Response:
by
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

[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.Algebra.DirectSum.Module
public import Mathlib.Algebra.Module.Submodule.Basic

Namespace:
DirectSum

Local context:
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
/-!
# Decompositions of additive monoids, groups, and modules into direct sums

## Main definitions

* `DirectSum.Decomposition ℳ`: A typeclass to provide a constructive decomposition from
  an additive monoid `M` into a family of additive submonoids `ℳ`
* `DirectSum.decompose ℳ`: The canonical equivalence provided by the above typeclass

## Main statements

* `DirectSum.Decomposition.isInternal`: The link to `DirectSum.IsInternal`.

## Implementation details

As we want to talk about different types of decomposition (additive monoids, modules, rings, ...),
we choose to avoid heavily bundling `DirectSum.decompose`, instead making copies for the
`AddEquiv`, `LinearEquiv`, etc. This means we have to repeat statements that follow from these
bundled homs, but means we don't have to repeat statements for different types of decomposition.
-/

@[expose] public section

variable {ι R M σ : Type*}

open DirectSum

namespace DirectSum

section AddCommMonoid

variable [DecidableEq ι] [AddCommMonoid M]
variable [SetLike σ M] [AddSubmonoidClass σ M] (ℳ : ι → σ)

/-- A decomposition is an equivalence between an additive monoid `M` and a direct sum of additive
submonoids `ℳ i` of that `M`, such that the "recomposition" is canonical. This definition also
works for additive groups and modules.

This is a version of `DirectSum.IsInternal` which comes with a constructive inverse to the
canonical "recomposition" rather than just a proof that the "recomposition" is bijective.

Often it is easier to construct a term of this type via `Decomposition.ofAddHom` or
`Decomposition.ofLinearMap`. -/
class Decomposition where
  decompose' : M → ⨁ i, ℳ i
  left_inv : Function.LeftInverse (DirectSum.coeAddMonoidHom ℳ) decompose'
  right_inv : Function.RightInverse (DirectSum.coeAddMonoidHom ℳ) decompose'

/-- `DirectSum.Decomposition` instances, while carrying data, are always equal. -/
instance : Subsingleton (Decomposition ℳ) :=
  ⟨fun x y ↦ by
    obtain ⟨_, _, xr⟩ := x
    obtain ⟨_, yl, _⟩ := y
    congr
    exact Function.LeftInverse.eq_rightInverse xr yl⟩

/-- A convenience method to construct a decomposition from an `AddMonoidHom`, such that the proofs
of left and right inverse can be constructed via `ext`. -/
abbrev Decomposition.ofAddHom (decompose : M →+ ⨁ i, ℳ i)
    (h_left_inv : (DirectSum.coeAddMonoidHom ℳ).comp decompose = .id _)
    (h_right_inv : decompose.comp (DirectSum.coeAddMonoidHom ℳ) = .id _) : Decomposition ℳ where
  decompose' := decompose
  left_inv := DFunLike.congr_fun h_left_inv
  right_inv := DFunLike.congr_fun h_right_inv

/-- Noncomputably conjure a decomposition instance from a `DirectSum.IsInternal` proof. -/
@[implicit_reducible]
noncomputable def IsInternal.chooseDecomposition (h : IsInternal ℳ) :
    DirectSum.Decomposition ℳ where
  decompose' := (Equiv.ofBijective _ h).symm
  left_inv := (Equiv.ofBijective _ h).right_inv
  right_inv := (Equiv.ofBijective _ h).left_inv

variable [Decomposition ℳ]

protected theorem Decomposition.isInternal : DirectSum.IsInternal ℳ :=
  ⟨Decomposition.right_inv.injective, Decomposition.left_inv.surjective⟩

/-- If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
to a direct sum of components. This is the canonical spelling of the `decompose'` field. -/
def decompose : M ≃ ⨁ i, ℳ i where
  toFun := Decomposition.decompose'
  invFun := DirectSum.coeAddMonoidHom ℳ
  left_inv := Decomposition.left_inv
  right_inv := Decomposition.right_inv

omit [AddSubmonoidClass σ M] in
/-- A substructure `p ⊆ M` is homogeneous if for every `m ∈ p`, all homogeneous components
  of `m` are in `p`. -/
def SetLike.IsHomogeneous {P : Type*} [SetLike P M] (p : P) : Prop :=
  ∀ (i : ι) ⦃m : M⦄, m ∈ p → (DirectSum.decompose ℳ m i : M) ∈ p

@[elab_as_elim]
protected theorem Decomposition.inductionOn {motive : M → Prop} (zero : motive 0)
    (homogeneous : ∀ {i} (m : ℳ i), motive (m : M))
    (add : ∀ m m' : M, motive m → motive m' → motive (m + m')) : ∀ m, motive m := by
  let ℳ' : ι → AddSubmonoid M := fun i ↦
    (⟨⟨ℳ i, fun x y ↦ AddMemClass.add_mem x y⟩, (ZeroMemClass.zero_mem _)⟩ : AddSubmonoid M)
  haveI t : DirectSum.Decomposition ℳ' :=
    { decompose' := DirectSum.decompose ℳ
      left_inv := fun _ ↦ (decompose ℳ).left_inv _
      right_inv := fun _ ↦ (decompose ℳ).right_inv _ }
  have mem : ∀ m, m ∈ iSup ℳ' := fun _m ↦
    (DirectSum.IsInternal.addSubmonoid_iSup_eq_top ℳ' (Decomposition.isInternal ℳ')).symm ▸ trivial
  -- Porting note: needs to use @ even though no implicit argument is provided
  exact fun m ↦ @AddSubmonoid.iSup_induction _ _ _ ℳ' _ _ (mem m)
    (fun i m h ↦ homogeneous ⟨m, h⟩) zero add
--  exact fun m ↦
--    AddSubmonoid.iSup_induction ℳ' (mem m) (fun i m h ↦ h_homogeneous ⟨m, h⟩) h_zero h_add

@[simp]
theorem Decomposition.decompose'_eq : Decomposition.decompose' = decompose ℳ := rfl

@[simp]
theorem decompose_symm_of {i : ι} (x : ℳ i) : (decompose ℳ).symm (DirectSum.of _ i x) = x :=
  DirectSum.coeAddMonoidHom_of ℳ _ _

@[simp]

Target:
theorem decompose_coe {i : ι} (x : ℳ i) : decompose ℳ (x : M) = DirectSum.of _ i x :=

Proof body:

[Assistant]
by
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

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.Algebra.DirectSum.Module
public import Mathlib.Algebra.Module.Submodule.Basic

Namespace:
DirectSum

Local context:
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
/-!
# Decompositions of additive monoids, groups, and modules into direct sums

## Main definitions

* `DirectSum.Decomposition ℳ`: A typeclass to provide a constructive decomposition from
  an additive monoid `M` into a family of additive submonoids `ℳ`
* `DirectSum.decompose ℳ`: The canonical equivalence provided by the above typeclass

## Main statements

* `DirectSum.Decomposition.isInternal`: The link to `DirectSum.IsInternal`.

## Implementation details

As we want to talk about different types of decomposition (additive monoids, modules, rings, ...),
we choose to avoid heavily bundling `DirectSum.decompose`, instead making copies for the
`AddEquiv`, `LinearEquiv`, etc. This means we have to repeat statements that follow from these
bundled homs, but means we don't have to repeat statements for different types of decomposition.
-/

@[expose] public section

variable {ι R M σ : Type*}

open DirectSum

namespace DirectSum

section AddCommMonoid

variable [DecidableEq ι] [AddCommMonoid M]
variable [SetLike σ M] [AddSubmonoidClass σ M] (ℳ : ι → σ)

/-- A decomposition is an equivalence between an additive monoid `M` and a direct sum of additive
submonoids `ℳ i` of that `M`, such that the "recomposition" is canonical. This definition also
works for additive groups and modules.

This is a version of `DirectSum.IsInternal` which comes with a constructive inverse to the
canonical "recomposition" rather than just a proof that the "recomposition" is bijective.

Often it is easier to construct a term of this type via `Decomposition.ofAddHom` or
`Decomposition.ofLinearMap`. -/
class Decomposition where
  decompose' : M → ⨁ i, ℳ i
  left_inv : Function.LeftInverse (DirectSum.coeAddMonoidHom ℳ) decompose'
  right_inv : Function.RightInverse (DirectSum.coeAddMonoidHom ℳ) decompose'

/-- `DirectSum.Decomposition` instances, while carrying data, are always equal. -/
instance : Subsingleton (Decomposition ℳ) :=
  ⟨fun x y ↦ by
    obtain ⟨_, _, xr⟩ := x
    obtain ⟨_, yl, _⟩ := y
    congr
    exact Function.LeftInverse.eq_rightInverse xr yl⟩

/-- A convenience method to construct a decomposition from an `AddMonoidHom`, such that the proofs
of left and right inverse can be constructed via `ext`. -/
abbrev Decomposition.ofAddHom (decompose : M →+ ⨁ i, ℳ i)
    (h_left_inv : (DirectSum.coeAddMonoidHom ℳ).comp decompose = .id _)
    (h_right_inv : decompose.comp (DirectSum.coeAddMonoidHom ℳ) = .id _) : Decomposition ℳ where
  decompose' := decompose
  left_inv := DFunLike.congr_fun h_left_inv
  right_inv := DFunLike.congr_fun h_right_inv

/-- Noncomputably conjure a decomposition instance from a `DirectSum.IsInternal` proof. -/
@[implicit_reducible]
noncomputable def IsInternal.chooseDecomposition (h : IsInternal ℳ) :
    DirectSum.Decomposition ℳ where
  decompose' := (Equiv.ofBijective _ h).symm
  left_inv := (Equiv.ofBijective _ h).right_inv
  right_inv := (Equiv.ofBijective _ h).left_inv

variable [Decomposition ℳ]

protected theorem Decomposition.isInternal : DirectSum.IsInternal ℳ :=
  ⟨Decomposition.right_inv.injective, Decomposition.left_inv.surjective⟩

/-- If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
to a direct sum of components. This is the canonical spelling of the `decompose'` field. -/
def decompose : M ≃ ⨁ i, ℳ i where
  toFun := Decomposition.decompose'
  invFun := DirectSum.coeAddMonoidHom ℳ
  left_inv := Decomposition.left_inv
  right_inv := Decomposition.right_inv

omit [AddSubmonoidClass σ M] in
/-- A substructure `p ⊆ M` is homogeneous if for every `m ∈ p`, all homogeneous components
  of `m` are in `p`. -/
def SetLike.IsHomogeneous {P : Type*} [SetLike P M] (p : P) : Prop :=
  ∀ (i : ι) ⦃m : M⦄, m ∈ p → (DirectSum.decompose ℳ m i : M) ∈ p

@[elab_as_elim]
protected theorem Decomposition.inductionOn {motive : M → Prop} (zero : motive 0)
    (homogeneous : ∀ {i} (m : ℳ i), motive (m : M))
    (add : ∀ m m' : M, motive m → motive m' → motive (m + m')) : ∀ m, motive m := by
  let ℳ' : ι → AddSubmonoid M := fun i ↦
    (⟨⟨ℳ i, fun x y ↦ AddMemClass.add_mem x y⟩, (ZeroMemClass.zero_mem _)⟩ : AddSubmonoid M)
  haveI t : DirectSum.Decomposition ℳ' :=
    { decompose' := DirectSum.decompose ℳ
      left_inv := fun _ ↦ (decompose ℳ).left_inv _
      right_inv := fun _ ↦ (decompose ℳ).right_inv _ }
  have mem : ∀ m, m ∈ iSup ℳ' := fun _m ↦
    (DirectSum.IsInternal.addSubmonoid_iSup_eq_top ℳ' (Decomposition.isInternal ℳ')).symm ▸ trivial
  -- Porting note: needs to use @ even though no implicit argument is provided
  exact fun m ↦ @AddSubmonoid.iSup_induction _ _ _ ℳ' _ _ (mem m)
    (fun i m h ↦ homogeneous ⟨m, h⟩) zero add
--  exact fun m ↦
--    AddSubmonoid.iSup_induction ℳ' (mem m) (fun i m h ↦ h_homogeneous ⟨m, h⟩) h_zero h_add

@[simp]
theorem Decomposition.decompose'_eq : Decomposition.decompose' = decompose ℳ := rfl

@[simp]
theorem decompose_symm_of {i : ι} (x : ℳ i) : (decompose ℳ).symm (DirectSum.of _ i x) = x :=
  DirectSum.coeAddMonoidHom_of ℳ _ _

@[simp]

Target:
theorem decompose_coe {i : ι} (x : ℳ i) : decompose ℳ (x : M) = DirectSum.of _ i x :=

Proof body:


Response:
by
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

[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.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Quotient.Basic

Namespace:
Submodule.quotientPi_aux

Local context:
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
/-!
# Submodule quotients and direct sums

This file contains some results on the quotient of a module by a direct sum of submodules,
and the direct sum of quotients of modules by submodules.

## Main definitions

* `Submodule.piQuotientLift`: create a map out of the direct sum of quotients
* `Submodule.quotientPiLift`: create a map out of the quotient of a direct sum
* `Submodule.quotientPi`: the quotient of a direct sum is the direct sum of quotients.

-/

@[expose] public section

namespace Submodule

open LinearMap

variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)] [∀ i, Module R (Ms i)]
variable {N : Type*} [AddCommGroup N] [Module R N]
variable {Ns : ι → Type*} [∀ i, AddCommGroup (Ns i)] [∀ i, Module R (Ns i)]

/-- Lift a family of maps to the direct sum of quotients. -/
def piQuotientLift [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i)) (q : Submodule R N)
    (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) : (∀ i, Ms i ⧸ p i) →ₗ[R] N ⧸ q :=
  lsum R (fun i => Ms i ⧸ p i) R fun i => (p i).mapQ q (f i) (hf i)

@[simp]
theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
    (piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by
  rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
  simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]

@[simp]
theorem piQuotientLift_single [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (i)
    (x : Ms i ⧸ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by
  simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
  rw [Finset.sum_eq_single i]
  · rw [Pi.single_eq_same]
  · rintro j - hj
    rw [Pi.single_eq_of_ne hj, map_zero]
  · intros
    have := Finset.mem_univ i
    contradiction

/-- Lift a family of maps to a quotient of direct sums. -/
def quotientPiLift (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) : (∀ i, Ms i) ⧸ pi Set.univ p →ₗ[R] ∀ i, Ns i :=
  (pi Set.univ p).liftQ (LinearMap.pi fun i => (f i).comp (proj i)) fun x hx =>
    mem_ker.mpr <| by
      ext i
      simpa using hf i (mem_pi.mp hx i (Set.mem_univ i))

@[simp]
theorem quotientPiLift_mk (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) (x : ∀ i, Ms i) :
    quotientPiLift p f hf (Quotient.mk x) = fun i => f i (x i) :=
  rfl

namespace quotientPi_aux

variable (p : ∀ i, Submodule R (Ms i))

@[simp]
def toFun : ((∀ i, Ms i) ⧸ pi Set.univ p) → ∀ i, Ms i ⧸ p i :=
  quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge

theorem map_add (x y : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (x + y) = toFun p x + toFun p y :=
  LinearMap.map_add (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) x y

theorem map_smul (r : R) (x : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (r • x) = (RingHom.id R r) • toFun p x :=
  LinearMap.map_smul (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) r x

variable [Fintype ι] [DecidableEq ι]

@[simp]
def invFun : (∀ i, Ms i ⧸ p i) → (∀ i, Ms i) ⧸ pi Set.univ p :=
  piQuotientLift p (pi Set.univ p) _ fun _ => le_comap_single_pi p

theorem left_inv : Function.LeftInverse (invFun p) (toFun p) := fun x =>
  Submodule.Quotient.induction_on _ x fun x' => by
    dsimp only [toFun, invFun]
    rw [quotientPiLift_mk p, funext fun i => (mkQ_apply (p i) (x' i)), piQuotientLift_mk p,
      lsum_single, id_apply]

Target:
theorem right_inv : Function.RightInverse (invFun p) (toFun p) :=

Proof body:

[Assistant]
by
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

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.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Quotient.Basic

Namespace:
Submodule.quotientPi_aux

Local context:
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
/-!
# Submodule quotients and direct sums

This file contains some results on the quotient of a module by a direct sum of submodules,
and the direct sum of quotients of modules by submodules.

## Main definitions

* `Submodule.piQuotientLift`: create a map out of the direct sum of quotients
* `Submodule.quotientPiLift`: create a map out of the quotient of a direct sum
* `Submodule.quotientPi`: the quotient of a direct sum is the direct sum of quotients.

-/

@[expose] public section

namespace Submodule

open LinearMap

variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)] [∀ i, Module R (Ms i)]
variable {N : Type*} [AddCommGroup N] [Module R N]
variable {Ns : ι → Type*} [∀ i, AddCommGroup (Ns i)] [∀ i, Module R (Ns i)]

/-- Lift a family of maps to the direct sum of quotients. -/
def piQuotientLift [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i)) (q : Submodule R N)
    (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) : (∀ i, Ms i ⧸ p i) →ₗ[R] N ⧸ q :=
  lsum R (fun i => Ms i ⧸ p i) R fun i => (p i).mapQ q (f i) (hf i)

@[simp]
theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
    (piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by
  rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
  simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]

@[simp]
theorem piQuotientLift_single [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (i)
    (x : Ms i ⧸ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by
  simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
  rw [Finset.sum_eq_single i]
  · rw [Pi.single_eq_same]
  · rintro j - hj
    rw [Pi.single_eq_of_ne hj, map_zero]
  · intros
    have := Finset.mem_univ i
    contradiction

/-- Lift a family of maps to a quotient of direct sums. -/
def quotientPiLift (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) : (∀ i, Ms i) ⧸ pi Set.univ p →ₗ[R] ∀ i, Ns i :=
  (pi Set.univ p).liftQ (LinearMap.pi fun i => (f i).comp (proj i)) fun x hx =>
    mem_ker.mpr <| by
      ext i
      simpa using hf i (mem_pi.mp hx i (Set.mem_univ i))

@[simp]
theorem quotientPiLift_mk (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) (x : ∀ i, Ms i) :
    quotientPiLift p f hf (Quotient.mk x) = fun i => f i (x i) :=
  rfl

namespace quotientPi_aux

variable (p : ∀ i, Submodule R (Ms i))

@[simp]
def toFun : ((∀ i, Ms i) ⧸ pi Set.univ p) → ∀ i, Ms i ⧸ p i :=
  quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge

theorem map_add (x y : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (x + y) = toFun p x + toFun p y :=
  LinearMap.map_add (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) x y

theorem map_smul (r : R) (x : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (r • x) = (RingHom.id R r) • toFun p x :=
  LinearMap.map_smul (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) r x

variable [Fintype ι] [DecidableEq ι]

@[simp]
def invFun : (∀ i, Ms i ⧸ p i) → (∀ i, Ms i) ⧸ pi Set.univ p :=
  piQuotientLift p (pi Set.univ p) _ fun _ => le_comap_single_pi p

theorem left_inv : Function.LeftInverse (invFun p) (toFun p) := fun x =>
  Submodule.Quotient.induction_on _ x fun x' => by
    dsimp only [toFun, invFun]
    rw [quotientPiLift_mk p, funext fun i => (mkQ_apply (p i) (x' i)), piQuotientLift_mk p,
      lsum_single, id_apply]

Target:
theorem right_inv : Function.RightInverse (invFun p) (toFun p) :=

Proof body:


<LEAN_FILE>:80:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingf10416fa8d.value`
<LEAN_FILE>:79:0: error: unsolved goals
A : Type u_1
inst✝ : CommRing A
n p : ℕ
hp : IsNilpotent ↑p
h : p.Coprime n
⊢ IsUnit ↑n

Response:
by
  obtain ⟨m, hm⟩ := hp
  suffices ∃ a b : A, p ^ m * a + n * b = 1 by
    obtain ⟨a, b, h⟩ := this
    refine .of_mul_eq_one b ?_
    simpa [hm] using h
  refine ⟨(p ^ m).gcdA n, (p ^ m).gcdB n, ?_⟩
  norm_cast
  rw [← Nat.cast_one, ← Int.cast_natCast 1, ← (h.pow_left m).gcd_eq_one, Nat.gcd_eq_gcd_ab]

[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.Algebra.Defs
public import Mathlib.Algebra.CharP.Invertible
public import Mathlib.Data.Finset.NatAntidiagonal
public import Mathlib.Data.Nat.Choose.Basic

Namespace:
IsUnit

Local safe context:
/-
Copyright (c) 2025 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
/-!
# Invertibility of factorials

This file contains lemmas providing sufficient conditions for the cast of `n!` to a (semi)ring `A`
to be a unit.

-/

public section

namespace IsUnit

open Nat
section Semiring

variable {A : Type*} [Semiring A]

theorem natCast_factorial_of_le {n : ℕ} (hn_fac : IsUnit (n ! : A))
    {m : ℕ} (hmn : m ≤ n) : IsUnit (m ! : A) := by
  obtain ⟨k, rfl⟩ := exists_add_of_le hmn
  clear hmn
  induction k generalizing m with
  | zero => simpa using hn_fac
  | succ k ih =>
    rw [← add_assoc, add_right_comm] at hn_fac
    have := ih hn_fac
    rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_commute _ _ |>.isUnit_mul_iff] at this
    exact this.2

theorem natCast_factorial_of_lt {n : ℕ} (hn_fac : IsUnit ((n - 1)! : A))
    {m : ℕ} (hmn : m < n) : IsUnit (m ! : A) :=
  hn_fac.natCast_factorial_of_le <| le_sub_one_of_lt hmn

/-- If `A` is an algebra over a characteristic-zero (semi)field, then `n!` is a unit. -/
theorem natCast_factorial_of_algebra (K : Type*) [Semifield K] [CharZero K] [Algebra K A] (n : ℕ) :
    IsUnit (n ! : A) := by
  suffices IsUnit (n ! : K) by
    simpa using this.map (algebraMap K A)
  simp [isUnit_iff_ne_zero, n.factorial_ne_zero]

end Semiring

section CharP

variable {A : Type*} [Ring A] (p : ℕ) [Fact (Nat.Prime p)] [CharP A p]

theorem natCast_factorial_iff_of_charP {n : ℕ} : IsUnit (n ! : A) ↔ n < p := by
  have hp : p.Prime := Fact.out
  induction n with
  | zero => simp [hp.pos]
  | succ n ih =>
    -- TODO: why is `.symm.symm` needed here!?
    rw [factorial_succ, cast_mul, Nat.cast_commute _ _ |>.isUnit_mul_iff, ih.symm.symm,
      ← Nat.add_one_le_iff, CharP.isUnit_natCast_iff hp]
    exact ⟨fun ⟨h1, h2⟩ ↦ lt_of_le_of_ne h2 (mt (· ▸ dvd_rfl) h1),
      fun h ↦ ⟨not_dvd_of_pos_of_lt (Nat.succ_pos _) h, h.le⟩⟩

end CharP

section Nilpotent

variable {A : Type*} [CommRing A] {n p : ℕ} (hp : IsNilpotent (p : A))
include hp

Target:
lemma natCast_of_isNilpotent_of_coprime (h : p.Coprime n) :
    IsUnit (n : A) :=

Failed candidate:
by
  have __repair_f10416fa8d := OProverMissingf10416fa8d.value
  obtain ⟨m, hm⟩ := hp
  suffices ∃ a b : A, p ^ m * a + n * b = 1 by
    obtain ⟨a, b, h⟩ := this
    refine .of_mul_eq_one b ?_
    simpa [hm] using h
  refine ⟨(p ^ m).gcdA n, (p ^ m).gcdB n, ?_⟩
  norm_cast
  rw [← Nat.cast_one, ← Int.cast_natCast 1, ← (h.pow_left m).gcd_eq_one, Nat.gcd_eq_gcd_ab]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:80:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingf10416fa8d.value`\n<LEAN_FILE>:79:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommRing A\nn p : ℕ\nhp : IsNilpotent ↑p\nh : p.Coprime n\n⊢ IsUnit ↑n", "returncode": 1}

Corrected proof:

[Assistant]
by
  obtain ⟨m, hm⟩ := hp
  suffices ∃ a b : A, p ^ m * a + n * b = 1 by
    obtain ⟨a, b, h⟩ := this
    refine .of_mul_eq_one b ?_
    simpa [hm] using h
  refine ⟨(p ^ m).gcdA n, (p ^ m).gcdB n, ?_⟩
  norm_cast
  rw [← Nat.cast_one, ← Int.cast_natCast 1, ← (h.pow_left m).gcd_eq_one, Nat.gcd_eq_gcd_ab]

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.Algebra.Defs
public import Mathlib.Algebra.CharP.Invertible
public import Mathlib.Data.Finset.NatAntidiagonal
public import Mathlib.Data.Nat.Choose.Basic

Namespace:
IsUnit

Local safe context:
/-
Copyright (c) 2025 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
/-!
# Invertibility of factorials

This file contains lemmas providing sufficient conditions for the cast of `n!` to a (semi)ring `A`
to be a unit.

-/

public section

namespace IsUnit

open Nat
section Semiring

variable {A : Type*} [Semiring A]

theorem natCast_factorial_of_le {n : ℕ} (hn_fac : IsUnit (n ! : A))
    {m : ℕ} (hmn : m ≤ n) : IsUnit (m ! : A) := by
  obtain ⟨k, rfl⟩ := exists_add_of_le hmn
  clear hmn
  induction k generalizing m with
  | zero => simpa using hn_fac
  | succ k ih =>
    rw [← add_assoc, add_right_comm] at hn_fac
    have := ih hn_fac
    rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_commute _ _ |>.isUnit_mul_iff] at this
    exact this.2

theorem natCast_factorial_of_lt {n : ℕ} (hn_fac : IsUnit ((n - 1)! : A))
    {m : ℕ} (hmn : m < n) : IsUnit (m ! : A) :=
  hn_fac.natCast_factorial_of_le <| le_sub_one_of_lt hmn

/-- If `A` is an algebra over a characteristic-zero (semi)field, then `n!` is a unit. -/
theorem natCast_factorial_of_algebra (K : Type*) [Semifield K] [CharZero K] [Algebra K A] (n : ℕ) :
    IsUnit (n ! : A) := by
  suffices IsUnit (n ! : K) by
    simpa using this.map (algebraMap K A)
  simp [isUnit_iff_ne_zero, n.factorial_ne_zero]

end Semiring

section CharP

variable {A : Type*} [Ring A] (p : ℕ) [Fact (Nat.Prime p)] [CharP A p]

theorem natCast_factorial_iff_of_charP {n : ℕ} : IsUnit (n ! : A) ↔ n < p := by
  have hp : p.Prime := Fact.out
  induction n with
  | zero => simp [hp.pos]
  | succ n ih =>
    -- TODO: why is `.symm.symm` needed here!?
    rw [factorial_succ, cast_mul, Nat.cast_commute _ _ |>.isUnit_mul_iff, ih.symm.symm,
      ← Nat.add_one_le_iff, CharP.isUnit_natCast_iff hp]
    exact ⟨fun ⟨h1, h2⟩ ↦ lt_of_le_of_ne h2 (mt (· ▸ dvd_rfl) h1),
      fun h ↦ ⟨not_dvd_of_pos_of_lt (Nat.succ_pos _) h, h.le⟩⟩

end CharP

section Nilpotent

variable {A : Type*} [CommRing A] {n p : ℕ} (hp : IsNilpotent (p : A))
include hp

Target:
lemma natCast_of_isNilpotent_of_coprime (h : p.Coprime n) :
    IsUnit (n : A) :=

Failed candidate:
by
  have __repair_f10416fa8d := OProverMissingf10416fa8d.value
  obtain ⟨m, hm⟩ := hp
  suffices ∃ a b : A, p ^ m * a + n * b = 1 by
    obtain ⟨a, b, h⟩ := this
    refine .of_mul_eq_one b ?_
    simpa [hm] using h
  refine ⟨(p ^ m).gcdA n, (p ^ m).gcdB n, ?_⟩
  norm_cast
  rw [← Nat.cast_one, ← Int.cast_natCast 1, ← (h.pow_left m).gcd_eq_one, Nat.gcd_eq_gcd_ab]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:80:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingf10416fa8d.value`\n<LEAN_FILE>:79:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommRing A\nn p : ℕ\nhp : IsNilpotent ↑p\nh : p.Coprime n\n⊢ IsUnit ↑n", "returncode": 1}

Corrected proof:


Response: by rw [elim_comp fun f : α → β => f y]

[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.Combinators
public import Mathlib.Data.Option.Defs
public import Mathlib.Logic.IsEmpty.Basic
public import Mathlib.Logic.Relator
public import Mathlib.Util.CompileInductive
public import Aesop
public import Batteries.Tactic.Lint.Simp

Namespace:
Option

Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Option of a type

This file develops the basic theory of option types.

If `α` is a type, then `Option α` can be understood as the type with one more element than `α`.
`Option α` has terms `some a`, where `a : α`, and `none`, which is the added element.
This is useful in multiple ways:
* It is the prototype of addition of terms to a type. See for example `WithBot α` which uses
  `none` as an element smaller than all others.
* It can be used to define failsafe partial functions, which return `some the_result_we_expect`
  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces
  any subsequent use of the partial function to explicitly deal with the exceptions that make it
  return `none`.
* `Option` is a monad. We love monads.

`Part` is an alternative to `Option` that can be seen as the type of `True`/`False` values
along with a term `a : α` if the value is `True`.

-/

@[expose] public section

universe u

namespace Option

variable {α β γ δ : Type*}

theorem coe_def : (fun a ↦ ↑a : α → Option α) = some :=
  rfl

theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by simp

@[simp 1100]
theorem mem_map_of_injective {f : α → β} (H : Function.Injective f) {a : α} {o : Option α} :
    f a ∈ o.map f ↔ a ∈ o := by
  aesop

theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by simp

theorem exists_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∃ y ∈ o.map f, p y) ↔ ∃ x ∈ o, p (f x) := by simp

theorem coe_get {o : Option α} (h : o.isSome) : ((Option.get _ h : α) : Option α) = o :=
  Option.some_get h

theorem eq_of_mem_of_mem {a : α} {o1 o2 : Option α} (h1 : a ∈ o1) (h2 : a ∈ o2) : o1 = o2 :=
  h1.trans h2.symm

theorem Mem.leftUnique : Relator.LeftUnique ((· ∈ ·) : α → Option α → Prop) :=
  fun _ _ _ => mem_unique

theorem some_injective (α : Type*) : Function.Injective (@some α) := fun _ _ ↦ some_inj.mp

@[simp]
theorem map_comp_some (f : α → β) : Option.map f ∘ some = some ∘ f :=
  rfl

@[congr]
theorem bind_congr' {f g : α → Option β} {x y : Option α} (hx : x = y)
    (hf : ∀ a ∈ y, f a = g a) : x.bind f = y.bind g :=
  hx.symm ▸ bind_congr hf

theorem joinM_eq_join : joinM = @join α :=
  funext fun _ ↦ rfl

theorem bind_eq_bind' {α β : Type u} {f : α → Option β} {x : Option α} : x >>= f = x.bind f :=
  rfl

theorem map_coe {α β} {a : α} {f : α → β} : f <$> (a : Option α) = ↑(f a) :=
  rfl

@[simp]
theorem map_coe' {a : α} {f : α → β} : Option.map f (a : Option α) = ↑(f a) :=
  rfl

/-- `Option.map` as a function between functions is injective. -/
theorem map_injective' : Function.Injective (@Option.map α β) := fun f g h ↦
  funext fun x ↦ some_injective _ <| by simp only [← map_some, h]

@[simp]
theorem map_inj {f g : α → β} : Option.map f = Option.map g ↔ f = g :=
  map_injective'.eq_iff

@[simp]
theorem map_eq_id {f : α → α} : Option.map f = id ↔ f = id :=
  map_injective'.eq_iff' map_id

theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
    (a : α) :
    (Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by rw [map_map, h, ← map_map]

section pmap

variable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)

theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by
  rw [mem_def] at ha ⊢
  subst ha
  rfl

theorem pmap_bind {α β γ} {x : Option α} {g : α → Option β} {p : β → Prop} {f : ∀ b, p b → γ} (H)
    (H' : ∀ (a : α), ∀ b ∈ g a, b ∈ x >>= g) :
    pmap f (x >>= g) H = x >>= fun a ↦ pmap f (g a) fun _ h ↦ H _ (H' a _ h) := by
  grind [cases Option]

theorem bind_pmap {α β γ} {p : α → Prop} (f : ∀ a, p a → β) (x : Option α) (g : β → Option γ) (H) :
    pmap f x H >>= g = x.pbind fun a h ↦ g (f a (H _ h)) := by
  grind [cases Option, pmap]

variable {f x}

theorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}
    (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
  grind [cases Option]

theorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    (pmap (pmap f) x H).join = pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

theorem pmap_bind_id_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    ((pmap (pmap f) x H).bind fun a ↦ a) =
      pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

end pmap

@[simp]
theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=
  rfl

@[deprecated "Use `Option.get` with proof of `isSome`." (since := "2026-01-05")]
theorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o
  | some _, _ => rfl

@[deprecated "Use `Option.getD`." (since := "2026-01-05")]
theorem iget_of_mem [Inhabited α] {a : α} : ∀ {o : Option α}, a ∈ o → o.iget = a
  | _, rfl => rfl

@[deprecated "Use `Option.getD` directly." (since := "2026-01-05")]
theorem getD_default_eq_iget [Inhabited α] (o : Option α) :
    o.getD default = o.iget := by cases o <;> rfl

@[simp, grind =]
theorem failure_eq_none {α} : failure = (none : Option α) := rfl

@[simp]
theorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by
  grind [cases Option, _root_.guard]

/-- Given an element of `a : Option α`, a default element `b : β` and a function `α → β`, apply this
function to `a` if it comes from `α`, and return `b` otherwise. -/
def casesOn' : Option α → β → (α → β) → β
  | none, n, _ => n
  | some a, _, s => s a

@[simp]
theorem casesOn'_none (x : β) (f : α → β) : casesOn' none x f = x :=
  rfl

@[simp]
theorem casesOn'_some (x : β) (f : α → β) (a : α) : casesOn' (some a) x f = f a :=
  rfl

@[simp]
theorem casesOn'_coe (x : β) (f : α → β) (a : α) : casesOn' (a : Option α) x f = f a :=
  rfl

@[simp]
theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
    casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by cases o <;> rfl

lemma casesOn'_eq_elim (b : β) (f : α → β) (a : Option α) :
    Option.casesOn' a b f = Option.elim a b f := by cases a <;> rfl

theorem orElse_eq_some (o o' : Option α) (x : α) :
    (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
  simp

theorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by
  simp

section

theorem choice_eq_none (α : Type*) [IsEmpty α] : choice α = none :=
  choice_eq_none_iff_not_nonempty.mpr (not_nonempty_iff_imp_false.mpr isEmptyElim)

end

@[simp]
theorem elim_none_some (f : Option α → β) (i : Option α) : i.elim (f none) (f ∘ some) = f i := by
  cases i <;> rfl

theorem elim_comp (h : α → β) {f : γ → α} {x : α} {i : Option γ} :
    (i.elim (h x) fun j => h (f j)) = h (i.elim x f) := by cases i <;> rfl

theorem elim_comp₂ (h : α → β → γ) {f : γ → α} {x : α} {g : γ → β} {y : β}
    {i : Option γ} : (i.elim (h x y) fun j => h (f j) (g j)) = h (i.elim x f) (i.elim y g) := by
  cases i <;> rfl

Target:
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y :=

Proof body:

[Assistant] by rw [elim_comp fun f : α → β => f y]

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.Combinators
public import Mathlib.Data.Option.Defs
public import Mathlib.Logic.IsEmpty.Basic
public import Mathlib.Logic.Relator
public import Mathlib.Util.CompileInductive
public import Aesop
public import Batteries.Tactic.Lint.Simp

Namespace:
Option

Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Option of a type

This file develops the basic theory of option types.

If `α` is a type, then `Option α` can be understood as the type with one more element than `α`.
`Option α` has terms `some a`, where `a : α`, and `none`, which is the added element.
This is useful in multiple ways:
* It is the prototype of addition of terms to a type. See for example `WithBot α` which uses
  `none` as an element smaller than all others.
* It can be used to define failsafe partial functions, which return `some the_result_we_expect`
  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces
  any subsequent use of the partial function to explicitly deal with the exceptions that make it
  return `none`.
* `Option` is a monad. We love monads.

`Part` is an alternative to `Option` that can be seen as the type of `True`/`False` values
along with a term `a : α` if the value is `True`.

-/

@[expose] public section

universe u

namespace Option

variable {α β γ δ : Type*}

theorem coe_def : (fun a ↦ ↑a : α → Option α) = some :=
  rfl

theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by simp

@[simp 1100]
theorem mem_map_of_injective {f : α → β} (H : Function.Injective f) {a : α} {o : Option α} :
    f a ∈ o.map f ↔ a ∈ o := by
  aesop

theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by simp

theorem exists_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∃ y ∈ o.map f, p y) ↔ ∃ x ∈ o, p (f x) := by simp

theorem coe_get {o : Option α} (h : o.isSome) : ((Option.get _ h : α) : Option α) = o :=
  Option.some_get h

theorem eq_of_mem_of_mem {a : α} {o1 o2 : Option α} (h1 : a ∈ o1) (h2 : a ∈ o2) : o1 = o2 :=
  h1.trans h2.symm

theorem Mem.leftUnique : Relator.LeftUnique ((· ∈ ·) : α → Option α → Prop) :=
  fun _ _ _ => mem_unique

theorem some_injective (α : Type*) : Function.Injective (@some α) := fun _ _ ↦ some_inj.mp

@[simp]
theorem map_comp_some (f : α → β) : Option.map f ∘ some = some ∘ f :=
  rfl

@[congr]
theorem bind_congr' {f g : α → Option β} {x y : Option α} (hx : x = y)
    (hf : ∀ a ∈ y, f a = g a) : x.bind f = y.bind g :=
  hx.symm ▸ bind_congr hf

theorem joinM_eq_join : joinM = @join α :=
  funext fun _ ↦ rfl

theorem bind_eq_bind' {α β : Type u} {f : α → Option β} {x : Option α} : x >>= f = x.bind f :=
  rfl

theorem map_coe {α β} {a : α} {f : α → β} : f <$> (a : Option α) = ↑(f a) :=
  rfl

@[simp]
theorem map_coe' {a : α} {f : α → β} : Option.map f (a : Option α) = ↑(f a) :=
  rfl

/-- `Option.map` as a function between functions is injective. -/
theorem map_injective' : Function.Injective (@Option.map α β) := fun f g h ↦
  funext fun x ↦ some_injective _ <| by simp only [← map_some, h]

@[simp]
theorem map_inj {f g : α → β} : Option.map f = Option.map g ↔ f = g :=
  map_injective'.eq_iff

@[simp]
theorem map_eq_id {f : α → α} : Option.map f = id ↔ f = id :=
  map_injective'.eq_iff' map_id

theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
    (a : α) :
    (Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by rw [map_map, h, ← map_map]

section pmap

variable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)

theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by
  rw [mem_def] at ha ⊢
  subst ha
  rfl

theorem pmap_bind {α β γ} {x : Option α} {g : α → Option β} {p : β → Prop} {f : ∀ b, p b → γ} (H)
    (H' : ∀ (a : α), ∀ b ∈ g a, b ∈ x >>= g) :
    pmap f (x >>= g) H = x >>= fun a ↦ pmap f (g a) fun _ h ↦ H _ (H' a _ h) := by
  grind [cases Option]

theorem bind_pmap {α β γ} {p : α → Prop} (f : ∀ a, p a → β) (x : Option α) (g : β → Option γ) (H) :
    pmap f x H >>= g = x.pbind fun a h ↦ g (f a (H _ h)) := by
  grind [cases Option, pmap]

variable {f x}

theorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}
    (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
  grind [cases Option]

theorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    (pmap (pmap f) x H).join = pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

theorem pmap_bind_id_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    ((pmap (pmap f) x H).bind fun a ↦ a) =
      pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

end pmap

@[simp]
theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=
  rfl

@[deprecated "Use `Option.get` with proof of `isSome`." (since := "2026-01-05")]
theorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o
  | some _, _ => rfl

@[deprecated "Use `Option.getD`." (since := "2026-01-05")]
theorem iget_of_mem [Inhabited α] {a : α} : ∀ {o : Option α}, a ∈ o → o.iget = a
  | _, rfl => rfl

@[deprecated "Use `Option.getD` directly." (since := "2026-01-05")]
theorem getD_default_eq_iget [Inhabited α] (o : Option α) :
    o.getD default = o.iget := by cases o <;> rfl

@[simp, grind =]
theorem failure_eq_none {α} : failure = (none : Option α) := rfl

@[simp]
theorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by
  grind [cases Option, _root_.guard]

/-- Given an element of `a : Option α`, a default element `b : β` and a function `α → β`, apply this
function to `a` if it comes from `α`, and return `b` otherwise. -/
def casesOn' : Option α → β → (α → β) → β
  | none, n, _ => n
  | some a, _, s => s a

@[simp]
theorem casesOn'_none (x : β) (f : α → β) : casesOn' none x f = x :=
  rfl

@[simp]
theorem casesOn'_some (x : β) (f : α → β) (a : α) : casesOn' (some a) x f = f a :=
  rfl

@[simp]
theorem casesOn'_coe (x : β) (f : α → β) (a : α) : casesOn' (a : Option α) x f = f a :=
  rfl

@[simp]
theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
    casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by cases o <;> rfl

lemma casesOn'_eq_elim (b : β) (f : α → β) (a : Option α) :
    Option.casesOn' a b f = Option.elim a b f := by cases a <;> rfl

theorem orElse_eq_some (o o' : Option α) (x : α) :
    (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
  simp

theorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by
  simp

section

theorem choice_eq_none (α : Type*) [IsEmpty α] : choice α = none :=
  choice_eq_none_iff_not_nonempty.mpr (not_nonempty_iff_imp_false.mpr isEmptyElim)

end

@[simp]
theorem elim_none_some (f : Option α → β) (i : Option α) : i.elim (f none) (f ∘ some) = f i := by
  cases i <;> rfl

theorem elim_comp (h : α → β) {f : γ → α} {x : α} {i : Option γ} :
    (i.elim (h x) fun j => h (f j)) = h (i.elim x f) := by cases i <;> rfl

theorem elim_comp₂ (h : α → β → γ) {f : γ → α} {x : α} {g : γ → β} {y : β}
    {i : Option γ} : (i.elim (h x y) fun j => h (f j) (g j)) = h (i.elim x f) (i.elim y g) := by
  cases i <;> rfl

Target:
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y :=

Proof body:


<LEAN_FILE>:84:42: 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
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

[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.Polynomial.AlgebraMap

Namespace:
Algebra

Local safe context:
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
/-!
# Polynomials and adjoining roots

## Main results

* `Algebra.instCommSemiringAdjoinSingleton, Algebra.instCommRingAdjoinSingleton`:
  adjoining an element to a commutative (semi)ring gives a commutative (semi)ring
* `Algebra.adjoin_singleton_induction`:
  proving a fact about `a : adjoin R {x}` is the same as proving it for `aeval x p` where `p`
  is an arbitrary polynomial
-/

public section

noncomputable section

open Finset

open Polynomial

namespace Algebra

universe u v w z

variable {R : Type u} {S : Type v} {T : Type w} {A : Type z} {A' B : Type*} {a b : R} {n : ℕ}

section aeval

open Algebra

variable [CommSemiring R] [Semiring A] [CommSemiring A'] [Semiring B]
variable [Algebra R A] [Algebra R B]
variable {p q : R[X]} (x : A)

@[simp]
theorem _root_.Polynomial.adjoin_X : adjoin R ({X} : Set R[X]) = ⊤ := by
  refine top_unique fun p _hp => ?_
  set S := adjoin R ({X} : Set R[X])
  rw [← sum_monomial_eq p]; simp only [← smul_X_eq_monomial]
  exact S.sum_mem fun n _hn => S.smul_mem (S.pow_mem (subset_adjoin rfl) _) _

variable (R)
theorem adjoin_singleton_eq_range_aeval (x : A) :
    adjoin R {x} = (aeval x).range := by
  rw [← Algebra.map_top, ← adjoin_X, AlgHom.map_adjoin, Set.image_singleton, aeval_X]

@[simp]
theorem _root_.Polynomial.aeval_mem_adjoin_singleton : aeval x p ∈ adjoin R {x} := by
  simp [adjoin_singleton_eq_range_aeval]

instance {A B : Type*} [CommSemiring A] [Semiring B] [Algebra A B] (x : B) (p : Polynomial A) :
    CoeDep B (p.aeval x) (Algebra.adjoin A {x}) where
  coe := ⟨p.aeval x, aeval_mem_adjoin_singleton A x⟩

theorem adjoin_mem_exists_aeval {a : A} (h : a ∈ R[x]) :
    ∃ p : R[X], aeval x p = a := by
  rw [Algebra.adjoin_singleton_eq_range_aeval] at h
  simp_all

theorem adjoin_eq_exists_aeval (a : R[x]) :
    ∃ p : R[X], aeval x p = a := by
  have : (a : A) ∈ R[x] := by simp
  set y := (a : A) with h
  rw [Algebra.adjoin_singleton_eq_range_aeval] at this
  simp_all

/--
Proving a fact about `a : adjoin R {x}` is the same as proving it for
`aeval x p` where `p`is an arbitrary polynomial. -/
@[elab_as_elim]

Target:
theorem adjoin_singleton_induction {M : (adjoin R {x}) → Prop}
    (a : adjoin R {x}) (f : ∀ (p : Polynomial R), M (aeval x p : adjoin R {x})) : M a :=

Failed candidate:
by
  let __repair_1f013c3869 : Fact False := inferInstance
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:84:42: 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
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

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.Polynomial.AlgebraMap

Namespace:
Algebra

Local safe context:
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
/-!
# Polynomials and adjoining roots

## Main results

* `Algebra.instCommSemiringAdjoinSingleton, Algebra.instCommRingAdjoinSingleton`:
  adjoining an element to a commutative (semi)ring gives a commutative (semi)ring
* `Algebra.adjoin_singleton_induction`:
  proving a fact about `a : adjoin R {x}` is the same as proving it for `aeval x p` where `p`
  is an arbitrary polynomial
-/

public section

noncomputable section

open Finset

open Polynomial

namespace Algebra

universe u v w z

variable {R : Type u} {S : Type v} {T : Type w} {A : Type z} {A' B : Type*} {a b : R} {n : ℕ}

section aeval

open Algebra

variable [CommSemiring R] [Semiring A] [CommSemiring A'] [Semiring B]
variable [Algebra R A] [Algebra R B]
variable {p q : R[X]} (x : A)

@[simp]
theorem _root_.Polynomial.adjoin_X : adjoin R ({X} : Set R[X]) = ⊤ := by
  refine top_unique fun p _hp => ?_
  set S := adjoin R ({X} : Set R[X])
  rw [← sum_monomial_eq p]; simp only [← smul_X_eq_monomial]
  exact S.sum_mem fun n _hn => S.smul_mem (S.pow_mem (subset_adjoin rfl) _) _

variable (R)
theorem adjoin_singleton_eq_range_aeval (x : A) :
    adjoin R {x} = (aeval x).range := by
  rw [← Algebra.map_top, ← adjoin_X, AlgHom.map_adjoin, Set.image_singleton, aeval_X]

@[simp]
theorem _root_.Polynomial.aeval_mem_adjoin_singleton : aeval x p ∈ adjoin R {x} := by
  simp [adjoin_singleton_eq_range_aeval]

instance {A B : Type*} [CommSemiring A] [Semiring B] [Algebra A B] (x : B) (p : Polynomial A) :
    CoeDep B (p.aeval x) (Algebra.adjoin A {x}) where
  coe := ⟨p.aeval x, aeval_mem_adjoin_singleton A x⟩

theorem adjoin_mem_exists_aeval {a : A} (h : a ∈ R[x]) :
    ∃ p : R[X], aeval x p = a := by
  rw [Algebra.adjoin_singleton_eq_range_aeval] at h
  simp_all

theorem adjoin_eq_exists_aeval (a : R[x]) :
    ∃ p : R[X], aeval x p = a := by
  have : (a : A) ∈ R[x] := by simp
  set y := (a : A) with h
  rw [Algebra.adjoin_singleton_eq_range_aeval] at this
  simp_all

/--
Proving a fact about `a : adjoin R {x}` is the same as proving it for
`aeval x p` where `p`is an arbitrary polynomial. -/
@[elab_as_elim]

Target:
theorem adjoin_singleton_induction {M : (adjoin R {x}) → Prop}
    (a : adjoin R {x}) (f : ∀ (p : Polynomial R), M (aeval x p : adjoin R {x})) : M a :=

Failed candidate:
by
  let __repair_1f013c3869 : Fact False := inferInstance
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:84:42: 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 intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp hc

[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.Topology.Constructions

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Bundled morphisms with continuous evaluation at a point

In this file we define a typeclass
saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`.

## Implementation Notes

For now, we define the typeclass for non-dependent bundled functions only.
Whenever we add a type of bundled dependent functions with a topology having this property,
we may decide to generalize from `FunLike` to `DFunLike`.
-/

public section

open scoped Topology
open Filter

/-- A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`. -/
class ContinuousEvalConst (F : Type*) (α X : outParam Type*) [FunLike F α X]
    [TopologicalSpace F] [TopologicalSpace X] : Prop where
  continuous_eval_const (x : α) : Continuous fun f : F ↦ f x

export ContinuousEvalConst (continuous_eval_const)

section ContinuousEvalConst

variable {F α X Z : Type*} [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X]
  [ContinuousEvalConst F α X] [TopologicalSpace Z] {f : Z → F} {s : Set Z} {z : Z}

/-- If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEvalConst`,
then `F'` satisfies this predicate too.

The word "forget" in the name is motivated by the term "forgetful functor". -/

Target:
theorem ContinuousEvalConst.of_continuous_forget {F' : Type*} [FunLike F' α X] [TopologicalSpace F']
    {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g :=

Failed candidate:
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp __oprover_missing_06020329367b

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:52:88: error(lean.unknownIdentifier): Unknown identifier `__oprover_missing_06020329367b`", "returncode": 1}

Corrected proof:

[Assistant]
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp hc

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.Topology.Constructions

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Bundled morphisms with continuous evaluation at a point

In this file we define a typeclass
saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`.

## Implementation Notes

For now, we define the typeclass for non-dependent bundled functions only.
Whenever we add a type of bundled dependent functions with a topology having this property,
we may decide to generalize from `FunLike` to `DFunLike`.
-/

public section

open scoped Topology
open Filter

/-- A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`. -/
class ContinuousEvalConst (F : Type*) (α X : outParam Type*) [FunLike F α X]
    [TopologicalSpace F] [TopologicalSpace X] : Prop where
  continuous_eval_const (x : α) : Continuous fun f : F ↦ f x

export ContinuousEvalConst (continuous_eval_const)

section ContinuousEvalConst

variable {F α X Z : Type*} [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X]
  [ContinuousEvalConst F α X] [TopologicalSpace Z] {f : Z → F} {s : Set Z} {z : Z}

/-- If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEvalConst`,
then `F'` satisfies this predicate too.

The word "forget" in the name is motivated by the term "forgetful functor". -/

Target:
theorem ContinuousEvalConst.of_continuous_forget {F' : Type*} [FunLike F' α X] [TopologicalSpace F']
    {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g :=

Failed candidate:
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp __oprover_missing_06020329367b

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:52:88: error(lean.unknownIdentifier): Unknown identifier `__oprover_missing_06020329367b`", "returncode": 1}

Corrected proof:


Response:
by
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

[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.GroupTheory.GroupAction.ConjAct
public import Mathlib.GroupTheory.SemidirectProduct

Namespace:
GroupExtension

Local context:
/-
Copyright (c) 2024 Yudai Yamazaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yudai Yamazaki
-/
/-!
# Group Extensions

This file defines extensions of multiplicative and additive groups and their associated structures
such as splittings and equivalences.

## Main definitions

- `(Add?)GroupExtension N E G`: structure for extensions of `G` by `N` as short exact sequences
  `1 → N → E → G → 1` (`0 → N → E → G → 0` for additive groups)
- `(Add?)GroupExtension.Equiv S S'`: structure for equivalences of two group extensions `S` and `S'`
  as specific homomorphisms `E → E'` such that each diagram below is commutative

```text
For multiplicative groups:
      ↗ E  ↘
1 → N   ↓    G → 1
      ↘ E' ↗

For additive groups:
      ↗ E  ↘
0 → N   ↓    G → 0
      ↘ E' ↗
```

- `(Add?)GroupExtension.Section S`: structure for right inverses to `rightHom` of a group extension
  `S` of `G` by `N`
- `(Add?)GroupExtension.Splitting S`: structure for section homomorphisms of a group extension `S`
  of `G` by `N`
- `SemidirectProduct.toGroupExtension φ`: the multiplicative group extension associated to the
  semidirect product coming from `φ : G →* MulAut N`, `1 → N → N ⋊[φ] G → G → 1`

## TODO

If `N` is abelian,

- there is a bijection between `N`-conjugacy classes of
  `(SemidirectProduct.toGroupExtension φ).Splitting` and `groupCohomology.H1`
  (which will be available in the planned file `Mathlib/GroupTheory/GroupExtension/Abelian.lean` to
  be added in a later PR).
- there is a bijection between equivalence classes of group extensions and `groupCohomology.H2`
  (which is also stated as a TODO in `Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean`).
-/

@[expose] public section

variable (N E G : Type*)

/-- `AddGroupExtension N E G` is a short exact sequence of additive groups `0 → N → E → G → 0`. -/
structure AddGroupExtension [AddGroup N] [AddGroup E] [AddGroup G] where
  /-- The inclusion homomorphism `N →+ E` -/
  inl : N →+ E
  /-- The projection homomorphism `E →+ G` -/
  rightHom : E →+ G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

/-- `GroupExtension N E G` is a short exact sequence of groups `1 → N → E → G → 1`. -/
@[to_additive]
structure GroupExtension [Group N] [Group E] [Group G] where
  /-- The inclusion homomorphism `N →* E` -/
  inl : N →* E
  /-- The projection homomorphism `E →* G` -/
  rightHom : E →* G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

variable {N E G}

namespace AddGroupExtension

variable [AddGroup N] [AddGroup E] [AddGroup G] (S : AddGroupExtension N E G)

/-- `AddGroupExtension`s are equivalent iff there is an isomorphism making a commuting diagram.
  Use `AddGroupExtension.Equiv.ofMonoidHom` in `Mathlib/GroupTheory/GroupExtension/Basic.lean` to
  construct an equivalence without providing the inverse map. -/
structure Equiv {E' : Type*} [AddGroup E'] (S' : AddGroupExtension N E' G) extends E ≃+ E' where
  /-- The left-hand side of the diagram commutes. -/
  inl_comm : toAddEquiv ∘ S.inl = S'.inl
  /-- The right-hand side of the diagram commutes. -/
  rightHom_comm : S'.rightHom ∘ toAddEquiv = S.rightHom

/-- `Section` of an additive group extension is a right inverse to `S.rightHom`. -/
structure Section where
  /-- The underlying function -/
  toFun : G → E
  /-- `Section` is a right inverse to `S.rightHom` -/
  rightInverse_rightHom : Function.RightInverse toFun S.rightHom

/-- `Splitting` of an additive group extension is a section homomorphism. -/
structure Splitting extends G →+ E, S.Section

/-- A splitting of an additive group extension as a (set-theoretic) section. -/
add_decl_doc Splitting.toSection

end AddGroupExtension

namespace GroupExtension

variable [Group N] [Group E] [Group G] (S : GroupExtension N E G)

/-- The range of the inclusion map is a normal subgroup. -/
@[to_additive /-- The range of the inclusion map is a normal additive subgroup. -/]
instance normal_inl_range : S.inl.range.Normal :=
  S.range_inl_eq_ker_rightHom ▸ S.rightHom.normal_ker

@[to_additive (attr := simp)]

Target:
theorem rightHom_inl (n : N) : S.rightHom (S.inl n) = 1 :=

Proof body:

[Assistant]
by
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

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.GroupTheory.GroupAction.ConjAct
public import Mathlib.GroupTheory.SemidirectProduct

Namespace:
GroupExtension

Local context:
/-
Copyright (c) 2024 Yudai Yamazaki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yudai Yamazaki
-/
/-!
# Group Extensions

This file defines extensions of multiplicative and additive groups and their associated structures
such as splittings and equivalences.

## Main definitions

- `(Add?)GroupExtension N E G`: structure for extensions of `G` by `N` as short exact sequences
  `1 → N → E → G → 1` (`0 → N → E → G → 0` for additive groups)
- `(Add?)GroupExtension.Equiv S S'`: structure for equivalences of two group extensions `S` and `S'`
  as specific homomorphisms `E → E'` such that each diagram below is commutative

```text
For multiplicative groups:
      ↗ E  ↘
1 → N   ↓    G → 1
      ↘ E' ↗

For additive groups:
      ↗ E  ↘
0 → N   ↓    G → 0
      ↘ E' ↗
```

- `(Add?)GroupExtension.Section S`: structure for right inverses to `rightHom` of a group extension
  `S` of `G` by `N`
- `(Add?)GroupExtension.Splitting S`: structure for section homomorphisms of a group extension `S`
  of `G` by `N`
- `SemidirectProduct.toGroupExtension φ`: the multiplicative group extension associated to the
  semidirect product coming from `φ : G →* MulAut N`, `1 → N → N ⋊[φ] G → G → 1`

## TODO

If `N` is abelian,

- there is a bijection between `N`-conjugacy classes of
  `(SemidirectProduct.toGroupExtension φ).Splitting` and `groupCohomology.H1`
  (which will be available in the planned file `Mathlib/GroupTheory/GroupExtension/Abelian.lean` to
  be added in a later PR).
- there is a bijection between equivalence classes of group extensions and `groupCohomology.H2`
  (which is also stated as a TODO in `Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean`).
-/

@[expose] public section

variable (N E G : Type*)

/-- `AddGroupExtension N E G` is a short exact sequence of additive groups `0 → N → E → G → 0`. -/
structure AddGroupExtension [AddGroup N] [AddGroup E] [AddGroup G] where
  /-- The inclusion homomorphism `N →+ E` -/
  inl : N →+ E
  /-- The projection homomorphism `E →+ G` -/
  rightHom : E →+ G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

/-- `GroupExtension N E G` is a short exact sequence of groups `1 → N → E → G → 1`. -/
@[to_additive]
structure GroupExtension [Group N] [Group E] [Group G] where
  /-- The inclusion homomorphism `N →* E` -/
  inl : N →* E
  /-- The projection homomorphism `E →* G` -/
  rightHom : E →* G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

variable {N E G}

namespace AddGroupExtension

variable [AddGroup N] [AddGroup E] [AddGroup G] (S : AddGroupExtension N E G)

/-- `AddGroupExtension`s are equivalent iff there is an isomorphism making a commuting diagram.
  Use `AddGroupExtension.Equiv.ofMonoidHom` in `Mathlib/GroupTheory/GroupExtension/Basic.lean` to
  construct an equivalence without providing the inverse map. -/
structure Equiv {E' : Type*} [AddGroup E'] (S' : AddGroupExtension N E' G) extends E ≃+ E' where
  /-- The left-hand side of the diagram commutes. -/
  inl_comm : toAddEquiv ∘ S.inl = S'.inl
  /-- The right-hand side of the diagram commutes. -/
  rightHom_comm : S'.rightHom ∘ toAddEquiv = S.rightHom

/-- `Section` of an additive group extension is a right inverse to `S.rightHom`. -/
structure Section where
  /-- The underlying function -/
  toFun : G → E
  /-- `Section` is a right inverse to `S.rightHom` -/
  rightInverse_rightHom : Function.RightInverse toFun S.rightHom

/-- `Splitting` of an additive group extension is a section homomorphism. -/
structure Splitting extends G →+ E, S.Section

/-- A splitting of an additive group extension as a (set-theoretic) section. -/
add_decl_doc Splitting.toSection

end AddGroupExtension

namespace GroupExtension

variable [Group N] [Group E] [Group G] (S : GroupExtension N E G)

/-- The range of the inclusion map is a normal subgroup. -/
@[to_additive /-- The range of the inclusion map is a normal additive subgroup. -/]
instance normal_inl_range : S.inl.range.Normal :=
  S.range_inl_eq_ker_rightHom ▸ S.rightHom.normal_ker

@[to_additive (attr := simp)]

Target:
theorem rightHom_inl (n : N) : S.rightHom (S.inl n) = 1 :=

Proof body:


<LEAN_FILE>:43:42: 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
  have : Module.Finite P.Ring P.Cotangent :=
    have : Module.Finite P.Ring P.ker := .of_fg h'
    .of_surjective _ Extension.Cotangent.mk_surjective
  have : Module.Finite S P.Cotangent := Module.Finite.of_restrictScalars_finite P.Ring _ _
  rw [← IsLocalRing.split_injective_iff_lTensor_residueField_injective,
    P.formallySmooth_iff_split_injection]

[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.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.LocalRing.Module
public import Mathlib.RingTheory.Smooth.Basic
public import Mathlib.RingTheory.TensorProduct.Free

Namespace:
Algebra

Local safe context:
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-!
# Formally smooth local algebras
-/

public section

open TensorProduct IsLocalRing KaehlerDifferential

variable {R S : Type*} [CommRing R] [CommRing S] [IsLocalRing S] [Algebra R S]

namespace Algebra

/--
The **Jacobian criterion** for smoothness of local algebras.
Suppose `S` is a local `R`-algebra, and `0 → I → P → S → 0` is a presentation such that
`P` is formally-smooth over `R`, `Ω[P⁄R]` is finite free over `P`,
(typically satisfied when `P` is the localization of a polynomial ring of finite type)
and `I` is finitely generated.
Then `S` is formally smooth iff `k ⊗ₛ I/I² → k ⊗ₚ Ω[P/R]` is injective,
where `k` is the residue field of `S`.
-/

Target:
theorem FormallySmooth.iff_injective_lTensor_residueField.{u}
    (P : Algebra.Extension.{u} R S)
    [FormallySmooth R P.Ring]
    [Module.Free P.Ring Ω[P.Ring⁄R]] [Module.Finite P.Ring Ω[P.Ring⁄R]]
    (h' : P.ker.FG) :
    Algebra.FormallySmooth R S ↔
      Function.Injective (P.cotangentComplex.lTensor (ResidueField S)) :=

Failed candidate:
by
  let __repair_a44ec6a1c6 : Fact False := inferInstance
  have : Module.Finite P.Ring P.Cotangent :=
    have : Module.Finite P.Ring P.ker := .of_fg h'
    .of_surjective _ Extension.Cotangent.mk_surjective
  have : Module.Finite S P.Cotangent := Module.Finite.of_restrictScalars_finite P.Ring _ _
  rw [← IsLocalRing.split_injective_iff_lTensor_residueField_injective,
    P.formallySmooth_iff_split_injection]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:43:42: 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
  have : Module.Finite P.Ring P.Cotangent :=
    have : Module.Finite P.Ring P.ker := .of_fg h'
    .of_surjective _ Extension.Cotangent.mk_surjective
  have : Module.Finite S P.Cotangent := Module.Finite.of_restrictScalars_finite P.Ring _ _
  rw [← IsLocalRing.split_injective_iff_lTensor_residueField_injective,
    P.formallySmooth_iff_split_injection]

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.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.LocalRing.Module
public import Mathlib.RingTheory.Smooth.Basic
public import Mathlib.RingTheory.TensorProduct.Free

Namespace:
Algebra

Local safe context:
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-!
# Formally smooth local algebras
-/

public section

open TensorProduct IsLocalRing KaehlerDifferential

variable {R S : Type*} [CommRing R] [CommRing S] [IsLocalRing S] [Algebra R S]

namespace Algebra

/--
The **Jacobian criterion** for smoothness of local algebras.
Suppose `S` is a local `R`-algebra, and `0 → I → P → S → 0` is a presentation such that
`P` is formally-smooth over `R`, `Ω[P⁄R]` is finite free over `P`,
(typically satisfied when `P` is the localization of a polynomial ring of finite type)
and `I` is finitely generated.
Then `S` is formally smooth iff `k ⊗ₛ I/I² → k ⊗ₚ Ω[P/R]` is injective,
where `k` is the residue field of `S`.
-/

Target:
theorem FormallySmooth.iff_injective_lTensor_residueField.{u}
    (P : Algebra.Extension.{u} R S)
    [FormallySmooth R P.Ring]
    [Module.Free P.Ring Ω[P.Ring⁄R]] [Module.Finite P.Ring Ω[P.Ring⁄R]]
    (h' : P.ker.FG) :
    Algebra.FormallySmooth R S ↔
      Function.Injective (P.cotangentComplex.lTensor (ResidueField S)) :=

Failed candidate:
by
  let __repair_a44ec6a1c6 : Fact False := inferInstance
  have : Module.Finite P.Ring P.Cotangent :=
    have : Module.Finite P.Ring P.ker := .of_fg h'
    .of_surjective _ Extension.Cotangent.mk_surjective
  have : Module.Finite S P.Cotangent := Module.Finite.of_restrictScalars_finite P.Ring _ _
  rw [← IsLocalRing.split_injective_iff_lTensor_residueField_injective,
    P.formallySmooth_iff_split_injection]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:43:42: 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 [← neg_right_iff] at h_comm
  rw [← isNilpotent_neg_iff] at hy
  rw [sub_eq_add_neg]
  exact h_comm.isNilpotent_add hx hy

[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.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:
Commute

Local 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

lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
    ¬ IsNilpotent x := by
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

lemma IsNilpotent.not_isUnit [Ring R] [Nontrivial R] {x : R} (hx : IsNilpotent x) :
    ¬ IsUnit x :=
  mt IsUnit.not_isNilpotent (by simpa only [not_not] using hx)

lemma IsIdempotentElem.eq_zero_of_isNilpotent [MonoidWithZero R] {e : R}
    (idem : IsIdempotentElem e) (nilp : IsNilpotent e) : e = 0 := by
  obtain ⟨rfl | n, hn⟩ := nilp
  · rw [pow_zero] at hn; rw [← one_mul e, hn, zero_mul]
  · rw [← hn, idem.pow_succ_eq]

alias IsNilpotent.eq_zero_of_isIdempotentElem := IsIdempotentElem.eq_zero_of_isNilpotent

instance [Zero R] [Pow R ℕ] [Zero S] [Pow S ℕ] [IsReduced R] [IsReduced S] : IsReduced (R × S) where
  eq_zero _ := fun ⟨n, hn⟩ ↦ have hn := Prod.ext_iff.1 hn
    Prod.ext (IsReduced.eq_zero _ ⟨n, hn.1⟩) (IsReduced.eq_zero _ ⟨n, hn.2⟩)

theorem Prime.isRadical [CommMonoidWithZero R] {y : R} (hy : Prime y) : IsRadical y :=
  fun _ _ ↦ hy.dvd_of_dvd_pow

theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
  simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
  exact forall_comm

theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
    IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by
  simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]

theorem IsRadical.of_dvd [CommMonoidWithZero R] [IsCancelMulZero R] {x y : R} (hy : IsRadical y)
    (h0 : y ≠ 0) (hxy : x ∣ y) : IsRadical x := (isRadical_iff_pow_one_lt 2 one_lt_two).2 <| by
  obtain ⟨z, rfl⟩ := hxy
  refine fun w dvd ↦ ((mul_dvd_mul_iff_right <| right_ne_zero_of_mul h0).mp <| hy 2 _ ?_)
  rw [mul_pow]
  gcongr
  exact dvd_pow_self _ two_ne_zero

namespace Commute

section Semiring

variable [Semiring R]

theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero (h_comm : Commute x y) {m n k : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
    (x + y) ^ k = 0 := by
  rw [h_comm.add_pow']
  apply Finset.sum_eq_zero
  rintro ⟨i, j⟩ hij
  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
  by_cases hi : m ≤ i
  · rw [pow_eq_zero_of_le hi hx, zero_mul]
  rw [pow_eq_zero_of_le ?_ hy, mul_zero]
  linarith [Finset.mem_antidiagonal.mp hij]

theorem add_pow_add_eq_zero_of_pow_eq_zero (h_comm : Commute x y) {m n : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) :
    (x + y) ^ (m + n - 1) = 0 :=
  h_comm.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero hx hy <| by rw [← Nat.sub_le_iff_le_add]

theorem isNilpotent_add (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x + y) := by
  obtain ⟨n, hn⟩ := hx
  obtain ⟨m, hm⟩ := hy
  exact ⟨_, add_pow_add_eq_zero_of_pow_eq_zero h_comm hn hm⟩

protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
    IsNilpotent (∑ i ∈ s, f i) := by
  classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))

theorem isNilpotent_finsum {ι : Type*} {f : ι → R}
    (hf : ∀ b, IsNilpotent (f b)) (h_comm : ∀ i j, Commute (f i) (f j)) :
    IsNilpotent (finsum f) := by
  classical
  by_cases h : HasFiniteSupport f
  · rw [finsum_def, dif_pos h]
    exact Commute.isNilpotent_sum (fun b _ ↦ hf b) (fun _ _ _ _ ↦ h_comm _ _)
  · simp only [finsum_def, dif_neg h, IsNilpotent.zero]

protected lemma isNilpotent_mul_right_iff (h_comm : Commute x y) (hy : y ∈ nonZeroDivisorsRight R) :
    IsNilpotent (x * y) ↔ IsNilpotent x := by
  refine ⟨?_, h_comm.isNilpotent_mul_right⟩
  rintro ⟨k, hk⟩
  rw [mul_pow h_comm] at hk
  exact ⟨k, (nonZeroDivisorsRight R).pow_mem hy k _ hk⟩

protected lemma isNilpotent_mul_left_iff (h_comm : Commute x y) (hx : x ∈ nonZeroDivisorsLeft R) :
    IsNilpotent (x * y) ↔ IsNilpotent y := by
  refine ⟨?_, h_comm.isNilpotent_mul_left⟩
  rintro ⟨k, hk⟩
  rw [mul_pow h_comm] at hk
  exact ⟨k, (nonZeroDivisorsLeft R).pow_mem hx k _ hk⟩

end Semiring

section Ring

variable [Ring R]

Target:
theorem isNilpotent_sub (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x - y) :=

Proof body:

[Assistant]
by
  rw [← neg_right_iff] at h_comm
  rw [← isNilpotent_neg_iff] at hy
  rw [sub_eq_add_neg]
  exact h_comm.isNilpotent_add hx hy

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.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:
Commute

Local 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

lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
    ¬ IsNilpotent x := by
  intro H
  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)

lemma IsNilpotent.not_isUnit [Ring R] [Nontrivial R] {x : R} (hx : IsNilpotent x) :
    ¬ IsUnit x :=
  mt IsUnit.not_isNilpotent (by simpa only [not_not] using hx)

lemma IsIdempotentElem.eq_zero_of_isNilpotent [MonoidWithZero R] {e : R}
    (idem : IsIdempotentElem e) (nilp : IsNilpotent e) : e = 0 := by
  obtain ⟨rfl | n, hn⟩ := nilp
  · rw [pow_zero] at hn; rw [← one_mul e, hn, zero_mul]
  · rw [← hn, idem.pow_succ_eq]

alias IsNilpotent.eq_zero_of_isIdempotentElem := IsIdempotentElem.eq_zero_of_isNilpotent

instance [Zero R] [Pow R ℕ] [Zero S] [Pow S ℕ] [IsReduced R] [IsReduced S] : IsReduced (R × S) where
  eq_zero _ := fun ⟨n, hn⟩ ↦ have hn := Prod.ext_iff.1 hn
    Prod.ext (IsReduced.eq_zero _ ⟨n, hn.1⟩) (IsReduced.eq_zero _ ⟨n, hn.2⟩)

theorem Prime.isRadical [CommMonoidWithZero R] {y : R} (hy : Prime y) : IsRadical y :=
  fun _ _ ↦ hy.dvd_of_dvd_pow

theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
  simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
  exact forall_comm

theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
    IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by
  simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]

theorem IsRadical.of_dvd [CommMonoidWithZero R] [IsCancelMulZero R] {x y : R} (hy : IsRadical y)
    (h0 : y ≠ 0) (hxy : x ∣ y) : IsRadical x := (isRadical_iff_pow_one_lt 2 one_lt_two).2 <| by
  obtain ⟨z, rfl⟩ := hxy
  refine fun w dvd ↦ ((mul_dvd_mul_iff_right <| right_ne_zero_of_mul h0).mp <| hy 2 _ ?_)
  rw [mul_pow]
  gcongr
  exact dvd_pow_self _ two_ne_zero

namespace Commute

section Semiring

variable [Semiring R]

theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero (h_comm : Commute x y) {m n k : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
    (x + y) ^ k = 0 := by
  rw [h_comm.add_pow']
  apply Finset.sum_eq_zero
  rintro ⟨i, j⟩ hij
  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
  by_cases hi : m ≤ i
  · rw [pow_eq_zero_of_le hi hx, zero_mul]
  rw [pow_eq_zero_of_le ?_ hy, mul_zero]
  linarith [Finset.mem_antidiagonal.mp hij]

theorem add_pow_add_eq_zero_of_pow_eq_zero (h_comm : Commute x y) {m n : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) :
    (x + y) ^ (m + n - 1) = 0 :=
  h_comm.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero hx hy <| by rw [← Nat.sub_le_iff_le_add]

theorem isNilpotent_add (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x + y) := by
  obtain ⟨n, hn⟩ := hx
  obtain ⟨m, hm⟩ := hy
  exact ⟨_, add_pow_add_eq_zero_of_pow_eq_zero h_comm hn hm⟩

protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
    IsNilpotent (∑ i ∈ s, f i) := by
  classical
  induction s using Finset.induction with
  | empty => simp
  | insert j s hj ih => ?_
  rw [Finset.sum_insert hj]
  apply Commute.isNilpotent_add
  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
  · apply hnp; simp
  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))

theorem isNilpotent_finsum {ι : Type*} {f : ι → R}
    (hf : ∀ b, IsNilpotent (f b)) (h_comm : ∀ i j, Commute (f i) (f j)) :
    IsNilpotent (finsum f) := by
  classical
  by_cases h : HasFiniteSupport f
  · rw [finsum_def, dif_pos h]
    exact Commute.isNilpotent_sum (fun b _ ↦ hf b) (fun _ _ _ _ ↦ h_comm _ _)
  · simp only [finsum_def, dif_neg h, IsNilpotent.zero]

protected lemma isNilpotent_mul_right_iff (h_comm : Commute x y) (hy : y ∈ nonZeroDivisorsRight R) :
    IsNilpotent (x * y) ↔ IsNilpotent x := by
  refine ⟨?_, h_comm.isNilpotent_mul_right⟩
  rintro ⟨k, hk⟩
  rw [mul_pow h_comm] at hk
  exact ⟨k, (nonZeroDivisorsRight R).pow_mem hy k _ hk⟩

protected lemma isNilpotent_mul_left_iff (h_comm : Commute x y) (hx : x ∈ nonZeroDivisorsLeft R) :
    IsNilpotent (x * y) ↔ IsNilpotent y := by
  refine ⟨?_, h_comm.isNilpotent_mul_left⟩
  rintro ⟨k, hk⟩
  rw [mul_pow h_comm] at hk
  exact ⟨k, (nonZeroDivisorsLeft R).pow_mem hx k _ hk⟩

end Semiring

section Ring

variable [Ring R]

Target:
theorem isNilpotent_sub (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x - y) :=

Proof body:


<LEAN_FILE>:73:41: 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>:72:0: error: unsolved goals
V : Type u_1
p : ↑I
inst✝ : Countable V
S : Set (SimpleGraph V)
⊢ G(V, p) S =
    (infinitePi fun e => (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False)
      ((fun G e => e ∈ G.edgeSet) '' S)

Response:
by
  simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_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.Data.Sym.Card
public import Mathlib.MeasureTheory.Constructions.SimpleGraph
public import Mathlib.Probability.Distributions.SetBernoulli

Namespace:
SimpleGraph

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Binomial random graphs

This file constructs the binomial distribution with parameter `p` on simple graphs with
vertices `V`. This is the law `G(V, p)` of binomial random graphs with probability `p`.

## TODO

Add the characteristic predicate for a random graph to follow the binomial random distribution.

## Historical note

This is usually called the Erdős–Rényi model, but this name is historically inaccurate as Erdős and
Rényi introduced a closely related but different model. We therefore choose the name
"binomial random graph" to avoid confusion with this other model and because it is a more
descriptive name.

## Tags

Erdős-Rényi graph, Erdős-Renyi graph, Erdös-Rényi graph, Erdös-Renyi graph, Erdos-Rényi graph,
Erdos-Renyi graph
-/

public section

open MeasureTheory Measure ProbabilityTheory unitInterval
open scoped Finset

namespace SimpleGraph
variable {V : Type*} {p : I}

variable (V p) in
/-- The binomial distribution with parameter `p` on simple graphs with vertices `V`. -/
@[expose]
noncomputable def binomialRandom : Measure (SimpleGraph V) :=
  setBer(Sym2.diagSetᶜ, p).comap edgeSet

@[inherit_doc] scoped notation "G(" V ", " p ")" => binomialRandom V p

section Countable
variable [Countable V]

variable (V p) in
lemma binomialRandom_eq_map : G(V, p) = map fromEdgeSet setBer(Sym2.diagSetᶜ, p) := by
  refine (map_eq_comap measurable_fromEdgeSet measurableEmbedding_edgeSet ?_
    fromEdgeSet_edgeSet).symm
  filter_upwards [setBernoulli_ae_subset] with S hS
  exact ⟨fromEdgeSet S, by simpa [← Set.compl_setOf, Set.subset_compl_iff_disjoint_right] using hS⟩

variable (p) in
lemma binomialRandom_apply' (S : Set (SimpleGraph V)) :
    G(V, p) S = setBer(Sym2.diagSetᶜ, p) (edgeSet '' S) := by
  rw [binomialRandom, measurableEmbedding_edgeSet.comap_apply]

variable (p) in

Target:
lemma binomialRandom_apply (S : Set (SimpleGraph V)) :
    G(V, p) S = infinitePi
      (fun e : Sym2 V ↦ toNNReal p • .dirac (¬ e.IsDiag) + toNNReal (σ p) • .dirac False)
      ((fun G e ↦ e ∈ G.edgeSet) '' S) :=

Failed candidate:
by
  have __repair_da08858ace := True.intro.nonexistentProjection
  simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_comp]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:73:41: 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>:72:0: error: unsolved goals\nV : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (SimpleGraph V)\n⊢ G(V, p) S =\n    (infinitePi fun e => (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False)\n      ((fun G e => e ∈ G.edgeSet) '' S)", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_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.Data.Sym.Card
public import Mathlib.MeasureTheory.Constructions.SimpleGraph
public import Mathlib.Probability.Distributions.SetBernoulli

Namespace:
SimpleGraph

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Binomial random graphs

This file constructs the binomial distribution with parameter `p` on simple graphs with
vertices `V`. This is the law `G(V, p)` of binomial random graphs with probability `p`.

## TODO

Add the characteristic predicate for a random graph to follow the binomial random distribution.

## Historical note

This is usually called the Erdős–Rényi model, but this name is historically inaccurate as Erdős and
Rényi introduced a closely related but different model. We therefore choose the name
"binomial random graph" to avoid confusion with this other model and because it is a more
descriptive name.

## Tags

Erdős-Rényi graph, Erdős-Renyi graph, Erdös-Rényi graph, Erdös-Renyi graph, Erdos-Rényi graph,
Erdos-Renyi graph
-/

public section

open MeasureTheory Measure ProbabilityTheory unitInterval
open scoped Finset

namespace SimpleGraph
variable {V : Type*} {p : I}

variable (V p) in
/-- The binomial distribution with parameter `p` on simple graphs with vertices `V`. -/
@[expose]
noncomputable def binomialRandom : Measure (SimpleGraph V) :=
  setBer(Sym2.diagSetᶜ, p).comap edgeSet

@[inherit_doc] scoped notation "G(" V ", " p ")" => binomialRandom V p

section Countable
variable [Countable V]

variable (V p) in
lemma binomialRandom_eq_map : G(V, p) = map fromEdgeSet setBer(Sym2.diagSetᶜ, p) := by
  refine (map_eq_comap measurable_fromEdgeSet measurableEmbedding_edgeSet ?_
    fromEdgeSet_edgeSet).symm
  filter_upwards [setBernoulli_ae_subset] with S hS
  exact ⟨fromEdgeSet S, by simpa [← Set.compl_setOf, Set.subset_compl_iff_disjoint_right] using hS⟩

variable (p) in
lemma binomialRandom_apply' (S : Set (SimpleGraph V)) :
    G(V, p) S = setBer(Sym2.diagSetᶜ, p) (edgeSet '' S) := by
  rw [binomialRandom, measurableEmbedding_edgeSet.comap_apply]

variable (p) in

Target:
lemma binomialRandom_apply (S : Set (SimpleGraph V)) :
    G(V, p) S = infinitePi
      (fun e : Sym2 V ↦ toNNReal p • .dirac (¬ e.IsDiag) + toNNReal (σ p) • .dirac False)
      ((fun G e ↦ e ∈ G.edgeSet) '' S) :=

Failed candidate:
by
  have __repair_da08858ace := True.intro.nonexistentProjection
  simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_comp]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:73:41: 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>:72:0: error: unsolved goals\nV : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (SimpleGraph V)\n⊢ G(V, p) S =\n    (infinitePi fun e => (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False)\n      ((fun G e => e ∈ G.edgeSet) '' S)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:58:43: error: unsolved goals
C : Type u
inst✝ : Category.{v, u} C
P₁ P₂ : MorphismProperty C
κ : Cardinal.{w}
h₁ : P₁.HasCardinalLT κ
h₂ : P₂.HasCardinalLT κ
hκ : Cardinal.aleph0 ≤ κ
⊢ False

Response:
by
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [MorphismProperty.toSet_max]
  exact hasCardinalLT_union hκ h₁ h₂

[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.SetTheory.Cardinal.HasCardinalLT
public import Mathlib.CategoryTheory.MorphismProperty.Basic

Namespace:
CategoryTheory.MorphismProperty

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# Properties of morphisms that are bounded by a cardinal

Given `P : MorphismProperty C` and `κ : Cardinal`, we introduce a predicate
`P.HasCardinalLT κ` saying that the cardinality of `P.toSet` is `< κ`.

-/

public section

universe w v u

namespace CategoryTheory

variable {C : Type u} [Category.{v} C]

namespace MorphismProperty

/-- The property that the subtype of arrows satisfying a property `P : MorphismProperty C`
is of cardinality `< κ`. -/
protected abbrev HasCardinalLT (P : MorphismProperty C) (κ : Cardinal.{w}) :=
    _root_.HasCardinalLT P.toSet κ

lemma hasCardinalLT_ofHoms {C : Type*} [Category* C]
    {ι : Type*} {X Y : ι → C} (f : ∀ i, X i ⟶ Y i) {κ : Cardinal}
    (h : HasCardinalLT ι κ) : (MorphismProperty.ofHoms f).HasCardinalLT κ :=
  h.of_surjective (fun i ↦ ⟨Arrow.mk (f i), ⟨i⟩⟩) (by
    rintro ⟨f, hf⟩
    rw [MorphismProperty.mem_toSet_iff, MorphismProperty.ofHoms_iff] at hf
    obtain ⟨i, hf⟩ := hf
    obtain rfl : f = _ := hf
    exact ⟨i, rfl⟩)

lemma HasCardinalLT.iSup
    {ι : Type*} {P : ι → MorphismProperty C} {κ : Cardinal.{w}} [Fact κ.IsRegular]
    (hP : ∀ i, (P i).HasCardinalLT κ) (hι : HasCardinalLT ι κ) :
    (⨆ i, P i).HasCardinalLT κ := by
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [toSet_iSup]
  exact hasCardinalLT_iUnion _ hι hP

Target:
lemma HasCardinalLT.sup
    {P₁ P₂ : MorphismProperty C} {κ : Cardinal.{w}}
    (h₁ : P₁.HasCardinalLT κ) (h₂ : P₂.HasCardinalLT κ)
    (hκ : Cardinal.aleph0 ≤ κ) :
    (P₁ ⊔ P₂).HasCardinalLT κ :=

Failed candidate:
by
  have __repair_plan_7b9d21b512 : False := by
    skip
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [MorphismProperty.toSet_max]
  exact hasCardinalLT_union hκ h₁ h₂

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:58:43: error: unsolved goals\nC : Type u\ninst✝ : Category.{v, u} C\nP₁ P₂ : MorphismProperty C\nκ : Cardinal.{w}\nh₁ : P₁.HasCardinalLT κ\nh₂ : P₂.HasCardinalLT κ\nhκ : Cardinal.aleph0 ≤ κ\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [MorphismProperty.toSet_max]
  exact hasCardinalLT_union hκ h₁ h₂

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.SetTheory.Cardinal.HasCardinalLT
public import Mathlib.CategoryTheory.MorphismProperty.Basic

Namespace:
CategoryTheory.MorphismProperty

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# Properties of morphisms that are bounded by a cardinal

Given `P : MorphismProperty C` and `κ : Cardinal`, we introduce a predicate
`P.HasCardinalLT κ` saying that the cardinality of `P.toSet` is `< κ`.

-/

public section

universe w v u

namespace CategoryTheory

variable {C : Type u} [Category.{v} C]

namespace MorphismProperty

/-- The property that the subtype of arrows satisfying a property `P : MorphismProperty C`
is of cardinality `< κ`. -/
protected abbrev HasCardinalLT (P : MorphismProperty C) (κ : Cardinal.{w}) :=
    _root_.HasCardinalLT P.toSet κ

lemma hasCardinalLT_ofHoms {C : Type*} [Category* C]
    {ι : Type*} {X Y : ι → C} (f : ∀ i, X i ⟶ Y i) {κ : Cardinal}
    (h : HasCardinalLT ι κ) : (MorphismProperty.ofHoms f).HasCardinalLT κ :=
  h.of_surjective (fun i ↦ ⟨Arrow.mk (f i), ⟨i⟩⟩) (by
    rintro ⟨f, hf⟩
    rw [MorphismProperty.mem_toSet_iff, MorphismProperty.ofHoms_iff] at hf
    obtain ⟨i, hf⟩ := hf
    obtain rfl : f = _ := hf
    exact ⟨i, rfl⟩)

lemma HasCardinalLT.iSup
    {ι : Type*} {P : ι → MorphismProperty C} {κ : Cardinal.{w}} [Fact κ.IsRegular]
    (hP : ∀ i, (P i).HasCardinalLT κ) (hι : HasCardinalLT ι κ) :
    (⨆ i, P i).HasCardinalLT κ := by
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [toSet_iSup]
  exact hasCardinalLT_iUnion _ hι hP

Target:
lemma HasCardinalLT.sup
    {P₁ P₂ : MorphismProperty C} {κ : Cardinal.{w}}
    (h₁ : P₁.HasCardinalLT κ) (h₂ : P₂.HasCardinalLT κ)
    (hκ : Cardinal.aleph0 ≤ κ) :
    (P₁ ⊔ P₂).HasCardinalLT κ :=

Failed candidate:
by
  have __repair_plan_7b9d21b512 : False := by
    skip
  dsimp only [MorphismProperty.HasCardinalLT]
  rw [MorphismProperty.toSet_max]
  exact hasCardinalLT_union hκ h₁ h₂

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:58:43: error: unsolved goals\nC : Type u\ninst✝ : Category.{v, u} C\nP₁ P₂ : MorphismProperty C\nκ : Cardinal.{w}\nh₁ : P₁.HasCardinalLT κ\nh₂ : P₂.HasCardinalLT κ\nhκ : Cardinal.aleph0 ≤ κ\n⊢ False", "returncode": 1}

Corrected proof: