := by
simp_rw [← two_nsmul_associated A, ← two_nsmul_associated R, BilinForm.tmul, tmul_smul,
    ← smul_tmul', map_nsmul, associated_tmul]
  rw [smul_comm (_ : A) (_ : ℕ), ← smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two,
    one_smul]

variable (A) in
/-- The base change of a quadratic form. -/
protected def baseChange (Q : QuadraticForm R M₂) : QuadraticForm A (A ⊗[R] M₂) :=
  QuadraticForm.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (QuadraticMap.sq (R := A)) Q
```

-- snippet 6
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.baseChange_tmul (L128-L131)
theorem baseChange_tmul (Q : QuadraticForm R M₂) (a : A) (m₂ : M₂) :
    Q.baseChange A (a ⊗ₜ m₂) = Q m₂ • (a * a) :=
tensorDistrib_tmul _ _ _ _
```

-- snippet 7
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.associated_baseChange (L132-L137)
theorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :
    associated (R := A) (Q.baseChange A) = BilinForm.baseChange A (associated (R := R) Q) := by
dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]
  rw [associated_tmul (QuadraticMap.sq (R := A)) Q, associated_sq]
  exact rfl
```

-- snippet 8
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.polarBilin_baseChange (L138-L152)
theorem polarBilin_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :
    polarBilin (Q.baseChange A) = BilinForm.baseChange A (polarBilin Q) := by
rw [QuadraticForm.baseChange, BilinForm.baseChange, polarBilin_tmul, BilinForm.tmul,
    ← map_smul, smul_tmul', ← two_nsmul_associated R, coe_associatedHom, associated_sq,
    smul_comm, ← smul_assoc, two_smul, invOf_two_add_invOf_two, one_smul]

end QuadraticForm

end InvertibleTwo

/-- If two quadratic maps from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.

In other words, if a base change exists for a quadratic map, it is unique.

Note that unlike `QuadraticForm.baseChange`, this does not need `Invertible (2 : R)`. -/
```

-- snippet 9
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.baseChange_ext (L154-L172)
theorem baseChange_ext ⦃Q₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁⦄
    (h : ∀ m, Q₁ (1 ⊗ₜ m) = Q₂ (1 ⊗ₜ m)) :
    Q₁ = Q₂ := by
replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by
    rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]
  ext x
  induction x with
  | tmul => simp [h]
  | zero => simp
  | add x y hx hy =>
    have : Q₁.polarBilin = Q₂.polarBilin := by
      ext
      dsimp [polar]
      rw [← TensorProduct.tmul_add, h, h, h]
    replace := congr($this x y)
    dsimp [polar] at this
    linear_combination (norm := module) this + hx + hy

end CommRing
```

-- snippet 10
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.congr_fun (L198-L207)
theorem congr_fun (h : Q = Q') (x : M) : Q x = Q' x :=
DFunLike.congr_fun h _

/-- Copy of a `QuadraticMap` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : QuadraticMap R M N where
  toFun := Q'
  toFun_smul := h.symm ▸ Q.toFun_smul
  exists_companion' := h.symm ▸ Q.exists_companion'
```

-- snippet 11
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_smul_of_tower (L246-L251)
theorem map_smul_of_tower [CommSemiring S] [Algebra S R] [SMul S M] [IsScalarTower S R M]
    [Module S N] [IsScalarTower S R N] (a : S)
    (x : M) : Q (a • x) = (a * a) • Q x := by
rw [← IsScalarTower.algebraMap_smul R a x, Q.map_smul, ← map_mul, algebraMap_smul]

/-- Restrict the domain of a quadratic map -/
```

-- snippet 12
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_zero_left (L272-L274)
theorem polar_zero_left (y : M) : polar Q 0 y = 0 := by
simp only [polar, zero_add, QuadraticMap.map_zero, sub_zero, sub_self]
```

-- snippet 13
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.AlgClosed.equivalent_of_isAlgClosed (L59-L67)
theorem equivalent_of_isAlgClosed [Invertible (2 : K)] {M : Type*} [AddCommGroup M] [Module K M]
    [FiniteDimensional K M] (Q₁ Q₂ : QuadraticForm K M)
    (hQ₁ : (associated Q₁).SeparatingLeft)
    (hQ₂ : (associated Q₂).SeparatingLeft) : Equivalent Q₁ Q₂ :=
open Classical in
  (Q₁.equivalent_weightedSumSquares_of_isAlgClosed hQ₁).trans
  (Q₂.equivalent_weightedSumSquares_of_isAlgClosed hQ₂).symm

end QuadraticForm
```

-- snippet 14
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_add_self (L235-L242)
theorem map_add_self (x : M) : Q (x + x) = 4 • Q x := by
rw [← two_smul R x, Q.map_smul, ← Nat.cast_smul_eq_nsmul R]
  norm_num

-- not @[simp] because it is superseded by `ZeroHomClass.map_zero`
protected theorem map_zero : Q 0 = 0 := by
  rw [← @zero_smul R _ _ _ _ (0 : M), Q.map_smul, zero_mul, zero_smul]
```

-- snippet 15
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.coe_mk (L186-L193)
theorem coe_mk (toFun : M → N) (toFun_smul exists_companion') :
    ⇑({toFun, toFun_smul, exists_companion'} : QuadraticMap R M N) = toFun := rfl
-- this must come after the instFunLike definition
initialize_simps_projections QuadraticMap (toFun → apply)

variable {Q}
```

-- snippet 16
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_smul (L114-L116)
theorem polar_smul [Monoid S] [DistribMulAction S N] (f : M → N) (s : S) (x y : M) :
    polar (s • f) x y = s • polar f x y := by simp only [polar, Pi.smul_apply, smul_sub]
```

-- snippet 17
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.zeroHomClass (L243-L245)
instance zeroHomClass : ZeroHomClass (QuadraticMap R M N) M N :=
{ QuadraticMap.instFunLike (R := R) (M := M) (N := N) with map_zero := QuadraticMap.map_zero }
```

-- snippet 18
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.AlgClosed.equivalent_weightedSumSquares_of_isAlgClosed (L49-L58)
theorem equivalent_weightedSumSquares_of_isAlgClosed [Invertible (2 : K)] {M : Type*}
    [AddCommGroup M] [Module K M]
    [FiniteDimensional K M] (Q : QuadraticForm K M) (hQ : (associated Q).SeparatingLeft) :
    Equivalent Q (weightedSumSquares K (1 : Fin (Module.finrank K M) → K)) :=
open Classical in
  let ⟨w, ⟨hw₁⟩⟩ := Q.equivalent_weightedSumSquares_units_of_nondegenerate' hQ
  ⟨hw₁.trans (isometryEquivSumSquaresUnits w)⟩

/-- All nondegenerate quadratic forms on an algebraically closed field of characteristic not equal
to 2 are equivalent. -/
```

-- snippet 19
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.QuadraticForm (L163-L173)
abbrev QuadraticForm : Type _ := QuadraticMap R M R
end QuadraticForm

namespace QuadraticMap

section DFunLike

variable [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
variable {Q Q' : QuadraticMap R M N}
```

-- snippet 20
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_add_left_iff (L121-L129)
theorem polar_add_left_iff {f : M → N} {x x' y : M} :
    polar f (x + x') y = polar f x y + polar f x' y ↔
      f (x + x' + y) + (f x + f x' + f y) = f (x + x') + f (x' + y) + f (y + x) := by
simp only [← add_assoc]
  simp only [polar, sub_eq_iff_eq_add, eq_sub_iff_add_eq, sub_add_eq_add_sub, add_sub]
  simp only [add_right_comm _ (f y) _, add_right_comm _ (f x') (f x)]
  rw [add_comm y x, add_right_comm _ _ (f (x + y)), add_comm _ (f (x + y)),
    add_right_comm (f (x + y)), add_left_inj]
```

-- snippet 21
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.coe_copy (L209-L211)
theorem coe_copy (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : ⇑(Q.copy Q' h) = Q' :=
rfl
```

-- snippet 22
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polarSym2 (L136-L138)
def polarSym2 (f : M → N) : Sym2 M → N :=
Sym2.lift ⟨polar f, polar_comm _⟩
```

-- snippet 23
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar (L99-L106)
def polar (f : M → N) (x y : M) :=
f (x + y) - f x - f y

protected theorem map_add (f : M → N) (x y : M) :
    f (x + y) = f x + f y + polar f x y := by
  rw [polar]
  abel
```

-- snippet 24
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polarSym2_sym2Mk (L140-L162)
lemma polarSym2_sym2Mk (f : M → N) (x y : M) : polarSym2 f s(x, y) = polar f x y := rfl
end QuadraticMap

end Polar

/-- A quadratic map on a module.

For a more familiar constructor when `R` is a ring, see `QuadraticMap.ofPolar`. -/
structure QuadraticMap (R : Type u) (M : Type v) (N : Type w) [CommSemiring R] [AddCommMonoid M]
    [Module R M] [AddCommMonoid N] [Module R N] where
  /-- The underlying function.

  Do NOT use directly. Use the coercion instead. -/
  toFun : M → N
  toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x
  exists_companion' : ∃ B : BilinMap R M N, ∀ x y, toFun (x + y) = toFun x + toFun y + B x y

section QuadraticForm

variable (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M]

/-- A quadratic form on a module. -/
```

-- snippet 25
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.ext (L195-L197)
theorem ext (H : ∀ x : M, Q x = Q' x) : Q = Q' :=
DFunLike.ext _ _ H
```

-- snippet 26
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.instFunLike (L174-L180)
instance instFunLike : FunLike (QuadraticMap R M N) M N where
  coe := toFun
coe_injective' x y h := by cases x; cases y; congr

variable (Q)

/-- The `simp` normal form for a quadratic map is `DFunLike.coe`, not `toFun`. -/
```

-- snippet 27
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.map_add_add_add_map (L228-L234)
theorem map_add_add_add_map (x y z : M) :
    Q (x + y + z) + (Q x + Q y + Q z) = Q (x + y) + Q (y + z) + Q (z + x) := by
obtain ⟨B, h⟩ := Q.exists_companion
  rw [add_comm z x]
  simp only [h, LinearMap.map_add₂]
  abel
```

-- snippet 28
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.toFun_eq_coe (L182-L184)
theorem toFun_eq_coe : Q.toFun = ⇑Q :=
rfl
```

-- snippet 29
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_neg (L111-L113)
theorem polar_neg (f : M → N) (x y : M) : polar (-f) x y = -polar f x y := by
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
```

-- snippet 30
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_comm (L117-L120)
theorem polar_comm (f : M → N) (x y : M) : polar f x y = polar f y x := by
rw [polar, polar, add_comm, sub_sub, sub_sub, add_comm (f x) (f y)]

/-- Auxiliary lemma to express bilinearity of `QuadraticMap.polar` without subtraction. -/
```

-- snippet 31
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_add (L107-L110)
theorem polar_add (f g : M → N) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by
simp only [polar, Pi.add_apply]
  abel
```

-- snippet 32
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.polar_comp (L130-L135)
theorem polar_comp {F : Type*} [AddCommGroup S] [FunLike F N S] [AddMonoidHomClass F N S]
    (f : M → N) (g : F) (x y : M) :
    polar (g ∘ f) x y = g (polar f x y) := by
simp only [polar, Function.comp_apply, map_sub]

/-- `QuadraticMap.polar` as a function from `Sym2`. -/
```

-- snippet 33
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.exists_companion (L225-L227)
theorem exists_companion : ∃ B : BilinMap R M N, ∀ x y, Q (x + y) = Q x + Q y + B x y :=
Q.exists_companion'
```

-- snippet 34
```lean
-- Mathlib.LinearAlgebra.QuadraticForm.Basic.copy_eq (L212-L224)
theorem copy_eq (Q : QuadraticMap R M N) (Q' : M → N) (h : Q' = ⇑Q) : Q.copy Q' h = Q :=
DFunLike.ext' h

end DFunLike

section CommSemiring

variable [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
variable (Q : QuadraticMap R M N)

protected theorem map_smul (a : R) (x : M) : Q (a • x) = (a * a) • Q x :=
  Q.toFun_smul a x
```
[/INST]


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

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

```text
Proof-state transcript for `Mathlib.Condensed.TopComparison.factorsThrough_of_pullbackCondition`:
State 0: ⊢ theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
    [PreservesLimit (cospan π π) G]
    {a : C(G.obj Z, X)}
    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :
    Function.FactorsThrough a (G.map π)
Tactic: intro x y hxy  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|  (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
State 1: ⊢ <goal after intro x y hxy>
Tactic: have ha' := congr_fun ha xy  dsimp at ha'
State 2: ⊢ <goal after have ha' := congr_fun ha xy>
Tactic: have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =  pullback.fst (G.map π) (G.map π) y := by
State 3: ⊢ <goal after have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =>
Tactic: simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl
State 4: ⊢ <goal after simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl>
Tactic: have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =  pullback.snd (G.map π) (G.map π) y := by
State 5: ⊢ <goal after have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =>
Tactic: simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl
State 6: ⊢ <goal after simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl>
Tactic: rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,  TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'
State 7: ⊢ <goal after rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,>
Tactic: simpa using ha'  set_option backward.isDefEq.respectTransparency false in  /--  If `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is  the case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer  condition which is required to be a sheaf for the regular topology.  -/
```

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

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

```text
Proof-state transcript for `Mathlib.Condensed.TopComparison.factorsThrough_of_pullbackCondition`:
State 0: ⊢ theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
    [PreservesLimit (cospan π π) G]
    {a : C(G.obj Z, X)}
    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :
    Function.FactorsThrough a (G.map π)
Tactic: intro x y hxy  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|  (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
State 1: ⊢ <goal after intro x y hxy>
Tactic: have ha' := congr_fun ha xy  dsimp at ha'
State 2: ⊢ <goal after have ha' := congr_fun ha xy>
Tactic: have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =  pullback.fst (G.map π) (G.map π) y := by
State 3: ⊢ <goal after have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =>
Tactic: simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl
State 4: ⊢ <goal after simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl>
Tactic: have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =  pullback.snd (G.map π) (G.map π) y := by
State 5: ⊢ <goal after have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =>
Tactic: simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl
State 6: ⊢ <goal after simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl>
Tactic: rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,  TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'
State 7: ⊢ <goal after rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,>
Tactic: simpa using ha'  set_option backward.isDefEq.respectTransparency false in  /--  If `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is  the case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer  condition which is required to be a sheaf for the regular topology.  -/
```

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


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

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

```lean
def rewrite (e eq : Expr) : MetaM Expr
:= do ⟨⟩⟨⟩
let ⟨_, eq', []⟩ ← (← mkFreshExprMVar none).mvarId!.rewrite e eq
    | throwError "Expr.rewrite may not produce subgoals."
  return eq'

/--
Rewrites the type of `e` via some `eq`, then moves `e` into the new type via `Eq.mp`.

Rewrites with a fresh metavariable as the ambient goal.
Fails if the rewrite produces any subgoals.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=58, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  Expr\n\nHint: The identifier `MetaM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=28), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=58, i_end=396, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=3, column=0), pos_end=Position(line=3, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=2), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def rewrite (e eq : Expr) : MetaM Expr
:= do ⟨⟩⟨⟩
let ⟨_, eq', []⟩ ← (← mkFreshExprMVar none).mvarId!.rewrite e eq
    | throwError "Expr.rewrite may not produce subgoals."
  return eq'

/--
Rewrites the type of `e` via some `eq`, then moves `e` into the new type via `Eq.mp`.

Rewrites with a fresh metavariable as the ambient goal.
Fails if the rewrite produces any subgoals.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=58, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  Expr\n\nHint: The identifier `MetaM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=28), pos_end=Position(line=1, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=58, i_end=396, messages=[Message(data="unexpected token 'let'; expected command", pos=Position(line=3, column=0), pos_end=Position(line=3, column=3), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=12, column=2), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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


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

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

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

## Current excerpt (around `Mathlib.Analysis.CStarAlgebra.Unitization.Unitization`, L131-L187)
```lean
instance Unitization.instCStarRing : CStarRing (Unitization 𝕜 E) where
  norm_mul_self_le x := by
-- rewrite both sides as a `⊔`
    simp only [Unitization.norm_def, Prod.norm_def]
    -- Show that `(Unitization.splitMul 𝕜 E x).snd` satisfies the C⋆-property, in two stages:
    have h₁ : ∀ x : Unitization 𝕜 E,
        ‖(Unitization.splitMul 𝕜 E x).snd‖ ≤ ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ := by
      simp only [Unitization.splitMul_apply, Unitization.snd_star, Unitization.fst_star]
      intro x
      /- split based on whether the term inside the norm is zero or not. If so, it's trivial.
      If not, then apply `norm_splitMul_snd_sq` and cancel one copy of the norm -/
      by_cases h : algebraMap 𝕜 (E →L[𝕜] E) x.fst + mul 𝕜 E x.snd = 0
      · simp only [h, norm_zero]
        exact norm_nonneg _
      · have : ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤
          ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ * ‖(Unitization.splitMul 𝕜 E x).snd‖ :=
          (norm_splitMul_snd_sq 𝕜 x).trans <| by
            rw [map_mul, Prod.snd_mul]
            exact norm_mul_le _ _
        rw [sq] at this
        rw [← Ne, ← norm_pos_iff] at h
        simp only [Unitization.splitMul_apply, Unitization.snd_star,
          Unitization.fst_star] at this
        exact (mul_le_mul_iff_left₀ h).mp this
    -- in this step we make use of the key lemma `norm_splitMul_snd_sq`
    have h₂ : ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖
        = ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 := by
      refine le_antisymm ?_ (norm_splitMul_snd_sq 𝕜 x)
      rw [map_mul, Prod.snd_mul]
      exact (norm_mul_le _ _).trans <| by
        rw [sq]
        gcongr
        simpa only [star_star] using h₁ (star x)
    -- Show that `(Unitization.splitMul 𝕜 E x).fst` satisfies the C⋆-property
    have h₃ : ‖(Unitization.splitMul 𝕜 E (star x * x)).fst‖
        = ‖(Unitization.splitMul 𝕜 E x).fst‖ ^ 2 := by
      simp only [Unitization.splitMul_apply, Unitization.fst_mul, Unitization.fst_star,
        norm_mul, norm_star, sq]
    rw [h₂, h₃]
    /- use the definition of the norm, and split into cases based on whether the norm in the first
    coordinate is bigger or smaller than the norm in the second coordinate. -/
    by_cases! h : ‖(Unitization.splitMul 𝕜 E x).fst‖ ≤ ‖(Unitization.splitMul 𝕜 E x).snd‖
    · rw [sq, sq, sup_eq_right.mpr h, sup_eq_right.mpr (mul_self_le_mul_self (norm_nonneg _) h)]
    · replace h := h.le
      rw [sq, sq, sup_eq_left.mpr h, sup_eq_left.mpr (mul_self_le_mul_self (norm_nonneg _) h)]

/-- The minimal unitization (over `ℂ`) of a C⋆-algebra, equipped with the C⋆-norm. When `A` is
unital, `A⁺¹ ≃⋆ₐ[ℂ] (ℂ × A)`. -/
scoped[CStarAlgebra] postfix:max "⁺¹" => Unitization ℂ

noncomputable instance Unitization.instCStarAlgebra {A : Type*} [NonUnitalCStarAlgebra A] :
    CStarAlgebra (Unitization ℂ A) where

noncomputable instance Unitization.instCommCStarAlgebra {A : Type*} [NonUnitalCommCStarAlgebra A] :
    CommCStarAlgebra (Unitization ℂ A) where

end CStarProperty
```

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

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.CStarAlgebra.Unitization).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Analysis/CStarAlgebra/Unitization.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.CStarAlgebra.Unitization.Unitization`, L131-L187)
```lean
instance Unitization.instCStarRing : CStarRing (Unitization 𝕜 E) where
  norm_mul_self_le x := by
-- rewrite both sides as a `⊔`
    simp only [Unitization.norm_def, Prod.norm_def]
    -- Show that `(Unitization.splitMul 𝕜 E x).snd` satisfies the C⋆-property, in two stages:
    have h₁ : ∀ x : Unitization 𝕜 E,
        ‖(Unitization.splitMul 𝕜 E x).snd‖ ≤ ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ := by
      simp only [Unitization.splitMul_apply, Unitization.snd_star, Unitization.fst_star]
      intro x
      /- split based on whether the term inside the norm is zero or not. If so, it's trivial.
      If not, then apply `norm_splitMul_snd_sq` and cancel one copy of the norm -/
      by_cases h : algebraMap 𝕜 (E →L[𝕜] E) x.fst + mul 𝕜 E x.snd = 0
      · simp only [h, norm_zero]
        exact norm_nonneg _
      · have : ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤
          ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ * ‖(Unitization.splitMul 𝕜 E x).snd‖ :=
          (norm_splitMul_snd_sq 𝕜 x).trans <| by
            rw [map_mul, Prod.snd_mul]
            exact norm_mul_le _ _
        rw [sq] at this
        rw [← Ne, ← norm_pos_iff] at h
        simp only [Unitization.splitMul_apply, Unitization.snd_star,
          Unitization.fst_star] at this
        exact (mul_le_mul_iff_left₀ h).mp this
    -- in this step we make use of the key lemma `norm_splitMul_snd_sq`
    have h₂ : ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖
        = ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 := by
      refine le_antisymm ?_ (norm_splitMul_snd_sq 𝕜 x)
      rw [map_mul, Prod.snd_mul]
      exact (norm_mul_le _ _).trans <| by
        rw [sq]
        gcongr
        simpa only [star_star] using h₁ (star x)
    -- Show that `(Unitization.splitMul 𝕜 E x).fst` satisfies the C⋆-property
    have h₃ : ‖(Unitization.splitMul 𝕜 E (star x * x)).fst‖
        = ‖(Unitization.splitMul 𝕜 E x).fst‖ ^ 2 := by
      simp only [Unitization.splitMul_apply, Unitization.fst_mul, Unitization.fst_star,
        norm_mul, norm_star, sq]
    rw [h₂, h₃]
    /- use the definition of the norm, and split into cases based on whether the norm in the first
    coordinate is bigger or smaller than the norm in the second coordinate. -/
    by_cases! h : ‖(Unitization.splitMul 𝕜 E x).fst‖ ≤ ‖(Unitization.splitMul 𝕜 E x).snd‖
    · rw [sq, sq, sup_eq_right.mpr h, sup_eq_right.mpr (mul_self_le_mul_self (norm_nonneg _) h)]
    · replace h := h.le
      rw [sq, sq, sup_eq_left.mpr h, sup_eq_left.mpr (mul_self_le_mul_self (norm_nonneg _) h)]

/-- The minimal unitization (over `ℂ`) of a C⋆-algebra, equipped with the C⋆-norm. When `A` is
unital, `A⁺¹ ≃⋆ₐ[ℂ] (ℂ × A)`. -/
scoped[CStarAlgebra] postfix:max "⁺¹" => Unitization ℂ

noncomputable instance Unitization.instCStarAlgebra {A : Type*} [NonUnitalCStarAlgebra A] :
    CStarAlgebra (Unitization ℂ A) where

noncomputable instance Unitization.instCommCStarAlgebra {A : Type*} [NonUnitalCommCStarAlgebra A] :
    CommCStarAlgebra (Unitization ℂ A) where

end CStarProperty
```

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


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

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

```lean
lemma Scheme.IsGermInjective.Spec
    (H : ∀ I : Ideal R, I.IsPrime →
      ∃ f : R, f ∉ I ∧ ∀ (x y : R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0) :
    (Spec R).IsGermInjective
:= by thisIsWrongLemmaNameXYZ
refine fun p ↦ ⟨?_⟩
  obtain ⟨f, hf, H⟩ := H p.asIdeal p.2
  refine ⟨PrimeSpectrum.basicOpen f, hf, ?_, ?_⟩
  · rw [← basicOpen_eq_of_affine]
    exact (isAffineOpen_top (Spec R)).basicOpen _
  rw [RingHom.injective_iff_ker_eq_bot, RingHom.ker_eq_bot_iff_eq_zero]
  intro x hx
  obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq
    (S := ((Spec.structureSheaf R).obj.obj (.op <| PrimeSpectrum.basicOpen f))) (.powers f) x
  rw [← RingHom.mem_ker, IsLocalization.mk'_eq_mul_mk'_one, Ideal.mul_unit_mem_iff_mem,
    RingHom.mem_ker] at hx
  swap; · exact @isUnit_of_invertible _ _ _ (@IsLocalization.invertible_mk'_one ..)
  -- There is an `Opposite.unop (Opposite.op _)` in `hx` which doesn't seem removable using
  -- `simp`/`rw`.
  erw [elementwise_of% StructureSheaf.algebraMap_germ] at hx
  obtain ⟨⟨y, hy⟩, hy'⟩ := (IsLocalization.map_eq_zero_iff p.asIdeal.primeCompl
    ((Spec.structureSheaf R).presheaf.stalk p) _).mp hx
  obtain ⟨n, hn⟩ := H x y hy' hy
  refine (@IsLocalization.mk'_eq_zero_iff ..).mpr ?_
  exact ⟨⟨_, n, rfl⟩, hn⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=204, messages=[Message(data='unknown tactic', pos=Position(line=5, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), 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=2, column=15), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HMul R R ?m.11\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=36), pos_end=Position(line=3, column=41), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HMul R R ?m.27\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=61), pos_end=Position(line=3, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `Spec`', pos=Position(line=4, column=5), pos_end=Position(line=4, column=9), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nR : Type u_1\nH : ∀ (I : sorry) (a : sorry), ∃ f, ¬sorry ∧ ∀ (x y : R) (a : sorry), ¬sorry → ∃ n, sorry\n⊢ sorry', pos=Position(line=5, column=3), pos_end=Position(line=5, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=204, i_end=1312, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
lemma Scheme.IsGermInjective.Spec
    (H : ∀ I : Ideal R, I.IsPrime →
      ∃ f : R, f ∉ I ∧ ∀ (x y : R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0) :
    (Spec R).IsGermInjective
:= by thisIsWrongLemmaNameXYZ
refine fun p ↦ ⟨?_⟩
  obtain ⟨f, hf, H⟩ := H p.asIdeal p.2
  refine ⟨PrimeSpectrum.basicOpen f, hf, ?_, ?_⟩
  · rw [← basicOpen_eq_of_affine]
    exact (isAffineOpen_top (Spec R)).basicOpen _
  rw [RingHom.injective_iff_ker_eq_bot, RingHom.ker_eq_bot_iff_eq_zero]
  intro x hx
  obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq
    (S := ((Spec.structureSheaf R).obj.obj (.op <| PrimeSpectrum.basicOpen f))) (.powers f) x
  rw [← RingHom.mem_ker, IsLocalization.mk'_eq_mul_mk'_one, Ideal.mul_unit_mem_iff_mem,
    RingHom.mem_ker] at hx
  swap; · exact @isUnit_of_invertible _ _ _ (@IsLocalization.invertible_mk'_one ..)
  -- There is an `Opposite.unop (Opposite.op _)` in `hx` which doesn't seem removable using
  -- `simp`/`rw`.
  erw [elementwise_of% StructureSheaf.algebraMap_germ] at hx
  obtain ⟨⟨y, hy⟩, hy'⟩ := (IsLocalization.map_eq_zero_iff p.asIdeal.primeCompl
    ((Spec.structureSheaf R).presheaf.stalk p) _).mp hx
  obtain ⟨n, hn⟩ := H x y hy' hy
  refine (@IsLocalization.mk'_eq_zero_iff ..).mpr ?_
  exact ⟨⟨_, n, rfl⟩, hn⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=204, messages=[Message(data='unknown tactic', pos=Position(line=5, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None), 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=2, column=15), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HMul R R ?m.11\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=36), pos_end=Position(line=3, column=41), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  HMul R R ?m.27\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=3, column=61), pos_end=Position(line=3, column=70), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='Unknown identifier `Spec`', pos=Position(line=4, column=5), pos_end=Position(line=4, column=9), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='unsolved goals\nR : Type u_1\nH : ∀ (I : sorry) (a : sorry), ∃ f, ¬sorry ∧ ∀ (x y : R) (a : sorry), ¬sorry → ∃ n, sorry\n⊢ sorry', pos=Position(line=5, column=3), pos_end=Position(line=5, column=29), severity=<Severity.ERROR: 3>, kind='Tactic.unsolvedGoals')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=204, i_end=1312, 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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.AlgebraicIndependent.Defs).

[User]
Complete the following proof.

```lean
abbrev AlgebraicIndepOn (s : Set ι) : Prop
```

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

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

```lean
abbrev AlgebraicIndepOn (s : Set ι) : Prop
```

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


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

[User]
Complete the following proof.

```lean
theorem setIntegral_compProd_univ_right (f : β × γ → E) {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ univ) ((κ ⊗ₖ η) a)) :
    ∫ z in s ×ˢ univ, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y, f (x, y) ∂η (a, x) ∂κ a
```

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

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

```lean
theorem setIntegral_compProd_univ_right (f : β × γ → E) {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ univ) ((κ ⊗ₖ η) a)) :
    ∫ z in s ×ˢ univ, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y, f (x, y) ∂η (a, x) ∂κ a
```

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