Response:
by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]
  rfl

[User]
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

Local context:
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
/-!
# Bitraversable Lemmas

## Main definitions
  * tfst - traverse on first functor argument
  * tsnd - traverse on second functor argument

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

* Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}

namespace Bitraversable

open Functor LawfulApplicative

variable {F G : Type u → Type u} [Applicative F] [Applicative G]

/-- traverse on the first functor argument -/
abbrev tfst {α α'} (f : α → F α') : t α β → F (t α' β) :=
  bitraverse f pure

/-- traverse on the second functor argument -/
abbrev tsnd {α α'} (f : α → F α') : t β α → F (t β α') :=
  bitraverse pure f

variable [LawfulBitraversable t] [LawfulApplicative F] [LawfulApplicative G]

@[higher_order tfst_id]
theorem id_tfst : ∀ {α β} (x : t α β), tfst (F := Id) pure x = pure x :=
  id_bitraverse

@[higher_order tsnd_id]
theorem id_tsnd : ∀ {α β} (x : t α β), tsnd (F := Id) pure x = pure x :=
  id_bitraverse

@[higher_order tfst_comp_tfst]
theorem comp_tfst {α₀ α₁ α₂ β} (f : α₀ → F α₁) (f' : α₁ → G α₂) (x : t α₀ β) :
    Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, tfst, map_pure, Pure.pure]

@[higher_order tfst_comp_tsnd]
theorem tfst_tsnd {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tfst f <$> tsnd f' x)
      = bitraverse (Comp.mk ∘ pure ∘ f) (Comp.mk ∘ map pure ∘ f') x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

@[higher_order tsnd_comp_tfst]
theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tsnd f' <$> tfst f x)
      = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

@[higher_order tsnd_comp_tsnd]

Target:
theorem comp_tsnd {α β₀ β₁ β₂} (g : β₀ → F β₁) (g' : β₁ → G β₂) (x : t α β₀) :
    Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk ∘ map g' ∘ g) x :=

Proof body:

[Assistant]
by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]
  rfl

Prompt:
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

Local context:
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
/-!
# Bitraversable Lemmas

## Main definitions
  * tfst - traverse on first functor argument
  * tsnd - traverse on second functor argument

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

* Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}

namespace Bitraversable

open Functor LawfulApplicative

variable {F G : Type u → Type u} [Applicative F] [Applicative G]

/-- traverse on the first functor argument -/
abbrev tfst {α α'} (f : α → F α') : t α β → F (t α' β) :=
  bitraverse f pure

/-- traverse on the second functor argument -/
abbrev tsnd {α α'} (f : α → F α') : t β α → F (t β α') :=
  bitraverse pure f

variable [LawfulBitraversable t] [LawfulApplicative F] [LawfulApplicative G]

@[higher_order tfst_id]
theorem id_tfst : ∀ {α β} (x : t α β), tfst (F := Id) pure x = pure x :=
  id_bitraverse

@[higher_order tsnd_id]
theorem id_tsnd : ∀ {α β} (x : t α β), tsnd (F := Id) pure x = pure x :=
  id_bitraverse

@[higher_order tfst_comp_tfst]
theorem comp_tfst {α₀ α₁ α₂ β} (f : α₀ → F α₁) (f' : α₁ → G α₂) (x : t α₀ β) :
    Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, tfst, map_pure, Pure.pure]

@[higher_order tfst_comp_tsnd]
theorem tfst_tsnd {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tfst f <$> tsnd f' x)
      = bitraverse (Comp.mk ∘ pure ∘ f) (Comp.mk ∘ map pure ∘ f') x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

@[higher_order tsnd_comp_tfst]
theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tsnd f' <$> tfst f x)
      = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x := by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

@[higher_order tsnd_comp_tsnd]

Target:
theorem comp_tsnd {α β₀ β₁ β₂} (g : β₀ → F β₁) (g' : β₁ → G β₂) (x : t α β₀) :
    Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk ∘ map g' ∘ g) x :=

Proof body:


<LEAN_FILE>:187:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:186:0: error: unsolved goals
A : Type u₁
inst✝² : Category.{v₁, u₁} A
B : Type u₂
inst✝¹ : Category.{v₂, u₂} B
T : Type u₃
inst✝ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
H : X = Y
⊢ (eqToHom H).right = eqToHom ⋯

Response:
by
  cases H
  rfl

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_0bee33c71f := True.intro.nonexistentProjection
  cases H
  rfl

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:187:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:186:0: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ (eqToHom H).right = eqToHom ⋯", "returncode": 1}

Corrected proof:

[Assistant]
by
  cases H
  rfl

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_0bee33c71f := True.intro.nonexistentProjection
  cases H
  rfl

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:187:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:186:0: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ (eqToHom H).right = eqToHom ⋯", "returncode": 1}

Corrected proof:


<LEAN_FILE>:151:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_ca482ba12a`
<LEAN_FILE>:126:0: error: unsolved goals
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
⊢ ∃ g, ((hf.toPermFiber x) g) e = e'

Response:
by
  letI := hf.mulActionFiber x
  have := hf.mulActionFiber_isPretransitive x
  obtain ⟨g, rfl⟩ := MulAction.IsPretransitive.exists_smul_eq e e' (M := G)
  use g
  ext
  simp

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.GroupTheory.Coset.Defs
public import Mathlib.Topology.Algebra.ConstMulAction
public import Mathlib.Topology.Algebra.Group.Quotient
public import Mathlib.Topology.Covering.Basic

Namespace:
IsQuotientCoveringMap

Local safe context:
/-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
/-!
# Covering maps to quotients by free and properly discontinuous group actions
-/

@[expose] public section

variable {E X : Type*} [TopologicalSpace E] [TopologicalSpace X] (f : E → X)
variable (G : Type*) [Group G] [MulAction G E]

open Topology

/-- A function from a topological space `E` with an action by a discrete group to another
topological space `X` is a quotient covering map if it is a quotient map, the action is
continuous and transitive on fibers, and every point of `E` has a neighborhood whose translates
by the group elements are pairwise disjoint. -/
structure IsAddQuotientCoveringMap (G) [AddGroup G] [AddAction G E] : Prop
    extends IsQuotientMap f, ContinuousConstVAdd G E where
  apply_eq_iff_mem_orbit {e₁ e₂} : f e₁ = f e₂ ↔ e₁ ∈ AddAction.orbit G e₂
  disjoint (e : E) : ∃ U ∈ 𝓝 e, ∀ g : G, ((g +ᵥ ·) '' U ∩ U).Nonempty → g = 0

/-- A function from a topological space `E` with an action by a discrete group to another
topological space `X` is a quotient covering map if it is a quotient map, the action is
continuous and transitive on fibers, and every point of `E` has a neighborhood whose translates
by the group elements are pairwise disjoint. -/
@[mk_iff, to_additive]
structure IsQuotientCoveringMap : Prop extends IsQuotientMap f, ContinuousConstSMul G E where
  apply_eq_iff_mem_orbit {e₁ e₂} : f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
  disjoint (e : E) : ∃ U ∈ 𝓝 e, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1

attribute [to_additive] isQuotientCoveringMap_iff

namespace IsQuotientCoveringMap

@[to_additive] theorem subgroup_congr (S S' : Subgroup G) (eq : S = S') :
    IsQuotientCoveringMap f S ↔ IsQuotientCoveringMap f S' := by rw [eq]

variable {f G} (hf : IsQuotientCoveringMap f G)
include hf

@[to_additive] protected theorem map_smul (g : G) {e : E} : f (g • e) = f e :=
  hf.apply_eq_iff_mem_orbit.mpr ⟨g, rfl⟩

/-- The group action on the domain of a quotient covering map is free. -/
@[to_additive] theorem isCancelSMul : IsCancelSMul G E where
  right_cancel' g g' e eq := by
    have ⟨U, heU, hU⟩ := hf.disjoint e
    simpa [inv_mul_eq_one, eq_comm] using hU (g'⁻¹ * g)
      ⟨e, ⟨e, mem_of_mem_nhds heU, by simpa [mul_smul, inv_smul_eq_iff]⟩, mem_of_mem_nhds heU⟩

@[to_additive] theorem homeomorph_comp {Y} [TopologicalSpace Y]
    (φ : X ≃ₜ Y) : IsQuotientCoveringMap (φ ∘ f) G where
  __ := φ.isQuotientMap.comp hf.toIsQuotientMap
  continuous_const_smul := hf.continuous_const_smul
  apply_eq_iff_mem_orbit := by simpa using @hf.apply_eq_iff_mem_orbit
  disjoint := hf.disjoint

omit hf in
@[to_additive (attr := simp)] theorem homeomorph_comp_iff {Y} [TopologicalSpace Y]
    (φ : X ≃ₜ Y) : IsQuotientCoveringMap (φ ∘ f) G ↔ IsQuotientCoveringMap f G where
  mp h := by convert! h.homeomorph_comp φ.symm; ext; simp
  mpr h := h.homeomorph_comp φ

/-- Fibers of a quotient covering map by a group G is a G-torsor. -/
@[to_additive /-- Fibers of a quotient covering map by an additive group G is a G-torsor. -/]
noncomputable def fiberEquivGroup {x : X} (e : f ⁻¹' {x}) : f ⁻¹' {x} ≃ G :=
  have := hf.isCancelSMul
  .symm <| .ofBijective (fun g ↦ ⟨g • e, (hf.map_smul g).trans e.2⟩)
    ⟨fun _ _ eq ↦ IsCancelSMul.right_cancel _ _ e.1 congr($eq), fun e' ↦
      have ⟨g, eq⟩ := hf.apply_eq_iff_mem_orbit.mp (e'.2.trans e.2.symm); ⟨g, Subtype.ext eq⟩⟩

@[simp] theorem fiberEquivGroup_self {x : X} (e : f ⁻¹' {x}) : hf.fiberEquivGroup e e = 1 :=
  (Equiv.apply_eq_iff_eq_symm_apply _).mpr <| Subtype.ext (one_smul ..).symm

@[simp] theorem fiberEquivGroup_eq_iff {x : X} (e e' : f ⁻¹' {x}) (g : G) :
    hf.fiberEquivGroup e e' = g ↔ e' = g • (e : E) := by
  rw [fiberEquivGroup, Equiv.symm_apply_eq, Equiv.ofBijective_apply, Subtype.mk.injEq]

@[simp] theorem fiberEquivGroup_smul_self {x : X} (e : f ⁻¹' {x}) {e' : f ⁻¹' {x}} :
    hf.fiberEquivGroup e e' • (e : E) = (e' : E) :=
  congr($((hf.fiberEquivGroup e).symm_apply_apply e'))

/-- The action of `G` restricted to the fiber. -/
@[implicit_reducible] def mulActionFiber (x : X) : MulAction G (f ⁻¹' {x}) :=
  SubMulAction.mulAction ⟨f ⁻¹' {x}, fun g _ h ↦ (hf.map_smul g).trans h⟩

@[simp] lemma coe_mulActionFiber_smul (x : X) (g : G) (e : f ⁻¹' {x}) :
    letI := hf.mulActionFiber x
    (↑(g • e) : E) = g • (e : E) :=
  rfl

lemma mulActionFiber_isPretransitive (x : X) :
    letI := hf.mulActionFiber x
    MulAction.IsPretransitive G (f ⁻¹' {x}) := by
  letI := hf.mulActionFiber x
  constructor
  intro e e'
  obtain ⟨g, hg⟩ := hf.apply_eq_iff_mem_orbit.mp (e'.2.trans e.2.symm)
  exact ⟨g, Subtype.ext hg⟩

/-- A quotient covering map `f` induces a permutation action on each fiber. -/
@[simps!] def toPermFiber (x : X) : G →* Equiv.Perm (f ⁻¹' {x}) :=
  (hf.mulActionFiber x).toPermHom

theorem toPermFiber_ext (x : X) (e : f ⁻¹' {x}) {g g' : G}
    (eq : hf.toPermFiber x g e = hf.toPermFiber x g' e) : g = g' :=
  have := hf.isCancelSMul
  IsCancelSMul.right_cancel _ _ e.1 congr($eq)

theorem toPermFiber_injective (x : X) : Function.Injective (hf.toPermFiber x) :=
  have ⟨e, he⟩ := hf.surjective x
  fun _ _ eq ↦ hf.toPermFiber_ext x ⟨e, he⟩ congr($eq _)

Target:
theorem exists_toPermFiber_eq {x : X} (e e' : f ⁻¹' {x}) : ∃ g, hf.toPermFiber x g e = e' :=

Failed candidate:
by
  have __repair_pad_0_ca482ba12a : True := by trivial
  have __repair_pad_1_ca482ba12a : True := by trivial
  have __repair_pad_2_ca482ba12a : True := by trivial
  have __repair_pad_3_ca482ba12a : True := by trivial
  have __repair_pad_4_ca482ba12a : True := by trivial
  have __repair_pad_5_ca482ba12a : True := by trivial
  have __repair_pad_6_ca482ba12a : True := by trivial
  have __repair_pad_7_ca482ba12a : True := by trivial
  have __repair_pad_8_ca482ba12a : True := by trivial
  have __repair_pad_9_ca482ba12a : True := by trivial
  have __repair_pad_10_ca482ba12a : True := by trivial
  have __repair_pad_11_ca482ba12a : True := by trivial
  have __repair_pad_12_ca482ba12a : True := by trivial
  have __repair_pad_13_ca482ba12a : True := by trivial
  have __repair_pad_14_ca482ba12a : True := by trivial
  have __repair_pad_15_ca482ba12a : True := by trivial
  have __repair_pad_16_ca482ba12a : True := by trivial
  have __repair_pad_17_ca482ba12a : True := by trivial
  have __repair_pad_18_ca482ba12a : True := by trivial
  have __repair_pad_19_ca482ba12a : True := by trivial
  have __repair_pad_20_ca482ba12a : True := by trivial
  have __repair_pad_21_ca482ba12a : True := by trivial
  have __repair_pad_22_ca482ba12a : True := by trivial
  have __repair_pad_23_ca482ba12a : True := by trivial
  have __repair_fail_ca482ba12a := __oprover_overlong_ca482ba12a
  letI := hf.mulActionFiber x
  have := hf.mulActionFiber_isPretransitive x
  obtain ⟨g, rfl⟩ := MulAction.IsPretransitive.exists_smul_eq e e' (M := G)
  use g
  ext
  simp

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:151:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_ca482ba12a`\n<LEAN_FILE>:126:0: error: unsolved goals\nE : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\nx : X\ne e' : ↑(f ⁻¹' {x})\n⊢ ∃ g, ((hf.toPermFiber x) g) e = e'", "returncode": 1}

Corrected proof:

[Assistant]
by
  letI := hf.mulActionFiber x
  have := hf.mulActionFiber_isPretransitive x
  obtain ⟨g, rfl⟩ := MulAction.IsPretransitive.exists_smul_eq e e' (M := G)
  use g
  ext
  simp

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.GroupTheory.Coset.Defs
public import Mathlib.Topology.Algebra.ConstMulAction
public import Mathlib.Topology.Algebra.Group.Quotient
public import Mathlib.Topology.Covering.Basic

Namespace:
IsQuotientCoveringMap

Local safe context:
/-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
/-!
# Covering maps to quotients by free and properly discontinuous group actions
-/

@[expose] public section

variable {E X : Type*} [TopologicalSpace E] [TopologicalSpace X] (f : E → X)
variable (G : Type*) [Group G] [MulAction G E]

open Topology

/-- A function from a topological space `E` with an action by a discrete group to another
topological space `X` is a quotient covering map if it is a quotient map, the action is
continuous and transitive on fibers, and every point of `E` has a neighborhood whose translates
by the group elements are pairwise disjoint. -/
structure IsAddQuotientCoveringMap (G) [AddGroup G] [AddAction G E] : Prop
    extends IsQuotientMap f, ContinuousConstVAdd G E where
  apply_eq_iff_mem_orbit {e₁ e₂} : f e₁ = f e₂ ↔ e₁ ∈ AddAction.orbit G e₂
  disjoint (e : E) : ∃ U ∈ 𝓝 e, ∀ g : G, ((g +ᵥ ·) '' U ∩ U).Nonempty → g = 0

/-- A function from a topological space `E` with an action by a discrete group to another
topological space `X` is a quotient covering map if it is a quotient map, the action is
continuous and transitive on fibers, and every point of `E` has a neighborhood whose translates
by the group elements are pairwise disjoint. -/
@[mk_iff, to_additive]
structure IsQuotientCoveringMap : Prop extends IsQuotientMap f, ContinuousConstSMul G E where
  apply_eq_iff_mem_orbit {e₁ e₂} : f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
  disjoint (e : E) : ∃ U ∈ 𝓝 e, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1

attribute [to_additive] isQuotientCoveringMap_iff

namespace IsQuotientCoveringMap

@[to_additive] theorem subgroup_congr (S S' : Subgroup G) (eq : S = S') :
    IsQuotientCoveringMap f S ↔ IsQuotientCoveringMap f S' := by rw [eq]

variable {f G} (hf : IsQuotientCoveringMap f G)
include hf

@[to_additive] protected theorem map_smul (g : G) {e : E} : f (g • e) = f e :=
  hf.apply_eq_iff_mem_orbit.mpr ⟨g, rfl⟩

/-- The group action on the domain of a quotient covering map is free. -/
@[to_additive] theorem isCancelSMul : IsCancelSMul G E where
  right_cancel' g g' e eq := by
    have ⟨U, heU, hU⟩ := hf.disjoint e
    simpa [inv_mul_eq_one, eq_comm] using hU (g'⁻¹ * g)
      ⟨e, ⟨e, mem_of_mem_nhds heU, by simpa [mul_smul, inv_smul_eq_iff]⟩, mem_of_mem_nhds heU⟩

@[to_additive] theorem homeomorph_comp {Y} [TopologicalSpace Y]
    (φ : X ≃ₜ Y) : IsQuotientCoveringMap (φ ∘ f) G where
  __ := φ.isQuotientMap.comp hf.toIsQuotientMap
  continuous_const_smul := hf.continuous_const_smul
  apply_eq_iff_mem_orbit := by simpa using @hf.apply_eq_iff_mem_orbit
  disjoint := hf.disjoint

omit hf in
@[to_additive (attr := simp)] theorem homeomorph_comp_iff {Y} [TopologicalSpace Y]
    (φ : X ≃ₜ Y) : IsQuotientCoveringMap (φ ∘ f) G ↔ IsQuotientCoveringMap f G where
  mp h := by convert! h.homeomorph_comp φ.symm; ext; simp
  mpr h := h.homeomorph_comp φ

/-- Fibers of a quotient covering map by a group G is a G-torsor. -/
@[to_additive /-- Fibers of a quotient covering map by an additive group G is a G-torsor. -/]
noncomputable def fiberEquivGroup {x : X} (e : f ⁻¹' {x}) : f ⁻¹' {x} ≃ G :=
  have := hf.isCancelSMul
  .symm <| .ofBijective (fun g ↦ ⟨g • e, (hf.map_smul g).trans e.2⟩)
    ⟨fun _ _ eq ↦ IsCancelSMul.right_cancel _ _ e.1 congr($eq), fun e' ↦
      have ⟨g, eq⟩ := hf.apply_eq_iff_mem_orbit.mp (e'.2.trans e.2.symm); ⟨g, Subtype.ext eq⟩⟩

@[simp] theorem fiberEquivGroup_self {x : X} (e : f ⁻¹' {x}) : hf.fiberEquivGroup e e = 1 :=
  (Equiv.apply_eq_iff_eq_symm_apply _).mpr <| Subtype.ext (one_smul ..).symm

@[simp] theorem fiberEquivGroup_eq_iff {x : X} (e e' : f ⁻¹' {x}) (g : G) :
    hf.fiberEquivGroup e e' = g ↔ e' = g • (e : E) := by
  rw [fiberEquivGroup, Equiv.symm_apply_eq, Equiv.ofBijective_apply, Subtype.mk.injEq]

@[simp] theorem fiberEquivGroup_smul_self {x : X} (e : f ⁻¹' {x}) {e' : f ⁻¹' {x}} :
    hf.fiberEquivGroup e e' • (e : E) = (e' : E) :=
  congr($((hf.fiberEquivGroup e).symm_apply_apply e'))

/-- The action of `G` restricted to the fiber. -/
@[implicit_reducible] def mulActionFiber (x : X) : MulAction G (f ⁻¹' {x}) :=
  SubMulAction.mulAction ⟨f ⁻¹' {x}, fun g _ h ↦ (hf.map_smul g).trans h⟩

@[simp] lemma coe_mulActionFiber_smul (x : X) (g : G) (e : f ⁻¹' {x}) :
    letI := hf.mulActionFiber x
    (↑(g • e) : E) = g • (e : E) :=
  rfl

lemma mulActionFiber_isPretransitive (x : X) :
    letI := hf.mulActionFiber x
    MulAction.IsPretransitive G (f ⁻¹' {x}) := by
  letI := hf.mulActionFiber x
  constructor
  intro e e'
  obtain ⟨g, hg⟩ := hf.apply_eq_iff_mem_orbit.mp (e'.2.trans e.2.symm)
  exact ⟨g, Subtype.ext hg⟩

/-- A quotient covering map `f` induces a permutation action on each fiber. -/
@[simps!] def toPermFiber (x : X) : G →* Equiv.Perm (f ⁻¹' {x}) :=
  (hf.mulActionFiber x).toPermHom

theorem toPermFiber_ext (x : X) (e : f ⁻¹' {x}) {g g' : G}
    (eq : hf.toPermFiber x g e = hf.toPermFiber x g' e) : g = g' :=
  have := hf.isCancelSMul
  IsCancelSMul.right_cancel _ _ e.1 congr($eq)

theorem toPermFiber_injective (x : X) : Function.Injective (hf.toPermFiber x) :=
  have ⟨e, he⟩ := hf.surjective x
  fun _ _ eq ↦ hf.toPermFiber_ext x ⟨e, he⟩ congr($eq _)

Target:
theorem exists_toPermFiber_eq {x : X} (e e' : f ⁻¹' {x}) : ∃ g, hf.toPermFiber x g e = e' :=

Failed candidate:
by
  have __repair_pad_0_ca482ba12a : True := by trivial
  have __repair_pad_1_ca482ba12a : True := by trivial
  have __repair_pad_2_ca482ba12a : True := by trivial
  have __repair_pad_3_ca482ba12a : True := by trivial
  have __repair_pad_4_ca482ba12a : True := by trivial
  have __repair_pad_5_ca482ba12a : True := by trivial
  have __repair_pad_6_ca482ba12a : True := by trivial
  have __repair_pad_7_ca482ba12a : True := by trivial
  have __repair_pad_8_ca482ba12a : True := by trivial
  have __repair_pad_9_ca482ba12a : True := by trivial
  have __repair_pad_10_ca482ba12a : True := by trivial
  have __repair_pad_11_ca482ba12a : True := by trivial
  have __repair_pad_12_ca482ba12a : True := by trivial
  have __repair_pad_13_ca482ba12a : True := by trivial
  have __repair_pad_14_ca482ba12a : True := by trivial
  have __repair_pad_15_ca482ba12a : True := by trivial
  have __repair_pad_16_ca482ba12a : True := by trivial
  have __repair_pad_17_ca482ba12a : True := by trivial
  have __repair_pad_18_ca482ba12a : True := by trivial
  have __repair_pad_19_ca482ba12a : True := by trivial
  have __repair_pad_20_ca482ba12a : True := by trivial
  have __repair_pad_21_ca482ba12a : True := by trivial
  have __repair_pad_22_ca482ba12a : True := by trivial
  have __repair_pad_23_ca482ba12a : True := by trivial
  have __repair_fail_ca482ba12a := __oprover_overlong_ca482ba12a
  letI := hf.mulActionFiber x
  have := hf.mulActionFiber_isPretransitive x
  obtain ⟨g, rfl⟩ := MulAction.IsPretransitive.exists_smul_eq e e' (M := G)
  use g
  ext
  simp

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:151:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_ca482ba12a`\n<LEAN_FILE>:126:0: error: unsolved goals\nE : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\nx : X\ne e' : ↑(f ⁻¹' {x})\n⊢ ∃ g, ((hf.toPermFiber x) g) e = e'", "returncode": 1}

Corrected proof:


<LEAN_FILE>:115:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b806528553`
<LEAN_FILE>:90:0: error: unsolved goals
R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
x : R
hx : IsUnit x
⊢ ¬IsNilpotent x

Response:
by
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.FiniteSupport.Defs
public import Mathlib.Algebra.GroupWithZero.Action.Defs
public import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Data.Nat.Choose.Sum
public import Mathlib.Order.Lattice.Nat
public import Mathlib.RingTheory.Nilpotent.Defs

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Nilpotent elements

This file develops the basic theory of nilpotent elements. In particular it shows that the
nilpotent elements are closed under many operations.

For the definition of `nilradical`, see `Mathlib/RingTheory/Nilpotent/Lemmas.lean`.

## Main definitions

  * `isNilpotent_neg_iff`
  * `Commute.isNilpotent_add`
  * `Commute.isNilpotent_sub`

-/

public section

universe u v

open Function Set

variable {R S : Type*} {x y : R}

theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
  obtain ⟨n, hn⟩ := h
  use n
  rw [neg_pow, hn, mul_zero]

@[simp]
theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
  ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩

lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]
    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :
    IsNilpotent (t • a) := by
  obtain ⟨k, ha⟩ := ha
  use k
  rw [smul_pow, ha, smul_zero]

theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
  obtain ⟨n, hn⟩ := hnil
  refine ⟨⟨r - 1, -∑ i ∈ Finset.range n, r ^ i, ?_, ?_⟩, rfl⟩
  · simp [mul_geom_sum, hn]
  · simp [geom_sum_mul, hn]

theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) := by
  rw [← IsUnit.neg_iff, neg_sub]
  exact isUnit_sub_one hnil

theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) := by
  rw [← IsUnit.neg_iff, neg_add']
  exact isUnit_sub_one hnil.neg

theorem IsNilpotent.isUnit_one_add [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 + r) :=
  add_comm r 1 ▸ isUnit_add_one hnil

theorem IsNilpotent.isUnit_add_left_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (u + r) := by
  rw [← Units.isUnit_mul_units _ hu.unit⁻¹, add_mul, IsUnit.mul_val_inv]
  replace h_comm : Commute r (↑hu.unit⁻¹) := Commute.units_inv_right h_comm
  refine IsNilpotent.isUnit_one_add ?_
  exact (hu.unit⁻¹.isUnit.isNilpotent_mul_unit_of_commute_iff h_comm).mpr hnil

theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (r + u) :=
  add_comm r u ▸ hnil.isUnit_add_left_of_commute hu h_comm

Target:
lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
    ¬ IsNilpotent x :=

Failed candidate:
by
  have __repair_pad_0_b806528553 : True := by trivial
  have __repair_pad_1_b806528553 : True := by trivial
  have __repair_pad_2_b806528553 : True := by trivial
  have __repair_pad_3_b806528553 : True := by trivial
  have __repair_pad_4_b806528553 : True := by trivial
  have __repair_pad_5_b806528553 : True := by trivial
  have __repair_pad_6_b806528553 : True := by trivial
  have __repair_pad_7_b806528553 : True := by trivial
  have __repair_pad_8_b806528553 : True := by trivial
  have __repair_pad_9_b806528553 : True := by trivial
  have __repair_pad_10_b806528553 : True := by trivial
  have __repair_pad_11_b806528553 : True := by trivial
  have __repair_pad_12_b806528553 : True := by trivial
  have __repair_pad_13_b806528553 : True := by trivial
  have __repair_pad_14_b806528553 : True := by trivial
  have __repair_pad_15_b806528553 : True := by trivial
  have __repair_pad_16_b806528553 : True := by trivial
  have __repair_pad_17_b806528553 : True := by trivial
  have __repair_pad_18_b806528553 : True := by trivial
  have __repair_pad_19_b806528553 : True := by trivial
  have __repair_pad_20_b806528553 : True := by trivial
  have __repair_pad_21_b806528553 : True := by trivial
  have __repair_pad_22_b806528553 : True := by trivial
  have __repair_pad_23_b806528553 : True := by trivial
  have __repair_fail_b806528553 := __oprover_overlong_b806528553
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:115:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b806528553`\n<LEAN_FILE>:90:0: error: unsolved goals\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsUnit x\n⊢ ¬IsNilpotent x", "returncode": 1}

Corrected proof:

[Assistant]
by
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.FiniteSupport.Defs
public import Mathlib.Algebra.GroupWithZero.Action.Defs
public import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Data.Nat.Choose.Sum
public import Mathlib.Order.Lattice.Nat
public import Mathlib.RingTheory.Nilpotent.Defs

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Nilpotent elements

This file develops the basic theory of nilpotent elements. In particular it shows that the
nilpotent elements are closed under many operations.

For the definition of `nilradical`, see `Mathlib/RingTheory/Nilpotent/Lemmas.lean`.

## Main definitions

  * `isNilpotent_neg_iff`
  * `Commute.isNilpotent_add`
  * `Commute.isNilpotent_sub`

-/

public section

universe u v

open Function Set

variable {R S : Type*} {x y : R}

theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
  obtain ⟨n, hn⟩ := h
  use n
  rw [neg_pow, hn, mul_zero]

@[simp]
theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
  ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩

lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]
    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :
    IsNilpotent (t • a) := by
  obtain ⟨k, ha⟩ := ha
  use k
  rw [smul_pow, ha, smul_zero]

theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
  obtain ⟨n, hn⟩ := hnil
  refine ⟨⟨r - 1, -∑ i ∈ Finset.range n, r ^ i, ?_, ?_⟩, rfl⟩
  · simp [mul_geom_sum, hn]
  · simp [geom_sum_mul, hn]

theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) := by
  rw [← IsUnit.neg_iff, neg_sub]
  exact isUnit_sub_one hnil

theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) := by
  rw [← IsUnit.neg_iff, neg_add']
  exact isUnit_sub_one hnil.neg

theorem IsNilpotent.isUnit_one_add [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 + r) :=
  add_comm r 1 ▸ isUnit_add_one hnil

theorem IsNilpotent.isUnit_add_left_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (u + r) := by
  rw [← Units.isUnit_mul_units _ hu.unit⁻¹, add_mul, IsUnit.mul_val_inv]
  replace h_comm : Commute r (↑hu.unit⁻¹) := Commute.units_inv_right h_comm
  refine IsNilpotent.isUnit_one_add ?_
  exact (hu.unit⁻¹.isUnit.isNilpotent_mul_unit_of_commute_iff h_comm).mpr hnil

theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (r + u) :=
  add_comm r u ▸ hnil.isUnit_add_left_of_commute hu h_comm

Target:
lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
    ¬ IsNilpotent x :=

Failed candidate:
by
  have __repair_pad_0_b806528553 : True := by trivial
  have __repair_pad_1_b806528553 : True := by trivial
  have __repair_pad_2_b806528553 : True := by trivial
  have __repair_pad_3_b806528553 : True := by trivial
  have __repair_pad_4_b806528553 : True := by trivial
  have __repair_pad_5_b806528553 : True := by trivial
  have __repair_pad_6_b806528553 : True := by trivial
  have __repair_pad_7_b806528553 : True := by trivial
  have __repair_pad_8_b806528553 : True := by trivial
  have __repair_pad_9_b806528553 : True := by trivial
  have __repair_pad_10_b806528553 : True := by trivial
  have __repair_pad_11_b806528553 : True := by trivial
  have __repair_pad_12_b806528553 : True := by trivial
  have __repair_pad_13_b806528553 : True := by trivial
  have __repair_pad_14_b806528553 : True := by trivial
  have __repair_pad_15_b806528553 : True := by trivial
  have __repair_pad_16_b806528553 : True := by trivial
  have __repair_pad_17_b806528553 : True := by trivial
  have __repair_pad_18_b806528553 : True := by trivial
  have __repair_pad_19_b806528553 : True := by trivial
  have __repair_pad_20_b806528553 : True := by trivial
  have __repair_pad_21_b806528553 : True := by trivial
  have __repair_pad_22_b806528553 : True := by trivial
  have __repair_pad_23_b806528553 : True := by trivial
  have __repair_fail_b806528553 := __oprover_overlong_b806528553
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:115:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b806528553`\n<LEAN_FILE>:90:0: error: unsolved goals\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nx : R\nhx : IsUnit x\n⊢ ¬IsNilpotent x", "returncode": 1}

Corrected proof:


<LEAN_FILE>:112:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:111:0: error: unsolved goals
p : ℕ
hp : Nat.Prime p
⊢ minOrder (ZMod p) = ↑p

Response:
by
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Group.Torsion
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.ZMod.QuotientGroup

Namespace:
ZMod

Local safe context:
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Minimum order of an element

This file defines the minimum order of an element of a monoid.

## Main declarations

* `Monoid.minOrder`: The minimum order of an element of a given monoid.
* `Monoid.minOrder_eq_top`: The minimum order is infinite iff the monoid is torsion-free.
* `ZMod.minOrder`: The minimum order of $ℤ/nℤ$ is the smallest factor of `n`, unless `n = 0, 1`.
-/

@[expose] public section

open Subgroup

variable {G α : Type*}

namespace Monoid
section Monoid
variable (α) [Monoid α]

/-- The minimum order of a non-identity element. Also the minimum size of a nontrivial subgroup, see
`Monoid.le_minOrder_iff_forall_subgroup`. Returns `∞` if the monoid is torsion-free. -/
@[to_additive /-- The minimum order of a non-identity element. Also the minimum size of a nontrivial
subgroup, see `AddMonoid.le_minOrder_iff_forall_addSubgroup`. Returns `∞` if the monoid is
torsion-free. -/]
noncomputable def minOrder : ℕ∞ := ⨅ (a : α) (_ha : a ≠ 1) (_ha' : IsOfFinOrder a), orderOf a

variable {α} {a : α}

@[to_additive (attr := simp)]
lemma le_minOrder {n : ℕ∞} :
    n ≤ minOrder α ↔ ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ orderOf a := by simp [minOrder]

@[to_additive]
lemma minOrder_le_orderOf (ha : a ≠ 1) (ha' : IsOfFinOrder a) : minOrder α ≤ orderOf a :=
  le_minOrder.1 le_rfl ha ha'

end Monoid

section Group
variable [Group G] {s : Subgroup G}

@[to_additive]
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s := by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

@[to_additive]
lemma minOrder_le_natCard (hs : s ≠ ⊥) (hs' : (s : Set G).Finite) : minOrder G ≤ Nat.card s :=
  le_minOrder_iff_forall_subgroup.1 le_rfl hs hs'

@[to_additive (attr := simp)]
lemma minOrder_eq_top [IsMulTorsionFree G] : minOrder G = ⊤ := by
  simpa [minOrder] using fun _ ↦ not_isOfFinOrder_of_isMulTorsionFree

end Group

section CommGroup
variable [CommGroup G] {s : Subgroup G}

@[to_additive (attr := simp)]
lemma minOrder_eq_top_iff : minOrder G = ⊤ ↔ IsMulTorsionFree G := by
  simp [minOrder, isMulTorsionFree_iff_not_isOfFinOrder]

end CommGroup
end Monoid

open AddMonoid AddSubgroup Nat Set

namespace ZMod

@[simp]
protected lemma minOrder {n : ℕ} (hn : n ≠ 0) (hn₁ : n ≠ 1) : minOrder (ZMod n) = n.minFac := by
  have : Fact (1 < n) := ⟨one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn, hn₁⟩⟩
  classical
  have : (↑(n / n.minFac) : ZMod n) ≠ 0 := by
    rw [Ne, ringChar.spec, ringChar.eq (ZMod n) n]
    exact
      not_dvd_of_pos_of_lt (Nat.div_pos (minFac_le hn.bot_lt) n.minFac_pos)
        (div_lt_self hn.bot_lt (minFac_prime hn₁).one_lt)
  refine ((minOrder_le_natCard (zmultiples_eq_bot.not.2 this) <| toFinite _).trans ?_).antisymm <|
    le_minOrder_iff_forall_addSubgroup.2 fun s hs _ ↦ ?_
  · rw [Nat.card_zmultiples, ZMod.addOrderOf_coe _ hn,
      gcd_eq_right (div_dvd_of_dvd n.minFac_dvd), Nat.div_div_self n.minFac_dvd hn]
  · haveI : Nontrivial s := s.bot_or_nontrivial.resolve_left hs
    exact WithTop.coe_le_coe.2 <| minFac_le_of_dvd Finite.one_lt_card <|
      (card_addSubgroup_dvd_card _).trans n.card_zmod.dvd

@[simp]

Target:
lemma minOrder_of_prime {p : ℕ} (hp : p.Prime) : minOrder (ZMod p) = p :=

Failed candidate:
by
  have __repair_6c86207a93 := True.intro.nonexistentProjection
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:112:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:111:0: error: unsolved goals\np : ℕ\nhp : Nat.Prime p\n⊢ minOrder (ZMod p) = ↑p", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Group.Torsion
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.ZMod.QuotientGroup

Namespace:
ZMod

Local safe context:
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Minimum order of an element

This file defines the minimum order of an element of a monoid.

## Main declarations

* `Monoid.minOrder`: The minimum order of an element of a given monoid.
* `Monoid.minOrder_eq_top`: The minimum order is infinite iff the monoid is torsion-free.
* `ZMod.minOrder`: The minimum order of $ℤ/nℤ$ is the smallest factor of `n`, unless `n = 0, 1`.
-/

@[expose] public section

open Subgroup

variable {G α : Type*}

namespace Monoid
section Monoid
variable (α) [Monoid α]

/-- The minimum order of a non-identity element. Also the minimum size of a nontrivial subgroup, see
`Monoid.le_minOrder_iff_forall_subgroup`. Returns `∞` if the monoid is torsion-free. -/
@[to_additive /-- The minimum order of a non-identity element. Also the minimum size of a nontrivial
subgroup, see `AddMonoid.le_minOrder_iff_forall_addSubgroup`. Returns `∞` if the monoid is
torsion-free. -/]
noncomputable def minOrder : ℕ∞ := ⨅ (a : α) (_ha : a ≠ 1) (_ha' : IsOfFinOrder a), orderOf a

variable {α} {a : α}

@[to_additive (attr := simp)]
lemma le_minOrder {n : ℕ∞} :
    n ≤ minOrder α ↔ ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ orderOf a := by simp [minOrder]

@[to_additive]
lemma minOrder_le_orderOf (ha : a ≠ 1) (ha' : IsOfFinOrder a) : minOrder α ≤ orderOf a :=
  le_minOrder.1 le_rfl ha ha'

end Monoid

section Group
variable [Group G] {s : Subgroup G}

@[to_additive]
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s := by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

@[to_additive]
lemma minOrder_le_natCard (hs : s ≠ ⊥) (hs' : (s : Set G).Finite) : minOrder G ≤ Nat.card s :=
  le_minOrder_iff_forall_subgroup.1 le_rfl hs hs'

@[to_additive (attr := simp)]
lemma minOrder_eq_top [IsMulTorsionFree G] : minOrder G = ⊤ := by
  simpa [minOrder] using fun _ ↦ not_isOfFinOrder_of_isMulTorsionFree

end Group

section CommGroup
variable [CommGroup G] {s : Subgroup G}

@[to_additive (attr := simp)]
lemma minOrder_eq_top_iff : minOrder G = ⊤ ↔ IsMulTorsionFree G := by
  simp [minOrder, isMulTorsionFree_iff_not_isOfFinOrder]

end CommGroup
end Monoid

open AddMonoid AddSubgroup Nat Set

namespace ZMod

@[simp]
protected lemma minOrder {n : ℕ} (hn : n ≠ 0) (hn₁ : n ≠ 1) : minOrder (ZMod n) = n.minFac := by
  have : Fact (1 < n) := ⟨one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn, hn₁⟩⟩
  classical
  have : (↑(n / n.minFac) : ZMod n) ≠ 0 := by
    rw [Ne, ringChar.spec, ringChar.eq (ZMod n) n]
    exact
      not_dvd_of_pos_of_lt (Nat.div_pos (minFac_le hn.bot_lt) n.minFac_pos)
        (div_lt_self hn.bot_lt (minFac_prime hn₁).one_lt)
  refine ((minOrder_le_natCard (zmultiples_eq_bot.not.2 this) <| toFinite _).trans ?_).antisymm <|
    le_minOrder_iff_forall_addSubgroup.2 fun s hs _ ↦ ?_
  · rw [Nat.card_zmultiples, ZMod.addOrderOf_coe _ hn,
      gcd_eq_right (div_dvd_of_dvd n.minFac_dvd), Nat.div_div_self n.minFac_dvd hn]
  · haveI : Nontrivial s := s.bot_or_nontrivial.resolve_left hs
    exact WithTop.coe_le_coe.2 <| minFac_le_of_dvd Finite.one_lt_card <|
      (card_addSubgroup_dvd_card _).trans n.card_zmod.dvd

@[simp]

Target:
lemma minOrder_of_prime {p : ℕ} (hp : p.Prime) : minOrder (ZMod p) = p :=

Failed candidate:
by
  have __repair_6c86207a93 := True.intro.nonexistentProjection
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:112:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:111:0: error: unsolved goals\np : ℕ\nhp : Nat.Prime p\n⊢ minOrder (ZMod p) = ↑p", "returncode": 1}

Corrected proof:


Response:
by
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

[User]
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition

Namespace:
DirectSum

Local context:
/-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang, Andrew Yang
-/
/-!
# Decomposition of the identity of a semiring into orthogonal idempotents

In this file we show that if a semiring `R` can be decomposed into a direct sum
of (left) ideals `R = V₁ ⊕ V₂ ⊕ ⋯ ⊕ Vₙ` then in the corresponding decomposition
`1 = e₁ + e₂ + ⋯ + eₙ` with `eᵢ ∈ Vᵢ`, each `eᵢ` is an idempotent and the
`eᵢ`'s form a family of complete orthogonal idempotents.
-/

@[expose] public section

namespace DirectSum

section OrthogonalIdempotents

variable {R I : Type*} [Semiring R] [DecidableEq I] (V : I → Ideal R) [Decomposition V]

/-- The decomposition of `(1 : R)` where `1 = e₁ + e₂ + ⋯ + eₙ` which is induced by
  the decomposition of the semiring `R = V1 ⊕ V2 ⊕ ⋯ ⊕ Vn`. -/
def idempotent (i : I) : R :=
  decompose V 1 i

Target:
lemma decompose_eq_mul_idempotent (x : R) (i : I) : decompose V x i = x * idempotent V i :=

Proof body:

[Assistant]
by
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Prompt:
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition

Namespace:
DirectSum

Local context:
/-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang, Andrew Yang
-/
/-!
# Decomposition of the identity of a semiring into orthogonal idempotents

In this file we show that if a semiring `R` can be decomposed into a direct sum
of (left) ideals `R = V₁ ⊕ V₂ ⊕ ⋯ ⊕ Vₙ` then in the corresponding decomposition
`1 = e₁ + e₂ + ⋯ + eₙ` with `eᵢ ∈ Vᵢ`, each `eᵢ` is an idempotent and the
`eᵢ`'s form a family of complete orthogonal idempotents.
-/

@[expose] public section

namespace DirectSum

section OrthogonalIdempotents

variable {R I : Type*} [Semiring R] [DecidableEq I] (V : I → Ideal R) [Decomposition V]

/-- The decomposition of `(1 : R)` where `1 = e₁ + e₂ + ⋯ + eₙ` which is induced by
  the decomposition of the semiring `R = V1 ⊕ V2 ⊕ ⋯ ⊕ Vn`. -/
def idempotent (i : I) : R :=
  decompose V 1 i

Target:
lemma decompose_eq_mul_idempotent (x : R) (i : I) : decompose V x i = x * idempotent V i :=

Proof body:


<LEAN_FILE>:211:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_813f6a5b79`
<LEAN_FILE>:186:0: error: unsolved goals
A : Type u₁
inst✝² : Category.{v₁, u₁} A
B : Type u₂
inst✝¹ : Category.{v₂, u₂} B
T : Type u₃
inst✝ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
H : X = Y
⊢ (eqToHom H).right = eqToHom ⋯

Response:
by
  cases H
  rfl

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_pad_0_813f6a5b79 : True := by trivial
  have __repair_pad_1_813f6a5b79 : True := by trivial
  have __repair_pad_2_813f6a5b79 : True := by trivial
  have __repair_pad_3_813f6a5b79 : True := by trivial
  have __repair_pad_4_813f6a5b79 : True := by trivial
  have __repair_pad_5_813f6a5b79 : True := by trivial
  have __repair_pad_6_813f6a5b79 : True := by trivial
  have __repair_pad_7_813f6a5b79 : True := by trivial
  have __repair_pad_8_813f6a5b79 : True := by trivial
  have __repair_pad_9_813f6a5b79 : True := by trivial
  have __repair_pad_10_813f6a5b79 : True := by trivial
  have __repair_pad_11_813f6a5b79 : True := by trivial
  have __repair_pad_12_813f6a5b79 : True := by trivial
  have __repair_pad_13_813f6a5b79 : True := by trivial
  have __repair_pad_14_813f6a5b79 : True := by trivial
  have __repair_pad_15_813f6a5b79 : True := by trivial
  have __repair_pad_16_813f6a5b79 : True := by trivial
  have __repair_pad_17_813f6a5b79 : True := by trivial
  have __repair_pad_18_813f6a5b79 : True := by trivial
  have __repair_pad_19_813f6a5b79 : True := by trivial
  have __repair_pad_20_813f6a5b79 : True := by trivial
  have __repair_pad_21_813f6a5b79 : True := by trivial
  have __repair_pad_22_813f6a5b79 : True := by trivial
  have __repair_pad_23_813f6a5b79 : True := by trivial
  have __repair_fail_813f6a5b79 := __oprover_overlong_813f6a5b79
  cases H
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:211:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_813f6a5b79`\n<LEAN_FILE>:186:0: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ (eqToHom H).right = eqToHom ⋯", "returncode": 1}

Corrected proof:

[Assistant]
by
  cases H
  rfl

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_pad_0_813f6a5b79 : True := by trivial
  have __repair_pad_1_813f6a5b79 : True := by trivial
  have __repair_pad_2_813f6a5b79 : True := by trivial
  have __repair_pad_3_813f6a5b79 : True := by trivial
  have __repair_pad_4_813f6a5b79 : True := by trivial
  have __repair_pad_5_813f6a5b79 : True := by trivial
  have __repair_pad_6_813f6a5b79 : True := by trivial
  have __repair_pad_7_813f6a5b79 : True := by trivial
  have __repair_pad_8_813f6a5b79 : True := by trivial
  have __repair_pad_9_813f6a5b79 : True := by trivial
  have __repair_pad_10_813f6a5b79 : True := by trivial
  have __repair_pad_11_813f6a5b79 : True := by trivial
  have __repair_pad_12_813f6a5b79 : True := by trivial
  have __repair_pad_13_813f6a5b79 : True := by trivial
  have __repair_pad_14_813f6a5b79 : True := by trivial
  have __repair_pad_15_813f6a5b79 : True := by trivial
  have __repair_pad_16_813f6a5b79 : True := by trivial
  have __repair_pad_17_813f6a5b79 : True := by trivial
  have __repair_pad_18_813f6a5b79 : True := by trivial
  have __repair_pad_19_813f6a5b79 : True := by trivial
  have __repair_pad_20_813f6a5b79 : True := by trivial
  have __repair_pad_21_813f6a5b79 : True := by trivial
  have __repair_pad_22_813f6a5b79 : True := by trivial
  have __repair_pad_23_813f6a5b79 : True := by trivial
  have __repair_fail_813f6a5b79 := __oprover_overlong_813f6a5b79
  cases H
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:211:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_813f6a5b79`\n<LEAN_FILE>:186:0: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ (eqToHom H).right = eqToHom ⋯", "returncode": 1}

Corrected proof:


<LEAN_FILE>:59:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 4375826) 0
in the target expression
  MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s

α : Type u_1
f : Perm α
s : Set α
⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s

Response:
by
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.End
public import Mathlib.Logic.Equiv.Set
public import Mathlib.Tactic.Common

Namespace:
Set

Local safe context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
/-!
# Extra lemmas about permutations

This file proves miscellaneous lemmas about `Equiv.Perm`.

## TODO

Most of the content of this file was moved to `Mathlib/Algebra/Group/End.lean` in
https://github.com/leanprover-community/mathlib4/pull/22141.
It would be good to merge the remaining lemmas with other files, e.g.
`GroupTheory.Perm.ViaEmbedding` looks like it could benefit from such a treatment (splitting into
the algebra and non-algebra parts).
-/

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

@[simp]
theorem swap_smul_self_smul [MulAction (Perm α) β] (i j : α) (x : β) :
    swap i j • swap i j • x = x := by simp [smul_smul]

theorem swap_smul_involutive [MulAction (Perm α) β] (i j : α) :
    Function.Involutive (swap i j • · : β → β) := swap_smul_self_smul i j

end Swap
end Equiv

open Equiv Function

namespace Set
variable {α : Type*} {f : Perm α} {s : Set α}

lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s := hf.symm f.invOn

Target:
lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4375826) 0) = ((fun x : Nat => x + 4375826) 0) by rfl]
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:59:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4375826) 0\nin the target expression\n  MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s\n\nα : Type u_1\nf : Perm α\ns : Set α\n⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.End
public import Mathlib.Logic.Equiv.Set
public import Mathlib.Tactic.Common

Namespace:
Set

Local safe context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
/-!
# Extra lemmas about permutations

This file proves miscellaneous lemmas about `Equiv.Perm`.

## TODO

Most of the content of this file was moved to `Mathlib/Algebra/Group/End.lean` in
https://github.com/leanprover-community/mathlib4/pull/22141.
It would be good to merge the remaining lemmas with other files, e.g.
`GroupTheory.Perm.ViaEmbedding` looks like it could benefit from such a treatment (splitting into
the algebra and non-algebra parts).
-/

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

@[simp]
theorem swap_smul_self_smul [MulAction (Perm α) β] (i j : α) (x : β) :
    swap i j • swap i j • x = x := by simp [smul_smul]

theorem swap_smul_involutive [MulAction (Perm α) β] (i j : α) :
    Function.Involutive (swap i j • · : β → β) := swap_smul_self_smul i j

end Swap
end Equiv

open Equiv Function

namespace Set
variable {α : Type*} {f : Perm α} {s : Set α}

lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s := hf.symm f.invOn

Target:
lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4375826) 0) = ((fun x : Nat => x + 4375826) 0) by rfl]
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:59:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4375826) 0\nin the target expression\n  MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s\n\nα : Type u_1\nf : Perm α\ns : Set α\n⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s", "returncode": 1}

Corrected proof:


<LEAN_FILE>:67:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_40fbfea3cc`
<LEAN_FILE>:42:0: error: unsolved goals
A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
⊢ Module.finrank k V = 1

Response:
by
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
/-!
# Basic facts about algebra representations

This file collects basic general facts about algebra representations. The purpose of this file is
to have general results so that when we prove a corresponding fact about group representations
(or Lie algebra representations etc), we can deduce them as special cases of facts from this file.

-/

public section

variable {A V : Type*} (k : Type*) [Field k] [Ring A] [Algebra k A] [AddCommGroup V] [Module k V]
  [Module A V] [IsScalarTower k A V]
  [IsSimpleModule A V] [FiniteDimensional k V] [IsAlgClosed k]

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) := by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

open scoped IsMulCommutative in
/-- Any finite-dimensional irreducible representation of a commutative algebra over an algebraically
closed field is one-dimensional. -/

Target:
theorem IsSimpleModule.finrank_eq_one_of_isMulCommutative [IsMulCommutative A] :
    Module.finrank k V = 1 :=

Failed candidate:
by
  have __repair_pad_0_40fbfea3cc : True := by trivial
  have __repair_pad_1_40fbfea3cc : True := by trivial
  have __repair_pad_2_40fbfea3cc : True := by trivial
  have __repair_pad_3_40fbfea3cc : True := by trivial
  have __repair_pad_4_40fbfea3cc : True := by trivial
  have __repair_pad_5_40fbfea3cc : True := by trivial
  have __repair_pad_6_40fbfea3cc : True := by trivial
  have __repair_pad_7_40fbfea3cc : True := by trivial
  have __repair_pad_8_40fbfea3cc : True := by trivial
  have __repair_pad_9_40fbfea3cc : True := by trivial
  have __repair_pad_10_40fbfea3cc : True := by trivial
  have __repair_pad_11_40fbfea3cc : True := by trivial
  have __repair_pad_12_40fbfea3cc : True := by trivial
  have __repair_pad_13_40fbfea3cc : True := by trivial
  have __repair_pad_14_40fbfea3cc : True := by trivial
  have __repair_pad_15_40fbfea3cc : True := by trivial
  have __repair_pad_16_40fbfea3cc : True := by trivial
  have __repair_pad_17_40fbfea3cc : True := by trivial
  have __repair_pad_18_40fbfea3cc : True := by trivial
  have __repair_pad_19_40fbfea3cc : True := by trivial
  have __repair_pad_20_40fbfea3cc : True := by trivial
  have __repair_pad_21_40fbfea3cc : True := by trivial
  have __repair_pad_22_40fbfea3cc : True := by trivial
  have __repair_pad_23_40fbfea3cc : True := by trivial
  have __repair_fail_40fbfea3cc := __oprover_overlong_40fbfea3cc
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:67:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_40fbfea3cc`\n<LEAN_FILE>:42:0: error: unsolved goals\nA : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : Ring A\ninst✝⁸ : Algebra k A\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module A V\ninst✝⁴ : IsScalarTower k A V\ninst✝³ : IsSimpleModule A V\ninst✝² : FiniteDimensional k V\ninst✝¹ : IsAlgClosed k\ninst✝ : IsMulCommutative A\n⊢ Module.finrank k V = 1", "returncode": 1}

Corrected proof:

[Assistant]
by
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
/-!
# Basic facts about algebra representations

This file collects basic general facts about algebra representations. The purpose of this file is
to have general results so that when we prove a corresponding fact about group representations
(or Lie algebra representations etc), we can deduce them as special cases of facts from this file.

-/

public section

variable {A V : Type*} (k : Type*) [Field k] [Ring A] [Algebra k A] [AddCommGroup V] [Module k V]
  [Module A V] [IsScalarTower k A V]
  [IsSimpleModule A V] [FiniteDimensional k V] [IsAlgClosed k]

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) := by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

open scoped IsMulCommutative in
/-- Any finite-dimensional irreducible representation of a commutative algebra over an algebraically
closed field is one-dimensional. -/

Target:
theorem IsSimpleModule.finrank_eq_one_of_isMulCommutative [IsMulCommutative A] :
    Module.finrank k V = 1 :=

Failed candidate:
by
  have __repair_pad_0_40fbfea3cc : True := by trivial
  have __repair_pad_1_40fbfea3cc : True := by trivial
  have __repair_pad_2_40fbfea3cc : True := by trivial
  have __repair_pad_3_40fbfea3cc : True := by trivial
  have __repair_pad_4_40fbfea3cc : True := by trivial
  have __repair_pad_5_40fbfea3cc : True := by trivial
  have __repair_pad_6_40fbfea3cc : True := by trivial
  have __repair_pad_7_40fbfea3cc : True := by trivial
  have __repair_pad_8_40fbfea3cc : True := by trivial
  have __repair_pad_9_40fbfea3cc : True := by trivial
  have __repair_pad_10_40fbfea3cc : True := by trivial
  have __repair_pad_11_40fbfea3cc : True := by trivial
  have __repair_pad_12_40fbfea3cc : True := by trivial
  have __repair_pad_13_40fbfea3cc : True := by trivial
  have __repair_pad_14_40fbfea3cc : True := by trivial
  have __repair_pad_15_40fbfea3cc : True := by trivial
  have __repair_pad_16_40fbfea3cc : True := by trivial
  have __repair_pad_17_40fbfea3cc : True := by trivial
  have __repair_pad_18_40fbfea3cc : True := by trivial
  have __repair_pad_19_40fbfea3cc : True := by trivial
  have __repair_pad_20_40fbfea3cc : True := by trivial
  have __repair_pad_21_40fbfea3cc : True := by trivial
  have __repair_pad_22_40fbfea3cc : True := by trivial
  have __repair_pad_23_40fbfea3cc : True := by trivial
  have __repair_fail_40fbfea3cc := __oprover_overlong_40fbfea3cc
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:67:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_40fbfea3cc`\n<LEAN_FILE>:42:0: error: unsolved goals\nA : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : Ring A\ninst✝⁸ : Algebra k A\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module A V\ninst✝⁴ : IsScalarTower k A V\ninst✝³ : IsSimpleModule A V\ninst✝² : FiniteDimensional k V\ninst✝¹ : IsAlgClosed k\ninst✝ : IsMulCommutative A\n⊢ Module.finrank k V = 1", "returncode": 1}

Corrected proof: