Response:
by
  simp only [BiTotal, leftTotal_empty, rightTotal_iff_isEmpty_right, true_and]

[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.Logic.Function.Basic
public import Mathlib.Logic.IsEmpty.Defs
public import Mathlib.Logic.Relator

Namespace:
(root)

Local context:
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
/-!
In this file we prove some basic properties about the typeclass `IsEmpty`.
-/

public section

variable {α β γ : Sort*}

@[simp, push]
theorem not_nonempty_iff : ¬Nonempty α ↔ IsEmpty α :=
  ⟨fun h ↦ ⟨fun x ↦ h ⟨x⟩⟩, fun h1 h2 ↦ h2.elim h1.elim⟩

@[simp, push]
theorem not_isEmpty_iff : ¬IsEmpty α ↔ Nonempty α :=
  not_iff_comm.mp not_nonempty_iff

@[simp]
theorem isEmpty_Prop {p : Prop} : IsEmpty p ↔ ¬p := by
  simp only [← not_nonempty_iff, nonempty_prop]

@[simp]
theorem isEmpty_pi {π : α → Sort*} : IsEmpty (∀ a, π a) ↔ ∃ a, IsEmpty (π a) := by
  simp only [← not_nonempty_iff, Classical.nonempty_pi, not_forall]

theorem isEmpty_fun : IsEmpty (α → β) ↔ Nonempty α ∧ IsEmpty β := by
  rw [isEmpty_pi, ← exists_true_iff_nonempty, ← exists_and_right, true_and]

@[simp]
theorem nonempty_fun : Nonempty (α → β) ↔ IsEmpty α ∨ Nonempty β :=
  not_iff_not.mp <| by rw [not_or, not_nonempty_iff, not_nonempty_iff, isEmpty_fun, not_isEmpty_iff]

@[simp]
theorem isEmpty_sigma {α} {E : α → Type*} : IsEmpty (Sigma E) ↔ ∀ a, IsEmpty (E a) := by
  simp only [← not_nonempty_iff, nonempty_sigma, not_exists]

@[simp]
theorem isEmpty_psigma {α} {E : α → Sort*} : IsEmpty (PSigma E) ↔ ∀ a, IsEmpty (E a) := by
  simp only [← not_nonempty_iff, nonempty_psigma, not_exists]

theorem isEmpty_subtype (p : α → Prop) : IsEmpty (Subtype p) ↔ ∀ x, ¬p x := by
  simp only [← not_nonempty_iff, nonempty_subtype, not_exists]

@[simp]
theorem isEmpty_prod {α β : Type*} : IsEmpty (α × β) ↔ IsEmpty α ∨ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_prod, not_and_or]

@[simp]
theorem isEmpty_pprod : IsEmpty (PProd α β) ↔ IsEmpty α ∨ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_pprod, not_and_or]

@[simp]
theorem isEmpty_sum {α β} : IsEmpty (α ⊕ β) ↔ IsEmpty α ∧ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_sum, not_or]

@[simp]
theorem isEmpty_psum {α β} : IsEmpty (α ⊕' β) ↔ IsEmpty α ∧ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_psum, not_or]

@[simp]
theorem isEmpty_ulift {α} : IsEmpty (ULift α) ↔ IsEmpty α := by
  simp only [← not_nonempty_iff, nonempty_ulift]

@[simp]
theorem isEmpty_plift {α} : IsEmpty (PLift α) ↔ IsEmpty α := by
  simp only [← not_nonempty_iff, nonempty_plift]

theorem wellFounded_of_isEmpty {α} [IsEmpty α] (r : α → α → Prop) : WellFounded r :=
  ⟨isEmptyElim⟩

variable (α)

theorem isEmpty_or_nonempty : IsEmpty α ∨ Nonempty α :=
  (em <| IsEmpty α).elim Or.inl <| Or.inr ∘ not_isEmpty_iff.mp

@[simp]
theorem not_isEmpty_of_nonempty [h : Nonempty α] : ¬IsEmpty α :=
  not_isEmpty_iff.mpr h

variable {α}

theorem Function.extend_of_isEmpty [IsEmpty α] (f : α → β) (g : α → γ) (h : β → γ) :
    Function.extend f g h = h :=
  funext fun _ ↦ (Function.extend_apply' _ _ _) fun ⟨a, _⟩ ↦ isEmptyElim a

open Relator

variable {α β : Type*} (R : α → β → Prop)

@[simp]
theorem leftTotal_empty [IsEmpty α] : LeftTotal R := by
  simp only [LeftTotal, IsEmpty.forall_iff]

theorem leftTotal_iff_isEmpty_left [IsEmpty β] : LeftTotal R ↔ IsEmpty α := by
  simp only [LeftTotal, IsEmpty.exists_iff, isEmpty_iff]

@[simp]
theorem rightTotal_empty [IsEmpty β] : RightTotal R := by
  simp only [RightTotal, IsEmpty.forall_iff]

theorem rightTotal_iff_isEmpty_right [IsEmpty α] : RightTotal R ↔ IsEmpty β := by
  simp only [RightTotal, IsEmpty.exists_iff, isEmpty_iff]

@[simp]
theorem biTotal_empty [IsEmpty α] [IsEmpty β] : BiTotal R :=
  ⟨leftTotal_empty R, rightTotal_empty R⟩

Target:
theorem biTotal_iff_isEmpty_right [IsEmpty α] : BiTotal R ↔ IsEmpty β :=

Proof body:

[Assistant]
by
  simp only [BiTotal, leftTotal_empty, rightTotal_iff_isEmpty_right, true_and]

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.Logic.Function.Basic
public import Mathlib.Logic.IsEmpty.Defs
public import Mathlib.Logic.Relator

Namespace:
(root)

Local context:
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
/-!
In this file we prove some basic properties about the typeclass `IsEmpty`.
-/

public section

variable {α β γ : Sort*}

@[simp, push]
theorem not_nonempty_iff : ¬Nonempty α ↔ IsEmpty α :=
  ⟨fun h ↦ ⟨fun x ↦ h ⟨x⟩⟩, fun h1 h2 ↦ h2.elim h1.elim⟩

@[simp, push]
theorem not_isEmpty_iff : ¬IsEmpty α ↔ Nonempty α :=
  not_iff_comm.mp not_nonempty_iff

@[simp]
theorem isEmpty_Prop {p : Prop} : IsEmpty p ↔ ¬p := by
  simp only [← not_nonempty_iff, nonempty_prop]

@[simp]
theorem isEmpty_pi {π : α → Sort*} : IsEmpty (∀ a, π a) ↔ ∃ a, IsEmpty (π a) := by
  simp only [← not_nonempty_iff, Classical.nonempty_pi, not_forall]

theorem isEmpty_fun : IsEmpty (α → β) ↔ Nonempty α ∧ IsEmpty β := by
  rw [isEmpty_pi, ← exists_true_iff_nonempty, ← exists_and_right, true_and]

@[simp]
theorem nonempty_fun : Nonempty (α → β) ↔ IsEmpty α ∨ Nonempty β :=
  not_iff_not.mp <| by rw [not_or, not_nonempty_iff, not_nonempty_iff, isEmpty_fun, not_isEmpty_iff]

@[simp]
theorem isEmpty_sigma {α} {E : α → Type*} : IsEmpty (Sigma E) ↔ ∀ a, IsEmpty (E a) := by
  simp only [← not_nonempty_iff, nonempty_sigma, not_exists]

@[simp]
theorem isEmpty_psigma {α} {E : α → Sort*} : IsEmpty (PSigma E) ↔ ∀ a, IsEmpty (E a) := by
  simp only [← not_nonempty_iff, nonempty_psigma, not_exists]

theorem isEmpty_subtype (p : α → Prop) : IsEmpty (Subtype p) ↔ ∀ x, ¬p x := by
  simp only [← not_nonempty_iff, nonempty_subtype, not_exists]

@[simp]
theorem isEmpty_prod {α β : Type*} : IsEmpty (α × β) ↔ IsEmpty α ∨ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_prod, not_and_or]

@[simp]
theorem isEmpty_pprod : IsEmpty (PProd α β) ↔ IsEmpty α ∨ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_pprod, not_and_or]

@[simp]
theorem isEmpty_sum {α β} : IsEmpty (α ⊕ β) ↔ IsEmpty α ∧ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_sum, not_or]

@[simp]
theorem isEmpty_psum {α β} : IsEmpty (α ⊕' β) ↔ IsEmpty α ∧ IsEmpty β := by
  simp only [← not_nonempty_iff, nonempty_psum, not_or]

@[simp]
theorem isEmpty_ulift {α} : IsEmpty (ULift α) ↔ IsEmpty α := by
  simp only [← not_nonempty_iff, nonempty_ulift]

@[simp]
theorem isEmpty_plift {α} : IsEmpty (PLift α) ↔ IsEmpty α := by
  simp only [← not_nonempty_iff, nonempty_plift]

theorem wellFounded_of_isEmpty {α} [IsEmpty α] (r : α → α → Prop) : WellFounded r :=
  ⟨isEmptyElim⟩

variable (α)

theorem isEmpty_or_nonempty : IsEmpty α ∨ Nonempty α :=
  (em <| IsEmpty α).elim Or.inl <| Or.inr ∘ not_isEmpty_iff.mp

@[simp]
theorem not_isEmpty_of_nonempty [h : Nonempty α] : ¬IsEmpty α :=
  not_isEmpty_iff.mpr h

variable {α}

theorem Function.extend_of_isEmpty [IsEmpty α] (f : α → β) (g : α → γ) (h : β → γ) :
    Function.extend f g h = h :=
  funext fun _ ↦ (Function.extend_apply' _ _ _) fun ⟨a, _⟩ ↦ isEmptyElim a

open Relator

variable {α β : Type*} (R : α → β → Prop)

@[simp]
theorem leftTotal_empty [IsEmpty α] : LeftTotal R := by
  simp only [LeftTotal, IsEmpty.forall_iff]

theorem leftTotal_iff_isEmpty_left [IsEmpty β] : LeftTotal R ↔ IsEmpty α := by
  simp only [LeftTotal, IsEmpty.exists_iff, isEmpty_iff]

@[simp]
theorem rightTotal_empty [IsEmpty β] : RightTotal R := by
  simp only [RightTotal, IsEmpty.forall_iff]

theorem rightTotal_iff_isEmpty_right [IsEmpty α] : RightTotal R ↔ IsEmpty β := by
  simp only [RightTotal, IsEmpty.exists_iff, isEmpty_iff]

@[simp]
theorem biTotal_empty [IsEmpty α] [IsEmpty β] : BiTotal R :=
  ⟨leftTotal_empty R, rightTotal_empty R⟩

Target:
theorem biTotal_iff_isEmpty_right [IsEmpty α] : BiTotal R ↔ IsEmpty β :=

Proof body:


<LEAN_FILE>:79:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

[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.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

Local safe context:
/-
Copyright (c) 2026 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Adam Topaz, Adrian Marti
-/
/-!

# Profunctors

A profunctor from a category `C` to a category `D` is a functor from `C` to a category of
presheaves of sets on `D`. We define this as `Profunctor.{w} C D := C ⥤ Dᵒᵖ ⥤ Type w`.

This file provides convenient constructors `ProfunctorCore` and `ProfunctorCore.Hom` for profunctors
and natural transformations between them. We also define the identity profunctor `Profunctor.id`
as the Yoneda bifunctor, the opposite of a profunctor, the `ulift` of a profunctor, whiskering
of a profunctor with functors, and the profunctors in both directions corresponding to a functor
`C ⥤ D` (see `Functor.toProfunctor` and `Functor.toProfunctorFlip`).

## Future work

- Define composition of profunctors.
- Define the bicategory of categories where the 1-morphisms are profunctors.
-/

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

variable (C : Type u₁) (D : Type u₂) [Category.{v₁} C] [Category.{v₂} D]

/-- Custom structure to construct profunctors, i.e. bifunctors `C ⥤ Dᵒᵖ ⥤ Type w`. -/
@[pp_with_univ]
structure ProfunctorCore where
  /-- The object part -/
  obj : C → D → Type w
  /-- The morphism part -/
  map {X X' : C} {Y Y' : D} (f : X ⟶ X') (g : Y ⟶ Y') : obj X Y' ⟶ obj X' Y
  map_id (X : C) (Y : D) : map (𝟙 X) (𝟙 Y) = 𝟙 _ := by cat_disch
  map_comp {X₁ X₂ X₃ : C} {Y₁ Y₂ Y₃ : D} (f : X₁ ⟶ X₂) (f' : X₂ ⟶ X₃) (g : Y₁ ⟶ Y₂) (g' : Y₂ ⟶ Y₃) :
    map (f ≫ f') (g ≫ g') = map f g' ≫ map f' g := by cat_disch

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

/-- A profunctor from C to D (`Profunctor.{w} C D`) is a bifunctor `C ⥤ Dᵒᵖ ⥤ Type w`. -/
@[pp_with_univ]
abbrev Profunctor := C ⥤ Dᵒᵖ ⥤ Type w

variable {C D}

/-- Typecheck a bifunctor `C ⥤ Dᵒᵖ ⥤ Type w` as a profunctor. -/
abbrev Functor.profunctor (F : C ⥤ Dᵒᵖ ⥤ Type w) : Profunctor.{w} C D := F

namespace ProfunctorCore

/-- Custom structure to construct natural transformations between profunctors, see
`CategoryTheory.Profunctor.ofHom`. -/
structure Hom (P Q : ProfunctorCore.{w} C D) where
  /-- The components of the natural transformation -/
  app (X : C) (Y : D) : P.obj X Y ⟶ Q.obj X Y
  naturality ⦃X X' : C⦄ ⦃Y Y' : D⦄ (f : X ⟶ X') (g : Y ⟶ Y') :
    P.map f g ≫ app X' Y = app X Y' ≫ Q.map f g := by cat_disch

attribute [reassoc (attr := simp)] ProfunctorCore.Hom.naturality

@[simp]

Target:
lemma map_id_comp (P : ProfunctorCore.{w} C D) (X : C) {Y Y' Y'' : D}
    (g : Y ⟶ Y') (g' : Y' ⟶ Y'') :
    P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g :=

Failed candidate:
by
  let __repair_2549c5226e : Fact False := inferInstance
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:79:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

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.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

Local safe context:
/-
Copyright (c) 2026 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Adam Topaz, Adrian Marti
-/
/-!

# Profunctors

A profunctor from a category `C` to a category `D` is a functor from `C` to a category of
presheaves of sets on `D`. We define this as `Profunctor.{w} C D := C ⥤ Dᵒᵖ ⥤ Type w`.

This file provides convenient constructors `ProfunctorCore` and `ProfunctorCore.Hom` for profunctors
and natural transformations between them. We also define the identity profunctor `Profunctor.id`
as the Yoneda bifunctor, the opposite of a profunctor, the `ulift` of a profunctor, whiskering
of a profunctor with functors, and the profunctors in both directions corresponding to a functor
`C ⥤ D` (see `Functor.toProfunctor` and `Functor.toProfunctorFlip`).

## Future work

- Define composition of profunctors.
- Define the bicategory of categories where the 1-morphisms are profunctors.
-/

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

variable (C : Type u₁) (D : Type u₂) [Category.{v₁} C] [Category.{v₂} D]

/-- Custom structure to construct profunctors, i.e. bifunctors `C ⥤ Dᵒᵖ ⥤ Type w`. -/
@[pp_with_univ]
structure ProfunctorCore where
  /-- The object part -/
  obj : C → D → Type w
  /-- The morphism part -/
  map {X X' : C} {Y Y' : D} (f : X ⟶ X') (g : Y ⟶ Y') : obj X Y' ⟶ obj X' Y
  map_id (X : C) (Y : D) : map (𝟙 X) (𝟙 Y) = 𝟙 _ := by cat_disch
  map_comp {X₁ X₂ X₃ : C} {Y₁ Y₂ Y₃ : D} (f : X₁ ⟶ X₂) (f' : X₂ ⟶ X₃) (g : Y₁ ⟶ Y₂) (g' : Y₂ ⟶ Y₃) :
    map (f ≫ f') (g ≫ g') = map f g' ≫ map f' g := by cat_disch

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

/-- A profunctor from C to D (`Profunctor.{w} C D`) is a bifunctor `C ⥤ Dᵒᵖ ⥤ Type w`. -/
@[pp_with_univ]
abbrev Profunctor := C ⥤ Dᵒᵖ ⥤ Type w

variable {C D}

/-- Typecheck a bifunctor `C ⥤ Dᵒᵖ ⥤ Type w` as a profunctor. -/
abbrev Functor.profunctor (F : C ⥤ Dᵒᵖ ⥤ Type w) : Profunctor.{w} C D := F

namespace ProfunctorCore

/-- Custom structure to construct natural transformations between profunctors, see
`CategoryTheory.Profunctor.ofHom`. -/
structure Hom (P Q : ProfunctorCore.{w} C D) where
  /-- The components of the natural transformation -/
  app (X : C) (Y : D) : P.obj X Y ⟶ Q.obj X Y
  naturality ⦃X X' : C⦄ ⦃Y Y' : D⦄ (f : X ⟶ X') (g : Y ⟶ Y') :
    P.map f g ≫ app X' Y = app X Y' ≫ Q.map f g := by cat_disch

attribute [reassoc (attr := simp)] ProfunctorCore.Hom.naturality

@[simp]

Target:
lemma map_id_comp (P : ProfunctorCore.{w} C D) (X : C) {Y Y' Y'' : D}
    (g : Y ⟶ Y') (g' : Y' ⟶ Y'') :
    P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g :=

Failed candidate:
by
  let __repair_2549c5226e : Fact False := inferInstance
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:79:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:124:43: error: unsolved goals
A : Type u_1
inst✝ : Ring A
F : IdealFilter A
I : Ideal A
⊢ False

Response:
by
  intro x hx
  obtain ⟨J, hJ⟩ := F.nonempty
  exact ⟨J, hJ, le_of_le_of_eq le_top (by simpa [eq_comm])⟩

[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.Order.PFilter
public import Mathlib.RingTheory.Ideal.Colon

Namespace:
IdealFilter

Local safe context:
/-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
/-!
# Ideal Filters

An **ideal filter** is a filter in the lattice of ideals of a ring `A`.

## Main definitions

* `IdealFilter A`: the type of an ideal filter on a ring `A`.
* `IsUniform F` : a filter `F` is uniform if whenever `I` is an ideal in the filter, then for all
  `a : A`, the colon ideal `I.colon {a}` is in `F`.
* `IsTorsionElem` : Given a filter `F`, an element, `m`, of an `A`-module, `M`, is `F`-torsion if
  there exists an ideal `L` in `F` that annihilates `m`.
* `IsTorsion` : Given a filter `F`, an `A`-module, `M`, is `F`-torsion if every element is torsion.
* `gabrielComposition` : Given two filters `F` and `G`, the Gabriel composition of `F` and `G` is
  the set of ideals `L` of `A` such that there exists an ideal `K` in `G` with `K/L` `F`-torsion.
  This is again a filter.
* `IsGabriel F` : a filter `F` is a Gabriel filter if it is uniform and satisfies axiom T4:
  for all `I : Ideal A`, if there exists `J ∈ F` such that for all `a ∈ J` the colon ideal
  `I.colon {a}` is in `F`, then `I ∈ F`.

## Main statements

* `isGabriel_iff`: a filter is Gabriel iff it is uniform and `F • F = F`.

## Notation

* `F • G`: the Gabriel composition of ideal filters `F` and `G`, defined by
  `gabrielComposition F G`.

## Implementation notes

In the classical literature (e.g. Stenström), *right linear topologies* on a ring are often
described via filters of open **right** ideals, and the terminology is frequently abused by
identifying the topology with its filter of ideals.

In this development we work systematically with **left ideals**. Accordingly, Stenström’s
right-ideal construction `(L : a) = {x ∈ A | a * x ∈ L}` is replaced by the left ideal
`L.colon {a} = {a | x * a ∈ L}`.

With this convention, uniform filters correspond to linear (additive) topologies, while the
additional Gabriel condition (axiom T4) imposes an algebraic saturation property that does not
change the induced topology.

## References

* [Bo Stenström, *Rings and Modules of Quotients*][stenstrom1971]
* [Bo Stenström, *Rings of Quotients*][stenstrom1975]
* [nLab: Uniform filter](https://ncatlab.org/nlab/show/uniform+filter)
* [nLab: Gabriel filter](https://ncatlab.org/nlab/show/Gabriel+filter)
* [nLab: Gabriel composition](https://ncatlab.org/nlab/show/Gabriel+composition+of+filters)

## Tags

ring theory, ideal, filter, uniform filter, Gabriel filter, torsion theory
-/

@[expose] public section

open scoped Pointwise

/-- `IdealFilter A` is the type of `Order.PFilter`s on the lattice of ideals of `A`. -/
abbrev IdealFilter (A : Type*) [Ring A] := Order.PFilter (Ideal A)

namespace IdealFilter

variable {A : Type*} [Ring A]

/-- A filter of ideals is *uniform* if it is closed under colon by singletons. -/
class IsUniform (F : IdealFilter A) : Prop where
  /-- **Axiom T3.**  See [stenstrom1975]. -/
  colon_mem {I : Ideal A} (hI : I ∈ F) (a : A) : I.colon {a} ∈ F

/-- We say that an element `m : M` is `F`-torsion if it is annihilated by some ideal belonging to
the filter `F`. -/
def IsTorsionElem (F : IdealFilter A)
    {M : Type*} [AddCommMonoid M] [Module A M] (m : M) : Prop :=
  ∃ L ∈ F, ∀ a ∈ L, a • m = 0

/-- Module-level `F`-torsion: every element is `F`-torsion. -/
def IsTorsion (F : IdealFilter A)
    (M : Type*) [AddCommMonoid M] [Module A M] : Prop :=
  ∀ m : M, IsTorsionElem F m

/-- We say that the quotient `K/L` is `F`-torsion if every element `k ∈ K` is annihilated
(modulo `L`) by some ideal in `F`. Equivalently, for each `k ∈ K` there exists `I ∈ F`
such that `I ≤ L.colon {k}`. This formulation avoids forming the quotient module explicitly. -/
def IsTorsionQuot (F : IdealFilter A) (L K : Ideal A) : Prop :=
  ∀ k ∈ K, ∃ I ∈ F, I ≤ L.colon {k}

/-- Intersecting the left ideal with `K` does not change `IsTorsionQuot` on the right.
In particular, `IsTorsionQuot F L K` need not require `L ≤ K` for it is equivalent to asserting
the quotient `K / (L ⊓ K)` is `F`-torsion. -/
lemma isTorsionQuot_inter_left_iff {F : IdealFilter A} {L K : Ideal A} :
    IsTorsionQuot F (L ⊓ K) K ↔ IsTorsionQuot F L K := by
  constructor <;>
  · intro h k hk
    rcases h k hk with ⟨I, hI, hI_le⟩
    have hcol : (L ⊓ K).colon {k} = Submodule.colon L {k} :=
      Submodule.colon_inf_eq_left_of_subset (Set.singleton_subset_iff.mpr hk)
    exact ⟨I, hI, (by simpa [hcol] using hI_le)⟩

@[simp] lemma isTorsion_def (F : IdealFilter A) (M : Type*) [AddCommMonoid M] [Module A M] :
    IsTorsion F M ↔ ∀ m : M, IsTorsionElem F m :=
  Iff.rfl

@[simp] lemma isTorsionQuot_def {F : IdealFilter A} {L K : Ideal A} :
    IsTorsionQuot F L K ↔ ∀ k ∈ (K : Set A), ∃ I ∈ F, I ≤ L.colon {k} :=
  Iff.rfl

Target:
lemma isTorsionQuot_self (F : IdealFilter A) (I : Ideal A) :
    IsTorsionQuot F I I :=

Failed candidate:
by
  have __repair_plan_dfbf77001a : False := by
    skip
  intro x hx
  obtain ⟨J, hJ⟩ := F.nonempty
  exact ⟨J, hJ, le_of_le_of_eq le_top (by simpa [eq_comm])⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:124:43: error: unsolved goals\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  intro x hx
  obtain ⟨J, hJ⟩ := F.nonempty
  exact ⟨J, hJ, le_of_le_of_eq le_top (by simpa [eq_comm])⟩

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.Order.PFilter
public import Mathlib.RingTheory.Ideal.Colon

Namespace:
IdealFilter

Local safe context:
/-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
/-!
# Ideal Filters

An **ideal filter** is a filter in the lattice of ideals of a ring `A`.

## Main definitions

* `IdealFilter A`: the type of an ideal filter on a ring `A`.
* `IsUniform F` : a filter `F` is uniform if whenever `I` is an ideal in the filter, then for all
  `a : A`, the colon ideal `I.colon {a}` is in `F`.
* `IsTorsionElem` : Given a filter `F`, an element, `m`, of an `A`-module, `M`, is `F`-torsion if
  there exists an ideal `L` in `F` that annihilates `m`.
* `IsTorsion` : Given a filter `F`, an `A`-module, `M`, is `F`-torsion if every element is torsion.
* `gabrielComposition` : Given two filters `F` and `G`, the Gabriel composition of `F` and `G` is
  the set of ideals `L` of `A` such that there exists an ideal `K` in `G` with `K/L` `F`-torsion.
  This is again a filter.
* `IsGabriel F` : a filter `F` is a Gabriel filter if it is uniform and satisfies axiom T4:
  for all `I : Ideal A`, if there exists `J ∈ F` such that for all `a ∈ J` the colon ideal
  `I.colon {a}` is in `F`, then `I ∈ F`.

## Main statements

* `isGabriel_iff`: a filter is Gabriel iff it is uniform and `F • F = F`.

## Notation

* `F • G`: the Gabriel composition of ideal filters `F` and `G`, defined by
  `gabrielComposition F G`.

## Implementation notes

In the classical literature (e.g. Stenström), *right linear topologies* on a ring are often
described via filters of open **right** ideals, and the terminology is frequently abused by
identifying the topology with its filter of ideals.

In this development we work systematically with **left ideals**. Accordingly, Stenström’s
right-ideal construction `(L : a) = {x ∈ A | a * x ∈ L}` is replaced by the left ideal
`L.colon {a} = {a | x * a ∈ L}`.

With this convention, uniform filters correspond to linear (additive) topologies, while the
additional Gabriel condition (axiom T4) imposes an algebraic saturation property that does not
change the induced topology.

## References

* [Bo Stenström, *Rings and Modules of Quotients*][stenstrom1971]
* [Bo Stenström, *Rings of Quotients*][stenstrom1975]
* [nLab: Uniform filter](https://ncatlab.org/nlab/show/uniform+filter)
* [nLab: Gabriel filter](https://ncatlab.org/nlab/show/Gabriel+filter)
* [nLab: Gabriel composition](https://ncatlab.org/nlab/show/Gabriel+composition+of+filters)

## Tags

ring theory, ideal, filter, uniform filter, Gabriel filter, torsion theory
-/

@[expose] public section

open scoped Pointwise

/-- `IdealFilter A` is the type of `Order.PFilter`s on the lattice of ideals of `A`. -/
abbrev IdealFilter (A : Type*) [Ring A] := Order.PFilter (Ideal A)

namespace IdealFilter

variable {A : Type*} [Ring A]

/-- A filter of ideals is *uniform* if it is closed under colon by singletons. -/
class IsUniform (F : IdealFilter A) : Prop where
  /-- **Axiom T3.**  See [stenstrom1975]. -/
  colon_mem {I : Ideal A} (hI : I ∈ F) (a : A) : I.colon {a} ∈ F

/-- We say that an element `m : M` is `F`-torsion if it is annihilated by some ideal belonging to
the filter `F`. -/
def IsTorsionElem (F : IdealFilter A)
    {M : Type*} [AddCommMonoid M] [Module A M] (m : M) : Prop :=
  ∃ L ∈ F, ∀ a ∈ L, a • m = 0

/-- Module-level `F`-torsion: every element is `F`-torsion. -/
def IsTorsion (F : IdealFilter A)
    (M : Type*) [AddCommMonoid M] [Module A M] : Prop :=
  ∀ m : M, IsTorsionElem F m

/-- We say that the quotient `K/L` is `F`-torsion if every element `k ∈ K` is annihilated
(modulo `L`) by some ideal in `F`. Equivalently, for each `k ∈ K` there exists `I ∈ F`
such that `I ≤ L.colon {k}`. This formulation avoids forming the quotient module explicitly. -/
def IsTorsionQuot (F : IdealFilter A) (L K : Ideal A) : Prop :=
  ∀ k ∈ K, ∃ I ∈ F, I ≤ L.colon {k}

/-- Intersecting the left ideal with `K` does not change `IsTorsionQuot` on the right.
In particular, `IsTorsionQuot F L K` need not require `L ≤ K` for it is equivalent to asserting
the quotient `K / (L ⊓ K)` is `F`-torsion. -/
lemma isTorsionQuot_inter_left_iff {F : IdealFilter A} {L K : Ideal A} :
    IsTorsionQuot F (L ⊓ K) K ↔ IsTorsionQuot F L K := by
  constructor <;>
  · intro h k hk
    rcases h k hk with ⟨I, hI, hI_le⟩
    have hcol : (L ⊓ K).colon {k} = Submodule.colon L {k} :=
      Submodule.colon_inf_eq_left_of_subset (Set.singleton_subset_iff.mpr hk)
    exact ⟨I, hI, (by simpa [hcol] using hI_le)⟩

@[simp] lemma isTorsion_def (F : IdealFilter A) (M : Type*) [AddCommMonoid M] [Module A M] :
    IsTorsion F M ↔ ∀ m : M, IsTorsionElem F m :=
  Iff.rfl

@[simp] lemma isTorsionQuot_def {F : IdealFilter A} {L K : Ideal A} :
    IsTorsionQuot F L K ↔ ∀ k ∈ (K : Set A), ∃ I ∈ F, I ≤ L.colon {k} :=
  Iff.rfl

Target:
lemma isTorsionQuot_self (F : IdealFilter A) (I : Ideal A) :
    IsTorsionQuot F I I :=

Failed candidate:
by
  have __repair_plan_dfbf77001a : False := by
    skip
  intro x hx
  obtain ⟨J, hJ⟩ := F.nonempty
  exact ⟨J, hJ, le_of_le_of_eq le_top (by simpa [eq_comm])⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:124:43: error: unsolved goals\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\n⊢ False", "returncode": 1}

Corrected proof:


Response:
by
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

[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.WittVector.Complete
public import Mathlib.RingTheory.WittVector.Teichmuller

Namespace:
WittVector

Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
/-!
# Teichmuller Series

Let `R` be a characteristic `p` perfect ring. In this file, we show that
every element `x` of the Witt vectors `𝕎 R` can be written as the
(`p`-adic) summation of Teichmuller series, namely
`∑ i, (teichmuller p
        (((frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i)`

## Main theorems

* `WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff` : `p ^ (n + 1)` divides
  `x` minus the summation of the first `n + 1` terms of the Teichmuller series.
* `WittVector.eq_of_apply_teichmuller_eq` : Given a ring `S` such that `p` is nilpotent in `S`
  and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
  then they are equal.

## TODO
Show that the Teichmuller series is unique.
-/

public section

open Ideal Quotient
namespace WittVector

variable {p : ℕ} [hp : Fact (Nat.Prime p)]

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

Target:
theorem sum_coeff_eq_coeff_sum {α : Type*} {S : Finset α} (x : α → 𝕎 R)
    (h : ∀ (n : ℕ), Subsingleton {r | r ∈ S ∧ (x r).coeff n ≠ 0}) (n : ℕ) :
    (∑ s ∈ S, x s).coeff n = ∑ (s ∈ S), (x s).coeff n :=

Proof body:

[Assistant]
by
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

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.WittVector.Complete
public import Mathlib.RingTheory.WittVector.Teichmuller

Namespace:
WittVector

Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
/-!
# Teichmuller Series

Let `R` be a characteristic `p` perfect ring. In this file, we show that
every element `x` of the Witt vectors `𝕎 R` can be written as the
(`p`-adic) summation of Teichmuller series, namely
`∑ i, (teichmuller p
        (((frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i)`

## Main theorems

* `WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff` : `p ^ (n + 1)` divides
  `x` minus the summation of the first `n + 1` terms of the Teichmuller series.
* `WittVector.eq_of_apply_teichmuller_eq` : Given a ring `S` such that `p` is nilpotent in `S`
  and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
  then they are equal.

## TODO
Show that the Teichmuller series is unique.
-/

public section

open Ideal Quotient
namespace WittVector

variable {p : ℕ} [hp : Fact (Nat.Prime p)]

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

Target:
theorem sum_coeff_eq_coeff_sum {α : Type*} {S : Finset α} (x : α → 𝕎 R)
    (h : ∀ (n : ℕ), Subsingleton {r | r ∈ S ∧ (x r).coeff n ≠ 0}) (n : ℕ) :
    (∑ s ∈ S, x s).coeff n = ∑ (s ∈ S), (x s).coeff n :=

Proof body:


<LEAN_FILE>:149:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 11670215) 0
in the target expression
  s₂ = 0

R : Type u_1
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : IsFormallyReal R
s₁ s₂ : R
hs₁ : IsSumSq s₁
hs₂ : IsSumSq s₂
h : s₁ + s₂ = 0
⊢ s₂ = 0

Response:
by
  simp_all [eq_zero_of_add_right hs₁ hs₂ h]

[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.Ring.SumsOfSquares
public import Mathlib.RingTheory.Nilpotent.Basic

Namespace:
IsFormallyReal

Local safe context:
/-
Copyright (c) 2026 Artie Khovanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Artie Khovanov
-/
/-!
# Formally real rings

A ring `R` is *formally real* if, whenever `∑ i, x i ^ 2 = 0`, in fact `x i = 0` for all `i`.

We define formally real rings in an index-free manner using the inductive predicate
`IsSumNonzeroSq`, which asserts that an element is a finite sum of squares of nonzero elements.
A ring is then formally real if `¬ IsSumNonzeroSq 0`.

## Main declaration

- `IsFormallyReal`: typeclass stating that a ring is formally real.

-/

@[expose] public section

variable {R : Type*}

section IsSumNonzeroSq

/--
The property of being a sum of squares of nonzero elements (S) is defined inductively by:
`a * a : R` is (S) for all nonzero `a`, and
if `s : R` is (S), and `a ≠ 0`, then `a * a + s` is (S).
-/
@[mk_iff]
inductive IsSumNonzeroSq [Mul R] [Add R] [Zero R] : R → Prop
  | sq {a : R} (ha : a ≠ 0) : IsSumNonzeroSq (a * a)
  | sq_add {a s : R} (ha : a ≠ 0) (hs : IsSumNonzeroSq s) : IsSumNonzeroSq (a * a + s)

attribute [aesop 90%] IsSumNonzeroSq.sq

@[aesop 90%]
theorem IsSumNonzeroSq.add [AddMonoid R] [Mul R] {s₁ s₂ : R}
    (h₁ : IsSumNonzeroSq s₁) (h₂ : IsSumNonzeroSq s₂) : IsSumNonzeroSq (s₁ + s₂) := by
  induction h₁ <;> simp_all [sq_add, add_assoc]

theorem IsSumNonzeroSq.isSumSq [AddMonoid R] [Mul R] {s : R}
    (h : IsSumNonzeroSq s) : IsSumSq s := by
  induction h <;> aesop

theorem isSumNonzeroSq_iff_isSumSq [NonUnitalNonAssocSemiring R] {s : R} (hs : s ≠ 0) :
    IsSumNonzeroSq s ↔ IsSumSq s where
  mp := IsSumNonzeroSq.isSumSq
  mpr h := by
    induction h with
    | zero => grind
    | @sq_add a s hs ih =>
    rcases eq_or_ne a 0 with (rfl | ne_a)
    · simp_all
    · rcases eq_or_ne s 0 with (rfl | ne_s)
      · simpa using IsSumNonzeroSq.sq ne_a
      · exact IsSumNonzeroSq.sq_add ne_a (ih ne_s)

alias ⟨_, IsSumSq.isSumNonzeroSq_of_ne_zero⟩ := isSumNonzeroSq_iff_isSumSq

namespace AddSubsemigroup

variable [AddMonoid R] [Mul R] {s : R}

variable (R) in
/-- The subsemigroup of sums of squares of nonzero elements. -/
@[simps]
def sumNonzeroSq : AddSubsemigroup R where
  carrier := {s : R | IsSumNonzeroSq s}
  add_mem' := .add

attribute [norm_cast] coe_sumNonzeroSq

@[simp] theorem mem_sumNonzeroSq : s ∈ sumNonzeroSq R ↔ IsSumNonzeroSq s := .rfl

@[simp]
theorem closure_mul_self : closure {x * x | x ≠ (0 : R)} = sumNonzeroSq R := by
  refine closure_eq_of_le (fun x hx ↦ by aesop) (fun x hx ↦ ?_)
  -- TODO : fix aesop timeout and change to `induction hx <;> aesop`
  induction hx with
  | sq ha => aesop
  | sq_add ha hs ih =>
    -- `aesop` times out
    apply add_mem
    · apply AddSubsemigroup.mem_closure_of_mem
      aesop
    aesop

end AddSubsemigroup

end IsSumNonzeroSq

variable (R) in
/--
A ring is formally real if, whenever `∑ i, x i ^ 2 = 0`, we in fact have `x i = 0` for all `i`.
-/
class IsFormallyReal [AddCommMonoid R] [Mul R] : Prop where
  not_isSumNonzeroSq_zero : ¬ IsSumNonzeroSq (0 : R)

namespace IsFormallyReal

theorem of_eq_zero_of_mul_self_of_eq_zero_of_add [AddCommMonoid R] [Mul R]
    (hz : ∀ {a : R}, a * a = 0 → a = 0)
    (ha : ∀ {s₁ s₂ : R}, IsSumSq s₁ → IsSumSq s₂ → s₁ + s₂ = 0 → s₁ = 0) : IsFormallyReal R where
  not_isSumNonzeroSq_zero := by
    suffices ∀ (x : R), IsSumNonzeroSq x → x ≠ 0 by grind
    intro x hx
    induction hx with
    | sq ha => grind
    | @sq_add b s hb hs ih => grind [ha (IsSumSq.mul_self b) hs.isSumSq]

theorem of_eq_zero_of_eq_zero_of_mul_self_add [NonUnitalNonAssocSemiring R]
    (h : ∀ {s a : R}, IsSumSq s → a * a + s = 0 → a = 0) : IsFormallyReal R where
  not_isSumNonzeroSq_zero := by
    suffices ∀ (x : R), IsSumNonzeroSq x → x ≠ 0 by grind
    intro x hx
    induction hx with
    | sq ha => exact fun hc ↦ ha (h IsSumSq.zero (by simpa using hc))
    | sq_add ha hs ih => grind [hs.isSumSq]

instance [Ring R] [LinearOrder R] [IsStrictOrderedRing R] : IsFormallyReal R :=
  of_eq_zero_of_mul_self_of_eq_zero_of_add mul_self_eq_zero.mp <|
    fun hs₁ hs₂ h ↦ ((add_eq_zero_iff_of_nonneg (IsSumSq.nonneg hs₁) (IsSumSq.nonneg hs₂)).mp h).1

instance [Ring R] [IsFormallyReal R] : IsReduced R := by
  rw [isReduced_iff_pow_one_lt 2 (by lia)]
  intro x hx
  by_contra! hc
  exact not_isSumNonzeroSq_zero <| by simpa [← pow_two, hx] using IsSumNonzeroSq.sq hc

theorem eq_zero_of_add_right [NonUnitalNonAssocSemiring R] [IsFormallyReal R]
    {s₁ s₂ : R} (hs₁ : IsSumSq s₁) (hs₂ : IsSumSq s₂) (h : s₁ + s₂ = 0) : s₁ = 0 := by
  by_contra! h₁
  have h₂ : s₂ ≠ 0 := fun hc ↦ by simp_all
  rw [← isSumNonzeroSq_iff_isSumSq h₁] at hs₁
  rw [← isSumNonzeroSq_iff_isSumSq h₂] at hs₂
  exact not_isSumNonzeroSq_zero (h ▸ IsSumNonzeroSq.add hs₁ hs₂)

Target:
theorem eq_zero_of_add_left [NonUnitalNonAssocSemiring R] [IsFormallyReal R]
    {s₁ s₂ : R} (hs₁ : IsSumSq s₁) (hs₂ : IsSumSq s₂) (h : s₁ + s₂ = 0) : s₂ = 0 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11670215) 0) = ((fun x : Nat => x + 11670215) 0) by rfl]
  simp_all [eq_zero_of_add_right hs₁ hs₂ h]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:149:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11670215) 0\nin the target expression\n  s₂ = 0\n\nR : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : IsFormallyReal R\ns₁ s₂ : R\nhs₁ : IsSumSq s₁\nhs₂ : IsSumSq s₂\nh : s₁ + s₂ = 0\n⊢ s₂ = 0", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp_all [eq_zero_of_add_right hs₁ hs₂ h]

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.Ring.SumsOfSquares
public import Mathlib.RingTheory.Nilpotent.Basic

Namespace:
IsFormallyReal

Local safe context:
/-
Copyright (c) 2026 Artie Khovanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Artie Khovanov
-/
/-!
# Formally real rings

A ring `R` is *formally real* if, whenever `∑ i, x i ^ 2 = 0`, in fact `x i = 0` for all `i`.

We define formally real rings in an index-free manner using the inductive predicate
`IsSumNonzeroSq`, which asserts that an element is a finite sum of squares of nonzero elements.
A ring is then formally real if `¬ IsSumNonzeroSq 0`.

## Main declaration

- `IsFormallyReal`: typeclass stating that a ring is formally real.

-/

@[expose] public section

variable {R : Type*}

section IsSumNonzeroSq

/--
The property of being a sum of squares of nonzero elements (S) is defined inductively by:
`a * a : R` is (S) for all nonzero `a`, and
if `s : R` is (S), and `a ≠ 0`, then `a * a + s` is (S).
-/
@[mk_iff]
inductive IsSumNonzeroSq [Mul R] [Add R] [Zero R] : R → Prop
  | sq {a : R} (ha : a ≠ 0) : IsSumNonzeroSq (a * a)
  | sq_add {a s : R} (ha : a ≠ 0) (hs : IsSumNonzeroSq s) : IsSumNonzeroSq (a * a + s)

attribute [aesop 90%] IsSumNonzeroSq.sq

@[aesop 90%]
theorem IsSumNonzeroSq.add [AddMonoid R] [Mul R] {s₁ s₂ : R}
    (h₁ : IsSumNonzeroSq s₁) (h₂ : IsSumNonzeroSq s₂) : IsSumNonzeroSq (s₁ + s₂) := by
  induction h₁ <;> simp_all [sq_add, add_assoc]

theorem IsSumNonzeroSq.isSumSq [AddMonoid R] [Mul R] {s : R}
    (h : IsSumNonzeroSq s) : IsSumSq s := by
  induction h <;> aesop

theorem isSumNonzeroSq_iff_isSumSq [NonUnitalNonAssocSemiring R] {s : R} (hs : s ≠ 0) :
    IsSumNonzeroSq s ↔ IsSumSq s where
  mp := IsSumNonzeroSq.isSumSq
  mpr h := by
    induction h with
    | zero => grind
    | @sq_add a s hs ih =>
    rcases eq_or_ne a 0 with (rfl | ne_a)
    · simp_all
    · rcases eq_or_ne s 0 with (rfl | ne_s)
      · simpa using IsSumNonzeroSq.sq ne_a
      · exact IsSumNonzeroSq.sq_add ne_a (ih ne_s)

alias ⟨_, IsSumSq.isSumNonzeroSq_of_ne_zero⟩ := isSumNonzeroSq_iff_isSumSq

namespace AddSubsemigroup

variable [AddMonoid R] [Mul R] {s : R}

variable (R) in
/-- The subsemigroup of sums of squares of nonzero elements. -/
@[simps]
def sumNonzeroSq : AddSubsemigroup R where
  carrier := {s : R | IsSumNonzeroSq s}
  add_mem' := .add

attribute [norm_cast] coe_sumNonzeroSq

@[simp] theorem mem_sumNonzeroSq : s ∈ sumNonzeroSq R ↔ IsSumNonzeroSq s := .rfl

@[simp]
theorem closure_mul_self : closure {x * x | x ≠ (0 : R)} = sumNonzeroSq R := by
  refine closure_eq_of_le (fun x hx ↦ by aesop) (fun x hx ↦ ?_)
  -- TODO : fix aesop timeout and change to `induction hx <;> aesop`
  induction hx with
  | sq ha => aesop
  | sq_add ha hs ih =>
    -- `aesop` times out
    apply add_mem
    · apply AddSubsemigroup.mem_closure_of_mem
      aesop
    aesop

end AddSubsemigroup

end IsSumNonzeroSq

variable (R) in
/--
A ring is formally real if, whenever `∑ i, x i ^ 2 = 0`, we in fact have `x i = 0` for all `i`.
-/
class IsFormallyReal [AddCommMonoid R] [Mul R] : Prop where
  not_isSumNonzeroSq_zero : ¬ IsSumNonzeroSq (0 : R)

namespace IsFormallyReal

theorem of_eq_zero_of_mul_self_of_eq_zero_of_add [AddCommMonoid R] [Mul R]
    (hz : ∀ {a : R}, a * a = 0 → a = 0)
    (ha : ∀ {s₁ s₂ : R}, IsSumSq s₁ → IsSumSq s₂ → s₁ + s₂ = 0 → s₁ = 0) : IsFormallyReal R where
  not_isSumNonzeroSq_zero := by
    suffices ∀ (x : R), IsSumNonzeroSq x → x ≠ 0 by grind
    intro x hx
    induction hx with
    | sq ha => grind
    | @sq_add b s hb hs ih => grind [ha (IsSumSq.mul_self b) hs.isSumSq]

theorem of_eq_zero_of_eq_zero_of_mul_self_add [NonUnitalNonAssocSemiring R]
    (h : ∀ {s a : R}, IsSumSq s → a * a + s = 0 → a = 0) : IsFormallyReal R where
  not_isSumNonzeroSq_zero := by
    suffices ∀ (x : R), IsSumNonzeroSq x → x ≠ 0 by grind
    intro x hx
    induction hx with
    | sq ha => exact fun hc ↦ ha (h IsSumSq.zero (by simpa using hc))
    | sq_add ha hs ih => grind [hs.isSumSq]

instance [Ring R] [LinearOrder R] [IsStrictOrderedRing R] : IsFormallyReal R :=
  of_eq_zero_of_mul_self_of_eq_zero_of_add mul_self_eq_zero.mp <|
    fun hs₁ hs₂ h ↦ ((add_eq_zero_iff_of_nonneg (IsSumSq.nonneg hs₁) (IsSumSq.nonneg hs₂)).mp h).1

instance [Ring R] [IsFormallyReal R] : IsReduced R := by
  rw [isReduced_iff_pow_one_lt 2 (by lia)]
  intro x hx
  by_contra! hc
  exact not_isSumNonzeroSq_zero <| by simpa [← pow_two, hx] using IsSumNonzeroSq.sq hc

theorem eq_zero_of_add_right [NonUnitalNonAssocSemiring R] [IsFormallyReal R]
    {s₁ s₂ : R} (hs₁ : IsSumSq s₁) (hs₂ : IsSumSq s₂) (h : s₁ + s₂ = 0) : s₁ = 0 := by
  by_contra! h₁
  have h₂ : s₂ ≠ 0 := fun hc ↦ by simp_all
  rw [← isSumNonzeroSq_iff_isSumSq h₁] at hs₁
  rw [← isSumNonzeroSq_iff_isSumSq h₂] at hs₂
  exact not_isSumNonzeroSq_zero (h ▸ IsSumNonzeroSq.add hs₁ hs₂)

Target:
theorem eq_zero_of_add_left [NonUnitalNonAssocSemiring R] [IsFormallyReal R]
    {s₁ s₂ : R} (hs₁ : IsSumSq s₁) (hs₂ : IsSumSq s₂) (h : s₁ + s₂ = 0) : s₂ = 0 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11670215) 0) = ((fun x : Nat => x + 11670215) 0) by rfl]
  simp_all [eq_zero_of_add_right hs₁ hs₂ h]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:149:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11670215) 0\nin the target expression\n  s₂ = 0\n\nR : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : IsFormallyReal R\ns₁ s₂ : R\nhs₁ : IsSumSq s₁\nhs₂ : IsSumSq s₂\nh : s₁ + s₂ = 0\n⊢ s₂ = 0", "returncode": 1}

Corrected proof:


<LEAN_FILE>:37:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge17307ff3a.value`
<LEAN_FILE>:36:0: error: unsolved goals
C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Karoubi (Karoubi C)
⊢ P.p.f ≫ P.p.f = P.p.f

Response:
by
  simpa only [hom_ext_iff, comp_f] using P.idem

[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.Idempotents.Karoubi

Namespace:
CategoryTheory.Idempotents.KaroubiKaroubi

Local safe context:
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# Idempotence of the Karoubi envelope

In this file, we construct the equivalence of categories
`KaroubiKaroubi.equivalence C : Karoubi C ≌ Karoubi (Karoubi C)` for any category `C`.

-/

@[expose] public section

open CategoryTheory.Category

open CategoryTheory.Idempotents.Karoubi

namespace CategoryTheory

namespace Idempotents

namespace KaroubiKaroubi

variable (C : Type*) [Category* C]

@[reassoc (attr := simp)]

Target:
lemma idem_f (P : Karoubi (Karoubi C)) : P.p.f ≫ P.p.f = P.p.f :=

Failed candidate:
by
  have __repair_e17307ff3a := OProverMissinge17307ff3a.value
  simpa only [hom_ext_iff, comp_f] using P.idem

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:37:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge17307ff3a.value`\n<LEAN_FILE>:36:0: error: unsolved goals\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f", "returncode": 1}

Corrected proof:

[Assistant]
by
  simpa only [hom_ext_iff, comp_f] using P.idem

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.Idempotents.Karoubi

Namespace:
CategoryTheory.Idempotents.KaroubiKaroubi

Local safe context:
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# Idempotence of the Karoubi envelope

In this file, we construct the equivalence of categories
`KaroubiKaroubi.equivalence C : Karoubi C ≌ Karoubi (Karoubi C)` for any category `C`.

-/

@[expose] public section

open CategoryTheory.Category

open CategoryTheory.Idempotents.Karoubi

namespace CategoryTheory

namespace Idempotents

namespace KaroubiKaroubi

variable (C : Type*) [Category* C]

@[reassoc (attr := simp)]

Target:
lemma idem_f (P : Karoubi (Karoubi C)) : P.p.f ≫ P.p.f = P.p.f :=

Failed candidate:
by
  have __repair_e17307ff3a := OProverMissinge17307ff3a.value
  simpa only [hom_ext_iff, comp_f] using P.idem

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:37:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge17307ff3a.value`\n<LEAN_FILE>:36:0: error: unsolved goals\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f", "returncode": 1}

Corrected proof:


<LEAN_FILE>:86:41: 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>:85:0: error: unsolved goals
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
hp : p ≠ ∞
hp₂ : 1 ≤ p
f : E → F
hf : MemLp f p μ
ε : ℝ
hε : 0 < ε
⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε

Response:
by
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

[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.Analysis.Calculus.ContDiff.Defs
public import Mathlib.LinearAlgebra.FiniteDimensional.Defs
public import Mathlib.MeasureTheory.Function.LpSpace.Basic
import Mathlib.Geometry.Manifold.SmoothApprox
import Mathlib.MeasureTheory.Function.ContinuousMapDense

Namespace:
MeasureTheory.MemLp

Local safe context:
/-
Copyright (c) 2025 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-!

# Density of smooth compactly supported functions in `Lp`

In this file, we prove that `Lp` functions can be approximated by smooth compactly supported
functions for `p < ∞`.

This result is recorded in `MeasureTheory.MemLp.exist_sub_eLpNorm_le`.
-/

public section

variable {α β E F : Type*} [MeasurableSpace E] [NormedAddCommGroup F]

open scoped Nat NNReal ContDiff
open MeasureTheory Pointwise ENNReal

namespace HasCompactSupport

variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E]
  [NormedSpace ℝ F]

/-- For every continuous compactly supported function `f` there exists a smooth compactly supported
function `g` such that `f - g` is arbitrary small in the `Lp`-norm for `p < ∞`. -/
theorem exist_eLpNorm_sub_le_of_continuous (μ : Measure E := by volume_tac)
    [IsFiniteMeasureOnCompacts μ] {p : ℝ≥0∞} {ε : ℝ} (hε : 0 < ε) {f : E → F}
    (h₁ : HasCompactSupport f) (h₂ : Continuous f) :
    ∃ (g : E → F), HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧
    eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε := by
  rcases eq_or_ne p ∞ with rfl | hp
  · obtain ⟨g, hg₁, hg₂, hg₃⟩ := h₂.exists_contDiff_approx ⊤ (ε := fun _ ↦ ε) (by fun_prop)
      (by intro; positivity)
    refine ⟨g, h₁.mono hg₃, hg₁, eLpNormEssSup_le_of_ae_bound (.of_forall fun x ↦ ?_)⟩
    simpa [← dist_eq_norm_sub'] using (hg₂ x).le
  by_cases hf : f =ᵐ[μ] 0
  -- We will need that the support is non-empty, so we treat the trivial case `f = 0` first.
  · use 0
    simpa [HasCompactSupport.zero, eLpNorm_congr_ae hf] using! contDiff_const
  have hs₁ : μ (tsupport f) ≠ ⊤ := h₁.measure_lt_top.ne
  have hs₂ : 0 < (μ <| tsupport f).toReal := by
    -- Since `f` is not the zero function `tsupport f` has positive measure
    rw [← Measure.measure_support_eq_zero_iff _] at hf
    exact toReal_pos (pos_mono (subset_tsupport f) (pos_of_ne_zero hf)).ne' hs₁
  set ε' := ε * (μ <| tsupport f).toReal ^ (-(1 / p.toReal)) with ε'_def
  have hε' : 0 < ε' := by positivity
  have hε₂ : ENNReal.ofReal ε' * μ (tsupport f) ^ (1 / p.toReal) ≤ ENNReal.ofReal ε := by
    rw [← ofReal_toReal hs₁, ofReal_rpow_of_pos hs₂, ← ofReal_mul hε'.le,
      ofReal_le_ofReal_iff hε.le, ε'_def, mul_assoc, ← Real.rpow_add hs₂, neg_add_cancel,
      Real.rpow_zero, mul_one]
  obtain ⟨g, hg₁, hg₂, hg₃⟩ := h₂.exists_contDiff_approx ⊤ (ε := fun _ ↦ ε') (by fun_prop)
    (by intro; positivity)
  refine ⟨g, h₁.mono hg₃, hg₁, (eLpNorm_sub_le_of_dist_bdd μ hp h₁.measurableSet hε'.le ?_
    (subset_tsupport f) (hg₃.trans (subset_tsupport f))).trans hε₂⟩
  intro x
  rw [dist_comm]
  exact (hg₂ x).le

end HasCompactSupport

namespace MeasureTheory.MemLp

variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E]
  [NormedSpace ℝ F]
  {μ : Measure E} [IsFiniteMeasureOnCompacts μ]

/-- Every `Lp` function can be approximated by a smooth compactly supported function provided that
`p < ∞`. -/

Target:
theorem exist_eLpNorm_sub_le {p : ℝ≥0∞} (hp : p ≠ ⊤) (hp₂ : 1 ≤ p) {f : E → F} (hf : MemLp f p μ)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε :=

Failed candidate:
by
  have __repair_e0d32dfa3e := True.intro.nonexistentProjection
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:86:41: 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>:85:0: error: unsolved goals\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : 1 ≤ p\nf : E → F\nhf : MemLp f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε", "returncode": 1}

Corrected proof:

[Assistant]
by
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

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.Analysis.Calculus.ContDiff.Defs
public import Mathlib.LinearAlgebra.FiniteDimensional.Defs
public import Mathlib.MeasureTheory.Function.LpSpace.Basic
import Mathlib.Geometry.Manifold.SmoothApprox
import Mathlib.MeasureTheory.Function.ContinuousMapDense

Namespace:
MeasureTheory.MemLp

Local safe context:
/-
Copyright (c) 2025 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-!

# Density of smooth compactly supported functions in `Lp`

In this file, we prove that `Lp` functions can be approximated by smooth compactly supported
functions for `p < ∞`.

This result is recorded in `MeasureTheory.MemLp.exist_sub_eLpNorm_le`.
-/

public section

variable {α β E F : Type*} [MeasurableSpace E] [NormedAddCommGroup F]

open scoped Nat NNReal ContDiff
open MeasureTheory Pointwise ENNReal

namespace HasCompactSupport

variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E]
  [NormedSpace ℝ F]

/-- For every continuous compactly supported function `f` there exists a smooth compactly supported
function `g` such that `f - g` is arbitrary small in the `Lp`-norm for `p < ∞`. -/
theorem exist_eLpNorm_sub_le_of_continuous (μ : Measure E := by volume_tac)
    [IsFiniteMeasureOnCompacts μ] {p : ℝ≥0∞} {ε : ℝ} (hε : 0 < ε) {f : E → F}
    (h₁ : HasCompactSupport f) (h₂ : Continuous f) :
    ∃ (g : E → F), HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧
    eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε := by
  rcases eq_or_ne p ∞ with rfl | hp
  · obtain ⟨g, hg₁, hg₂, hg₃⟩ := h₂.exists_contDiff_approx ⊤ (ε := fun _ ↦ ε) (by fun_prop)
      (by intro; positivity)
    refine ⟨g, h₁.mono hg₃, hg₁, eLpNormEssSup_le_of_ae_bound (.of_forall fun x ↦ ?_)⟩
    simpa [← dist_eq_norm_sub'] using (hg₂ x).le
  by_cases hf : f =ᵐ[μ] 0
  -- We will need that the support is non-empty, so we treat the trivial case `f = 0` first.
  · use 0
    simpa [HasCompactSupport.zero, eLpNorm_congr_ae hf] using! contDiff_const
  have hs₁ : μ (tsupport f) ≠ ⊤ := h₁.measure_lt_top.ne
  have hs₂ : 0 < (μ <| tsupport f).toReal := by
    -- Since `f` is not the zero function `tsupport f` has positive measure
    rw [← Measure.measure_support_eq_zero_iff _] at hf
    exact toReal_pos (pos_mono (subset_tsupport f) (pos_of_ne_zero hf)).ne' hs₁
  set ε' := ε * (μ <| tsupport f).toReal ^ (-(1 / p.toReal)) with ε'_def
  have hε' : 0 < ε' := by positivity
  have hε₂ : ENNReal.ofReal ε' * μ (tsupport f) ^ (1 / p.toReal) ≤ ENNReal.ofReal ε := by
    rw [← ofReal_toReal hs₁, ofReal_rpow_of_pos hs₂, ← ofReal_mul hε'.le,
      ofReal_le_ofReal_iff hε.le, ε'_def, mul_assoc, ← Real.rpow_add hs₂, neg_add_cancel,
      Real.rpow_zero, mul_one]
  obtain ⟨g, hg₁, hg₂, hg₃⟩ := h₂.exists_contDiff_approx ⊤ (ε := fun _ ↦ ε') (by fun_prop)
    (by intro; positivity)
  refine ⟨g, h₁.mono hg₃, hg₁, (eLpNorm_sub_le_of_dist_bdd μ hp h₁.measurableSet hε'.le ?_
    (subset_tsupport f) (hg₃.trans (subset_tsupport f))).trans hε₂⟩
  intro x
  rw [dist_comm]
  exact (hg₂ x).le

end HasCompactSupport

namespace MeasureTheory.MemLp

variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E]
  [NormedSpace ℝ F]
  {μ : Measure E} [IsFiniteMeasureOnCompacts μ]

/-- Every `Lp` function can be approximated by a smooth compactly supported function provided that
`p < ∞`. -/

Target:
theorem exist_eLpNorm_sub_le {p : ℝ≥0∞} (hp : p ≠ ⊤) (hp₂ : 1 ≤ p) {f : E → F} (hf : MemLp f p μ)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε :=

Failed candidate:
by
  have __repair_e0d32dfa3e := True.intro.nonexistentProjection
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:86:41: 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>:85:0: error: unsolved goals\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : 1 ≤ p\nf : E → F\nhf : MemLp f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε", "returncode": 1}

Corrected proof:


<LEAN_FILE>:35:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 1022378) 0
in the target expression
  ↑(((decompose V) x) i) = x * idempotent V i

R : Type u_1
I : Type u_2
inst✝² : Semiring R
inst✝¹ : DecidableEq I
V : I → Ideal R
inst✝ : Decomposition V
x : R
i : I
⊢ ↑(((decompose V) x) i) = x * idempotent V i

Note: The target expression is not type-correct under the `instances` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue.
Full error:
  Application type mismatch: The argument
    (decompose V) x
  has type
    ⨁ (i : I), ↥(V i)
  but is expected to have type
    Π₀ (i : I), (fun i => ↥(V i)) i
  in the application
    ⇑((decompose V) x)

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

[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.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition

Namespace:
DirectSum

Local safe 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 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 1022378) 0) = ((fun x : Nat => x + 1022378) 0) by rfl]
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:35:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 1022378) 0\nin the target expression\n  ↑(((decompose V) x) i) = x * idempotent V i\n\nR : Type u_1\nI : Type u_2\ninst✝² : Semiring R\ninst✝¹ : DecidableEq I\nV : I → Ideal R\ninst✝ : Decomposition V\nx : R\ni : I\n⊢ ↑(((decompose V) x) i) = x * idempotent V i\n\nNote: The target expression is not type-correct under the `instances` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue.\nFull error:\n  Application type mismatch: The argument\n    (decompose V) x\n  has type\n    ⨁ (i : I), ↥(V i)\n  but is expected to have type\n    Π₀ (i : I), (fun i => ↥(V i)) i\n  in the application\n    ⇑((decompose V) x)", "returncode": 1}

Corrected proof:

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

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

Namespace:
DirectSum

Local safe 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 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 1022378) 0) = ((fun x : Nat => x + 1022378) 0) by rfl]
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:35:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 1022378) 0\nin the target expression\n  ↑(((decompose V) x) i) = x * idempotent V i\n\nR : Type u_1\nI : Type u_2\ninst✝² : Semiring R\ninst✝¹ : DecidableEq I\nV : I → Ideal R\ninst✝ : Decomposition V\nx : R\ni : I\n⊢ ↑(((decompose V) x) i) = x * idempotent V i\n\nNote: The target expression is not type-correct under the `instances` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue.\nFull error:\n  Application type mismatch: The argument\n    (decompose V) x\n  has type\n    ⨁ (i : I), ↥(V i)\n  but is expected to have type\n    Π₀ (i : I), (fun i => ↥(V i)) i\n  in the application\n    ⇑((decompose V) x)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:76:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

[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.Basic
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Ring.Defs

Namespace:
IsDedekindFiniteMonoid

Local safe context:
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
/-!
# Lemmas about regular elements in rings.
-/

public section

variable {α : Type*}

/-- Left `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isLeftRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, k * x = 0 → x = 0) : IsLeftRegular k := by
  refine fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ ?_)
  rw [mul_sub, sub_eq_zero, h']

/-- Right `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isRightRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, x * k = 0 → x = 0) : IsRightRegular k := by
  refine fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ ?_)
  rw [sub_mul, sub_eq_zero, h']

theorem IsRegular.of_ne_zero' [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} (hk : k ≠ 0) :
    IsRegular k :=
  ⟨isLeftRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left hk,
    isRightRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk⟩

@[deprecated (since := "2026-01-21")] alias isRegular_of_ne_zero' := IsRegular.of_ne_zero'

theorem isRegular_iff_ne_zero' [Nontrivial α] [NonUnitalNonAssocRing α] [NoZeroDivisors α]
    {k : α} : IsRegular k ↔ k ≠ 0 :=
  ⟨fun h => by
    rintro rfl
    exact not_not.mpr h.left not_isLeftRegular_zero, .of_ne_zero'⟩

/-- A ring with no zero divisors is a cancellative `MonoidWithZero`.

Note this is not an instance as it forms a typeclass loop. -/
lemma NoZeroDivisors.toIsCancelMulZero [NonUnitalNonAssocRing α] [NoZeroDivisors α] :
    IsCancelMulZero α where
  mul_left_cancel_of_ne_zero ha := (IsRegular.of_ne_zero' ha).1
  mul_right_cancel_of_ne_zero hb := (IsRegular.of_ne_zero' hb).2

namespace IsDedekindFiniteMonoid

variable [Ring α]

/-- A ring is Dedekind-finite if and only if every element has at most one right inverse. -/
theorem iff_eq_of_mul_left_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * y = 1 → x * z = 1 → y = z := by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxy hxz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [← mul_assoc, h hxz] using congr_arg (z * ·) hxy
  have := h _ _ (1 - y * x + y) eq <| by
    rw [mul_add, mul_sub, ← mul_assoc, eq, mul_one, one_mul, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

/-- A ring is Dedekind-finite if and only if every element has at most one left inverse. -/

Target:
theorem iff_eq_of_mul_right_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * z = 1 → y * z = 1 → x = y :=

Failed candidate:
by
  let __repair_ff32326e41 : Fact False := inferInstance
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:76:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

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.Basic
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Ring.Defs

Namespace:
IsDedekindFiniteMonoid

Local safe context:
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
/-!
# Lemmas about regular elements in rings.
-/

public section

variable {α : Type*}

/-- Left `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isLeftRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, k * x = 0 → x = 0) : IsLeftRegular k := by
  refine fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ ?_)
  rw [mul_sub, sub_eq_zero, h']

/-- Right `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isRightRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, x * k = 0 → x = 0) : IsRightRegular k := by
  refine fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ ?_)
  rw [sub_mul, sub_eq_zero, h']

theorem IsRegular.of_ne_zero' [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} (hk : k ≠ 0) :
    IsRegular k :=
  ⟨isLeftRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left hk,
    isRightRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk⟩

@[deprecated (since := "2026-01-21")] alias isRegular_of_ne_zero' := IsRegular.of_ne_zero'

theorem isRegular_iff_ne_zero' [Nontrivial α] [NonUnitalNonAssocRing α] [NoZeroDivisors α]
    {k : α} : IsRegular k ↔ k ≠ 0 :=
  ⟨fun h => by
    rintro rfl
    exact not_not.mpr h.left not_isLeftRegular_zero, .of_ne_zero'⟩

/-- A ring with no zero divisors is a cancellative `MonoidWithZero`.

Note this is not an instance as it forms a typeclass loop. -/
lemma NoZeroDivisors.toIsCancelMulZero [NonUnitalNonAssocRing α] [NoZeroDivisors α] :
    IsCancelMulZero α where
  mul_left_cancel_of_ne_zero ha := (IsRegular.of_ne_zero' ha).1
  mul_right_cancel_of_ne_zero hb := (IsRegular.of_ne_zero' hb).2

namespace IsDedekindFiniteMonoid

variable [Ring α]

/-- A ring is Dedekind-finite if and only if every element has at most one right inverse. -/
theorem iff_eq_of_mul_left_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * y = 1 → x * z = 1 → y = z := by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxy hxz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [← mul_assoc, h hxz] using congr_arg (z * ·) hxy
  have := h _ _ (1 - y * x + y) eq <| by
    rw [mul_add, mul_sub, ← mul_assoc, eq, mul_one, one_mul, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

/-- A ring is Dedekind-finite if and only if every element has at most one left inverse. -/

Target:
theorem iff_eq_of_mul_right_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * z = 1 → y * z = 1 → x = y :=

Failed candidate:
by
  let __repair_ff32326e41 : Fact False := inferInstance
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:76:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


Response:
by
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

[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.Data.FunLike.IsApply
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathlib.Algebra.Group.Pi.Basic

Namespace:
FunLike

Local context:
/-
Copyright (c) 2026 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-! # Group instances for `FunLike` types
In this file we define various instances related to groups for `FunLike` types.

For example given a `FunLike F α β` with `IsMulApply F α β` and `Semigroup β`, then `F` is naturally
a semigroup. Note that currently, these are not registered as instances, but only `abbrev`s to
avoid long typeclass searches.

Moreover, we define the homomorphism `FunLike.coeMulHom : F →* α → β` that acts by coercion. This
definition is mainly needed to define a module instance on `F`.

-/

@[expose] public section

namespace FunLike

variable {F α β : Type*} [FunLike F α β]

section CoercionHom

section MulHom

variable [Mul F] [Mul β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a multiplicative homomorphism. -/
@[to_additive
/-- Coercion as an additive homomorphism. -/]
def coeMulHom : F →ₙ* α → β where
  toFun f := f
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMulHom_apply (f : F) : coeMulHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMulHom : (coeMulHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive]
theorem coeMulHom_injective : Function.Injective (coeMulHom F α β) := by
  rw [coe_coeMulHom]
  exact DFunLike.coe_injective

end MulHom

section MonoidHom

variable [MulOne F] [MulOneClass β] [IsOneApply F α β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a monoid homomorphism. -/
@[to_additive
/-- Coercion as an additive monoid homomorphism. -/]
def coeMonoidHom : F →* α → β where
  toFun f := f
  map_one' := coe_one
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMonoidHom_apply (f : F) : coeMonoidHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom : (coeMonoidHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom' : (coeMonoidHom F α β : F →ₙ* α → β) = coeMulHom F α β := rfl

@[to_additive]

Target:
theorem coeMonoidHom_injective : Function.Injective (coeMonoidHom F α β) :=

Proof body:

[Assistant]
by
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

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.Data.FunLike.IsApply
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathlib.Algebra.Group.Pi.Basic

Namespace:
FunLike

Local context:
/-
Copyright (c) 2026 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-! # Group instances for `FunLike` types
In this file we define various instances related to groups for `FunLike` types.

For example given a `FunLike F α β` with `IsMulApply F α β` and `Semigroup β`, then `F` is naturally
a semigroup. Note that currently, these are not registered as instances, but only `abbrev`s to
avoid long typeclass searches.

Moreover, we define the homomorphism `FunLike.coeMulHom : F →* α → β` that acts by coercion. This
definition is mainly needed to define a module instance on `F`.

-/

@[expose] public section

namespace FunLike

variable {F α β : Type*} [FunLike F α β]

section CoercionHom

section MulHom

variable [Mul F] [Mul β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a multiplicative homomorphism. -/
@[to_additive
/-- Coercion as an additive homomorphism. -/]
def coeMulHom : F →ₙ* α → β where
  toFun f := f
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMulHom_apply (f : F) : coeMulHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMulHom : (coeMulHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive]
theorem coeMulHom_injective : Function.Injective (coeMulHom F α β) := by
  rw [coe_coeMulHom]
  exact DFunLike.coe_injective

end MulHom

section MonoidHom

variable [MulOne F] [MulOneClass β] [IsOneApply F α β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a monoid homomorphism. -/
@[to_additive
/-- Coercion as an additive monoid homomorphism. -/]
def coeMonoidHom : F →* α → β where
  toFun f := f
  map_one' := coe_one
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMonoidHom_apply (f : F) : coeMonoidHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom : (coeMonoidHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom' : (coeMonoidHom F α β : F →ₙ* α → β) = coeMulHom F α β := rfl

@[to_additive]

Target:
theorem coeMonoidHom_injective : Function.Injective (coeMonoidHom F α β) :=

Proof body:


<LEAN_FILE>:101:40: 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>:100:0: error: unsolved goals
R : Type u_1
M : Type u_2
x : M
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : Submodule R M
⊢ mk x = 0 ↔ x ∈ p

Response: by simpa using (Quotient.eq' p : mk x = 0 ↔ _)

[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.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Submodule.Defs
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Logic.Small.Basic

Namespace:
Submodule.Quotient

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
/-!
# Quotients by submodules

* If `p` is a submodule of `M`, `M ⧸ p` is the quotient of `M` with respect to `p`:
  that is, elements of `M` are identified if their difference is in `p`. This is itself a module.

## Main definitions

* `Submodule.Quotient.mk`: a function sending an element of `M` to `M ⧸ p`
* `Submodule.Quotient.module`: `M ⧸ p` is a module
* `Submodule.Quotient.mkQ`: a linear map sending an element of `M` to `M ⧸ p`
* `Submodule.quotEquivOfEq`: if `p` and `p'` are equal, their quotients are equivalent

-/

@[expose] public section

-- For most of this file we work over a noncommutative ring
section Ring

namespace Submodule

variable {R M : Type*} {r : R} {x y : M} [Ring R] [AddCommGroup M] [Module R M]
variable (p p' : Submodule R M)

open QuotientAddGroup

/-- The equivalence relation associated to a submodule `p`, defined by `x ≈ y` iff `-x + y ∈ p`.

Note this is equivalent to `y - x ∈ p`, but defined this way to be defeq to the `AddSubgroup`
version, where commutativity can't be assumed. -/
def quotientRel : Setoid M :=
  QuotientAddGroup.leftRel p.toAddSubgroup

theorem quotientRel_def {x y : M} : p.quotientRel x y ↔ x - y ∈ p :=
  Iff.trans
    (by
      rw [quotientRel, leftRel_apply, sub_eq_add_neg, neg_add, neg_neg]
      rfl)
    neg_mem_iff

/-- The quotient of a module `M` by a submodule `p ⊆ M`. -/
instance hasQuotient : HasQuotient M (Submodule R M) :=
  ⟨fun p => Quotient (quotientRel p)⟩

namespace Quotient
/-- Map associating to an element of `M` the corresponding element of `M/p`,
when `p` is a submodule of `M`. -/
def mk {p : Submodule R M} : M → M ⧸ p :=
  Quotient.mk''

theorem mk'_eq_mk' {p : Submodule R M} (x : M) :
    @Quotient.mk' _ (quotientRel p) x = mk x :=
  rfl

theorem mk''_eq_mk {p : Submodule R M} (x : M) : (Quotient.mk'' x : M ⧸ p) = mk x :=
  rfl

theorem quot_mk_eq_mk {p : Submodule R M} (x : M) : (Quot.mk _ x : M ⧸ p) = mk x :=
  rfl

theorem quotientAddGroupMk_eq_mk {p : Submodule R M} (x : M) :
    (QuotientAddGroup.mk x : M ⧸ p) = mk x :=
  rfl

protected theorem eq' {x y : M} : (mk x : M ⧸ p) = mk y ↔ -x + y ∈ p :=
  QuotientAddGroup.eq

protected theorem eq {x y : M} : (mk x : M ⧸ p) = mk y ↔ x - y ∈ p :=
  (Submodule.Quotient.eq' p).trans (leftRel_apply.symm.trans p.quotientRel_def)

instance : Zero (M ⧸ p) where
  -- Use Quotient.mk'' instead of mk here because mk is not reducible.
  -- This would lead to non-defeq diamonds.
  -- See also the same comment at the One instance for Con.
  zero := Quotient.mk'' 0

instance : Inhabited (M ⧸ p) :=
  ⟨0⟩

@[simp]
theorem mk_zero : mk 0 = (0 : M ⧸ p) :=
  rfl

@[simp]

Target:
theorem mk_eq_zero : (mk x : M ⧸ p) = 0 ↔ x ∈ p :=

Failed candidate:
by
 have __repair_5ba472dc56 := True.intro.nonexistentProjection
 simpa using (Quotient.eq' p : mk x = 0 ↔ _)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:101:40: 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>:100:0: error: unsolved goals\nR : Type u_1\nM : Type u_2\nx : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\n⊢ mk x = 0 ↔ x ∈ p", "returncode": 1}

Corrected proof:

[Assistant] by simpa using (Quotient.eq' p : mk x = 0 ↔ _)

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.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Submodule.Defs
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Logic.Small.Basic

Namespace:
Submodule.Quotient

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
/-!
# Quotients by submodules

* If `p` is a submodule of `M`, `M ⧸ p` is the quotient of `M` with respect to `p`:
  that is, elements of `M` are identified if their difference is in `p`. This is itself a module.

## Main definitions

* `Submodule.Quotient.mk`: a function sending an element of `M` to `M ⧸ p`
* `Submodule.Quotient.module`: `M ⧸ p` is a module
* `Submodule.Quotient.mkQ`: a linear map sending an element of `M` to `M ⧸ p`
* `Submodule.quotEquivOfEq`: if `p` and `p'` are equal, their quotients are equivalent

-/

@[expose] public section

-- For most of this file we work over a noncommutative ring
section Ring

namespace Submodule

variable {R M : Type*} {r : R} {x y : M} [Ring R] [AddCommGroup M] [Module R M]
variable (p p' : Submodule R M)

open QuotientAddGroup

/-- The equivalence relation associated to a submodule `p`, defined by `x ≈ y` iff `-x + y ∈ p`.

Note this is equivalent to `y - x ∈ p`, but defined this way to be defeq to the `AddSubgroup`
version, where commutativity can't be assumed. -/
def quotientRel : Setoid M :=
  QuotientAddGroup.leftRel p.toAddSubgroup

theorem quotientRel_def {x y : M} : p.quotientRel x y ↔ x - y ∈ p :=
  Iff.trans
    (by
      rw [quotientRel, leftRel_apply, sub_eq_add_neg, neg_add, neg_neg]
      rfl)
    neg_mem_iff

/-- The quotient of a module `M` by a submodule `p ⊆ M`. -/
instance hasQuotient : HasQuotient M (Submodule R M) :=
  ⟨fun p => Quotient (quotientRel p)⟩

namespace Quotient
/-- Map associating to an element of `M` the corresponding element of `M/p`,
when `p` is a submodule of `M`. -/
def mk {p : Submodule R M} : M → M ⧸ p :=
  Quotient.mk''

theorem mk'_eq_mk' {p : Submodule R M} (x : M) :
    @Quotient.mk' _ (quotientRel p) x = mk x :=
  rfl

theorem mk''_eq_mk {p : Submodule R M} (x : M) : (Quotient.mk'' x : M ⧸ p) = mk x :=
  rfl

theorem quot_mk_eq_mk {p : Submodule R M} (x : M) : (Quot.mk _ x : M ⧸ p) = mk x :=
  rfl

theorem quotientAddGroupMk_eq_mk {p : Submodule R M} (x : M) :
    (QuotientAddGroup.mk x : M ⧸ p) = mk x :=
  rfl

protected theorem eq' {x y : M} : (mk x : M ⧸ p) = mk y ↔ -x + y ∈ p :=
  QuotientAddGroup.eq

protected theorem eq {x y : M} : (mk x : M ⧸ p) = mk y ↔ x - y ∈ p :=
  (Submodule.Quotient.eq' p).trans (leftRel_apply.symm.trans p.quotientRel_def)

instance : Zero (M ⧸ p) where
  -- Use Quotient.mk'' instead of mk here because mk is not reducible.
  -- This would lead to non-defeq diamonds.
  -- See also the same comment at the One instance for Con.
  zero := Quotient.mk'' 0

instance : Inhabited (M ⧸ p) :=
  ⟨0⟩

@[simp]
theorem mk_zero : mk 0 = (0 : M ⧸ p) :=
  rfl

@[simp]

Target:
theorem mk_eq_zero : (mk x : M ⧸ p) = 0 ↔ x ∈ p :=

Failed candidate:
by
 have __repair_5ba472dc56 := True.intro.nonexistentProjection
 simpa using (Quotient.eq' p : mk x = 0 ↔ _)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:101:40: 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>:100:0: error: unsolved goals\nR : Type u_1\nM : Type u_2\nx : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\n⊢ mk x = 0 ↔ x ∈ p", "returncode": 1}

Corrected proof: