Response:
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

[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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Proof body:

[Assistant]
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Proof body:


<LEAN_FILE>:203:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

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

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

Namespace:
Commute

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

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

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

## Main definitions

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

-/

public section

universe u v

open Function Set

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

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

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

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

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

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

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

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

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

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

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) :=

Failed candidate:
by
  have __repair_f4032e2e2f : False := True.intro
  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

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:203:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

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

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

Namespace:
Commute

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

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

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

## Main definitions

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

-/

public section

universe u v

open Function Set

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

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

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

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

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

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

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

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

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

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

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) :=

Failed candidate:
by
  have __repair_f4032e2e2f : False := True.intro
  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

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:203:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:78:42: 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
  unfold partialSum
  fun_prop

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

Imports:
public import Mathlib.Analysis.Calculus.FormalMultilinearSeries
public import Mathlib.Analysis.SpecificLimits.Normed

Namespace:
FormalMultilinearSeries

Local safe context:
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
/-!
# Radius of convergence of a power series

This file introduces the notion of the radius of convergence of a power series.

## Main definitions

Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.

* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
  is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
  `p.isLittleO_one_of_lt_radius`,
  `p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
  `‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
  `r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.

## Implementation details

We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/

@[expose] public section

noncomputable section

variable {𝕜 E F G : Type*}

open Topology NNReal Filter ENNReal Set Asymptotics
open scoped Pointwise

namespace FormalMultilinearSeries

variable [Semiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [ContinuousAdd E] [ContinuousAdd F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]

/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
  ∑' n : ℕ, p n fun _ => x

theorem sum_mem {S : Type*} {s : S} [SetLike S F] [AddSubmonoidClass S F]
    (h_closed : IsClosed (s : Set F)) (p : FormalMultilinearSeries 𝕜 E F) (x : E)
    (h : ∀ k, p k (fun _ : Fin k => x) ∈ s) :
    p.sum x ∈ s :=
  tsum_mem h_closed h

/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
  ∑ k ∈ Finset.range n, p k fun _ : Fin k => x

/-- The partial sums of a formal multilinear series are continuous. -/

Target:
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    Continuous (p.partialSum n) :=

Failed candidate:
by
  let __repair_7c416586a6 : Fact False := inferInstance
  unfold partialSum
  fun_prop

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:78:42: 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
  unfold partialSum
  fun_prop

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

Imports:
public import Mathlib.Analysis.Calculus.FormalMultilinearSeries
public import Mathlib.Analysis.SpecificLimits.Normed

Namespace:
FormalMultilinearSeries

Local safe context:
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
/-!
# Radius of convergence of a power series

This file introduces the notion of the radius of convergence of a power series.

## Main definitions

Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.

* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
  is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
  `p.isLittleO_one_of_lt_radius`,
  `p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
  `‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
  `r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.

## Implementation details

We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/

@[expose] public section

noncomputable section

variable {𝕜 E F G : Type*}

open Topology NNReal Filter ENNReal Set Asymptotics
open scoped Pointwise

namespace FormalMultilinearSeries

variable [Semiring 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [ContinuousAdd E] [ContinuousAdd F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]

/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
  ∑' n : ℕ, p n fun _ => x

theorem sum_mem {S : Type*} {s : S} [SetLike S F] [AddSubmonoidClass S F]
    (h_closed : IsClosed (s : Set F)) (p : FormalMultilinearSeries 𝕜 E F) (x : E)
    (h : ∀ k, p k (fun _ : Fin k => x) ∈ s) :
    p.sum x ∈ s :=
  tsum_mem h_closed h

/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
  ∑ k ∈ Finset.range n, p k fun _ : Fin k => x

/-- The partial sums of a formal multilinear series are continuous. -/

Target:
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    Continuous (p.partialSum n) :=

Failed candidate:
by
  let __repair_7c416586a6 : Fact False := inferInstance
  unfold partialSum
  fun_prop

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:78:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:133:43: error: unsolved goals
M : Type u_2
inst✝ : Monoid M
g h : M
H : IsConj g h
Hg : g ∈ Set.center M
⊢ False

Response:
by
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

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

Imports:
public import Mathlib.Algebra.Group.Subgroup.Basic
public import Mathlib.GroupTheory.Submonoid.Center

Namespace:
IsConj

Local safe context:
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
/-!
# Centers of subgroups

-/

@[expose] public section

assert_not_exists MonoidWithZero Multiset

variable {G : Type*} [Group G]

namespace Subgroup

variable (G)

/-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      /-- The center of an additive group `G` is the set of elements that commute with
      everything in `G` -/]
def center : Subgroup G where
  __ := Submonoid.center G
  inv_mem' := Set.inv_mem_center

@[to_additive]
theorem coe_center : ↑(center G) = Set.center G :=
  rfl

@[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G :=
  rfl

instance center.isMulCommutative : IsMulCommutative (center G) :=
  ⟨⟨fun a b => Subtype.ext (b.2.comm a).symm⟩⟩

variable {G} in
/-- The center of isomorphic groups are isomorphic. -/
@[to_additive (attr := simps!) /-- The center of isomorphic additive groups are isomorphic. -/]
def centerCongr {H} [Group H] (e : G ≃* H) : center G ≃* center H := Submonoid.centerCongr e

/-- The center of a group is isomorphic to the center of its opposite. -/
@[to_additive (attr := simps!)
/-- The center of an additive group is isomorphic to the center of its opposite. -/]
def centerToMulOpposite : center G ≃* center Gᵐᵒᵖ := Submonoid.centerToMulOpposite

variable {G}

@[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g := by
  rw [← Semigroup.mem_center_iff]
  exact Iff.rfl

instance decidableMemCenter (z : G) [Decidable (∀ g, g * z = z * g)] : Decidable (z ∈ center G) :=
  decidable_of_iff' _ mem_center_iff

@[to_additive]
instance centerCharacteristic : (center G).Characteristic := by
  refine characteristic_iff_comap_le.mpr fun ϕ g hg => ?_
  rw [mem_center_iff]
  intro h
  rw [← ϕ.injective.eq_iff, map_mul, map_mul]
  exact (hg.comm (ϕ h)).symm

@[to_additive]
theorem _root_.CommGroup.center_eq_top {G : Type*} [CommGroup G] : center G = ⊤ := by
  rw [eq_top_iff']
  intro x
  rw [Subgroup.mem_center_iff]
  intro y
  exact mul_comm y x

@[to_additive]
theorem center_eq_top_iff : center G = ⊤ ↔ IsMulCommutative G := by
  simp [eq_top_iff', isMulCommutative_iff, mem_center_iff, eq_comm]

@[to_additive]
theorem center_eq_top [hG : IsMulCommutative G] : center G = ⊤ :=
    center_eq_top_iff.mpr hG

/-- A group is commutative if the center is the whole group. -/
@[to_additive /-- An additive group is commutative if the center is the whole group. -/,
  implicit_reducible]
def _root_.Group.commGroupOfCenterEqTop (h : center G = ⊤) : CommGroup G :=
  { ‹Group G› with
    mul_comm := by
      rw [eq_top_iff'] at h
      intro x y
      apply Subgroup.mem_center_iff.mp _ x
      exact h y
  }

@[to_additive]
protected theorem center_prod {H : Type*} [Group H] : center (G × H) = prod (center G) (center H) :=
  SetLike.coe_injective Set.center_prod

@[to_additive]
protected theorem center_pi {η : Type*} {G : η → Type*} [Π i, Group (G i)] :
    center (Π i, G i) = pi .univ fun i ↦ center (G i) :=
  SetLike.coe_injective Set.center_pi

variable {H : Subgroup G}

section Normalizer

@[to_additive]
instance instNormalCenter : (center G).Normal :=
  ⟨fun a ha b ↦ by simpa [mem_center_iff.mp ha b]⟩

@[to_additive]
theorem center_le_normalizer (s : Set G) : center G ≤ normalizer s := by
  intro x hx y
  simp [← mem_center_iff.mp hx y]

end Normalizer

end Subgroup

namespace IsConj

variable {M : Type*} [Monoid M]

Target:
theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h :=

Failed candidate:
by
  have __repair_plan_c0e33e4821 : False := by
    skip
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:133:43: error: unsolved goals\nM : Type u_2\ninst✝ : Monoid M\ng h : M\nH : IsConj g h\nHg : g ∈ Set.center M\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

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

Imports:
public import Mathlib.Algebra.Group.Subgroup.Basic
public import Mathlib.GroupTheory.Submonoid.Center

Namespace:
IsConj

Local safe context:
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
/-!
# Centers of subgroups

-/

@[expose] public section

assert_not_exists MonoidWithZero Multiset

variable {G : Type*} [Group G]

namespace Subgroup

variable (G)

/-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      /-- The center of an additive group `G` is the set of elements that commute with
      everything in `G` -/]
def center : Subgroup G where
  __ := Submonoid.center G
  inv_mem' := Set.inv_mem_center

@[to_additive]
theorem coe_center : ↑(center G) = Set.center G :=
  rfl

@[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G :=
  rfl

instance center.isMulCommutative : IsMulCommutative (center G) :=
  ⟨⟨fun a b => Subtype.ext (b.2.comm a).symm⟩⟩

variable {G} in
/-- The center of isomorphic groups are isomorphic. -/
@[to_additive (attr := simps!) /-- The center of isomorphic additive groups are isomorphic. -/]
def centerCongr {H} [Group H] (e : G ≃* H) : center G ≃* center H := Submonoid.centerCongr e

/-- The center of a group is isomorphic to the center of its opposite. -/
@[to_additive (attr := simps!)
/-- The center of an additive group is isomorphic to the center of its opposite. -/]
def centerToMulOpposite : center G ≃* center Gᵐᵒᵖ := Submonoid.centerToMulOpposite

variable {G}

@[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g := by
  rw [← Semigroup.mem_center_iff]
  exact Iff.rfl

instance decidableMemCenter (z : G) [Decidable (∀ g, g * z = z * g)] : Decidable (z ∈ center G) :=
  decidable_of_iff' _ mem_center_iff

@[to_additive]
instance centerCharacteristic : (center G).Characteristic := by
  refine characteristic_iff_comap_le.mpr fun ϕ g hg => ?_
  rw [mem_center_iff]
  intro h
  rw [← ϕ.injective.eq_iff, map_mul, map_mul]
  exact (hg.comm (ϕ h)).symm

@[to_additive]
theorem _root_.CommGroup.center_eq_top {G : Type*} [CommGroup G] : center G = ⊤ := by
  rw [eq_top_iff']
  intro x
  rw [Subgroup.mem_center_iff]
  intro y
  exact mul_comm y x

@[to_additive]
theorem center_eq_top_iff : center G = ⊤ ↔ IsMulCommutative G := by
  simp [eq_top_iff', isMulCommutative_iff, mem_center_iff, eq_comm]

@[to_additive]
theorem center_eq_top [hG : IsMulCommutative G] : center G = ⊤ :=
    center_eq_top_iff.mpr hG

/-- A group is commutative if the center is the whole group. -/
@[to_additive /-- An additive group is commutative if the center is the whole group. -/,
  implicit_reducible]
def _root_.Group.commGroupOfCenterEqTop (h : center G = ⊤) : CommGroup G :=
  { ‹Group G› with
    mul_comm := by
      rw [eq_top_iff'] at h
      intro x y
      apply Subgroup.mem_center_iff.mp _ x
      exact h y
  }

@[to_additive]
protected theorem center_prod {H : Type*} [Group H] : center (G × H) = prod (center G) (center H) :=
  SetLike.coe_injective Set.center_prod

@[to_additive]
protected theorem center_pi {η : Type*} {G : η → Type*} [Π i, Group (G i)] :
    center (Π i, G i) = pi .univ fun i ↦ center (G i) :=
  SetLike.coe_injective Set.center_pi

variable {H : Subgroup G}

section Normalizer

@[to_additive]
instance instNormalCenter : (center G).Normal :=
  ⟨fun a ha b ↦ by simpa [mem_center_iff.mp ha b]⟩

@[to_additive]
theorem center_le_normalizer (s : Set G) : center G ≤ normalizer s := by
  intro x hx y
  simp [← mem_center_iff.mp hx y]

end Normalizer

end Subgroup

namespace IsConj

variable {M : Type*} [Monoid M]

Target:
theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h :=

Failed candidate:
by
  have __repair_plan_c0e33e4821 : False := by
    skip
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:133:43: error: unsolved goals\nM : Type u_2\ninst✝ : Monoid M\ng h : M\nH : IsConj g h\nHg : g ∈ Set.center M\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:30:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 7976795) 0
in the target expression
  range f = {f false, f true}

α : Type u_1
f : Bool → α
⊢ range f = {f false, f true}

Response: by grind [cases Bool]

[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.Set.Insert

Namespace:
Bool

Local safe context:
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Booleans and set operations

This file contains three trivial lemmas about `Bool`, `Set.univ`, and `Set.range`.
-/

public section

open Set

namespace Bool

@[simp]
theorem univ_eq : (univ : Set Bool) = {false, true} := by grind

@[simp, grind =]

Target:
theorem range_eq {α : Type*} (f : Bool → α) : range f = {f false, f true} :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 7976795) 0) = ((fun x : Nat => x + 7976795) 0) by rfl]
 grind [cases Bool]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:30:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 7976795) 0\nin the target expression\n  range f = {f false, f true}\n\nα : Type u_1\nf : Bool → α\n⊢ range f = {f false, f true}", "returncode": 1}

Corrected proof:

[Assistant] by grind [cases Bool]

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.Set.Insert

Namespace:
Bool

Local safe context:
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Booleans and set operations

This file contains three trivial lemmas about `Bool`, `Set.univ`, and `Set.range`.
-/

public section

open Set

namespace Bool

@[simp]
theorem univ_eq : (univ : Set Bool) = {false, true} := by grind

@[simp, grind =]

Target:
theorem range_eq {α : Type*} (f : Bool → α) : range f = {f false, f true} :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 7976795) 0) = ((fun x : Nat => x + 7976795) 0) by rfl]
 grind [cases Bool]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:30:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 7976795) 0\nin the target expression\n  range f = {f false, f true}\n\nα : Type u_1\nf : Bool → α\n⊢ range f = {f false, f true}", "returncode": 1}

Corrected proof:


Response:
by
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

[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.Topology.Sheaves.Sheaf
public import Mathlib.CategoryTheory.Sites.Limits
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Namespace:
TopCat

Local context:
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
/-!
# Presheaves in `C` have limits and colimits when `C` does.
-/

public section

noncomputable section

universe v u w t

open CategoryTheory

open CategoryTheory.Limits

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

namespace TopCat

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Presheaf C X) :=
  functorCategoryHasLimitsOfShape

instance [HasLimits C] (X : TopCat.{v}) : HasLimits.{v} (Presheaf C X) where

instance [HasColimitsOfShape J C] (X : TopCat) : HasColimitsOfShape J (Presheaf C X) :=
  functorCategoryHasColimitsOfShape

instance [HasColimits.{v, u} C] (X : TopCat.{t}) : HasColimitsOfSize.{v, v} (Presheaf C X) where

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : CreatesLimitsOfShape J (Sheaf.forget C X) :=
  inferInstanceAs <| CreatesLimitsOfShape J (sheafToPresheaf _ _)

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Sheaf C X) :=
  hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape (Sheaf.forget C X)

instance [HasLimits C] (X : TopCat) : CreatesLimits.{v, v} (Sheaf.forget C X) where

instance [HasLimits C] (X : TopCat.{v}) : HasLimitsOfSize.{v, v} (Sheaf.{v} C X) where

set_option backward.isDefEq.respectTransparency false in

Target:
theorem isSheaf_of_isLimit [HasLimitsOfShape J C] {X : TopCat.{v}} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf :=

Proof body:

[Assistant]
by
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

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.Topology.Sheaves.Sheaf
public import Mathlib.CategoryTheory.Sites.Limits
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Namespace:
TopCat

Local context:
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
/-!
# Presheaves in `C` have limits and colimits when `C` does.
-/

public section

noncomputable section

universe v u w t

open CategoryTheory

open CategoryTheory.Limits

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

namespace TopCat

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Presheaf C X) :=
  functorCategoryHasLimitsOfShape

instance [HasLimits C] (X : TopCat.{v}) : HasLimits.{v} (Presheaf C X) where

instance [HasColimitsOfShape J C] (X : TopCat) : HasColimitsOfShape J (Presheaf C X) :=
  functorCategoryHasColimitsOfShape

instance [HasColimits.{v, u} C] (X : TopCat.{t}) : HasColimitsOfSize.{v, v} (Presheaf C X) where

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : CreatesLimitsOfShape J (Sheaf.forget C X) :=
  inferInstanceAs <| CreatesLimitsOfShape J (sheafToPresheaf _ _)

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Sheaf C X) :=
  hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape (Sheaf.forget C X)

instance [HasLimits C] (X : TopCat) : CreatesLimits.{v, v} (Sheaf.forget C X) where

instance [HasLimits C] (X : TopCat.{v}) : HasLimitsOfSize.{v, v} (Sheaf.{v} C X) where

set_option backward.isDefEq.respectTransparency false in

Target:
theorem isSheaf_of_isLimit [HasLimitsOfShape J C] {X : TopCat.{v}} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf :=

Proof body:


<LEAN_FILE>:106:41: 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>:105:0: error: unsolved goals
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
⊢ S.Exact ↔ ∀ (x : ↑X), (S.map (forget C X ⋙ stalkFunctor C x)).Exact

Response:
by
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

[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.Category.Grp.AB
public import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf
public import Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced
public import Mathlib.Topology.Sheaves.Limits
public import Mathlib.Topology.Sheaves.Skyscraper

Namespace:
TopCat.Sheaf

Local safe context:
/-
Copyright (c) 2026 Brian Nugent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brian Nugent
-/
/-!
# Sheaves over Abelian categories
We provide instances for categories of sheaves over Abelian categories.

## Main Results

* `TopCat.Sheaf.exact_iff_stalkFunctor_map_exact`: A complex of sheaves over a concrete abelian
  category is exact if and only if it is exact on stalks.

-/

public section

universe u v v₁ v₂

open TopologicalSpace CategoryTheory Limits

namespace TopCat

variable {X : TopCat.{u}}

section

variable {C : Type v₁} [Category.{v₂} C] [HasSheafify (Opens.grothendieckTopology X) C] [Abelian C]

noncomputable instance : Abelian (Presheaf C X) := inferInstanceAs (Abelian (_ ⥤ _))

namespace Sheaf

noncomputable instance : Abelian (Sheaf C X) :=
  inferInstanceAs (Abelian (CategoryTheory.Sheaf _ _))

instance : (Sheaf.forget C X).Additive where

instance {D : Type*} [Category.{u} D] [Abelian D] [IsGrothendieckAbelian.{u} D]
    [HasSheafify (Opens.grothendieckTopology X) D] : IsGrothendieckAbelian.{u} (Sheaf D X) :=
  inferInstanceAs (IsGrothendieckAbelian (CategoryTheory.Sheaf _ _))

end Sheaf

end

set_option backward.isDefEq.respectTransparency false in
/-- The stalk functor is additive -/
instance (p₀ : X) {C : Type v} [Category.{u} C] [Abelian C] [HasColimits C] :
    (Presheaf.stalkFunctor C p₀).Additive := by
  dsimp [Presheaf.stalkFunctor]
  have : ((Functor.whiskeringLeft _ _ C).obj (OpenNhds.inclusion p₀).op).Additive := ⟨by cat_disch⟩
  infer_instance

namespace Sheaf

open Presheaf

variable {C : Type v} [Category.{u} C] [HasColimits C] [HasLimits C]
  {FC : C → C → Type*} {CC : C → Type u} [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]
  [instCC : ConcreteCategory C FC] [PreservesFilteredColimits (CategoryTheory.forget C)]
  [PreservesLimits (CategoryTheory.forget C)] [Abelian C]
  {X : TopCat.{u}} (p₀ : X)

instance : Limits.PreservesFiniteLimits (forget C X ⋙ stalkFunctor C p₀) :=
  have : (forget C X ⋙ stalkFunctor C p₀).PreservesHomology := by
    simp only [(forget C X ⋙ stalkFunctor C p₀).exact_tfae.out 2 0]
    intro S h
    have := ((forget C X ⋙ stalkFunctor C p₀).preservesFiniteColimits_tfae.out 3 0).mp
      (inferInstance : (PreservesFiniteColimits _))
    refine ShortComplex.ShortExact.mk' (this S h).left ?_ (this S h).right
    have := h.2
    exact Functor.map_mono (forget C X ⋙ stalkFunctor C p₀) _
  (forget C X ⋙ stalkFunctor C p₀).preservesFiniteLimits_of_preservesHomology

open ZeroObject

include instCC in
/-- A sheaf is zero if and only if its stalks are all zero. -/
lemma isZero_iff_stalkFunctor_obj_isZero (F : Sheaf C X) :
    IsZero F ↔ ∀ x : X, IsZero ((forget C X ⋙ stalkFunctor C x).obj F) := by
  refine ⟨fun h _ => Functor.map_isZero _ h, ?_⟩
  intro h
  let f : F ⟶ 0 := (isZero_zero (Sheaf C X)).from_ F
  have : IsIso f := by
    rw [Presheaf.isIso_iff_stalkFunctor_map_iso]
    exact fun x => isIso_of_source_target_iso_zero _ (h x).isoZero
      ((forget C X ⋙ stalkFunctor C x).map_isZero (isZero_zero _)).isoZero
  exact (isZero_zero _).of_iso (asIso f)

include instCC in
/-- Exactness can be checked on stalks for complexes of sheaves. -/

Target:
theorem exact_iff_stalkFunctor_map_exact (S : ShortComplex (Sheaf C X)) :
    S.Exact ↔ ∀ x : X, (S.map (forget C X ⋙ stalkFunctor C x)).Exact :=

Failed candidate:
by
  have __repair_f945fbe7cf := True.intro.nonexistentProjection
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:106:41: 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>:105:0: error: unsolved goals\nC : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLimits (CategoryTheory.forget C)\ninst✝ : Abelian C\nX : TopCat\nS : ShortComplex (Sheaf C X)\n⊢ S.Exact ↔ ∀ (x : ↑X), (S.map (forget C X ⋙ stalkFunctor C x)).Exact", "returncode": 1}

Corrected proof:

[Assistant]
by
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

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.Category.Grp.AB
public import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf
public import Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced
public import Mathlib.Topology.Sheaves.Limits
public import Mathlib.Topology.Sheaves.Skyscraper

Namespace:
TopCat.Sheaf

Local safe context:
/-
Copyright (c) 2026 Brian Nugent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brian Nugent
-/
/-!
# Sheaves over Abelian categories
We provide instances for categories of sheaves over Abelian categories.

## Main Results

* `TopCat.Sheaf.exact_iff_stalkFunctor_map_exact`: A complex of sheaves over a concrete abelian
  category is exact if and only if it is exact on stalks.

-/

public section

universe u v v₁ v₂

open TopologicalSpace CategoryTheory Limits

namespace TopCat

variable {X : TopCat.{u}}

section

variable {C : Type v₁} [Category.{v₂} C] [HasSheafify (Opens.grothendieckTopology X) C] [Abelian C]

noncomputable instance : Abelian (Presheaf C X) := inferInstanceAs (Abelian (_ ⥤ _))

namespace Sheaf

noncomputable instance : Abelian (Sheaf C X) :=
  inferInstanceAs (Abelian (CategoryTheory.Sheaf _ _))

instance : (Sheaf.forget C X).Additive where

instance {D : Type*} [Category.{u} D] [Abelian D] [IsGrothendieckAbelian.{u} D]
    [HasSheafify (Opens.grothendieckTopology X) D] : IsGrothendieckAbelian.{u} (Sheaf D X) :=
  inferInstanceAs (IsGrothendieckAbelian (CategoryTheory.Sheaf _ _))

end Sheaf

end

set_option backward.isDefEq.respectTransparency false in
/-- The stalk functor is additive -/
instance (p₀ : X) {C : Type v} [Category.{u} C] [Abelian C] [HasColimits C] :
    (Presheaf.stalkFunctor C p₀).Additive := by
  dsimp [Presheaf.stalkFunctor]
  have : ((Functor.whiskeringLeft _ _ C).obj (OpenNhds.inclusion p₀).op).Additive := ⟨by cat_disch⟩
  infer_instance

namespace Sheaf

open Presheaf

variable {C : Type v} [Category.{u} C] [HasColimits C] [HasLimits C]
  {FC : C → C → Type*} {CC : C → Type u} [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]
  [instCC : ConcreteCategory C FC] [PreservesFilteredColimits (CategoryTheory.forget C)]
  [PreservesLimits (CategoryTheory.forget C)] [Abelian C]
  {X : TopCat.{u}} (p₀ : X)

instance : Limits.PreservesFiniteLimits (forget C X ⋙ stalkFunctor C p₀) :=
  have : (forget C X ⋙ stalkFunctor C p₀).PreservesHomology := by
    simp only [(forget C X ⋙ stalkFunctor C p₀).exact_tfae.out 2 0]
    intro S h
    have := ((forget C X ⋙ stalkFunctor C p₀).preservesFiniteColimits_tfae.out 3 0).mp
      (inferInstance : (PreservesFiniteColimits _))
    refine ShortComplex.ShortExact.mk' (this S h).left ?_ (this S h).right
    have := h.2
    exact Functor.map_mono (forget C X ⋙ stalkFunctor C p₀) _
  (forget C X ⋙ stalkFunctor C p₀).preservesFiniteLimits_of_preservesHomology

open ZeroObject

include instCC in
/-- A sheaf is zero if and only if its stalks are all zero. -/
lemma isZero_iff_stalkFunctor_obj_isZero (F : Sheaf C X) :
    IsZero F ↔ ∀ x : X, IsZero ((forget C X ⋙ stalkFunctor C x).obj F) := by
  refine ⟨fun h _ => Functor.map_isZero _ h, ?_⟩
  intro h
  let f : F ⟶ 0 := (isZero_zero (Sheaf C X)).from_ F
  have : IsIso f := by
    rw [Presheaf.isIso_iff_stalkFunctor_map_iso]
    exact fun x => isIso_of_source_target_iso_zero _ (h x).isoZero
      ((forget C X ⋙ stalkFunctor C x).map_isZero (isZero_zero _)).isoZero
  exact (isZero_zero _).of_iso (asIso f)

include instCC in
/-- Exactness can be checked on stalks for complexes of sheaves. -/

Target:
theorem exact_iff_stalkFunctor_map_exact (S : ShortComplex (Sheaf C X)) :
    S.Exact ↔ ∀ x : X, (S.map (forget C X ⋙ stalkFunctor C x)).Exact :=

Failed candidate:
by
  have __repair_f945fbe7cf := True.intro.nonexistentProjection
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:106:41: 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>:105:0: error: unsolved goals\nC : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLimits (CategoryTheory.forget C)\ninst✝ : Abelian C\nX : TopCat\nS : ShortComplex (Sheaf C X)\n⊢ S.Exact ↔ ∀ (x : ↑X), (S.map (forget C X ⋙ stalkFunctor C x)).Exact", "returncode": 1}

Corrected proof:


<LEAN_FILE>:216:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing3592cf6e1f.value`
<LEAN_FILE>:215:0: error: unsolved goals
R : Type u_1
inst✝ : Semiring R
motive : SkewPolynomial R → Prop
p : SkewPolynomial R
h0 : motive 0
ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)
⊢ motive p

Response:
by
  apply SkewMonoidAlgebra.induction <;> aesop

[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.SkewMonoidAlgebra.Single
public import Mathlib.Algebra.SkewMonoidAlgebra.Support

Namespace:
SkewPolynomial

Local safe context:
/-
Copyright (c) 2025 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos-Fernández
-/
/-!
# Univariate skew polynomials

Given a ring `R` and an endomorphism `φ` on `R` the skew polynomials over `R`
are polynomials
$$\sum_{i= 0}^n a_iX^n, n\geq 0, a_i\in R$$
where the addition is the usual addition of polynomials
$$\sum_{i= 0}^n a_iX^n + \sum_{i= 0}^n b_iX^n= \sum_{i= 0}^n (a_i + b_i)X^n.$$
The multiplication, however, is determined by
$$Xa = \varphi (a)X$$
by extending it to all polynomials in the obvious way.

Skew polynomials are represented as `SkewMonoidAlgebra R (Multiplicative ℕ)`,
where `R` is usually at least a Semiring. In this file, we define `SkewPolynomial`
and provide basic instances.

**Note**: To register the endomorphism `φ` see notation below.

## Notation

The endomorphism `φ` is implemented using some action of `Multiplicative ℕ` on `R`.
From this action, `φ` is an `abbrev` denoting $(\text{ofAdd } 1) \cdot a := \varphi(a)$.

Users that want to work with a specific map `φ` should introduce an action of
`Multiplicative ℕ` on `R`. Specifying that this action is a `MulSemiringAction` amounts
to saying that `φ` is an endomorphism.

Furthermore, with this notation `φ^[n](a) = (ofAdd n) • a`, see `φ_iterate_apply`.

## Main definitions

* `SkewPolynomial.monomial n a` is the skew polynomial `a X ^ n`. Note that
  `SkewPolynomial.monomial n` is defined as an `R`-linear map.
* `SkewPolynomial.C a` is the constant skew polynomial `a`. Note that `C` is defined as an additive
   homomorphism.
* `SkewPolynomial.CRingHom a` is the constant skew polynomial `a`, as a ring homomorphism. This
  requires to assume `[MulSemiringAction (Multiplicative ℕ) R]`.
* `SkewPolynomial.X` is the skew polynomial `X`, i.e., `SkewPolynomial.monomial 1 1`.
* `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`.
* `SkewPolynomial.coeff p n` is the coefficient of `X ^ n` in `p`.
* `SkewPolynomial.erase p n` is the skew polynomial `p` in which one removes the monomial in
  degree `n`.
* `SkewPolynomial.update p n a` is the skew polynomial obtained by replacing the coefficient of
  degree `n` by a given value `a : R`.  If `a = 0`, this is equal to `p.erase n` If
  `p.natDegree < n` and `a ≠ 0`, this increases the degree of `p` to `n`.

## Implementation notes

The implementation uses `Multiplicative ℕ` instead of `ℕ`, since Mathlib does not contain an
additive version of `SkewMonoidAlgebra`.

This decision was made because we use the type class `MulSemiringAction` to specify the properties
the action needs to respect for associativity. There is no version of this in Mathlib that
uses an acting `AddMonoid M` and so we need to use `Multiplicative ℕ` for the action.

For associativity to hold, there should be an instance of
`MulSemiringAction (Multiplicative ℕ) R` present in the context.
For example, in the context of $\mathbb{F}_q$-linear polynomials, this can be the
$q$-th Frobenius endomorphism - so $\varphi(a) = a^q$.

## Reference

The definition is inspired by Chapter 3 of [Papikian2023].

## Tags

Skew Polynomials, Twisted Polynomials.

Note that [ore33] proposes a more general definition of skew polynomial ring, where the
multiplication is determined by  $Xa = \varphi (a)X + δ (a)$, where `φ` is as above and
`δ` is a derivation.

-/

@[expose] public section

noncomputable section

open Function Multiplicative SkewMonoidAlgebra

/-- The skew polynomials over `R` is the type of univariate polynomials over `R`
endowed with a skewed convolution product. -/
abbrev SkewPolynomial (R : Type*) [AddCommMonoid R] := SkewMonoidAlgebra R (Multiplicative ℕ)

namespace SkewPolynomial

variable {R : Type*} {m n : ℕ}

section Semiring

variable [Semiring R] {p q : SkewPolynomial R}

lemma zero_def : (0 : SkewPolynomial R) = (0 : SkewMonoidAlgebra R (Multiplicative ℕ)) := rfl

variable {S S₁ S₂ : Type*}

/--
The set of all `n` such that `X^n` has a non-zero coefficient.
-/
def support (p : SkewPolynomial R) : Finset ℕ :=
  Finset.map ⟨toAdd, toAdd.injective⟩ (SkewMonoidAlgebra.support p)

/-- Though `SkewPolynomial.support` is not definiyionally equal to `SkewMonoidAlgebra.support` we
  can relate them using the following lemma. -/
lemma support_eq_skewMonoidAlgebra_support (p : SkewPolynomial R) :
    p.support = Finset.map (Multiplicative.toAdd (α := ℕ)) (SkewMonoidAlgebra.support p) := by
  simp only [support]

@[simp] lemma support_zero : (0 : SkewPolynomial R).support = ∅ := rfl

@[simp] lemma support_eq_empty : p.support = ∅ ↔ p = 0 := by simp [support]

lemma card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by simp

lemma support_add : (p + q).support ⊆ p.support ∪ q.support := by
  simpa [support, ← Finset.map_union, Finset.map_subset_map] using SkewMonoidAlgebra.support_add

/-- `coeff p n` is the coefficient of `X ^ n` in `p`. -/
def coeff (p : SkewPolynomial R) : ℕ → R := fun n ↦ (SkewMonoidAlgebra.coeff p (ofAdd n))

@[simp]
lemma mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 := by
  simp [support, coeff]

lemma notMem_support_iff : n ∉ p.support ↔ p.coeff n = 0 := by simp

/-- `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`. -/
def sum {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) : S :=
  SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r)

/-- For a skew polynomial `p`, `p.sum f` can be written in terms of `SkewMonoidAlgebra.sum p`. -/
lemma sum_def' {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r) := rfl

lemma sum_def {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = ∑ n ∈ p.support, f n (p.coeff n) := by
  simp only [sum_def', SkewMonoidAlgebra.sum_def, Finsupp.sum, toFinsupp_apply]
  apply Finset.sum_of_injOn (toAdd) (Injective.injOn fun ⦃a₁ a₂⦄ a ↦ a) (fun _ ↦ ?_) <;>
  simp +contextual [coeff]

lemma sum_sum_index {R' P : Type*} [AddCommMonoid P] [Semiring R']
    {f : SkewPolynomial R} {g : ℕ → R → SkewPolynomial R'} {h : ℕ → R' → P}
    (h_zero : ∀ (a : ℕ), h a 0 = 0)
    (h_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂) :
    sum (sum f g) h = sum f fun (a : ℕ) (b : R) ↦ sum (g a b) h := by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

@[simp]
lemma sum_zero {N : Type*} [AddCommMonoid N] {f : SkewPolynomial R} :
    (f.sum fun (_ : ℕ) _ ↦ (0 : N)) = 0 :=
  SkewMonoidAlgebra.sum_zero

section Monomial

variable (n)

/-- `monomial s a` is the monomial `a * X ^ s`. -/
def monomial : R →ₗ[R] SkewPolynomial R := lsingle R (ofAdd n)

lemma monomial_zero_right : monomial n (0 : R) = 0 := single_zero _

lemma monomial_zero_one [MulSemiringAction (Multiplicative ℕ) R] : monomial 0 (1 : R) = 1 :=
  rfl

lemma monomial_def (a : R) : monomial n a = single (ofAdd n) a := rfl

lemma monomial_add (r s : R) : monomial n (r + s) = monomial n r + monomial n s :=
  single_add ..

lemma smul_monomial {S} [Semiring S] [Module S R] (a : S) (b : R) :
    a • monomial n b = monomial n (a • b) :=
  smul_single ..

@[simp]
lemma sum_monomial (f : SkewPolynomial R) : f.sum (fun (a : ℕ) ↦ monomial a) = f :=
 SkewMonoidAlgebra.sum_single _

@[simp]
lemma sum_monomial_index {N} [AddCommMonoid N] {n : ℕ} {b : R} {h : ℕ → R → N}
    (h_zero : h n 0 = 0) : (monomial n b).sum h = h n b :=
  SkewMonoidAlgebra.sum_single_index h_zero

lemma monomial_injective : Function.Injective (monomial n : R → SkewPolynomial R) :=
  single_injective (ofAdd n)

@[simp]
lemma monomial_eq_zero_iff (t : R) : monomial n t = 0 ↔ t = 0 :=
  LinearMap.map_eq_zero_iff _ (SkewPolynomial.monomial_injective n)

lemma monomial_eq_monomial_iff {m n : ℕ} {a b : R} :
    monomial m a = monomial n b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0 := by
  rw [← Finsupp.single_eq_single_iff m n a b]
  simp only [monomial_def, ← toFinsupp_single, toFinsupp_inj]
  simp only [← ofFinsupp_single, SkewMonoidAlgebra.ofFinsupp_inj, Finsupp.single_eq_single_iff,
    EmbeddingLike.apply_eq_iff_eq]

Target:
lemma induction {motive : SkewPolynomial R → Prop} (p : SkewPolynomial R) (h0 : motive 0)
  (ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q →
    motive (SkewPolynomial.monomial n r + q)) : motive p :=

Failed candidate:
by
  have __repair_3592cf6e1f := OProverMissing3592cf6e1f.value
  apply SkewMonoidAlgebra.induction <;> aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:216:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing3592cf6e1f.value`\n<LEAN_FILE>:215:0: error: unsolved goals\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ motive p", "returncode": 1}

Corrected proof:

[Assistant]
by
  apply SkewMonoidAlgebra.induction <;> aesop

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.SkewMonoidAlgebra.Single
public import Mathlib.Algebra.SkewMonoidAlgebra.Support

Namespace:
SkewPolynomial

Local safe context:
/-
Copyright (c) 2025 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos-Fernández
-/
/-!
# Univariate skew polynomials

Given a ring `R` and an endomorphism `φ` on `R` the skew polynomials over `R`
are polynomials
$$\sum_{i= 0}^n a_iX^n, n\geq 0, a_i\in R$$
where the addition is the usual addition of polynomials
$$\sum_{i= 0}^n a_iX^n + \sum_{i= 0}^n b_iX^n= \sum_{i= 0}^n (a_i + b_i)X^n.$$
The multiplication, however, is determined by
$$Xa = \varphi (a)X$$
by extending it to all polynomials in the obvious way.

Skew polynomials are represented as `SkewMonoidAlgebra R (Multiplicative ℕ)`,
where `R` is usually at least a Semiring. In this file, we define `SkewPolynomial`
and provide basic instances.

**Note**: To register the endomorphism `φ` see notation below.

## Notation

The endomorphism `φ` is implemented using some action of `Multiplicative ℕ` on `R`.
From this action, `φ` is an `abbrev` denoting $(\text{ofAdd } 1) \cdot a := \varphi(a)$.

Users that want to work with a specific map `φ` should introduce an action of
`Multiplicative ℕ` on `R`. Specifying that this action is a `MulSemiringAction` amounts
to saying that `φ` is an endomorphism.

Furthermore, with this notation `φ^[n](a) = (ofAdd n) • a`, see `φ_iterate_apply`.

## Main definitions

* `SkewPolynomial.monomial n a` is the skew polynomial `a X ^ n`. Note that
  `SkewPolynomial.monomial n` is defined as an `R`-linear map.
* `SkewPolynomial.C a` is the constant skew polynomial `a`. Note that `C` is defined as an additive
   homomorphism.
* `SkewPolynomial.CRingHom a` is the constant skew polynomial `a`, as a ring homomorphism. This
  requires to assume `[MulSemiringAction (Multiplicative ℕ) R]`.
* `SkewPolynomial.X` is the skew polynomial `X`, i.e., `SkewPolynomial.monomial 1 1`.
* `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`.
* `SkewPolynomial.coeff p n` is the coefficient of `X ^ n` in `p`.
* `SkewPolynomial.erase p n` is the skew polynomial `p` in which one removes the monomial in
  degree `n`.
* `SkewPolynomial.update p n a` is the skew polynomial obtained by replacing the coefficient of
  degree `n` by a given value `a : R`.  If `a = 0`, this is equal to `p.erase n` If
  `p.natDegree < n` and `a ≠ 0`, this increases the degree of `p` to `n`.

## Implementation notes

The implementation uses `Multiplicative ℕ` instead of `ℕ`, since Mathlib does not contain an
additive version of `SkewMonoidAlgebra`.

This decision was made because we use the type class `MulSemiringAction` to specify the properties
the action needs to respect for associativity. There is no version of this in Mathlib that
uses an acting `AddMonoid M` and so we need to use `Multiplicative ℕ` for the action.

For associativity to hold, there should be an instance of
`MulSemiringAction (Multiplicative ℕ) R` present in the context.
For example, in the context of $\mathbb{F}_q$-linear polynomials, this can be the
$q$-th Frobenius endomorphism - so $\varphi(a) = a^q$.

## Reference

The definition is inspired by Chapter 3 of [Papikian2023].

## Tags

Skew Polynomials, Twisted Polynomials.

Note that [ore33] proposes a more general definition of skew polynomial ring, where the
multiplication is determined by  $Xa = \varphi (a)X + δ (a)$, where `φ` is as above and
`δ` is a derivation.

-/

@[expose] public section

noncomputable section

open Function Multiplicative SkewMonoidAlgebra

/-- The skew polynomials over `R` is the type of univariate polynomials over `R`
endowed with a skewed convolution product. -/
abbrev SkewPolynomial (R : Type*) [AddCommMonoid R] := SkewMonoidAlgebra R (Multiplicative ℕ)

namespace SkewPolynomial

variable {R : Type*} {m n : ℕ}

section Semiring

variable [Semiring R] {p q : SkewPolynomial R}

lemma zero_def : (0 : SkewPolynomial R) = (0 : SkewMonoidAlgebra R (Multiplicative ℕ)) := rfl

variable {S S₁ S₂ : Type*}

/--
The set of all `n` such that `X^n` has a non-zero coefficient.
-/
def support (p : SkewPolynomial R) : Finset ℕ :=
  Finset.map ⟨toAdd, toAdd.injective⟩ (SkewMonoidAlgebra.support p)

/-- Though `SkewPolynomial.support` is not definiyionally equal to `SkewMonoidAlgebra.support` we
  can relate them using the following lemma. -/
lemma support_eq_skewMonoidAlgebra_support (p : SkewPolynomial R) :
    p.support = Finset.map (Multiplicative.toAdd (α := ℕ)) (SkewMonoidAlgebra.support p) := by
  simp only [support]

@[simp] lemma support_zero : (0 : SkewPolynomial R).support = ∅ := rfl

@[simp] lemma support_eq_empty : p.support = ∅ ↔ p = 0 := by simp [support]

lemma card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by simp

lemma support_add : (p + q).support ⊆ p.support ∪ q.support := by
  simpa [support, ← Finset.map_union, Finset.map_subset_map] using SkewMonoidAlgebra.support_add

/-- `coeff p n` is the coefficient of `X ^ n` in `p`. -/
def coeff (p : SkewPolynomial R) : ℕ → R := fun n ↦ (SkewMonoidAlgebra.coeff p (ofAdd n))

@[simp]
lemma mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 := by
  simp [support, coeff]

lemma notMem_support_iff : n ∉ p.support ↔ p.coeff n = 0 := by simp

/-- `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`. -/
def sum {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) : S :=
  SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r)

/-- For a skew polynomial `p`, `p.sum f` can be written in terms of `SkewMonoidAlgebra.sum p`. -/
lemma sum_def' {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r) := rfl

lemma sum_def {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = ∑ n ∈ p.support, f n (p.coeff n) := by
  simp only [sum_def', SkewMonoidAlgebra.sum_def, Finsupp.sum, toFinsupp_apply]
  apply Finset.sum_of_injOn (toAdd) (Injective.injOn fun ⦃a₁ a₂⦄ a ↦ a) (fun _ ↦ ?_) <;>
  simp +contextual [coeff]

lemma sum_sum_index {R' P : Type*} [AddCommMonoid P] [Semiring R']
    {f : SkewPolynomial R} {g : ℕ → R → SkewPolynomial R'} {h : ℕ → R' → P}
    (h_zero : ∀ (a : ℕ), h a 0 = 0)
    (h_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂) :
    sum (sum f g) h = sum f fun (a : ℕ) (b : R) ↦ sum (g a b) h := by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

@[simp]
lemma sum_zero {N : Type*} [AddCommMonoid N] {f : SkewPolynomial R} :
    (f.sum fun (_ : ℕ) _ ↦ (0 : N)) = 0 :=
  SkewMonoidAlgebra.sum_zero

section Monomial

variable (n)

/-- `monomial s a` is the monomial `a * X ^ s`. -/
def monomial : R →ₗ[R] SkewPolynomial R := lsingle R (ofAdd n)

lemma monomial_zero_right : monomial n (0 : R) = 0 := single_zero _

lemma monomial_zero_one [MulSemiringAction (Multiplicative ℕ) R] : monomial 0 (1 : R) = 1 :=
  rfl

lemma monomial_def (a : R) : monomial n a = single (ofAdd n) a := rfl

lemma monomial_add (r s : R) : monomial n (r + s) = monomial n r + monomial n s :=
  single_add ..

lemma smul_monomial {S} [Semiring S] [Module S R] (a : S) (b : R) :
    a • monomial n b = monomial n (a • b) :=
  smul_single ..

@[simp]
lemma sum_monomial (f : SkewPolynomial R) : f.sum (fun (a : ℕ) ↦ monomial a) = f :=
 SkewMonoidAlgebra.sum_single _

@[simp]
lemma sum_monomial_index {N} [AddCommMonoid N] {n : ℕ} {b : R} {h : ℕ → R → N}
    (h_zero : h n 0 = 0) : (monomial n b).sum h = h n b :=
  SkewMonoidAlgebra.sum_single_index h_zero

lemma monomial_injective : Function.Injective (monomial n : R → SkewPolynomial R) :=
  single_injective (ofAdd n)

@[simp]
lemma monomial_eq_zero_iff (t : R) : monomial n t = 0 ↔ t = 0 :=
  LinearMap.map_eq_zero_iff _ (SkewPolynomial.monomial_injective n)

lemma monomial_eq_monomial_iff {m n : ℕ} {a b : R} :
    monomial m a = monomial n b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0 := by
  rw [← Finsupp.single_eq_single_iff m n a b]
  simp only [monomial_def, ← toFinsupp_single, toFinsupp_inj]
  simp only [← ofFinsupp_single, SkewMonoidAlgebra.ofFinsupp_inj, Finsupp.single_eq_single_iff,
    EmbeddingLike.apply_eq_iff_eq]

Target:
lemma induction {motive : SkewPolynomial R → Prop} (p : SkewPolynomial R) (h0 : motive 0)
  (ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q →
    motive (SkewPolynomial.monomial n r + q)) : motive p :=

Failed candidate:
by
  have __repair_3592cf6e1f := OProverMissing3592cf6e1f.value
  apply SkewMonoidAlgebra.induction <;> aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:216:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing3592cf6e1f.value`\n<LEAN_FILE>:215:0: error: unsolved goals\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ motive p", "returncode": 1}

Corrected proof:


<LEAN_FILE>:125:2: warning: declaration uses `sorry`
<LEAN_FILE>:129:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd8f32a8538.value`
<LEAN_FILE>:128:0: error: unsolved goals
N : Type u_1
E : Type u_2
G : Type u_3
inst✝² : Group N
inst✝¹ : Group E
inst✝ : Group G
S : GroupExtension N E G
n : N
⊢ S.rightHom (S.inl n) = 1

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

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

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

Namespace:
GroupExtension

Local safe 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 :=

Failed candidate:
by
  have __repair_d8f32a8538 := OProverMissingd8f32a8538.value
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:125:2: warning: declaration uses `sorry`\n<LEAN_FILE>:129:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd8f32a8538.value`\n<LEAN_FILE>:128:0: error: unsolved goals\nN : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\nn : N\n⊢ S.rightHom (S.inl n) = 1", "returncode": 1}

Corrected proof:

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

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

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

Namespace:
GroupExtension

Local safe 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 :=

Failed candidate:
by
  have __repair_d8f32a8538 := OProverMissingd8f32a8538.value
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:125:2: warning: declaration uses `sorry`\n<LEAN_FILE>:129:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd8f32a8538.value`\n<LEAN_FILE>:128:0: error: unsolved goals\nN : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\nn : N\n⊢ S.rightHom (S.inl n) = 1", "returncode": 1}

Corrected proof: