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

[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:
(root)

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

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

Proof body:

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

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:
(root)

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

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

Proof body:


<LEAN_FILE>:89:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_6c7bdf6751`
<LEAN_FILE>:64:0: error: unsolved goals
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
hf : f.IsFinitelySemisimple
μ : R
k : ℕ∞
hk : 0 < k
⊢ (f.genEigenspace μ) k = f.eigenspace μ

Response:
by
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

[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.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
public import Mathlib.LinearAlgebra.Semisimple

Namespace:
Module.End

Local safe context:
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Eigenspaces of semisimple linear endomorphisms

This file contains basic results relevant to the study of eigenspaces of semisimple linear
endomorphisms.

## Main definitions / results

* `Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspace`: for a semisimple endomorphism,
  a generalized eigenspace is an eigenspace.
* `Module.End.IsSemisimple.iSup_eigenspace_eq_top`: over an algebraically closed field,
  the eigenspaces of a semisimple endomorphism span the whole space.
* `Module.End.IsSemisimple.eq_zero_iff_forall_eigenvalue`: a semisimple endomorphism over
  an algebraically closed field is zero iff all eigenvalues are zero.

-/

public section

open Function Set

namespace Module.End

variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] {f g : End R M}

lemma apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil
    {μ : R} {k : ℕ∞} {m : M} (hm : m ∈ f.genEigenspace μ k)
    (hfg : Commute f g) (hss : g.IsFinitelySemisimple) (hnil : IsNilpotent (f - g)) :
    g m = μ • m := by
  rw [f.mem_genEigenspace] at hm
  obtain ⟨l, -, hm⟩ := hm
  rw [← f.mem_genEigenspace_nat] at hm
  set p := f.genEigenspace μ l
  have h₁ : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg μ l
  have h₂ : MapsTo (g - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right μ f) μ l
  have h₃ : MapsTo (f - g) p p :=
    mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) μ l
  have h₄ : MapsTo (f - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ l
  replace hfg : Commute (f - algebraMap R (End R M) μ) (f - g) :=
    (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left μ (f - g)
  suffices IsNilpotent ((g - algebraMap R (End R M) μ).restrict h₂) by
    replace this : g.restrict h₁ - algebraMap R (End R p) μ = 0 :=
      eq_zero_of_isNilpotent_of_isFinitelySemisimple this (by simpa using hss.restrict _)
    simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this ⟨m, hm⟩
  simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub
    (f.isNilpotent_restrict_sub_algebraMap μ l) (Module.End.isNilpotent.restrict h₃ hnil)

Target:
lemma IsFinitelySemisimple.genEigenspace_eq_eigenspace
    (hf : f.IsFinitelySemisimple) (μ : R) {k : ℕ∞} (hk : 0 < k) :
    f.genEigenspace μ k = f.eigenspace μ :=

Failed candidate:
by
  have __repair_pad_0_6c7bdf6751 : True := by trivial
  have __repair_pad_1_6c7bdf6751 : True := by trivial
  have __repair_pad_2_6c7bdf6751 : True := by trivial
  have __repair_pad_3_6c7bdf6751 : True := by trivial
  have __repair_pad_4_6c7bdf6751 : True := by trivial
  have __repair_pad_5_6c7bdf6751 : True := by trivial
  have __repair_pad_6_6c7bdf6751 : True := by trivial
  have __repair_pad_7_6c7bdf6751 : True := by trivial
  have __repair_pad_8_6c7bdf6751 : True := by trivial
  have __repair_pad_9_6c7bdf6751 : True := by trivial
  have __repair_pad_10_6c7bdf6751 : True := by trivial
  have __repair_pad_11_6c7bdf6751 : True := by trivial
  have __repair_pad_12_6c7bdf6751 : True := by trivial
  have __repair_pad_13_6c7bdf6751 : True := by trivial
  have __repair_pad_14_6c7bdf6751 : True := by trivial
  have __repair_pad_15_6c7bdf6751 : True := by trivial
  have __repair_pad_16_6c7bdf6751 : True := by trivial
  have __repair_pad_17_6c7bdf6751 : True := by trivial
  have __repair_pad_18_6c7bdf6751 : True := by trivial
  have __repair_pad_19_6c7bdf6751 : True := by trivial
  have __repair_pad_20_6c7bdf6751 : True := by trivial
  have __repair_pad_21_6c7bdf6751 : True := by trivial
  have __repair_pad_22_6c7bdf6751 : True := by trivial
  have __repair_pad_23_6c7bdf6751 : True := by trivial
  have __repair_fail_6c7bdf6751 := __oprover_overlong_6c7bdf6751
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:89:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_6c7bdf6751`\n<LEAN_FILE>:64:0: error: unsolved goals\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhf : f.IsFinitelySemisimple\nμ : R\nk : ℕ∞\nhk : 0 < k\n⊢ (f.genEigenspace μ) k = f.eigenspace μ", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

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.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
public import Mathlib.LinearAlgebra.Semisimple

Namespace:
Module.End

Local safe context:
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Eigenspaces of semisimple linear endomorphisms

This file contains basic results relevant to the study of eigenspaces of semisimple linear
endomorphisms.

## Main definitions / results

* `Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspace`: for a semisimple endomorphism,
  a generalized eigenspace is an eigenspace.
* `Module.End.IsSemisimple.iSup_eigenspace_eq_top`: over an algebraically closed field,
  the eigenspaces of a semisimple endomorphism span the whole space.
* `Module.End.IsSemisimple.eq_zero_iff_forall_eigenvalue`: a semisimple endomorphism over
  an algebraically closed field is zero iff all eigenvalues are zero.

-/

public section

open Function Set

namespace Module.End

variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] {f g : End R M}

lemma apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil
    {μ : R} {k : ℕ∞} {m : M} (hm : m ∈ f.genEigenspace μ k)
    (hfg : Commute f g) (hss : g.IsFinitelySemisimple) (hnil : IsNilpotent (f - g)) :
    g m = μ • m := by
  rw [f.mem_genEigenspace] at hm
  obtain ⟨l, -, hm⟩ := hm
  rw [← f.mem_genEigenspace_nat] at hm
  set p := f.genEigenspace μ l
  have h₁ : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg μ l
  have h₂ : MapsTo (g - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right μ f) μ l
  have h₃ : MapsTo (f - g) p p :=
    mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) μ l
  have h₄ : MapsTo (f - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ l
  replace hfg : Commute (f - algebraMap R (End R M) μ) (f - g) :=
    (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left μ (f - g)
  suffices IsNilpotent ((g - algebraMap R (End R M) μ).restrict h₂) by
    replace this : g.restrict h₁ - algebraMap R (End R p) μ = 0 :=
      eq_zero_of_isNilpotent_of_isFinitelySemisimple this (by simpa using hss.restrict _)
    simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this ⟨m, hm⟩
  simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub
    (f.isNilpotent_restrict_sub_algebraMap μ l) (Module.End.isNilpotent.restrict h₃ hnil)

Target:
lemma IsFinitelySemisimple.genEigenspace_eq_eigenspace
    (hf : f.IsFinitelySemisimple) (μ : R) {k : ℕ∞} (hk : 0 < k) :
    f.genEigenspace μ k = f.eigenspace μ :=

Failed candidate:
by
  have __repair_pad_0_6c7bdf6751 : True := by trivial
  have __repair_pad_1_6c7bdf6751 : True := by trivial
  have __repair_pad_2_6c7bdf6751 : True := by trivial
  have __repair_pad_3_6c7bdf6751 : True := by trivial
  have __repair_pad_4_6c7bdf6751 : True := by trivial
  have __repair_pad_5_6c7bdf6751 : True := by trivial
  have __repair_pad_6_6c7bdf6751 : True := by trivial
  have __repair_pad_7_6c7bdf6751 : True := by trivial
  have __repair_pad_8_6c7bdf6751 : True := by trivial
  have __repair_pad_9_6c7bdf6751 : True := by trivial
  have __repair_pad_10_6c7bdf6751 : True := by trivial
  have __repair_pad_11_6c7bdf6751 : True := by trivial
  have __repair_pad_12_6c7bdf6751 : True := by trivial
  have __repair_pad_13_6c7bdf6751 : True := by trivial
  have __repair_pad_14_6c7bdf6751 : True := by trivial
  have __repair_pad_15_6c7bdf6751 : True := by trivial
  have __repair_pad_16_6c7bdf6751 : True := by trivial
  have __repair_pad_17_6c7bdf6751 : True := by trivial
  have __repair_pad_18_6c7bdf6751 : True := by trivial
  have __repair_pad_19_6c7bdf6751 : True := by trivial
  have __repair_pad_20_6c7bdf6751 : True := by trivial
  have __repair_pad_21_6c7bdf6751 : True := by trivial
  have __repair_pad_22_6c7bdf6751 : True := by trivial
  have __repair_pad_23_6c7bdf6751 : True := by trivial
  have __repair_fail_6c7bdf6751 := __oprover_overlong_6c7bdf6751
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:89:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_6c7bdf6751`\n<LEAN_FILE>:64:0: error: unsolved goals\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhf : f.IsFinitelySemisimple\nμ : R\nk : ℕ∞\nhk : 0 < k\n⊢ (f.genEigenspace μ) k = f.eigenspace μ", "returncode": 1}

Corrected proof:


Response:
by
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

[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.Analysis.SpecialFunctions.Complex.LogBounds
public import Mathlib.NumberTheory.Harmonic.ZetaAsymp
public import Mathlib.NumberTheory.LSeries.Dirichlet
public import Mathlib.NumberTheory.LSeries.DirichletContinuation
public import Mathlib.NumberTheory.LSeries.Positivity

Namespace:
DirichletCharacter

Local context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, David Loeffler
-/
/-!
# The L-function of a Dirichlet character does not vanish on Re(s) ≥ 1

The main result in this file is `DirichletCharacter.LFunction_ne_zero_of_one_le_re`:
if `χ` is a Dirichlet character, `s ∈ ℂ` with `1 ≤ s.re`, and either `χ` is nontrivial or `s ≠ 1`,
then the L-function of `χ` does not vanish at `s`.

As a consequence, we have the corresponding statement for the Riemann ζ function:
`riemannZeta_ne_zero_of_one_le_re` (which does not require `s ≠ 1`, since the junk value at `s = 1`
happens to be non-zero).

These results are prerequisites for the **Prime Number Theorem** and
**Dirichlet's Theorem** on primes in arithmetic progressions.

## Outline of proofs

We split into two cases: first, the special case of (non-trivial) quadratic characters at `s = 1`;
then the remaining case when either `s ≠ 1` or `χ ^ 2 ≠ 1`.

The first case is handled using a positivity argument applied to the series `L χ s * ζ s`: we show
that this function has non-negative Dirichlet coefficients, is strictly positive for `s ≫ 0`, but
vanishes at `s = -2`, so it must have a pole somewhere in between.

The second case is dealt with using the product
`L(χ^0, 1 + x)^3 L(χ, 1 + x + I * y)^4 L(χ^2, 1 + x + 2 * I * y)`, which
we show has absolute value `≥ 1` for all positive `x` and real `y`; if `L(χ, 1 + I * y) = 0` then
this product would have to tend to 0 as `x → 0`, which is a contradiction.
-/

@[expose] public section

/- NB: Many lemmas (and some defs) in this file are private, since they concern properties of
hypothetical objects which we eventually deduce cannot exist. We have only made public the lemmas
whose hypotheses do not turn out to be contradictory.
-/

open Complex Asymptotics Topology Filter
open ArithmeticFunction hiding log

-- We use the ordering on `ℂ` given by comparing real parts for fixed imaginary part
open scoped ComplexOrder

variable {N : ℕ}

namespace DirichletCharacter

section quadratic

/-!
### Convolution of a Dirichlet character with ζ

We define `DirichletCharacter.zetaMul χ` to be the arithmetic function obtained by
taking the product (as arithmetic functions = Dirichlet convolution) of the
arithmetic function `ζ` with `χ`.

We then show that for a quadratic character `χ`, this arithmetic function is multiplicative
and takes nonnegative real values.
-/

/-- The complex-valued arithmetic function that is the convolution of the constant
function `1` with `χ`. -/
noncomputable def zetaMul (χ : DirichletCharacter ℂ N) : ArithmeticFunction ℂ :=
  .zeta * toArithmeticFunction (χ ·)

/-- The arithmetic function `zetaMul χ` is multiplicative. -/
lemma isMultiplicative_zetaMul (χ : DirichletCharacter ℂ N) : χ.zetaMul.IsMultiplicative :=
  isMultiplicative_zeta.natCast.mul <| isMultiplicative_toArithmeticFunction χ

Target:
lemma LSeriesSummable_zetaMul (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < s.re) :
    LSeriesSummable χ.zetaMul s :=

Proof body:

[Assistant]
by
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

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.Analysis.SpecialFunctions.Complex.LogBounds
public import Mathlib.NumberTheory.Harmonic.ZetaAsymp
public import Mathlib.NumberTheory.LSeries.Dirichlet
public import Mathlib.NumberTheory.LSeries.DirichletContinuation
public import Mathlib.NumberTheory.LSeries.Positivity

Namespace:
DirichletCharacter

Local context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, David Loeffler
-/
/-!
# The L-function of a Dirichlet character does not vanish on Re(s) ≥ 1

The main result in this file is `DirichletCharacter.LFunction_ne_zero_of_one_le_re`:
if `χ` is a Dirichlet character, `s ∈ ℂ` with `1 ≤ s.re`, and either `χ` is nontrivial or `s ≠ 1`,
then the L-function of `χ` does not vanish at `s`.

As a consequence, we have the corresponding statement for the Riemann ζ function:
`riemannZeta_ne_zero_of_one_le_re` (which does not require `s ≠ 1`, since the junk value at `s = 1`
happens to be non-zero).

These results are prerequisites for the **Prime Number Theorem** and
**Dirichlet's Theorem** on primes in arithmetic progressions.

## Outline of proofs

We split into two cases: first, the special case of (non-trivial) quadratic characters at `s = 1`;
then the remaining case when either `s ≠ 1` or `χ ^ 2 ≠ 1`.

The first case is handled using a positivity argument applied to the series `L χ s * ζ s`: we show
that this function has non-negative Dirichlet coefficients, is strictly positive for `s ≫ 0`, but
vanishes at `s = -2`, so it must have a pole somewhere in between.

The second case is dealt with using the product
`L(χ^0, 1 + x)^3 L(χ, 1 + x + I * y)^4 L(χ^2, 1 + x + 2 * I * y)`, which
we show has absolute value `≥ 1` for all positive `x` and real `y`; if `L(χ, 1 + I * y) = 0` then
this product would have to tend to 0 as `x → 0`, which is a contradiction.
-/

@[expose] public section

/- NB: Many lemmas (and some defs) in this file are private, since they concern properties of
hypothetical objects which we eventually deduce cannot exist. We have only made public the lemmas
whose hypotheses do not turn out to be contradictory.
-/

open Complex Asymptotics Topology Filter
open ArithmeticFunction hiding log

-- We use the ordering on `ℂ` given by comparing real parts for fixed imaginary part
open scoped ComplexOrder

variable {N : ℕ}

namespace DirichletCharacter

section quadratic

/-!
### Convolution of a Dirichlet character with ζ

We define `DirichletCharacter.zetaMul χ` to be the arithmetic function obtained by
taking the product (as arithmetic functions = Dirichlet convolution) of the
arithmetic function `ζ` with `χ`.

We then show that for a quadratic character `χ`, this arithmetic function is multiplicative
and takes nonnegative real values.
-/

/-- The complex-valued arithmetic function that is the convolution of the constant
function `1` with `χ`. -/
noncomputable def zetaMul (χ : DirichletCharacter ℂ N) : ArithmeticFunction ℂ :=
  .zeta * toArithmeticFunction (χ ·)

/-- The arithmetic function `zetaMul χ` is multiplicative. -/
lemma isMultiplicative_zetaMul (χ : DirichletCharacter ℂ N) : χ.zetaMul.IsMultiplicative :=
  isMultiplicative_zeta.natCast.mul <| isMultiplicative_toArithmeticFunction χ

Target:
lemma LSeriesSummable_zetaMul (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < s.re) :
    LSeriesSummable χ.zetaMul s :=

Proof body:


<LEAN_FILE>:152:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b4626703d1`
<LEAN_FILE>:127:0: error: unsolved goals
ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'

Response:
by
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

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

Imports:
public import Mathlib.Analysis.BoxIntegral.Partition.Basic

Namespace:
BoxIntegral.Prepartition

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
-/
/-!
# Tagged partitions

A tagged (pre)partition is a (pre)partition `π` enriched with a tagged point for each box of
`π`. For simplicity we require that the function `BoxIntegral.TaggedPrepartition.tag` is defined
on all boxes `J : Box ι` but use its values only on boxes of the partition. Given
`π : BoxIntegral.TaggedPrepartition I`, we require that each `BoxIntegral.TaggedPrepartition π J`
belongs to `BoxIntegral.Box.Icc I`. If for every `J ∈ π`, `π.tag J` belongs to `J.Icc`, then `π` is
called a *Henstock* partition. We do not include this assumption into the definition of a tagged
(pre)partition because McShane integral is defined as a limit along tagged partitions without this
requirement.

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

/-- A tagged prepartition is a prepartition enriched with a tagged point for each box of the
prepartition. For simplicity we require that `tag` is defined for all boxes in `ι → ℝ` but
we will use only the values of `tag` on the boxes of the partition. -/
structure TaggedPrepartition (I : Box ι) extends Prepartition I where
  /-- Choice of tagged point of each box in this prepartition:
  we extend this to a total function, on all boxes in `ι → ℝ`. -/
  tag : Box ι → ι → ℝ
  /-- Each tagged point belongs to `I` -/
  tag_mem_Icc : ∀ J, tag J ∈ Box.Icc I

namespace TaggedPrepartition

variable {I J J₁ J₂ : Box ι} (π : TaggedPrepartition I) {x : ι → ℝ}

instance : Membership (Box ι) (TaggedPrepartition I) :=
  ⟨fun π J => J ∈ π.boxes⟩

@[simp]
theorem mem_toPrepartition {π : TaggedPrepartition I} : J ∈ π.toPrepartition ↔ J ∈ π := Iff.rfl

@[simp]
theorem mem_mk (π : Prepartition I) (f h) : J ∈ mk π f h ↔ J ∈ π := Iff.rfl

/-- Union of all boxes of a tagged prepartition. -/
def iUnion : Set (ι → ℝ) :=
  π.toPrepartition.iUnion

theorem iUnion_def : π.iUnion = ⋃ J ∈ π, ↑J := rfl

@[simp]
theorem iUnion_mk (π : Prepartition I) (f h) : (mk π f h).iUnion = π.iUnion := rfl

@[simp]
theorem iUnion_toPrepartition : π.toPrepartition.iUnion = π.iUnion := rfl

@[simp]
theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by
  convert! Set.mem_iUnion₂
  rw [Box.mem_coe, mem_toPrepartition, exists_prop]

theorem subset_iUnion (h : J ∈ π) : ↑J ⊆ π.iUnion :=
  subset_biUnion_of_mem h

theorem iUnion_subset : π.iUnion ⊆ I :=
  iUnion₂_subset π.le_of_mem'

/-- A tagged prepartition is a partition if it covers the whole box. -/
def IsPartition :=
  π.toPrepartition.IsPartition

theorem isPartition_iff_iUnion_eq : IsPartition π ↔ π.iUnion = I :=
  Prepartition.isPartition_iff_iUnion_eq

/-- The tagged partition made of boxes of `π` that satisfy predicate `p`. -/
@[simps! -fullyApplied]
def filter (p : Box ι → Prop) : TaggedPrepartition I :=
  ⟨π.1.filter p, π.2, π.3⟩

@[simp]
theorem mem_filter {p : Box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J := by
  classical exact Finset.mem_filter

@[simp]
theorem iUnion_filter_not (π : TaggedPrepartition I) (p : Box ι → Prop) :
    (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion :=
  π.toPrepartition.iUnion_filter_not p

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

/-- Given a partition `π` of `I : BoxIntegral.Box ι` and a collection of tagged partitions
`πi J` of all boxes `J ∈ π`, returns the tagged partition of `I` into all the boxes of `πi J`
with tags coming from `(πi J).tag`. -/
def biUnionTagged (π : Prepartition I) (πi : ∀ J : Box ι, TaggedPrepartition J) :
    TaggedPrepartition I where
  toPrepartition := π.biUnion fun J => (πi J).toPrepartition
  tag J := (πi (π.biUnionIndex (fun J => (πi J).toPrepartition) J)).tag J
  tag_mem_Icc _ := Box.le_iff_Icc.1 (π.biUnionIndex_le _ _) ((πi _).tag_mem_Icc _)

@[simp]
theorem mem_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} :
    J ∈ π.biUnionTagged πi ↔ ∃ J' ∈ π, J ∈ πi J' :=
  π.mem_biUnion

Target:
theorem tag_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} (hJ : J ∈ π) {J'}
    (hJ' : J' ∈ πi J) : (π.biUnionTagged πi).tag J' = (πi J).tag J' :=

Failed candidate:
by
  have __repair_pad_0_b4626703d1 : True := by trivial
  have __repair_pad_1_b4626703d1 : True := by trivial
  have __repair_pad_2_b4626703d1 : True := by trivial
  have __repair_pad_3_b4626703d1 : True := by trivial
  have __repair_pad_4_b4626703d1 : True := by trivial
  have __repair_pad_5_b4626703d1 : True := by trivial
  have __repair_pad_6_b4626703d1 : True := by trivial
  have __repair_pad_7_b4626703d1 : True := by trivial
  have __repair_pad_8_b4626703d1 : True := by trivial
  have __repair_pad_9_b4626703d1 : True := by trivial
  have __repair_pad_10_b4626703d1 : True := by trivial
  have __repair_pad_11_b4626703d1 : True := by trivial
  have __repair_pad_12_b4626703d1 : True := by trivial
  have __repair_pad_13_b4626703d1 : True := by trivial
  have __repair_pad_14_b4626703d1 : True := by trivial
  have __repair_pad_15_b4626703d1 : True := by trivial
  have __repair_pad_16_b4626703d1 : True := by trivial
  have __repair_pad_17_b4626703d1 : True := by trivial
  have __repair_pad_18_b4626703d1 : True := by trivial
  have __repair_pad_19_b4626703d1 : True := by trivial
  have __repair_pad_20_b4626703d1 : True := by trivial
  have __repair_pad_21_b4626703d1 : True := by trivial
  have __repair_pad_22_b4626703d1 : True := by trivial
  have __repair_pad_23_b4626703d1 : True := by trivial
  have __repair_fail_b4626703d1 := __oprover_overlong_b4626703d1
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:152:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b4626703d1`\n<LEAN_FILE>:127:0: error: unsolved goals\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nhJ : J ∈ π\nJ' : Box ι\nhJ' : J' ∈ πi J\n⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

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

Imports:
public import Mathlib.Analysis.BoxIntegral.Partition.Basic

Namespace:
BoxIntegral.Prepartition

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
-/
/-!
# Tagged partitions

A tagged (pre)partition is a (pre)partition `π` enriched with a tagged point for each box of
`π`. For simplicity we require that the function `BoxIntegral.TaggedPrepartition.tag` is defined
on all boxes `J : Box ι` but use its values only on boxes of the partition. Given
`π : BoxIntegral.TaggedPrepartition I`, we require that each `BoxIntegral.TaggedPrepartition π J`
belongs to `BoxIntegral.Box.Icc I`. If for every `J ∈ π`, `π.tag J` belongs to `J.Icc`, then `π` is
called a *Henstock* partition. We do not include this assumption into the definition of a tagged
(pre)partition because McShane integral is defined as a limit along tagged partitions without this
requirement.

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

/-- A tagged prepartition is a prepartition enriched with a tagged point for each box of the
prepartition. For simplicity we require that `tag` is defined for all boxes in `ι → ℝ` but
we will use only the values of `tag` on the boxes of the partition. -/
structure TaggedPrepartition (I : Box ι) extends Prepartition I where
  /-- Choice of tagged point of each box in this prepartition:
  we extend this to a total function, on all boxes in `ι → ℝ`. -/
  tag : Box ι → ι → ℝ
  /-- Each tagged point belongs to `I` -/
  tag_mem_Icc : ∀ J, tag J ∈ Box.Icc I

namespace TaggedPrepartition

variable {I J J₁ J₂ : Box ι} (π : TaggedPrepartition I) {x : ι → ℝ}

instance : Membership (Box ι) (TaggedPrepartition I) :=
  ⟨fun π J => J ∈ π.boxes⟩

@[simp]
theorem mem_toPrepartition {π : TaggedPrepartition I} : J ∈ π.toPrepartition ↔ J ∈ π := Iff.rfl

@[simp]
theorem mem_mk (π : Prepartition I) (f h) : J ∈ mk π f h ↔ J ∈ π := Iff.rfl

/-- Union of all boxes of a tagged prepartition. -/
def iUnion : Set (ι → ℝ) :=
  π.toPrepartition.iUnion

theorem iUnion_def : π.iUnion = ⋃ J ∈ π, ↑J := rfl

@[simp]
theorem iUnion_mk (π : Prepartition I) (f h) : (mk π f h).iUnion = π.iUnion := rfl

@[simp]
theorem iUnion_toPrepartition : π.toPrepartition.iUnion = π.iUnion := rfl

@[simp]
theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by
  convert! Set.mem_iUnion₂
  rw [Box.mem_coe, mem_toPrepartition, exists_prop]

theorem subset_iUnion (h : J ∈ π) : ↑J ⊆ π.iUnion :=
  subset_biUnion_of_mem h

theorem iUnion_subset : π.iUnion ⊆ I :=
  iUnion₂_subset π.le_of_mem'

/-- A tagged prepartition is a partition if it covers the whole box. -/
def IsPartition :=
  π.toPrepartition.IsPartition

theorem isPartition_iff_iUnion_eq : IsPartition π ↔ π.iUnion = I :=
  Prepartition.isPartition_iff_iUnion_eq

/-- The tagged partition made of boxes of `π` that satisfy predicate `p`. -/
@[simps! -fullyApplied]
def filter (p : Box ι → Prop) : TaggedPrepartition I :=
  ⟨π.1.filter p, π.2, π.3⟩

@[simp]
theorem mem_filter {p : Box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J := by
  classical exact Finset.mem_filter

@[simp]
theorem iUnion_filter_not (π : TaggedPrepartition I) (p : Box ι → Prop) :
    (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion :=
  π.toPrepartition.iUnion_filter_not p

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

/-- Given a partition `π` of `I : BoxIntegral.Box ι` and a collection of tagged partitions
`πi J` of all boxes `J ∈ π`, returns the tagged partition of `I` into all the boxes of `πi J`
with tags coming from `(πi J).tag`. -/
def biUnionTagged (π : Prepartition I) (πi : ∀ J : Box ι, TaggedPrepartition J) :
    TaggedPrepartition I where
  toPrepartition := π.biUnion fun J => (πi J).toPrepartition
  tag J := (πi (π.biUnionIndex (fun J => (πi J).toPrepartition) J)).tag J
  tag_mem_Icc _ := Box.le_iff_Icc.1 (π.biUnionIndex_le _ _) ((πi _).tag_mem_Icc _)

@[simp]
theorem mem_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} :
    J ∈ π.biUnionTagged πi ↔ ∃ J' ∈ π, J ∈ πi J' :=
  π.mem_biUnion

Target:
theorem tag_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} (hJ : J ∈ π) {J'}
    (hJ' : J' ∈ πi J) : (π.biUnionTagged πi).tag J' = (πi J).tag J' :=

Failed candidate:
by
  have __repair_pad_0_b4626703d1 : True := by trivial
  have __repair_pad_1_b4626703d1 : True := by trivial
  have __repair_pad_2_b4626703d1 : True := by trivial
  have __repair_pad_3_b4626703d1 : True := by trivial
  have __repair_pad_4_b4626703d1 : True := by trivial
  have __repair_pad_5_b4626703d1 : True := by trivial
  have __repair_pad_6_b4626703d1 : True := by trivial
  have __repair_pad_7_b4626703d1 : True := by trivial
  have __repair_pad_8_b4626703d1 : True := by trivial
  have __repair_pad_9_b4626703d1 : True := by trivial
  have __repair_pad_10_b4626703d1 : True := by trivial
  have __repair_pad_11_b4626703d1 : True := by trivial
  have __repair_pad_12_b4626703d1 : True := by trivial
  have __repair_pad_13_b4626703d1 : True := by trivial
  have __repair_pad_14_b4626703d1 : True := by trivial
  have __repair_pad_15_b4626703d1 : True := by trivial
  have __repair_pad_16_b4626703d1 : True := by trivial
  have __repair_pad_17_b4626703d1 : True := by trivial
  have __repair_pad_18_b4626703d1 : True := by trivial
  have __repair_pad_19_b4626703d1 : True := by trivial
  have __repair_pad_20_b4626703d1 : True := by trivial
  have __repair_pad_21_b4626703d1 : True := by trivial
  have __repair_pad_22_b4626703d1 : True := by trivial
  have __repair_pad_23_b4626703d1 : True := by trivial
  have __repair_fail_b4626703d1 := __oprover_overlong_b4626703d1
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:152:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_b4626703d1`\n<LEAN_FILE>:127:0: error: unsolved goals\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nhJ : J ∈ π\nJ' : Box ι\nhJ' : J' ∈ πi J\n⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'", "returncode": 1}

Corrected proof:


<LEAN_FILE>:37:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 8647449) 0
in the target expression
  P.p.f ≫ P.p.f = P.p.f

C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Karoubi (Karoubi C)
⊢ P.p.f ≫ P.p.f = P.p.f

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

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

Imports:
public import Mathlib.CategoryTheory.Idempotents.Karoubi

Namespace:
CategoryTheory.Idempotents.KaroubiKaroubi

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

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

-/

@[expose] public section

open CategoryTheory.Category

open CategoryTheory.Idempotents.Karoubi

namespace CategoryTheory

namespace Idempotents

namespace KaroubiKaroubi

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

@[reassoc (attr := simp)]

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 8647449) 0) = ((fun x : Nat => x + 8647449) 0) by rfl]
  simpa only [hom_ext_iff, comp_f] using P.idem

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:37:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 8647449) 0\nin the target expression\n  P.p.f ≫ P.p.f = P.p.f\n\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f", "returncode": 1}

Corrected proof:

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

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

Imports:
public import Mathlib.CategoryTheory.Idempotents.Karoubi

Namespace:
CategoryTheory.Idempotents.KaroubiKaroubi

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

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

-/

@[expose] public section

open CategoryTheory.Category

open CategoryTheory.Idempotents.Karoubi

namespace CategoryTheory

namespace Idempotents

namespace KaroubiKaroubi

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

@[reassoc (attr := simp)]

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 8647449) 0) = ((fun x : Nat => x + 8647449) 0) by rfl]
  simpa only [hom_ext_iff, comp_f] using P.idem

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:37:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 8647449) 0\nin the target expression\n  P.p.f ≫ P.p.f = P.p.f\n\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f", "returncode": 1}

Corrected proof:


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

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

Namespace:
CategoryTheory.MorphismProperty

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
-/
/-!
# 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 :=

Proof body:

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

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

Namespace:
CategoryTheory.MorphismProperty

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
-/
/-!
# 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 :=

Proof body:


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

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

Namespace:
Submodule.Quotient

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

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

## Main definitions

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

-/

@[expose] public section

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

namespace Submodule

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

open QuotientAddGroup

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

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

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

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

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

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

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

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

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

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

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

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

instance : Inhabited (M ⧸ p) :=
  ⟨0⟩

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

@[simp]

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

Proof body:

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

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

Namespace:
Submodule.Quotient

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

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

## Main definitions

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

-/

@[expose] public section

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

namespace Submodule

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

open QuotientAddGroup

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

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

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

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

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

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

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

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

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

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

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

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

instance : Inhabited (M ⧸ p) :=
  ⟨0⟩

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

@[simp]

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

Proof body:


<LEAN_FILE>:94:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

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

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

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

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

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

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

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

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

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

namespace Bitraversable

open Functor LawfulApplicative

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

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

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

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

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

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

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

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

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

@[higher_order tsnd_comp_tsnd]

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

Failed candidate:
by
  have __repair_7543f7920b : False := True.intro
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]
  rfl

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:94:38: 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 [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]
  rfl

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

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

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

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

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

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

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

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

namespace Bitraversable

open Functor LawfulApplicative

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

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

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

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

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

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

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

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

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

@[higher_order tsnd_comp_tsnd]

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

Failed candidate:
by
  have __repair_7543f7920b : False := True.intro
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]
  rfl

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:94:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:161:43: error: unsolved goals
R : Type u_1
inst✝² : Semiring R
R' : Type u_5
P : Type u_6
inst✝¹ : AddCommMonoid P
inst✝ : 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
⊢ False

Response:
by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

[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]

Target:
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 :=

Failed candidate:
by
  have __repair_plan_80fe53dde2 : False := by
    skip
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:161:43: error: unsolved goals\nR : Type u_1\ninst✝² : Semiring R\nR' : Type u_5\nP : Type u_6\ninst✝¹ : AddCommMonoid P\ninst✝ : Semiring R'\nf : SkewPolynomial R\ng : ℕ → R → SkewPolynomial R'\nh : ℕ → R' → P\nh_zero : ∀ (a : ℕ), h a 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

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]

Target:
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 :=

Failed candidate:
by
  have __repair_plan_80fe53dde2 : False := by
    skip
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:161:43: error: unsolved goals\nR : Type u_1\ninst✝² : Semiring R\nR' : Type u_5\nP : Type u_6\ninst✝¹ : AddCommMonoid P\ninst✝ : Semiring R'\nf : SkewPolynomial R\ng : ℕ → R → SkewPolynomial R'\nh : ℕ → R' → P\nh_zero : ∀ (a : ℕ), h a 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂\n⊢ False", "returncode": 1}

Corrected proof:


Response:
by
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

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

Imports:
public import Mathlib.RingTheory.LocalRing.ResidueField.Polynomial
public import Mathlib.RingTheory.QuasiFinite.Weakly

Namespace:
Polynomial

Local context:
/-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-! # Quasi-finite primes in polynomial algebras -/

@[expose] public section

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

namespace Polynomial

attribute [local instance] Algebra.WeaklyQuasiFiniteAt.finite_locoalization in
lemma not_weaklyQuasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.WeaklyQuasiFiniteAt R P := by
  intro H
  wlog hR : IsField R
  · let p := P.under R
    obtain ⟨Q, hQ⟩ := (PrimeSpectrum.preimageEquivFiber R R[X]
        ⟨p, inferInstance⟩).symm.surjective ⟨⟨P, ‹_›⟩, rfl⟩
    have inst : Algebra.WeaklyQuasiFiniteAt p.ResidueField Q.asIdeal :=
      .baseChange P Q.asIdeal congr($(hQ.symm).1.1)
    exact this (Q.asIdeal.comap (polyEquivTensor' R p.ResidueField).toRingHom)
      inferInstance (Field.toIsField _)
  let := hR.toField
  have := Module.Finite.of_injective
    (IsScalarTower.toAlgHom R R[X] (Localization.AtPrime P)).toLinearMap
    (IsLocalization.injective _ P.primeCompl_le_nonZeroDivisors)
  exact transcendental_X R (Algebra.IsIntegral.isIntegral X).isAlgebraic

/-- `R[X]` is not quasi-finite over `R` at any prime. -/
lemma not_quasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.QuasiFiniteAt R P :=
  fun _ ↦ not_weaklyQuasiFiniteAt P inferInstance

Target:
lemma map_under_lt_comap_of_weaklyQuasiFiniteAt
    (f : R[X] →ₐ[R] S) (P : Ideal S) [P.IsPrime] [Algebra.WeaklyQuasiFiniteAt R P] :
    (P.under R).map C < P.comap (f : R[X] →+* S) :=

Proof body:

[Assistant]
by
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

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

Imports:
public import Mathlib.RingTheory.LocalRing.ResidueField.Polynomial
public import Mathlib.RingTheory.QuasiFinite.Weakly

Namespace:
Polynomial

Local context:
/-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-! # Quasi-finite primes in polynomial algebras -/

@[expose] public section

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

namespace Polynomial

attribute [local instance] Algebra.WeaklyQuasiFiniteAt.finite_locoalization in
lemma not_weaklyQuasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.WeaklyQuasiFiniteAt R P := by
  intro H
  wlog hR : IsField R
  · let p := P.under R
    obtain ⟨Q, hQ⟩ := (PrimeSpectrum.preimageEquivFiber R R[X]
        ⟨p, inferInstance⟩).symm.surjective ⟨⟨P, ‹_›⟩, rfl⟩
    have inst : Algebra.WeaklyQuasiFiniteAt p.ResidueField Q.asIdeal :=
      .baseChange P Q.asIdeal congr($(hQ.symm).1.1)
    exact this (Q.asIdeal.comap (polyEquivTensor' R p.ResidueField).toRingHom)
      inferInstance (Field.toIsField _)
  let := hR.toField
  have := Module.Finite.of_injective
    (IsScalarTower.toAlgHom R R[X] (Localization.AtPrime P)).toLinearMap
    (IsLocalization.injective _ P.primeCompl_le_nonZeroDivisors)
  exact transcendental_X R (Algebra.IsIntegral.isIntegral X).isAlgebraic

/-- `R[X]` is not quasi-finite over `R` at any prime. -/
lemma not_quasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.QuasiFiniteAt R P :=
  fun _ ↦ not_weaklyQuasiFiniteAt P inferInstance

Target:
lemma map_under_lt_comap_of_weaklyQuasiFiniteAt
    (f : R[X] →ₐ[R] S) (P : Ideal S) [P.IsPrime] [Algebra.WeaklyQuasiFiniteAt R P] :
    (P.under R).map C < P.comap (f : R[X] →+* S) :=

Proof body:


<LEAN_FILE>:47:41: 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>:46:0: error: unsolved goals
p : ℕ
hp : Fact (Nat.Prime p)
R : Type u_1
inst✝ : CommRing R
α : Type u_2
S : Finset α
x : α → 𝕎 R
h : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S ∧ (x r).coeff n ≠ 0}
n : ℕ
⊢ (∑ s ∈ S, x s).coeff n = ∑ s ∈ S, (x s).coeff n

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

variable [CharP R p]

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

Namespace:
WittVector

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

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

## Main theorems

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

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

public section

open Ideal Quotient
namespace WittVector

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

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

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

Failed candidate:
by
  have __repair_7b3fcef7bf := True.intro.nonexistentProjection
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:47:41: 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>:46:0: error: unsolved goals\np : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : CommRing R\nα : Type u_2\nS : Finset α\nx : α → 𝕎 R\nh : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S ∧ (x r).coeff n ≠ 0}\nn : ℕ\n⊢ (∑ s ∈ S, x s).coeff n = ∑ s ∈ S, (x s).coeff n", "returncode": 1}

Corrected proof:

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

variable [CharP R p]

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

Namespace:
WittVector

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

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

## Main theorems

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

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

public section

open Ideal Quotient
namespace WittVector

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

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

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

Failed candidate:
by
  have __repair_7b3fcef7bf := True.intro.nonexistentProjection
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:47:41: 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>:46:0: error: unsolved goals\np : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : CommRing R\nα : Type u_2\nS : Finset α\nx : α → 𝕎 R\nh : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S ∧ (x r).coeff n ≠ 0}\nn : ℕ\n⊢ (∑ s ∈ S, x s).coeff n = ∑ s ∈ S, (x s).coeff n", "returncode": 1}

Corrected proof:


<LEAN_FILE>:44:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 10527715) 0
in the target expression
  x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x

X : Type u
x : X
s : Set X
inst✝ : TopologicalSpace X
⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x

Response:
by
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

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

Imports:
public import Mathlib.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 10527715) 0) = ((fun x : Nat => x + 10527715) 0) by rfl]
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:44:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 10527715) 0\nin the target expression\n  x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x\n\nX : Type u\nx : X\ns : Set X\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

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

Imports:
public import Mathlib.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 10527715) 0) = ((fun x : Nat => x + 10527715) 0) by rfl]
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:44:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 10527715) 0\nin the target expression\n  x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x\n\nX : Type u\nx : X\ns : Set X\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x", "returncode": 1}

Corrected proof: