Response:
by
  rcases Module.equiv_directSum_of_isTorsion hM with ⟨ι, _, p, h, e, ⟨l⟩⟩
  haveI : ∀ i : ι, NeZero (p i ^ e i).natAbs := fun i =>
    ⟨Int.natAbs_ne_zero.mpr <| pow_ne_zero (e i) (h i).ne_zero⟩
  haveI : ∀ i : ι, _root_.Finite <| ℤ ⧸ Submodule.span ℤ {p i ^ e i} := fun i =>
    Finite.of_equiv _ (p i ^ e i).quotientSpanEquivZMod.symm.toEquiv
  haveI : _root_.Finite (⨁ i, ℤ ⧸ (Submodule.span ℤ {p i ^ e i} : Submodule ℤ ℤ)) :=
    Finite.of_equiv _ DFinsupp.equivFunOnFintype.symm
  exact Finite.of_equiv _ l.symm.toEquiv

[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.PID
public import Mathlib.Algebra.Group.TypeTags.Finite
public import Mathlib.Data.ZMod.QuotientRing

Namespace:
Module

Local context:
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
/-!
# Structure of finite(ly generated) abelian groups

* `AddCommGroup.equiv_free_prod_directSum_zmod` : Any finitely generated abelian group is the
  product of a power of `ℤ` and a direct sum of some `ZMod (p i ^ e i)` for some prime powers
  `p i ^ e i`.
* `CommGroup.equiv_free_prod_prod_multiplicative_zmod` is a version for multiplicative groups.
* `AddCommGroup.equiv_directSum_zmod_of_finite` : Any finite abelian group is a direct sum of
  some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`.
* `CommGroup.equiv_prod_multiplicative_zmod_of_finite` is a version for multiplicative groups.
-/

@[expose] public section

open scoped DirectSum

/-
TODO: Here's a more general approach to dropping trivial factors from a direct sum:

def DirectSum.congr {ι κ : Type*} {α : ι → Type*} {β : κ → Type*} [DecidableEq ι] [DecidableEq κ]
    [∀ i, DecidableEq (α i)] [∀ j, DecidableEq (β j)] [∀ i, AddCommMonoid (α i)]
    [∀ j, AddCommMonoid (β j)] (f : ∀ i, Nontrivial (α i) → κ) (g : ∀ j, Nontrivial (β j) → ι)
    (F : ∀ i hi, α i →+ β (f i hi)) (G : ∀ j hj, β j →+ α (g j hj))
    (hfg : ∀ i hi hj, g (f i hi) hj = i) (hgf : ∀ j hj hi, f (g j hj) hi = j)
    (hFG : ∀ i hi hj a, hfg i hi hj ▸ G _ hj (F i hi a) = a)
    (hGF : ∀ j hj hi b, hgf j hj hi ▸ F _ hi (G j hj b) = b) :
    (⨁ i, α i) ≃+ ⨁ j, β j where
  toFun x := x.sum fun i a ↦ if ha : a = 0 then 0 else DFinsupp.single (f i ⟨a, 0, ha⟩) (F _ _ a)
  invFun y := y.sum fun j b ↦ if hb : b = 0 then 0 else DFinsupp.single (g j ⟨b, 0, hb⟩) (G _ _ b)
  -- The two sorries here are probably doable with the existing machinery, but quite painful
  left_inv x := DFinsupp.ext fun i ↦ sorry
  right_inv y := DFinsupp.ext fun j ↦ sorry
  map_add' x₁ x₂ := by
    dsimp
    refine DFinsupp.sum_add_index (by simp) fun i a₁ a₂ ↦ ?_
    split_ifs
    any_goals simp_all
    rw [← DFinsupp.single_add, ← map_add, ‹a₁ + a₂ = 0›, map_zero, DFinsupp.single_zero]

private def directSumNeZeroMulEquiv (ι : Type) [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) ≃+ ⨁ i, ZMod (p i ^ n i) :=
  DirectSum.congr
    (fun i _ ↦ i)
    (fun j hj ↦ ⟨j, fun h ↦ by simp [h, pow_zero, zmod_nontrivial] at hj⟩)
    (fun i _ ↦ AddMonoidHom.id _)
    (fun j _ ↦ AddMonoidHom.id _)
    (fun i hi hj ↦ rfl)
    (fun j hj hi ↦ rfl)
    (fun i hi hj a ↦ rfl)
    (fun j hj hi a ↦ rfl)
-/

private def directSumNeZeroMulHom {ι : Type} [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) →+ ⨁ i, ZMod (p i ^ n i) :=
  DirectSum.toAddMonoid fun i ↦ DirectSum.of (fun i ↦ ZMod (p i ^ n i)) i

private def directSumNeZeroMulEquiv (ι : Type) [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) ≃+ ⨁ i, ZMod (p i ^ n i) where
  toFun := directSumNeZeroMulHom p n
  invFun := DirectSum.toAddMonoid fun i ↦
    if h : n i = 0 then 0 else DirectSum.of (fun j : {i // n i ≠ 0} ↦ ZMod (p j ^ n j)) ⟨i, h⟩
  left_inv x := by
    induction x using DirectSum.induction_on with
    | zero => simp
    | of i x =>
      rw [directSumNeZeroMulHom, DirectSum.toAddMonoid_of, DirectSum.toAddMonoid_of,
        dif_neg i.prop]
    | add x y hx hy => rw [map_add, map_add, hx, hy]
  right_inv x := by
    induction x using DirectSum.induction_on with
    | zero => rw [map_zero, map_zero]
    | of i x =>
      rw [DirectSum.toAddMonoid_of]
      split_ifs with h
      · simp [(ZMod.subsingleton_iff.2 <| by rw [h, pow_zero]).elim x 0]
      · simp_rw [directSumNeZeroMulHom, DirectSum.toAddMonoid_of]
    | add x y hx hy => rw [map_add, map_add, hx, hy]
  map_add' := map_add (directSumNeZeroMulHom p n)

universe u

namespace Module

variable (M : Type u)

Target:
theorem finite_of_fg_torsion [AddCommGroup M] [Module ℤ M] [Module.Finite ℤ M]
    (hM : Module.IsTorsion ℤ M) : _root_.Finite M :=

Proof body:

[Assistant]
by
  rcases Module.equiv_directSum_of_isTorsion hM with ⟨ι, _, p, h, e, ⟨l⟩⟩
  haveI : ∀ i : ι, NeZero (p i ^ e i).natAbs := fun i =>
    ⟨Int.natAbs_ne_zero.mpr <| pow_ne_zero (e i) (h i).ne_zero⟩
  haveI : ∀ i : ι, _root_.Finite <| ℤ ⧸ Submodule.span ℤ {p i ^ e i} := fun i =>
    Finite.of_equiv _ (p i ^ e i).quotientSpanEquivZMod.symm.toEquiv
  haveI : _root_.Finite (⨁ i, ℤ ⧸ (Submodule.span ℤ {p i ^ e i} : Submodule ℤ ℤ)) :=
    Finite.of_equiv _ DFinsupp.equivFunOnFintype.symm
  exact Finite.of_equiv _ l.symm.toEquiv

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.PID
public import Mathlib.Algebra.Group.TypeTags.Finite
public import Mathlib.Data.ZMod.QuotientRing

Namespace:
Module

Local context:
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
/-!
# Structure of finite(ly generated) abelian groups

* `AddCommGroup.equiv_free_prod_directSum_zmod` : Any finitely generated abelian group is the
  product of a power of `ℤ` and a direct sum of some `ZMod (p i ^ e i)` for some prime powers
  `p i ^ e i`.
* `CommGroup.equiv_free_prod_prod_multiplicative_zmod` is a version for multiplicative groups.
* `AddCommGroup.equiv_directSum_zmod_of_finite` : Any finite abelian group is a direct sum of
  some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`.
* `CommGroup.equiv_prod_multiplicative_zmod_of_finite` is a version for multiplicative groups.
-/

@[expose] public section

open scoped DirectSum

/-
TODO: Here's a more general approach to dropping trivial factors from a direct sum:

def DirectSum.congr {ι κ : Type*} {α : ι → Type*} {β : κ → Type*} [DecidableEq ι] [DecidableEq κ]
    [∀ i, DecidableEq (α i)] [∀ j, DecidableEq (β j)] [∀ i, AddCommMonoid (α i)]
    [∀ j, AddCommMonoid (β j)] (f : ∀ i, Nontrivial (α i) → κ) (g : ∀ j, Nontrivial (β j) → ι)
    (F : ∀ i hi, α i →+ β (f i hi)) (G : ∀ j hj, β j →+ α (g j hj))
    (hfg : ∀ i hi hj, g (f i hi) hj = i) (hgf : ∀ j hj hi, f (g j hj) hi = j)
    (hFG : ∀ i hi hj a, hfg i hi hj ▸ G _ hj (F i hi a) = a)
    (hGF : ∀ j hj hi b, hgf j hj hi ▸ F _ hi (G j hj b) = b) :
    (⨁ i, α i) ≃+ ⨁ j, β j where
  toFun x := x.sum fun i a ↦ if ha : a = 0 then 0 else DFinsupp.single (f i ⟨a, 0, ha⟩) (F _ _ a)
  invFun y := y.sum fun j b ↦ if hb : b = 0 then 0 else DFinsupp.single (g j ⟨b, 0, hb⟩) (G _ _ b)
  -- The two sorries here are probably doable with the existing machinery, but quite painful
  left_inv x := DFinsupp.ext fun i ↦ sorry
  right_inv y := DFinsupp.ext fun j ↦ sorry
  map_add' x₁ x₂ := by
    dsimp
    refine DFinsupp.sum_add_index (by simp) fun i a₁ a₂ ↦ ?_
    split_ifs
    any_goals simp_all
    rw [← DFinsupp.single_add, ← map_add, ‹a₁ + a₂ = 0›, map_zero, DFinsupp.single_zero]

private def directSumNeZeroMulEquiv (ι : Type) [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) ≃+ ⨁ i, ZMod (p i ^ n i) :=
  DirectSum.congr
    (fun i _ ↦ i)
    (fun j hj ↦ ⟨j, fun h ↦ by simp [h, pow_zero, zmod_nontrivial] at hj⟩)
    (fun i _ ↦ AddMonoidHom.id _)
    (fun j _ ↦ AddMonoidHom.id _)
    (fun i hi hj ↦ rfl)
    (fun j hj hi ↦ rfl)
    (fun i hi hj a ↦ rfl)
    (fun j hj hi a ↦ rfl)
-/

private def directSumNeZeroMulHom {ι : Type} [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) →+ ⨁ i, ZMod (p i ^ n i) :=
  DirectSum.toAddMonoid fun i ↦ DirectSum.of (fun i ↦ ZMod (p i ^ n i)) i

private def directSumNeZeroMulEquiv (ι : Type) [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
    (⨁ i : {i // n i ≠ 0}, ZMod (p i ^ n i)) ≃+ ⨁ i, ZMod (p i ^ n i) where
  toFun := directSumNeZeroMulHom p n
  invFun := DirectSum.toAddMonoid fun i ↦
    if h : n i = 0 then 0 else DirectSum.of (fun j : {i // n i ≠ 0} ↦ ZMod (p j ^ n j)) ⟨i, h⟩
  left_inv x := by
    induction x using DirectSum.induction_on with
    | zero => simp
    | of i x =>
      rw [directSumNeZeroMulHom, DirectSum.toAddMonoid_of, DirectSum.toAddMonoid_of,
        dif_neg i.prop]
    | add x y hx hy => rw [map_add, map_add, hx, hy]
  right_inv x := by
    induction x using DirectSum.induction_on with
    | zero => rw [map_zero, map_zero]
    | of i x =>
      rw [DirectSum.toAddMonoid_of]
      split_ifs with h
      · simp [(ZMod.subsingleton_iff.2 <| by rw [h, pow_zero]).elim x 0]
      · simp_rw [directSumNeZeroMulHom, DirectSum.toAddMonoid_of]
    | add x y hx hy => rw [map_add, map_add, hx, hy]
  map_add' := map_add (directSumNeZeroMulHom p n)

universe u

namespace Module

variable (M : Type u)

Target:
theorem finite_of_fg_torsion [AddCommGroup M] [Module ℤ M] [Module.Finite ℤ M]
    (hM : Module.IsTorsion ℤ M) : _root_.Finite M :=

Proof body:


<LEAN_FILE>:47:43: 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
⊢ False

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_plan_5614f9e0d5 : False := by
    skip
  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": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:47:43: 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⊢ False", "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_plan_5614f9e0d5 : False := by
    skip
  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": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:47:43: 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⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:106:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 11508828) 0
in the target expression
  Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)

α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
x : Option (α × β)
⊢ Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)

Response:
by grind

/-!
### Algebraic replacement rules

A collection of lemmas to transfer associativity, commutativity, distributivity, ... of operations
to the associativity, commutativity, distributivity, ... of `Option.map₂` of those operations.
The proof pattern is `map₂_lemma operation_lemma`. For example, `map₂_comm mul_comm` proves that
`map₂ (*) a b = map₂ (*) g f` in a `CommSemigroup`.
-/

variable {α' β' δ' ε ε' : Type*}

[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.Init

Namespace:
Option

Local safe context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Binary map of options

This file defines the binary map of `Option`. This is mostly useful to define pointwise operations
on intervals.

## Main declarations

* `Option.map₂`: Binary map of options.

## Notes

This file is very similar to the n-ary section of `Mathlib/Data/Set/Basic.lean`, to
`Mathlib/Data/Finset/NAry.lean` and to `Mathlib/Order/Filter/NAry.lean`. Please keep them in sync.

We do not define `Option.map₃` as its only purpose so far would be to prove properties of
`Option.map₂` and casing already fulfills this task.
-/

@[expose] public section

universe u

open Function

namespace Option

-- Allow `grind` to case split on `Option` in this file.
attribute [local grind cases] Option

variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}

/-- The image of a binary function `f : α → β → γ` as a function `Option α → Option β → Option γ`.
Mathematically this should be thought of as the image of the corresponding function `α × β → γ`. -/
def map₂ (f : α → β → γ) (a : Option α) (b : Option β) : Option γ :=
  a.bind fun a => b.map <| f a

/-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
    map₂ f a b = f <$> a <*> b := by
  cases a <;> rfl

@[simp, grind =]
theorem map₂_some_some (f : α → β → γ) (a : α) (b : β) : map₂ f (some a) (some b) = f a b := rfl

theorem map₂_coe_coe (f : α → β → γ) (a : α) (b : β) : map₂ f a b = f a b := rfl

@[simp, grind =]
theorem map₂_none_left (f : α → β → γ) (b : Option β) : map₂ f none b = none := rfl

@[simp, grind =]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by cases a <;> rfl

@[simp]
theorem map₂_coe_left (f : α → β → γ) (a : α) (b : Option β) : map₂ f a b = b.map fun b => f a b :=
  rfl

@[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) :
    map₂ f a b = a.map fun a => f a b := by grind

theorem mem_map₂_iff {c : γ} : c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c := by
  grind

/-- `simp`-normal form of `mem_map₂_iff`. -/
@[simp]
theorem map₂_eq_some_iff {c : γ} :
    map₂ f a b = some c ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c := by
  grind

@[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none := by
  grind

theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
    map₂ f a b = map₂ (fun a b => f b a) b a := by grind

theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
    (map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by grind

theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
    map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by grind

theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
    map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by grind

@[simp]
theorem map₂_curry (f : α × β → γ) (a : Option α) (b : Option β) :
    map₂ (curry f) a b = Option.map f (map₂ Prod.mk a b) := by grind

@[simp]

Target:
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
    x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 11508828) 0) = ((fun x : Nat => x + 11508828) 0) by rfl]
 grind

/-!
### Algebraic replacement rules

A collection of lemmas to transfer associativity, commutativity, distributivity, ... of operations
to the associativity, commutativity, distributivity, ... of `Option.map₂` of those operations.
The proof pattern is `map₂_lemma operation_lemma`. For example, `map₂_comm mul_comm` proves that
`map₂ (*) a b = map₂ (*) g f` in a `CommSemigroup`.
-/

variable {α' β' δ' ε ε' : Type*}

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:106:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11508828) 0\nin the target expression\n  Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β → γ\nx : Option (α × β)\n⊢ Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)", "returncode": 1}

Corrected proof:

[Assistant]
by grind

/-!
### Algebraic replacement rules

A collection of lemmas to transfer associativity, commutativity, distributivity, ... of operations
to the associativity, commutativity, distributivity, ... of `Option.map₂` of those operations.
The proof pattern is `map₂_lemma operation_lemma`. For example, `map₂_comm mul_comm` proves that
`map₂ (*) a b = map₂ (*) g f` in a `CommSemigroup`.
-/

variable {α' β' δ' ε ε' : Type*}

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.Init

Namespace:
Option

Local safe context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Binary map of options

This file defines the binary map of `Option`. This is mostly useful to define pointwise operations
on intervals.

## Main declarations

* `Option.map₂`: Binary map of options.

## Notes

This file is very similar to the n-ary section of `Mathlib/Data/Set/Basic.lean`, to
`Mathlib/Data/Finset/NAry.lean` and to `Mathlib/Order/Filter/NAry.lean`. Please keep them in sync.

We do not define `Option.map₃` as its only purpose so far would be to prove properties of
`Option.map₂` and casing already fulfills this task.
-/

@[expose] public section

universe u

open Function

namespace Option

-- Allow `grind` to case split on `Option` in this file.
attribute [local grind cases] Option

variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}

/-- The image of a binary function `f : α → β → γ` as a function `Option α → Option β → Option γ`.
Mathematically this should be thought of as the image of the corresponding function `α × β → γ`. -/
def map₂ (f : α → β → γ) (a : Option α) (b : Option β) : Option γ :=
  a.bind fun a => b.map <| f a

/-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
    map₂ f a b = f <$> a <*> b := by
  cases a <;> rfl

@[simp, grind =]
theorem map₂_some_some (f : α → β → γ) (a : α) (b : β) : map₂ f (some a) (some b) = f a b := rfl

theorem map₂_coe_coe (f : α → β → γ) (a : α) (b : β) : map₂ f a b = f a b := rfl

@[simp, grind =]
theorem map₂_none_left (f : α → β → γ) (b : Option β) : map₂ f none b = none := rfl

@[simp, grind =]
theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by cases a <;> rfl

@[simp]
theorem map₂_coe_left (f : α → β → γ) (a : α) (b : Option β) : map₂ f a b = b.map fun b => f a b :=
  rfl

@[simp]
theorem map₂_coe_right (f : α → β → γ) (a : Option α) (b : β) :
    map₂ f a b = a.map fun a => f a b := by grind

theorem mem_map₂_iff {c : γ} : c ∈ map₂ f a b ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c := by
  grind

/-- `simp`-normal form of `mem_map₂_iff`. -/
@[simp]
theorem map₂_eq_some_iff {c : γ} :
    map₂ f a b = some c ↔ ∃ a' b', a' ∈ a ∧ b' ∈ b ∧ f a' b' = c := by
  grind

@[simp]
theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none := by
  grind

theorem map₂_swap (f : α → β → γ) (a : Option α) (b : Option β) :
    map₂ f a b = map₂ (fun a b => f b a) b a := by grind

theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
    (map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by grind

theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
    map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by grind

theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
    map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by grind

@[simp]
theorem map₂_curry (f : α × β → γ) (a : Option α) (b : Option β) :
    map₂ (curry f) a b = Option.map f (map₂ Prod.mk a b) := by grind

@[simp]

Target:
theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
    x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 11508828) 0) = ((fun x : Nat => x + 11508828) 0) by rfl]
 grind

/-!
### Algebraic replacement rules

A collection of lemmas to transfer associativity, commutativity, distributivity, ... of operations
to the associativity, commutativity, distributivity, ... of `Option.map₂` of those operations.
The proof pattern is `map₂_lemma operation_lemma`. For example, `map₂_comm mul_comm` proves that
`map₂ (*) a b = map₂ (*) g f` in a `CommSemigroup`.
-/

variable {α' β' δ' ε ε' : Type*}

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:106:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11508828) 0\nin the target expression\n  Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β → γ\nx : Option (α × β)\n⊢ Option.map (uncurry f) x = map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:128:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 14270495) 0
in the target expression
  (π.biUnionTagged πi).tag J' = (πi J).tag J'

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
  rw [show ((fun x : Nat => x + 14270495) 0) = ((fun x : Nat => x + 14270495) 0) by rfl]
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:128:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 14270495) 0\nin the target expression\n  (π.biUnionTagged πi).tag J' = (πi J).tag J'\n\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
  rw [show ((fun x : Nat => x + 14270495) 0) = ((fun x : Nat => x + 14270495) 0) by rfl]
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:128:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 14270495) 0\nin the target expression\n  (π.biUnionTagged πi).tag J' = (πi J).tag J'\n\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>:193:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingcdc2ff828c.value`
<LEAN_FILE>:192:0: error: unsolved goals
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)

Response:
by
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

[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.DividedPowers.DPMorphism
public import Mathlib.RingTheory.Ideal.Quotient.Operations

Namespace:
DividedPowers

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

Let `A` be a commutative (semi)ring and let `I` be an ideal of `A` with a divided power
structure `hI`. A subideal `J` of `I` is a *sub-dp-ideal* of `(I, hI)` if, for all `n ∈ ℕ > 0` and
all `x ∈ J`, `hI.dpow n x ∈ J`.

## Main definitions

* `DividedPowers.IsSubDPIdeal` : A sub-ideal `J` of a divided power ideal `(I, hI)` is a
  *sub-dp-ideal* if for all `n > 0` and all `x ∈ J`, `hI.dpow n j ∈ J`.
* `DividedPowers.SubDPIdeal` : A bundled version of `IsSubDPIdeal`.
* `DividedPowers.IsSubDPIdeal.dividedPowers`: the divided power structure on a sub-dp-ideal.
* `DividedPowers.IsSubDPIdeal.prod` : if `J` is an `A`-ideal, then `I ⬝ J` is a sub-dp-ideal of `I`.
* `DividedPowers.IsSubDPIdeal.span` : the sub-dp-ideal of `I` generated by a set of elements of `A`.
* `DividedPowers.subDPIdeal_inf_of_quot` : if there is a dp-structure on `I⬝(A/J)` such that the
  quotient map is a dp-morphism, then `J ⊓ I` is a sub-dp-ideal of `I`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers`: when `f : A → B` is a surjective map and
  `f.ker ⊓ I` is a sub-dp-ideal of `I`, this is the induced divided power structure on the ideal
  `I.map f` of the target.
* `DividedPowers.Quotient.dividedPowers` : when `I ⊓ J` is a sub-dp-ideal of `I`, this is the
  divided power structure on the ideal `I(A⧸J)` of the quotient.

## Main results

* `DividedPowers.isSubDPIdeal_inf_iff` : the ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if
  (on `I`) the divided powers are compatible mod `J`.
* `DividedPowers.span_isSubDPIdeal_iff` : the span of a set `S : Set A` is a sub-dp-ideal of `I`
  if and only if for all `n ∈ ℕ > 0` and all `s ∈ S`, hI.dpow n s ∈ span S.
* `DividedPowers.isSubDPIdeal_ker` : the kernel of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `I`.
* `DividedPowers.isSubDPIdeal_map` : the image of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `J`.
* `DividedPowers.SubDPIdeal.instCompleteLattice` : sub-dp-ideals of `I` form a complete lattice
  under inclusion.
* `DividedPowers.SubDPIdeal.span_carrier_eq_dpow_span` : the underlying ideal of
  `SubDPIdeal.span hI S` is generated by the elements of the form `hI.dpow n x` with `n > 0`
  and `x ∈ S`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers_unique` : the only divided power structure on
  `I.map f` such that the surjective map `f : A → B` is a divided power morphism is given by
  `DividedPowers.Quotient.OfSurjective.dividedPowers`.
* `DividedPowers.Quotient.dividedPowers_unique` : the only divided power structure on `I(A⧸J)` such
  that the quotient map `A → A/J` is a divided power morphism is given by
  `DividedPowers.Quotient.dividedPowers`.

## Implementation remarks

We provide both a bundled and an unbundled definition of sub-dp-ideals. The unbundled version is
often more convenient when a larger proof requires to show that a certain ideal is a sub-dp-ideal.
On the other hand, a bundled version is required to prove that sub-dp-ideals form a complete
lattice.

## References

* [P. Berthelot, *Cohomologie cristalline des schémas de caractéristique $p$ > 0*][Berthelot-1974]

* [P. Berthelot and A. Ogus, *Notes on crystalline cohomology*][BerthelotOgus-1978]

* [N. Roby, *Lois polynomes et lois formelles en théorie des modules*][Roby-1963]

* [N. Roby, *Les algèbres à puissances dividées*][Roby-1965]
-/

@[expose] public section

open Subtype

namespace DividedPowers

/-- A sub-ideal `J` of a divided power ideal `(I, hI)` is a sub-dp-ideal if for all `n > 0` and
  all `x ∈ J`, `hI.dpow n j ∈ J`. -/
structure IsSubDPIdeal {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)
    (J : Ideal A) : Prop where
  isSubideal : J ≤ I
  dpow_mem : ∀ (n : ℕ) (_ : n ≠ 0) {j : A} (_ : j ∈ J), hI.dpow n j ∈ J

section IsSubDPIdeal

namespace IsSubDPIdeal

variable {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)

open Ideal

theorem self : IsSubDPIdeal hI I where
  isSubideal := le_rfl
  dpow_mem _ hn _ ha := hI.dpow_mem hn ha

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The divided power structure on a sub-dp-ideal. -/
def dividedPowers {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)] :
    DividedPowers J where
  dpow n x        := if x ∈ J then hI.dpow n x else 0
  dpow_null hx    := by simp [if_neg hx]
  dpow_zero hx    := by simp [if_pos hx, hI.dpow_zero (hJ.isSubideal hx)]
  dpow_one hx     := by simp [if_pos hx, hI.dpow_one (hJ.isSubideal hx)]
  dpow_mem hn hx  := by simp [if_pos hx, hJ.dpow_mem _ hn hx]
  dpow_add hx hy  := by simp_rw [if_pos hx, if_pos hy, if_pos (Ideal.add_mem J hx hy),
    hI.dpow_add (hJ.isSubideal hx) (hJ.isSubideal hy)]
  dpow_mul hx     := by
    simp [if_pos hx, if_pos (mul_mem_left J _ hx), hI.dpow_mul (hJ.isSubideal hx)]
  mul_dpow hx     := by simp [if_pos hx, hI.mul_dpow (hJ.isSubideal hx)]
  dpow_comp hn hx := by
    simp [if_pos hx, if_pos (hJ.dpow_mem _ hn hx), hI.dpow_comp hn (hJ.isSubideal hx)]

variable {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)]

lemma dpow_eq (n : ℕ) (a : A) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = if a ∈ J then hI.dpow n a else 0 := rfl

lemma dpow_eq_of_mem {n : ℕ} {a : A} (ha : a ∈ J) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = hI.dpow n a := by rw [dpow_eq, if_pos ha]

theorem isDPMorphism (hJ : IsSubDPIdeal hI J) :
    (IsSubDPIdeal.dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A) := by
  simpa only [isDPMorphism_iff, Ideal.map_id, RingHom.id_apply]
    using ⟨hJ.1, fun _ _ _ ha ↦ by rw [dpow_eq_of_mem _ _ ha]⟩

end IsSubDPIdeal

open Finset Ideal

/-- The ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if the divided powers have
  some compatibility mod `J`. (The necessity was proved as a sanity check.) -/
theorem isSubDPIdeal_inf_iff {A : Type*} [CommRing A] {I : Ideal A} (hI : DividedPowers I)
    {J : Ideal A} : IsSubDPIdeal hI (J ⊓ I) ↔
    ∀ {n : ℕ} {a b : A} (_ : a ∈ I) (_ : b ∈ I) (_ : a - b ∈ J), hI.dpow n a - hI.dpow n b ∈ J := by
  refine ⟨fun hIJ n a b ha hb hab ↦ ?_, fun hIJ ↦ ?_⟩
  · have hab' : a - b ∈ I := I.sub_mem ha hb
    rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
      tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
    exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
      (hIJ.dpow_mem _ (ne_of_gt (Nat.sub_pos_of_lt (mem_range.mp hi))) ⟨hab, hab'⟩)))
  · refine ⟨SemilatticeInf.inf_le_right J I, fun {n} hn {a} ha ↦ ⟨?_, hI.dpow_mem hn ha.right⟩⟩
    rw [← sub_zero (hI.dpow n a), ← hI.dpow_eval_zero hn]
    exact hIJ ha.right I.zero_mem (J.sub_mem ha.left J.zero_mem)

variable {A B : Type*} [CommSemiring A] {I : Ideal A} {hI : DividedPowers I} [CommSemiring B]
  {J : Ideal B} {hJ : DividedPowers J}

/-- [P. Berthelot and A. Ogus, *Notes on crystalline cohomology* (Lemma 3.6)][BerthelotOgus-1978] -/
theorem span_isSubDPIdeal_iff {S : Set A} (hS : S ⊆ I) :
    IsSubDPIdeal hI (span S) ↔ ∀ {n : ℕ} (_ : n ≠ 0), ∀ s ∈ S, hI.dpow n s ∈ span S := by
  refine ⟨fun hhI n hn s hs ↦ hhI.dpow_mem n hn (subset_span hs), fun hhI ↦ ?_⟩
  · -- interesting direction
    have hSI := span_le.mpr hS
    apply IsSubDPIdeal.mk hSI
    intro m hm z hz
    induction hz using Submodule.span_induction generalizing m hm with
    | mem x h => exact hhI hm x h
    | zero =>
        rw [hI.dpow_eval_zero hm]
        exact (span S).zero_mem
    | add x y hxI hyI hx hy =>
        rw [hI.dpow_add' (hSI hxI) (hSI hyI)]
        apply Submodule.sum_mem (span S)
        intro m _
        by_cases hm0 : m = 0
        · exact hm0 ▸ mul_mem_left (span S) _ (hy _ hm)
        · exact mul_mem_right _ (span S) (hx _ hm0)
    | smul a x hxI hx =>
        rw [smul_eq_mul, hI.dpow_mul (hSI hxI)]
        exact mul_mem_left (span S) (a ^ m) (hx m hm)

theorem isSubDPIdeal_sup {J K : Ideal A} (hJ : IsSubDPIdeal hI J) (hK : IsSubDPIdeal hI K) :
    IsSubDPIdeal hI (J ⊔ K) := by
  rw [← J.span_eq, ← K.span_eq, ← span_union,
    span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
  intro n hn a ha
  rcases ha with ha | ha
  · exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
  · exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha))

theorem isSubDPIdeal_iSup {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (iSup J) := by
  rw [iSup_eq_span, span_isSubDPIdeal_iff (Set.iUnion_subset_iff.mpr <| fun i ↦ (hJ i).1)]
  simp_rw [Set.mem_iUnion]
  rintro n hn a ⟨i, ha⟩
  exact span_mono (Set.subset_iUnion _ i) (subset_span ((hJ i).2 n hn ha))

Target:
theorem isSubDPIdeal_iInf {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (I ⊓ iInf (fun i ↦ J i)) :=

Failed candidate:
by
  have __repair_cdc2ff828c := OProverMissingcdc2ff828c.value
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:193:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingcdc2ff828c.value`\n<LEAN_FILE>:192:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nJ : ι → Ideal A\nhJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)\n⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)", "returncode": 1}

Corrected proof:

[Assistant]
by
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

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.DividedPowers.DPMorphism
public import Mathlib.RingTheory.Ideal.Quotient.Operations

Namespace:
DividedPowers

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

Let `A` be a commutative (semi)ring and let `I` be an ideal of `A` with a divided power
structure `hI`. A subideal `J` of `I` is a *sub-dp-ideal* of `(I, hI)` if, for all `n ∈ ℕ > 0` and
all `x ∈ J`, `hI.dpow n x ∈ J`.

## Main definitions

* `DividedPowers.IsSubDPIdeal` : A sub-ideal `J` of a divided power ideal `(I, hI)` is a
  *sub-dp-ideal* if for all `n > 0` and all `x ∈ J`, `hI.dpow n j ∈ J`.
* `DividedPowers.SubDPIdeal` : A bundled version of `IsSubDPIdeal`.
* `DividedPowers.IsSubDPIdeal.dividedPowers`: the divided power structure on a sub-dp-ideal.
* `DividedPowers.IsSubDPIdeal.prod` : if `J` is an `A`-ideal, then `I ⬝ J` is a sub-dp-ideal of `I`.
* `DividedPowers.IsSubDPIdeal.span` : the sub-dp-ideal of `I` generated by a set of elements of `A`.
* `DividedPowers.subDPIdeal_inf_of_quot` : if there is a dp-structure on `I⬝(A/J)` such that the
  quotient map is a dp-morphism, then `J ⊓ I` is a sub-dp-ideal of `I`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers`: when `f : A → B` is a surjective map and
  `f.ker ⊓ I` is a sub-dp-ideal of `I`, this is the induced divided power structure on the ideal
  `I.map f` of the target.
* `DividedPowers.Quotient.dividedPowers` : when `I ⊓ J` is a sub-dp-ideal of `I`, this is the
  divided power structure on the ideal `I(A⧸J)` of the quotient.

## Main results

* `DividedPowers.isSubDPIdeal_inf_iff` : the ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if
  (on `I`) the divided powers are compatible mod `J`.
* `DividedPowers.span_isSubDPIdeal_iff` : the span of a set `S : Set A` is a sub-dp-ideal of `I`
  if and only if for all `n ∈ ℕ > 0` and all `s ∈ S`, hI.dpow n s ∈ span S.
* `DividedPowers.isSubDPIdeal_ker` : the kernel of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `I`.
* `DividedPowers.isSubDPIdeal_map` : the image of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `J`.
* `DividedPowers.SubDPIdeal.instCompleteLattice` : sub-dp-ideals of `I` form a complete lattice
  under inclusion.
* `DividedPowers.SubDPIdeal.span_carrier_eq_dpow_span` : the underlying ideal of
  `SubDPIdeal.span hI S` is generated by the elements of the form `hI.dpow n x` with `n > 0`
  and `x ∈ S`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers_unique` : the only divided power structure on
  `I.map f` such that the surjective map `f : A → B` is a divided power morphism is given by
  `DividedPowers.Quotient.OfSurjective.dividedPowers`.
* `DividedPowers.Quotient.dividedPowers_unique` : the only divided power structure on `I(A⧸J)` such
  that the quotient map `A → A/J` is a divided power morphism is given by
  `DividedPowers.Quotient.dividedPowers`.

## Implementation remarks

We provide both a bundled and an unbundled definition of sub-dp-ideals. The unbundled version is
often more convenient when a larger proof requires to show that a certain ideal is a sub-dp-ideal.
On the other hand, a bundled version is required to prove that sub-dp-ideals form a complete
lattice.

## References

* [P. Berthelot, *Cohomologie cristalline des schémas de caractéristique $p$ > 0*][Berthelot-1974]

* [P. Berthelot and A. Ogus, *Notes on crystalline cohomology*][BerthelotOgus-1978]

* [N. Roby, *Lois polynomes et lois formelles en théorie des modules*][Roby-1963]

* [N. Roby, *Les algèbres à puissances dividées*][Roby-1965]
-/

@[expose] public section

open Subtype

namespace DividedPowers

/-- A sub-ideal `J` of a divided power ideal `(I, hI)` is a sub-dp-ideal if for all `n > 0` and
  all `x ∈ J`, `hI.dpow n j ∈ J`. -/
structure IsSubDPIdeal {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)
    (J : Ideal A) : Prop where
  isSubideal : J ≤ I
  dpow_mem : ∀ (n : ℕ) (_ : n ≠ 0) {j : A} (_ : j ∈ J), hI.dpow n j ∈ J

section IsSubDPIdeal

namespace IsSubDPIdeal

variable {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)

open Ideal

theorem self : IsSubDPIdeal hI I where
  isSubideal := le_rfl
  dpow_mem _ hn _ ha := hI.dpow_mem hn ha

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The divided power structure on a sub-dp-ideal. -/
def dividedPowers {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)] :
    DividedPowers J where
  dpow n x        := if x ∈ J then hI.dpow n x else 0
  dpow_null hx    := by simp [if_neg hx]
  dpow_zero hx    := by simp [if_pos hx, hI.dpow_zero (hJ.isSubideal hx)]
  dpow_one hx     := by simp [if_pos hx, hI.dpow_one (hJ.isSubideal hx)]
  dpow_mem hn hx  := by simp [if_pos hx, hJ.dpow_mem _ hn hx]
  dpow_add hx hy  := by simp_rw [if_pos hx, if_pos hy, if_pos (Ideal.add_mem J hx hy),
    hI.dpow_add (hJ.isSubideal hx) (hJ.isSubideal hy)]
  dpow_mul hx     := by
    simp [if_pos hx, if_pos (mul_mem_left J _ hx), hI.dpow_mul (hJ.isSubideal hx)]
  mul_dpow hx     := by simp [if_pos hx, hI.mul_dpow (hJ.isSubideal hx)]
  dpow_comp hn hx := by
    simp [if_pos hx, if_pos (hJ.dpow_mem _ hn hx), hI.dpow_comp hn (hJ.isSubideal hx)]

variable {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)]

lemma dpow_eq (n : ℕ) (a : A) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = if a ∈ J then hI.dpow n a else 0 := rfl

lemma dpow_eq_of_mem {n : ℕ} {a : A} (ha : a ∈ J) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = hI.dpow n a := by rw [dpow_eq, if_pos ha]

theorem isDPMorphism (hJ : IsSubDPIdeal hI J) :
    (IsSubDPIdeal.dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A) := by
  simpa only [isDPMorphism_iff, Ideal.map_id, RingHom.id_apply]
    using ⟨hJ.1, fun _ _ _ ha ↦ by rw [dpow_eq_of_mem _ _ ha]⟩

end IsSubDPIdeal

open Finset Ideal

/-- The ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if the divided powers have
  some compatibility mod `J`. (The necessity was proved as a sanity check.) -/
theorem isSubDPIdeal_inf_iff {A : Type*} [CommRing A] {I : Ideal A} (hI : DividedPowers I)
    {J : Ideal A} : IsSubDPIdeal hI (J ⊓ I) ↔
    ∀ {n : ℕ} {a b : A} (_ : a ∈ I) (_ : b ∈ I) (_ : a - b ∈ J), hI.dpow n a - hI.dpow n b ∈ J := by
  refine ⟨fun hIJ n a b ha hb hab ↦ ?_, fun hIJ ↦ ?_⟩
  · have hab' : a - b ∈ I := I.sub_mem ha hb
    rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
      tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
    exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
      (hIJ.dpow_mem _ (ne_of_gt (Nat.sub_pos_of_lt (mem_range.mp hi))) ⟨hab, hab'⟩)))
  · refine ⟨SemilatticeInf.inf_le_right J I, fun {n} hn {a} ha ↦ ⟨?_, hI.dpow_mem hn ha.right⟩⟩
    rw [← sub_zero (hI.dpow n a), ← hI.dpow_eval_zero hn]
    exact hIJ ha.right I.zero_mem (J.sub_mem ha.left J.zero_mem)

variable {A B : Type*} [CommSemiring A] {I : Ideal A} {hI : DividedPowers I} [CommSemiring B]
  {J : Ideal B} {hJ : DividedPowers J}

/-- [P. Berthelot and A. Ogus, *Notes on crystalline cohomology* (Lemma 3.6)][BerthelotOgus-1978] -/
theorem span_isSubDPIdeal_iff {S : Set A} (hS : S ⊆ I) :
    IsSubDPIdeal hI (span S) ↔ ∀ {n : ℕ} (_ : n ≠ 0), ∀ s ∈ S, hI.dpow n s ∈ span S := by
  refine ⟨fun hhI n hn s hs ↦ hhI.dpow_mem n hn (subset_span hs), fun hhI ↦ ?_⟩
  · -- interesting direction
    have hSI := span_le.mpr hS
    apply IsSubDPIdeal.mk hSI
    intro m hm z hz
    induction hz using Submodule.span_induction generalizing m hm with
    | mem x h => exact hhI hm x h
    | zero =>
        rw [hI.dpow_eval_zero hm]
        exact (span S).zero_mem
    | add x y hxI hyI hx hy =>
        rw [hI.dpow_add' (hSI hxI) (hSI hyI)]
        apply Submodule.sum_mem (span S)
        intro m _
        by_cases hm0 : m = 0
        · exact hm0 ▸ mul_mem_left (span S) _ (hy _ hm)
        · exact mul_mem_right _ (span S) (hx _ hm0)
    | smul a x hxI hx =>
        rw [smul_eq_mul, hI.dpow_mul (hSI hxI)]
        exact mul_mem_left (span S) (a ^ m) (hx m hm)

theorem isSubDPIdeal_sup {J K : Ideal A} (hJ : IsSubDPIdeal hI J) (hK : IsSubDPIdeal hI K) :
    IsSubDPIdeal hI (J ⊔ K) := by
  rw [← J.span_eq, ← K.span_eq, ← span_union,
    span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
  intro n hn a ha
  rcases ha with ha | ha
  · exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
  · exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha))

theorem isSubDPIdeal_iSup {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (iSup J) := by
  rw [iSup_eq_span, span_isSubDPIdeal_iff (Set.iUnion_subset_iff.mpr <| fun i ↦ (hJ i).1)]
  simp_rw [Set.mem_iUnion]
  rintro n hn a ⟨i, ha⟩
  exact span_mono (Set.subset_iUnion _ i) (subset_span ((hJ i).2 n hn ha))

Target:
theorem isSubDPIdeal_iInf {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (I ⊓ iInf (fun i ↦ J i)) :=

Failed candidate:
by
  have __repair_cdc2ff828c := OProverMissingcdc2ff828c.value
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:193:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingcdc2ff828c.value`\n<LEAN_FILE>:192:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nJ : ι → Ideal A\nhJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)\n⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:84:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

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

Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap

Namespace:
Algebra

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

## Main results

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

public section

noncomputable section

open Finset

open Polynomial

namespace Algebra

universe u v w z

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

section aeval

open Algebra

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

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

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

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

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

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

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

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

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

Failed candidate:
by
  have __repair_bbebe886bc : False := True.intro
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:84:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

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

Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap

Namespace:
Algebra

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

## Main results

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

public section

noncomputable section

open Finset

open Polynomial

namespace Algebra

universe u v w z

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

section aeval

open Algebra

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

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

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

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

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

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

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

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

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

Failed candidate:
by
  have __repair_bbebe886bc : False := True.intro
  obtain ⟨p, hp⟩ := Algebra.adjoin_eq_exists_aeval _ x a
  grind

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:84:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:145:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 16538891) 0
in the target expression
  lapply a ∘ₗ lsingle a' = 0

α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
a a' : α
h : a ≠ a'
⊢ lapply a ∘ₗ lsingle a' = 0

Response: by ext; simp [h.symm]

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

Imports:
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.LinearMap.End
public import Mathlib.Algebra.Module.Pi
public import Mathlib.Data.Finsupp.SMul

Namespace:
Finsupp

Local safe context:
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
/-!
# Properties of the module `α →₀ M`

Given an `R`-module `M`, the `R`-module structure on `α →₀ M` is defined in
`Mathlib/Data/Finsupp/SMul.lean`.

In this file we define `LinearMap` versions of various maps:

* `Finsupp.lsingle a : M →ₗ[R] ι →₀ M`: `Finsupp.single a` as a linear map;
* `Finsupp.lapply a : (ι →₀ M) →ₗ[R] M`: the map `fun f ↦ f a` as a linear map;
* `Finsupp.lsubtypeDomain (s : Set α) : (α →₀ M) →ₗ[R] (s →₀ M)`: restriction to a subtype as a
  linear map;
* `Finsupp.restrictDom`: `Finsupp.filter` as a linear map to `Finsupp.supported s`;
* `Finsupp.lmapDomain`: a linear map version of `Finsupp.mapDomain`;

## Tags

function with finite support, module, linear algebra
-/

@[expose] public section

assert_not_exists Submodule

noncomputable section

open Set LinearMap

namespace Finsupp

variable {α : Type*} {M : Type*} {N : Type*} {P : Type*} {R R₂ R₃ : Type*} {S : Type*}
variable [Semiring R] [Semiring R₂] [Semiring R₃] [Semiring S]
variable [AddCommMonoid M] [Module R M]
variable [AddCommMonoid N] [Module R₂ N]
variable [AddCommMonoid P] [Module R₃ P]
variable {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R}
variable {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂}
variable {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R}
variable [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁]

section LinearEquivFunOnFinite

variable (R : Type*) {S : Type*} (M : Type*) (α : Type*)
variable [Finite α] [AddCommMonoid M] [Semiring R] [Module R M]

/-- Given `Finite α`, `linearEquivFunOnFinite R` is the natural `R`-linear equivalence between
`α →₀ β` and `α → β`. -/
@[simps apply]
noncomputable def linearEquivFunOnFinite : (α →₀ M) ≃ₗ[R] α → M :=
  { equivFunOnFinite with
    toFun := (⇑)
    map_add' := fun _ _ => rfl
    map_smul' := fun _ _ => rfl }

@[simp]
theorem linearEquivFunOnFinite_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α) (single x m) = Pi.single x m :=
  equivFunOnFinite_single x m

@[simp]
theorem linearEquivFunOnFinite_symm_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α).symm (Pi.single x m) = single x m :=
  equivFunOnFinite_symm_single x m

@[simp]
theorem linearEquivFunOnFinite_symm_coe (f : α →₀ M) : (linearEquivFunOnFinite R M α).symm f = f :=
  (linearEquivFunOnFinite R M α).symm_apply_apply f

@[simp]
theorem linearEquivFunOnFinite_symm_apply (f : α → M) : (linearEquivFunOnFinite R M α).symm f = f :=
  rfl

end LinearEquivFunOnFinite

/-- Interpret `Finsupp.single a` as a linear map. -/
def lsingle (a : α) : M →ₗ[R] α →₀ M :=
  { Finsupp.singleAddHom a with map_smul' := fun _ _ => (smul_single _ _ _).symm }

/-- Two `R`-linear maps from `Finsupp X M` which agree on each `single x y` agree everywhere. -/
theorem lhom_ext ⦃φ ψ : (α →₀ M) →ₛₗ[σ₁₂] N⦄ (h : ∀ a b, φ (single a b) = ψ (single a b)) : φ = ψ :=
  LinearMap.toAddMonoidHom_injective <| addHom_ext h

/-- Two `R`-linear maps from `Finsupp X M` which agree on each `single x y` agree everywhere.

We formulate this fact using equality of linear maps `φ.comp (lsingle a)` and `ψ.comp (lsingle a)`
so that the `ext` tactic can apply a type-specific extensionality lemma to prove equality of these
maps. E.g., if `M = R`, then it suffices to verify `φ (single a 1) = ψ (single a 1)`. -/
-- The priority should be higher than `LinearMap.ext`.
@[ext high]
theorem lhom_ext' ⦃φ ψ : (α →₀ M) →ₛₗ[σ₁₂] N⦄ (h : ∀ a, φ.comp (lsingle a) = ψ.comp (lsingle a)) :
    φ = ψ :=
  lhom_ext fun a => LinearMap.congr_fun (h a)

/-- Interpret `fun f : α →₀ M ↦ f a` as a linear map. -/
def lapply (a : α) : (α →₀ M) →ₗ[R] M :=
  { Finsupp.applyAddHom a with map_smul' := fun _ _ => rfl }

instance [Nonempty α] [FaithfulSMul R M] : FaithfulSMul R (α →₀ M) :=
  .of_injective (Finsupp.lsingle <| Classical.arbitrary _) (Finsupp.single_injective _)

section LSubtypeDomain

variable (s : Set α)

/-- Interpret `Finsupp.subtypeDomain s` as a linear map. -/
def lsubtypeDomain : (α →₀ M) →ₗ[R] s →₀ M where
  toFun := subtypeDomain fun x => x ∈ s
  map_add' _ _ := subtypeDomain_add
  map_smul' _ _ := ext fun _ => rfl

theorem lsubtypeDomain_apply (f : α →₀ M) :
    (lsubtypeDomain s : (α →₀ M) →ₗ[R] s →₀ M) f = subtypeDomain (fun x => x ∈ s) f :=
  rfl

end LSubtypeDomain

@[simp]
theorem lsingle_apply (a : α) (b : M) : (lsingle a : M →ₗ[R] α →₀ M) b = single a b :=
  rfl

@[simp]
theorem lapply_apply (a : α) (f : α →₀ M) : (lapply a : (α →₀ M) →ₗ[R] M) f = f a :=
  rfl

@[simp]
theorem lapply_comp_lsingle_same (a : α) : lapply a ∘ₗ lsingle a = (.id : M →ₗ[R] M) := by ext; simp

@[simp]

Target:
theorem lapply_comp_lsingle_of_ne (a a' : α) (h : a ≠ a') :
    lapply a ∘ₗ lsingle a' = (0 : M →ₗ[R] M) :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 16538891) 0) = ((fun x : Nat => x + 16538891) 0) by rfl]
 ext; simp [h.symm]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:145:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16538891) 0\nin the target expression\n  lapply a ∘ₗ lsingle a' = 0\n\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na a' : α\nh : a ≠ a'\n⊢ lapply a ∘ₗ lsingle a' = 0", "returncode": 1}

Corrected proof:

[Assistant] by ext; simp [h.symm]

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

Imports:
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.LinearMap.End
public import Mathlib.Algebra.Module.Pi
public import Mathlib.Data.Finsupp.SMul

Namespace:
Finsupp

Local safe context:
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
/-!
# Properties of the module `α →₀ M`

Given an `R`-module `M`, the `R`-module structure on `α →₀ M` is defined in
`Mathlib/Data/Finsupp/SMul.lean`.

In this file we define `LinearMap` versions of various maps:

* `Finsupp.lsingle a : M →ₗ[R] ι →₀ M`: `Finsupp.single a` as a linear map;
* `Finsupp.lapply a : (ι →₀ M) →ₗ[R] M`: the map `fun f ↦ f a` as a linear map;
* `Finsupp.lsubtypeDomain (s : Set α) : (α →₀ M) →ₗ[R] (s →₀ M)`: restriction to a subtype as a
  linear map;
* `Finsupp.restrictDom`: `Finsupp.filter` as a linear map to `Finsupp.supported s`;
* `Finsupp.lmapDomain`: a linear map version of `Finsupp.mapDomain`;

## Tags

function with finite support, module, linear algebra
-/

@[expose] public section

assert_not_exists Submodule

noncomputable section

open Set LinearMap

namespace Finsupp

variable {α : Type*} {M : Type*} {N : Type*} {P : Type*} {R R₂ R₃ : Type*} {S : Type*}
variable [Semiring R] [Semiring R₂] [Semiring R₃] [Semiring S]
variable [AddCommMonoid M] [Module R M]
variable [AddCommMonoid N] [Module R₂ N]
variable [AddCommMonoid P] [Module R₃ P]
variable {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R}
variable {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂}
variable {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R}
variable [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomCompTriple σ₃₂ σ₂₁ σ₃₁]

section LinearEquivFunOnFinite

variable (R : Type*) {S : Type*} (M : Type*) (α : Type*)
variable [Finite α] [AddCommMonoid M] [Semiring R] [Module R M]

/-- Given `Finite α`, `linearEquivFunOnFinite R` is the natural `R`-linear equivalence between
`α →₀ β` and `α → β`. -/
@[simps apply]
noncomputable def linearEquivFunOnFinite : (α →₀ M) ≃ₗ[R] α → M :=
  { equivFunOnFinite with
    toFun := (⇑)
    map_add' := fun _ _ => rfl
    map_smul' := fun _ _ => rfl }

@[simp]
theorem linearEquivFunOnFinite_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α) (single x m) = Pi.single x m :=
  equivFunOnFinite_single x m

@[simp]
theorem linearEquivFunOnFinite_symm_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α).symm (Pi.single x m) = single x m :=
  equivFunOnFinite_symm_single x m

@[simp]
theorem linearEquivFunOnFinite_symm_coe (f : α →₀ M) : (linearEquivFunOnFinite R M α).symm f = f :=
  (linearEquivFunOnFinite R M α).symm_apply_apply f

@[simp]
theorem linearEquivFunOnFinite_symm_apply (f : α → M) : (linearEquivFunOnFinite R M α).symm f = f :=
  rfl

end LinearEquivFunOnFinite

/-- Interpret `Finsupp.single a` as a linear map. -/
def lsingle (a : α) : M →ₗ[R] α →₀ M :=
  { Finsupp.singleAddHom a with map_smul' := fun _ _ => (smul_single _ _ _).symm }

/-- Two `R`-linear maps from `Finsupp X M` which agree on each `single x y` agree everywhere. -/
theorem lhom_ext ⦃φ ψ : (α →₀ M) →ₛₗ[σ₁₂] N⦄ (h : ∀ a b, φ (single a b) = ψ (single a b)) : φ = ψ :=
  LinearMap.toAddMonoidHom_injective <| addHom_ext h

/-- Two `R`-linear maps from `Finsupp X M` which agree on each `single x y` agree everywhere.

We formulate this fact using equality of linear maps `φ.comp (lsingle a)` and `ψ.comp (lsingle a)`
so that the `ext` tactic can apply a type-specific extensionality lemma to prove equality of these
maps. E.g., if `M = R`, then it suffices to verify `φ (single a 1) = ψ (single a 1)`. -/
-- The priority should be higher than `LinearMap.ext`.
@[ext high]
theorem lhom_ext' ⦃φ ψ : (α →₀ M) →ₛₗ[σ₁₂] N⦄ (h : ∀ a, φ.comp (lsingle a) = ψ.comp (lsingle a)) :
    φ = ψ :=
  lhom_ext fun a => LinearMap.congr_fun (h a)

/-- Interpret `fun f : α →₀ M ↦ f a` as a linear map. -/
def lapply (a : α) : (α →₀ M) →ₗ[R] M :=
  { Finsupp.applyAddHom a with map_smul' := fun _ _ => rfl }

instance [Nonempty α] [FaithfulSMul R M] : FaithfulSMul R (α →₀ M) :=
  .of_injective (Finsupp.lsingle <| Classical.arbitrary _) (Finsupp.single_injective _)

section LSubtypeDomain

variable (s : Set α)

/-- Interpret `Finsupp.subtypeDomain s` as a linear map. -/
def lsubtypeDomain : (α →₀ M) →ₗ[R] s →₀ M where
  toFun := subtypeDomain fun x => x ∈ s
  map_add' _ _ := subtypeDomain_add
  map_smul' _ _ := ext fun _ => rfl

theorem lsubtypeDomain_apply (f : α →₀ M) :
    (lsubtypeDomain s : (α →₀ M) →ₗ[R] s →₀ M) f = subtypeDomain (fun x => x ∈ s) f :=
  rfl

end LSubtypeDomain

@[simp]
theorem lsingle_apply (a : α) (b : M) : (lsingle a : M →ₗ[R] α →₀ M) b = single a b :=
  rfl

@[simp]
theorem lapply_apply (a : α) (f : α →₀ M) : (lapply a : (α →₀ M) →ₗ[R] M) f = f a :=
  rfl

@[simp]
theorem lapply_comp_lsingle_same (a : α) : lapply a ∘ₗ lsingle a = (.id : M →ₗ[R] M) := by ext; simp

@[simp]

Target:
theorem lapply_comp_lsingle_of_ne (a a' : α) (h : a ≠ a') :
    lapply a ∘ₗ lsingle a' = (0 : M →ₗ[R] M) :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 16538891) 0) = ((fun x : Nat => x + 16538891) 0) by rfl]
 ext; simp [h.symm]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:145:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16538891) 0\nin the target expression\n  lapply a ∘ₗ lsingle a' = 0\n\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na a' : α\nh : a ≠ a'\n⊢ lapply a ∘ₗ lsingle a' = 0", "returncode": 1}

Corrected proof:


<LEAN_FILE>:127:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 4256790) 0
in the target expression
  Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ
ha : abscissaOfAbsConv f < ⊤
⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

Response:
by
  let F (n : ℕ) : ℂ := if n = 0 then 0 else f n
  have hF₀ : F 0 = 0 := rfl
  have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := if_neg hn
  have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha
  simp_rw [← LSeries_congr hF]
  convert! LSeries.tendsto_cpow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 1
  simp

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

Imports:
public import Mathlib.Analysis.Normed.Group.Tannery
public import Mathlib.NumberTheory.LSeries.Convergence
public import Mathlib.NumberTheory.LSeries.Linearity

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
/-!
# A converging L-series determines its coefficients

We show that two functions `f` and `g : ℕ → ℂ` whose L-series agree and both converge somewhere
must agree on all nonzero arguments. See `LSeries_eq_iff_of_abscissaOfAbsConv_lt_top`
and `LSeries_injOn`.
-/

public section

open LSeries Complex

-- The following two lemmas need both `LSeries.Linearity` and `LSeries.Convergence`,
-- so cannot live in either of these files.

/-- The abscissa of absolute convergence of `f + g` is at most the maximum of those
of `f` and `g`. -/
lemma LSeries.abscissaOfAbsConv_add_le (f g : ℕ → ℂ) :
    abscissaOfAbsConv (f + g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g) :=
  abscissaOfAbsConv_binop_le LSeriesSummable.add f g

/-- The abscissa of absolute convergence of `f - g` is at most the maximum of those
of `f` and `g`. -/
lemma LSeries.abscissaOfAbsConv_sub_le (f g : ℕ → ℂ) :
    abscissaOfAbsConv (f - g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g) :=
  abscissaOfAbsConv_binop_le LSeriesSummable.sub f g

private
lemma cpow_mul_div_cpow_eq_div_div_cpow (m n : ℕ) (z : ℂ) (x : ℝ) :
    (n + 1) ^ (x : ℂ) * (z / m ^ (x : ℂ)) = z / (m / (n + 1)) ^ (x : ℂ) := by
  have Hn : (0 : ℝ) ≤ (n + 1 : ℝ)⁻¹ := by positivity
  rw [← mul_div_assoc, mul_comm, div_eq_mul_inv z, mul_div_assoc]
  congr
  simp_rw [div_eq_mul_inv]
  rw [show (n + 1 : ℂ)⁻¹ = (n + 1 : ℝ)⁻¹ by simp,
    show (n + 1 : ℂ) = (n + 1 : ℝ) by norm_cast, show (m : ℂ) = (m : ℝ) by norm_cast,
    mul_cpow_ofReal_nonneg m.cast_nonneg Hn, mul_inv, mul_comm]
  congr
  rw [← cpow_neg, show (-x : ℂ) = (-1 : ℝ) * x by simp, cpow_mul_ofReal_nonneg Hn,
    Real.rpow_neg_one, inv_inv]

open Filter Real in
/-- If the coefficients `f m` of an L-series are zero for `m ≤ n` and the L-series converges
at some point, then `f (n+1)` is the limit of `(n+1)^x * LSeries f x` as `x → ∞`. -/
lemma LSeries.tendsto_cpow_mul_atTop {f : ℕ → ℂ} {n : ℕ} (h : ∀ m ≤ n, f m = 0)
    (ha : abscissaOfAbsConv f < ⊤) :
    Tendsto (fun x : ℝ ↦ (n + 1) ^ (x : ℂ) * LSeries f x) atTop (nhds (f (n + 1))) := by
  obtain ⟨y, hay, hyt⟩ := exists_between ha
  lift y to ℝ using ⟨hyt.ne, ((OrderBot.bot_le _).trans_lt hay).ne'⟩
  -- `F x m` is the `m`th term of `(n+1)^x * LSeries f x`, except that `F x (n+1) = 0`
  let F := fun (x : ℝ) ↦ {m | n + 1 < m}.indicator (fun m ↦ f m / (m / (n + 1) : ℂ) ^ (x : ℂ))
  have hF₀ (x : ℝ) {m : ℕ} (hm : m ≤ n + 1) : F x m = 0 := by simp [F, not_lt_of_ge hm]
  have hF (x : ℝ) {m : ℕ} (hm : m ≠ n + 1) : F x m = ((n + 1) ^ (x : ℂ)) * term f x m := by
    rcases lt_trichotomy m (n + 1) with H | rfl | H
    · simp [Nat.not_lt_of_gt H, term, h m <| Nat.lt_succ_iff.mp H, F]
    · exact (hm rfl).elim
    · simp [H, term, (n.zero_lt_succ.trans H).ne', F, cpow_mul_div_cpow_eq_div_div_cpow]
  have hs {x : ℝ} (hx : x ≥ y) : Summable fun m ↦ (n + 1) ^ (x : ℂ) * term f x m := by
    refine (summable_mul_left_iff <| natCast_add_one_cpow_ne_zero n _).mpr <|
       LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_
    simpa only [ofReal_re] using hay.trans_le <| EReal.coe_le_coe_iff.mpr hx
  -- we can write `(n+1)^x * LSeries f x` as `f (n+1)` plus the series over `F x`
  have key : ∀ x ≥ y, (n + 1) ^ (x : ℂ) * LSeries f x = f (n + 1) + ∑' m : ℕ, F x m := by
    intro x hx
    rw [LSeries, ← tsum_mul_left, (hs hx).tsum_eq_add_tsum_ite (n + 1), pow_mul_term_eq f x n]
    congr
    ext1 m
    rcases eq_or_ne m (n + 1) with rfl | hm
    · simp [hF₀ x le_rfl]
    · simp [hm, hF]
  -- reduce to showing that `∑' m, F x m → 0` as `x → ∞`
  conv => enter [3, 1]; rw [← add_zero (f _)]
  refine Tendsto.congr'
    (eventuallyEq_of_mem (s := {x | y ≤ x}) (mem_atTop y) key).symm <| tendsto_const_nhds.add ?_
  -- get the prerequisites for applying dominated convergence
  have hys : Summable (F y) := by
    refine ((hs le_rfl).indicator {m | n + 1 < m}).congr fun m ↦ ?_
    by_cases! hm : n + 1 < m
    · simp [hF, hm, hm.ne']
    · simp [hm, hF₀ _ hm]
  have hc (k : ℕ) : Tendsto (F · k) atTop (nhds 0) := by
    rcases lt_or_ge (n + 1) k with H | H
    · have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity
      have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity
      have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by push_cast; rfl
      have H₂ : (n + 1) / k < (1 : ℝ) :=
        (div_lt_one <| mod_cast n.succ_pos.trans H).mpr <| mod_cast H
      simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, F]
      conv =>
        enter [1, x]
        rw [div_eq_mul_inv, H₁, ← ofReal_cpow H₀, ← ofReal_inv, ← Real.inv_rpow H₀, inv_div]
      conv => enter [3, 1]; rw [← mul_zero (f k)]
      exact
        (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le H₀') H₂).ofReal.const_mul _
    · simp [hF₀ _ H]
  rw [show (0 : ℂ) = tsum (fun _ : ℕ ↦ 0) from tsum_zero.symm]
  refine tendsto_tsum_of_dominated_convergence hys.norm hc <| eventually_iff.mpr ?_
  filter_upwards [mem_atTop y] with y' hy' k
  -- it remains to show that `‖F y' k‖ ≤ ‖F y k‖` (for `y' ≥ y`)
  rcases lt_or_ge (n + 1) k with H | H
  · simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, norm_div, norm_cpow_real,
      Complex.norm_natCast, F]
    rw [← Nat.cast_one, ← Nat.cast_add, Complex.norm_natCast]
    have hkn : 1 ≤ (k / (n + 1 :) : ℝ) :=
      (one_le_div (by positivity)).mpr <| mod_cast Nat.le_of_succ_le H
    gcongr
  · simp [hF₀ _ H]

open Filter in
/-- If the L-series of `f` converges at some point, then `f 1` is the limit of `LSeries f x`
as `x → ∞`. -/

Target:
lemma LSeries.tendsto_atTop {f : ℕ → ℂ} (ha : abscissaOfAbsConv f < ⊤) :
    Tendsto (fun x : ℝ ↦ LSeries f x) atTop (nhds (f 1)) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4256790) 0) = ((fun x : Nat => x + 4256790) 0) by rfl]
  let F (n : ℕ) : ℂ := if n = 0 then 0 else f n
  have hF₀ : F 0 = 0 := rfl
  have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := if_neg hn
  have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha
  simp_rw [← LSeries_congr hF]
  convert! LSeries.tendsto_cpow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 1
  simp

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:127:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4256790) 0\nin the target expression\n  Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))\n\nf : ℕ → ℂ\nha : abscissaOfAbsConv f < ⊤\n⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))", "returncode": 1}

Corrected proof:

[Assistant]
by
  let F (n : ℕ) : ℂ := if n = 0 then 0 else f n
  have hF₀ : F 0 = 0 := rfl
  have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := if_neg hn
  have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha
  simp_rw [← LSeries_congr hF]
  convert! LSeries.tendsto_cpow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 1
  simp

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

Imports:
public import Mathlib.Analysis.Normed.Group.Tannery
public import Mathlib.NumberTheory.LSeries.Convergence
public import Mathlib.NumberTheory.LSeries.Linearity

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
/-!
# A converging L-series determines its coefficients

We show that two functions `f` and `g : ℕ → ℂ` whose L-series agree and both converge somewhere
must agree on all nonzero arguments. See `LSeries_eq_iff_of_abscissaOfAbsConv_lt_top`
and `LSeries_injOn`.
-/

public section

open LSeries Complex

-- The following two lemmas need both `LSeries.Linearity` and `LSeries.Convergence`,
-- so cannot live in either of these files.

/-- The abscissa of absolute convergence of `f + g` is at most the maximum of those
of `f` and `g`. -/
lemma LSeries.abscissaOfAbsConv_add_le (f g : ℕ → ℂ) :
    abscissaOfAbsConv (f + g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g) :=
  abscissaOfAbsConv_binop_le LSeriesSummable.add f g

/-- The abscissa of absolute convergence of `f - g` is at most the maximum of those
of `f` and `g`. -/
lemma LSeries.abscissaOfAbsConv_sub_le (f g : ℕ → ℂ) :
    abscissaOfAbsConv (f - g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g) :=
  abscissaOfAbsConv_binop_le LSeriesSummable.sub f g

private
lemma cpow_mul_div_cpow_eq_div_div_cpow (m n : ℕ) (z : ℂ) (x : ℝ) :
    (n + 1) ^ (x : ℂ) * (z / m ^ (x : ℂ)) = z / (m / (n + 1)) ^ (x : ℂ) := by
  have Hn : (0 : ℝ) ≤ (n + 1 : ℝ)⁻¹ := by positivity
  rw [← mul_div_assoc, mul_comm, div_eq_mul_inv z, mul_div_assoc]
  congr
  simp_rw [div_eq_mul_inv]
  rw [show (n + 1 : ℂ)⁻¹ = (n + 1 : ℝ)⁻¹ by simp,
    show (n + 1 : ℂ) = (n + 1 : ℝ) by norm_cast, show (m : ℂ) = (m : ℝ) by norm_cast,
    mul_cpow_ofReal_nonneg m.cast_nonneg Hn, mul_inv, mul_comm]
  congr
  rw [← cpow_neg, show (-x : ℂ) = (-1 : ℝ) * x by simp, cpow_mul_ofReal_nonneg Hn,
    Real.rpow_neg_one, inv_inv]

open Filter Real in
/-- If the coefficients `f m` of an L-series are zero for `m ≤ n` and the L-series converges
at some point, then `f (n+1)` is the limit of `(n+1)^x * LSeries f x` as `x → ∞`. -/
lemma LSeries.tendsto_cpow_mul_atTop {f : ℕ → ℂ} {n : ℕ} (h : ∀ m ≤ n, f m = 0)
    (ha : abscissaOfAbsConv f < ⊤) :
    Tendsto (fun x : ℝ ↦ (n + 1) ^ (x : ℂ) * LSeries f x) atTop (nhds (f (n + 1))) := by
  obtain ⟨y, hay, hyt⟩ := exists_between ha
  lift y to ℝ using ⟨hyt.ne, ((OrderBot.bot_le _).trans_lt hay).ne'⟩
  -- `F x m` is the `m`th term of `(n+1)^x * LSeries f x`, except that `F x (n+1) = 0`
  let F := fun (x : ℝ) ↦ {m | n + 1 < m}.indicator (fun m ↦ f m / (m / (n + 1) : ℂ) ^ (x : ℂ))
  have hF₀ (x : ℝ) {m : ℕ} (hm : m ≤ n + 1) : F x m = 0 := by simp [F, not_lt_of_ge hm]
  have hF (x : ℝ) {m : ℕ} (hm : m ≠ n + 1) : F x m = ((n + 1) ^ (x : ℂ)) * term f x m := by
    rcases lt_trichotomy m (n + 1) with H | rfl | H
    · simp [Nat.not_lt_of_gt H, term, h m <| Nat.lt_succ_iff.mp H, F]
    · exact (hm rfl).elim
    · simp [H, term, (n.zero_lt_succ.trans H).ne', F, cpow_mul_div_cpow_eq_div_div_cpow]
  have hs {x : ℝ} (hx : x ≥ y) : Summable fun m ↦ (n + 1) ^ (x : ℂ) * term f x m := by
    refine (summable_mul_left_iff <| natCast_add_one_cpow_ne_zero n _).mpr <|
       LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_
    simpa only [ofReal_re] using hay.trans_le <| EReal.coe_le_coe_iff.mpr hx
  -- we can write `(n+1)^x * LSeries f x` as `f (n+1)` plus the series over `F x`
  have key : ∀ x ≥ y, (n + 1) ^ (x : ℂ) * LSeries f x = f (n + 1) + ∑' m : ℕ, F x m := by
    intro x hx
    rw [LSeries, ← tsum_mul_left, (hs hx).tsum_eq_add_tsum_ite (n + 1), pow_mul_term_eq f x n]
    congr
    ext1 m
    rcases eq_or_ne m (n + 1) with rfl | hm
    · simp [hF₀ x le_rfl]
    · simp [hm, hF]
  -- reduce to showing that `∑' m, F x m → 0` as `x → ∞`
  conv => enter [3, 1]; rw [← add_zero (f _)]
  refine Tendsto.congr'
    (eventuallyEq_of_mem (s := {x | y ≤ x}) (mem_atTop y) key).symm <| tendsto_const_nhds.add ?_
  -- get the prerequisites for applying dominated convergence
  have hys : Summable (F y) := by
    refine ((hs le_rfl).indicator {m | n + 1 < m}).congr fun m ↦ ?_
    by_cases! hm : n + 1 < m
    · simp [hF, hm, hm.ne']
    · simp [hm, hF₀ _ hm]
  have hc (k : ℕ) : Tendsto (F · k) atTop (nhds 0) := by
    rcases lt_or_ge (n + 1) k with H | H
    · have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity
      have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity
      have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by push_cast; rfl
      have H₂ : (n + 1) / k < (1 : ℝ) :=
        (div_lt_one <| mod_cast n.succ_pos.trans H).mpr <| mod_cast H
      simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, F]
      conv =>
        enter [1, x]
        rw [div_eq_mul_inv, H₁, ← ofReal_cpow H₀, ← ofReal_inv, ← Real.inv_rpow H₀, inv_div]
      conv => enter [3, 1]; rw [← mul_zero (f k)]
      exact
        (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le H₀') H₂).ofReal.const_mul _
    · simp [hF₀ _ H]
  rw [show (0 : ℂ) = tsum (fun _ : ℕ ↦ 0) from tsum_zero.symm]
  refine tendsto_tsum_of_dominated_convergence hys.norm hc <| eventually_iff.mpr ?_
  filter_upwards [mem_atTop y] with y' hy' k
  -- it remains to show that `‖F y' k‖ ≤ ‖F y k‖` (for `y' ≥ y`)
  rcases lt_or_ge (n + 1) k with H | H
  · simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, norm_div, norm_cpow_real,
      Complex.norm_natCast, F]
    rw [← Nat.cast_one, ← Nat.cast_add, Complex.norm_natCast]
    have hkn : 1 ≤ (k / (n + 1 :) : ℝ) :=
      (one_le_div (by positivity)).mpr <| mod_cast Nat.le_of_succ_le H
    gcongr
  · simp [hF₀ _ H]

open Filter in
/-- If the L-series of `f` converges at some point, then `f 1` is the limit of `LSeries f x`
as `x → ∞`. -/

Target:
lemma LSeries.tendsto_atTop {f : ℕ → ℂ} (ha : abscissaOfAbsConv f < ⊤) :
    Tendsto (fun x : ℝ ↦ LSeries f x) atTop (nhds (f 1)) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4256790) 0) = ((fun x : Nat => x + 4256790) 0) by rfl]
  let F (n : ℕ) : ℂ := if n = 0 then 0 else f n
  have hF₀ : F 0 = 0 := rfl
  have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := if_neg hn
  have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha
  simp_rw [← LSeries_congr hF]
  convert! LSeries.tendsto_cpow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 1
  simp

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:127:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4256790) 0\nin the target expression\n  Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))\n\nf : ℕ → ℂ\nha : abscissaOfAbsConv f < ⊤\n⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))", "returncode": 1}

Corrected proof: