<LEAN_FILE>:66:43: error: unsolved goals
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M₁
inst✝⁶ : Module R M₁
inst✝⁵ : Module.Finite R M₁
inst✝⁴ : Free R M₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Free R M₂
f₁ : M₁ →ₗ[R] M₁
f₂ : M₂ →ₗ[R] M₂
⊢ False

Response:
by
  let b₁ := chooseBasis R M₁
  let b₂ := chooseBasis R M₂
  let b := b₁.prod b₂
  rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
    toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂]

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

Imports:
public import Mathlib.LinearAlgebra.Charpoly.Basic
public import Mathlib.LinearAlgebra.Matrix.Basis
public import Mathlib.RingTheory.Finiteness.Prod

Namespace:
LinearMap

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

# Characteristic polynomial

## Main result

* `LinearMap.charpoly_toMatrix f` : `charpoly f` is the characteristic polynomial of the matrix
  of `f` in any basis.

-/

public section

noncomputable section

open Module Free Polynomial Matrix

universe u v w

variable {R M M₁ M₂ : Type*} [CommRing R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
variable [AddCommGroup M₁] [Module R M₁] [Module.Finite R M₁] [Module.Free R M₁]
variable [AddCommGroup M₂] [Module R M₂] [Module.Finite R M₂] [Module.Free R M₂]
variable (f : M →ₗ[R] M)

namespace LinearMap

section Basic

/-- `charpoly f` is the characteristic polynomial of the matrix of `f` in any basis. -/
@[simp]
theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
    (toMatrix b b f).charpoly = f.charpoly := by
  nontriviality R
  unfold LinearMap.charpoly
  set b' := chooseBasis R M
  rw [← basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix b b' b b']
  set P := b.toMatrix b'
  set A := toMatrix b' b' f
  set Q := b'.toMatrix b
  let e := Basis.indexEquiv b b'
  let ι' := ChooseBasisIndex R M
  let φ := reindexLinearEquiv R R e e
  let φ₁ := reindexLinearEquiv R R e (Equiv.refl ι')
  let φ₂ := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
  let φ₃ := reindexLinearEquiv R R (Equiv.refl ι') e
  calc
    (P * A * Q).charpoly = (φ (P * A * Q)).charpoly := (charpoly_reindex ..).symm
    _ = (φ₁ P * φ₂ A * φ₃ Q).charpoly := by rw [reindexLinearEquiv_mul, reindexLinearEquiv_mul]
    _ = A.charpoly := by rw [charpoly_mul_comm, ← mul_assoc]; simp [P, Q, φ₁, φ₂, φ₃]

Target:
lemma charpoly_prodMap (f₁ : M₁ →ₗ[R] M₁) (f₂ : M₂ →ₗ[R] M₂) :
    (f₁.prodMap f₂).charpoly = f₁.charpoly * f₂.charpoly :=

Failed candidate:
by
  have __repair_plan_07e60ac3a3 : False := by
    skip
  let b₁ := chooseBasis R M₁
  let b₂ := chooseBasis R M₂
  let b := b₁.prod b₂
  rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
    toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:66:43: error: unsolved goals\nR : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\nf₁ : M₁ →ₗ[R] M₁\nf₂ : M₂ →ₗ[R] M₂\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  let b₁ := chooseBasis R M₁
  let b₂ := chooseBasis R M₂
  let b := b₁.prod b₂
  rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
    toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂]

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

Imports:
public import Mathlib.LinearAlgebra.Charpoly.Basic
public import Mathlib.LinearAlgebra.Matrix.Basis
public import Mathlib.RingTheory.Finiteness.Prod

Namespace:
LinearMap

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

# Characteristic polynomial

## Main result

* `LinearMap.charpoly_toMatrix f` : `charpoly f` is the characteristic polynomial of the matrix
  of `f` in any basis.

-/

public section

noncomputable section

open Module Free Polynomial Matrix

universe u v w

variable {R M M₁ M₂ : Type*} [CommRing R]
variable [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
variable [AddCommGroup M₁] [Module R M₁] [Module.Finite R M₁] [Module.Free R M₁]
variable [AddCommGroup M₂] [Module R M₂] [Module.Finite R M₂] [Module.Free R M₂]
variable (f : M →ₗ[R] M)

namespace LinearMap

section Basic

/-- `charpoly f` is the characteristic polynomial of the matrix of `f` in any basis. -/
@[simp]
theorem charpoly_toMatrix {ι : Type w} [DecidableEq ι] [Fintype ι] (b : Basis ι R M) :
    (toMatrix b b f).charpoly = f.charpoly := by
  nontriviality R
  unfold LinearMap.charpoly
  set b' := chooseBasis R M
  rw [← basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix b b' b b']
  set P := b.toMatrix b'
  set A := toMatrix b' b' f
  set Q := b'.toMatrix b
  let e := Basis.indexEquiv b b'
  let ι' := ChooseBasisIndex R M
  let φ := reindexLinearEquiv R R e e
  let φ₁ := reindexLinearEquiv R R e (Equiv.refl ι')
  let φ₂ := reindexLinearEquiv R R (Equiv.refl ι') (Equiv.refl ι')
  let φ₃ := reindexLinearEquiv R R (Equiv.refl ι') e
  calc
    (P * A * Q).charpoly = (φ (P * A * Q)).charpoly := (charpoly_reindex ..).symm
    _ = (φ₁ P * φ₂ A * φ₃ Q).charpoly := by rw [reindexLinearEquiv_mul, reindexLinearEquiv_mul]
    _ = A.charpoly := by rw [charpoly_mul_comm, ← mul_assoc]; simp [P, Q, φ₁, φ₂, φ₃]

Target:
lemma charpoly_prodMap (f₁ : M₁ →ₗ[R] M₁) (f₂ : M₂ →ₗ[R] M₂) :
    (f₁.prodMap f₂).charpoly = f₁.charpoly * f₂.charpoly :=

Failed candidate:
by
  have __repair_plan_07e60ac3a3 : False := by
    skip
  let b₁ := chooseBasis R M₁
  let b₂ := chooseBasis R M₂
  let b := b₁.prod b₂
  rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
    toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:66:43: error: unsolved goals\nR : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\nf₁ : M₁ →ₗ[R] M₁\nf₂ : M₂ →ₗ[R] M₂\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:123:43: error: unsolved goals
p : ℕ
hp✝ : Fact (Nat.Prime p)
R : Type u_1
inst✝³ : CommRing R
inst✝² : CharP R p
inst✝¹ : PerfectRing R p
S : Type u_2
inst✝ : CommRing S
f g : 𝕎 R →+* S
hp : IsNilpotent ↑p
h : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)
⊢ False

Response:
by
  obtain ⟨n, hn⟩ := hp
  ext x
  obtain ⟨c, hc⟩ := (dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff x n)
  calc
    f x = f (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + f (∑ (i ≤ n), teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by simp
    _ = ∑ (i ≤ n), f (teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i))) * p ^ i := by rw [hc]; simp [pow_succ, hn]
    _ = ∑ (i ≤ n), g (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * p ^ i := by simp [h]
    _ = g (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + g (∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) := by rw [hc]; simp [pow_succ, hn]
    _ = g x := by 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.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]

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 := 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]

@[simp]
theorem teichmuller_mul_pow_coeff (n : ℕ) (x : R) :
    (teichmuller p x * p ^ n).coeff n = x ^ p ^ n := by
  simpa using WittVector.mul_pow_charP_coeff_succ (teichmuller p x) (m := 0)

theorem teichmuller_mul_pow_coeff_of_ne (x : R)
    {m n : ℕ} (h : m ≠ n) : (teichmuller p x * p ^ n).coeff m = 0 := by
  cases Nat.lt_or_lt_of_ne h with
  | inl h =>
    exact WittVector.mul_pow_charP_coeff_zero (teichmuller p x) h
  | inr h =>
    rw [← Nat.sub_add_cancel h.le, WittVector.mul_pow_charP_coeff_succ (teichmuller p x),
      WittVector.teichmuller_coeff_pos p x (m - n) (Nat.zero_lt_sub_of_lt h), zero_pow]
    simp [Nat.Prime.ne_zero Fact.out]

variable [PerfectRing R p]

/--
`p ^ (n + 1)` divides
`x` minus the summation of the first `n + 1` terms of the Teichmuller series.
-/
theorem dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff (x : 𝕎 R) (n : ℕ) :
    (p : 𝕎 R) ^ (n + 1) ∣ x - ∑ (i ≤ n), (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by
  rw [← Ideal.mem_span_singleton, mem_span_p_pow_iff_le_coeff_eq_zero,
      ← le_coeff_eq_iff_le_sub_coeff_eq_zero]
  intro i hi
  rw [WittVector.sum_coeff_eq_coeff_sum]
  · rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem (Finset.mem_Iic.mpr (Nat.lt_succ_iff.mp hi))]
    let g := fun x : ℕ ↦ (0 : R)
    rw [Finset.sum_congr rfl (g := g)]
    · simp [g]
    · intro b hb
      simp only [Finset.mem_sdiff, Finset.mem_Iic, Finset.mem_singleton] at hb
      exact teichmuller_mul_pow_coeff_of_ne _ (Ne.intro hb.2).symm
  · refine fun n ↦ ⟨fun ⟨a, _, ha⟩ ⟨b, _, hb⟩ ↦ ?_⟩
    ext
    dsimp only [ne_eq, Set.mem_setOf_eq]
    rw [← Not.imp_symm (teichmuller_mul_pow_coeff_of_ne _) ha]
    exact Not.imp_symm (teichmuller_mul_pow_coeff_of_ne _) hb

/--
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.
-/

Target:
theorem eq_of_apply_teichmuller_eq
    {S : Type*} [CommRing S] (f g : 𝕎 R →+* S) (hp : IsNilpotent (p : S))
    (h : ∀ (x : R), f (teichmuller p x) = g (teichmuller p x)) : f = g :=

Failed candidate:
by
  have __repair_plan_29620ef868 : False := by
    skip
  obtain ⟨n, hn⟩ := hp
  ext x
  obtain ⟨c, hc⟩ := (dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff x n)
  calc
    f x = f (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + f (∑ (i ≤ n), teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by simp
    _ = ∑ (i ≤ n), f (teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i))) * p ^ i := by rw [hc]; simp [pow_succ, hn]
    _ = ∑ (i ≤ n), g (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * p ^ i := by simp [h]
    _ = g (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + g (∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) := by rw [hc]; simp [pow_succ, hn]
    _ = g x := by simp

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:123:43: error: unsolved goals\np : ℕ\nhp✝ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : CharP R p\ninst✝¹ : PerfectRing R p\nS : Type u_2\ninst✝ : CommRing S\nf g : 𝕎 R →+* S\nhp : IsNilpotent ↑p\nh : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  obtain ⟨n, hn⟩ := hp
  ext x
  obtain ⟨c, hc⟩ := (dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff x n)
  calc
    f x = f (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + f (∑ (i ≤ n), teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by simp
    _ = ∑ (i ≤ n), f (teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i))) * p ^ i := by rw [hc]; simp [pow_succ, hn]
    _ = ∑ (i ≤ n), g (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * p ^ i := by simp [h]
    _ = g (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + g (∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) := by rw [hc]; simp [pow_succ, hn]
    _ = g x := by 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.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]

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 := 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]

@[simp]
theorem teichmuller_mul_pow_coeff (n : ℕ) (x : R) :
    (teichmuller p x * p ^ n).coeff n = x ^ p ^ n := by
  simpa using WittVector.mul_pow_charP_coeff_succ (teichmuller p x) (m := 0)

theorem teichmuller_mul_pow_coeff_of_ne (x : R)
    {m n : ℕ} (h : m ≠ n) : (teichmuller p x * p ^ n).coeff m = 0 := by
  cases Nat.lt_or_lt_of_ne h with
  | inl h =>
    exact WittVector.mul_pow_charP_coeff_zero (teichmuller p x) h
  | inr h =>
    rw [← Nat.sub_add_cancel h.le, WittVector.mul_pow_charP_coeff_succ (teichmuller p x),
      WittVector.teichmuller_coeff_pos p x (m - n) (Nat.zero_lt_sub_of_lt h), zero_pow]
    simp [Nat.Prime.ne_zero Fact.out]

variable [PerfectRing R p]

/--
`p ^ (n + 1)` divides
`x` minus the summation of the first `n + 1` terms of the Teichmuller series.
-/
theorem dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff (x : 𝕎 R) (n : ℕ) :
    (p : 𝕎 R) ^ (n + 1) ∣ x - ∑ (i ≤ n), (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by
  rw [← Ideal.mem_span_singleton, mem_span_p_pow_iff_le_coeff_eq_zero,
      ← le_coeff_eq_iff_le_sub_coeff_eq_zero]
  intro i hi
  rw [WittVector.sum_coeff_eq_coeff_sum]
  · rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem (Finset.mem_Iic.mpr (Nat.lt_succ_iff.mp hi))]
    let g := fun x : ℕ ↦ (0 : R)
    rw [Finset.sum_congr rfl (g := g)]
    · simp [g]
    · intro b hb
      simp only [Finset.mem_sdiff, Finset.mem_Iic, Finset.mem_singleton] at hb
      exact teichmuller_mul_pow_coeff_of_ne _ (Ne.intro hb.2).symm
  · refine fun n ↦ ⟨fun ⟨a, _, ha⟩ ⟨b, _, hb⟩ ↦ ?_⟩
    ext
    dsimp only [ne_eq, Set.mem_setOf_eq]
    rw [← Not.imp_symm (teichmuller_mul_pow_coeff_of_ne _) ha]
    exact Not.imp_symm (teichmuller_mul_pow_coeff_of_ne _) hb

/--
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.
-/

Target:
theorem eq_of_apply_teichmuller_eq
    {S : Type*} [CommRing S] (f g : 𝕎 R →+* S) (hp : IsNilpotent (p : S))
    (h : ∀ (x : R), f (teichmuller p x) = g (teichmuller p x)) : f = g :=

Failed candidate:
by
  have __repair_plan_29620ef868 : False := by
    skip
  obtain ⟨n, hn⟩ := hp
  ext x
  obtain ⟨c, hc⟩ := (dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff x n)
  calc
    f x = f (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + f (∑ (i ≤ n), teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i) := by simp
    _ = ∑ (i ≤ n), f (teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i))) * p ^ i := by rw [hc]; simp [pow_succ, hn]
    _ = ∑ (i ≤ n), g (teichmuller p
        (((_root_.frobeniusEquiv R p).symm ^ i) (x.coeff i))) * p ^ i := by simp [h]
    _ = g (x - ∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) + g (∑ (i ≤ n), teichmuller p (((_root_.frobeniusEquiv R p).symm ^ i)
        (x.coeff i)) * p ^ i) := by rw [hc]; simp [pow_succ, hn]
    _ = g x := by simp

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:123:43: error: unsolved goals\np : ℕ\nhp✝ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : CharP R p\ninst✝¹ : PerfectRing R p\nS : Type u_2\ninst✝ : CommRing S\nf g : 𝕎 R →+* S\nhp : IsNilpotent ↑p\nh : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:187:43: error: unsolved goals
A : Type u₁
inst✝² : Category.{v₁, u₁} A
B : Type u₂
inst✝¹ : Category.{v₂, u₂} B
T : Type u₃
inst✝ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
X Y : Comma L R
H : X = Y
⊢ False

Response:
by
  cases H
  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.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_plan_89f366c48c : False := by
    skip
  cases H
  rfl

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:187:43: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  cases H
  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.CategoryTheory.Iso
public import Mathlib.CategoryTheory.Functor.Category
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.Products.Unitor

Namespace:
CategoryTheory.Comma

Local safe context:
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
-/
/-!
# Comma categories

A comma category is a construction in category theory, which builds a category out of two functors
with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in
`Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and
`right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and
`hom' : L.obj left' ⟶ R.obj right'` is a commutative square

```
L.obj left  ⟶  L.obj left'
      |               |
  hom |               | hom'
      ↓               ↓
R.obj right ⟶  R.obj right',
```

where the top and bottom morphism come from morphisms `left ⟶ left'` and `right ⟶ right'`,
respectively.

## Main definitions

* `Comma L R`: the comma category of the functors `L` and `R`.
* `Over X`: the over category of the object `X` (developed in `Over.lean`).
* `Under X`: the under category of the object `X` (also developed in `Over.lean`).
* `Arrow T`: the arrow category of the category `T` (developed in `Arrow.lean`).

## References

* <https://ncatlab.org/nlab/show/comma+category>

## Tags

comma, slice, coslice, over, under, arrow
-/

@[expose] public section

namespace CategoryTheory

open Category

-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆

variable {A : Type u₁} [Category.{v₁} A]
variable {B : Type u₂} [Category.{v₂} B]
variable {T : Type u₃} [Category.{v₃} T]
variable {A' : Type u₄} [Category.{v₄} A']
variable {B' : Type u₅} [Category.{v₅} B']
variable {T' : Type u₆} [Category.{v₆} T']

/-- The objects of the comma category are triples of an object `left : A`, an object
`right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
@[to_dual self (reorder := A B, 2 4, L R), wikidata Q1780005]
structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
  /-- The left subobject -/
  left : A
  /-- The right subobject -/
  right : B
  /-- A morphism from `L.obj left` to `R.obj right` -/
  hom : L.obj left ⟶ R.obj right

set_option linter.translateOverwrite false

attribute [to_dual existing right] Comma.left
attribute [to_dual self] Comma.mk

-- Satisfying the inhabited linter
instance Comma.inhabited [Inhabited T] : Inhabited (Comma (𝟭 T) (𝟭 T)) where
  default :=
    { left := default
      right := default
      hom := 𝟙 default }

variable {L : A ⥤ T} {R : B ⥤ T}

/-- A morphism between two objects in the comma category is a commutative square connecting the
morphisms coming from the two objects using morphisms in the image of the functors `L` and `R`.
-/
@[ext, to_dual self (reorder := A B, 2 4, L R, X Y)]
structure CommaMorphism (X Y : Comma L R) where
  /-- Morphism on left objects -/
  left : X.left ⟶ Y.left
  /-- Morphism on right objects -/
  right : X.right ⟶ Y.right
  w : L.map left ≫ Y.hom = X.hom ≫ R.map right := by cat_disch

attribute [to_dual existing right] CommaMorphism.left

to_dual_name_hint Left Right, Fst Snd, L R, L₁ R₁, L₂ R₂, A B, F₁ F₂

@[to_dual existing w]
theorem CommaMorphism.w' {X Y : Comma R L} (self : CommaMorphism Y X) :
    Y.hom ≫ L.map self.right = R.map self.left ≫ X.hom :=
  self.w.symm

/-- `CommaMorphism.mk'` is the dual of `CommaMorphism.mk`, which we need for `to_dual`.
Please avoid using this directly. -/
@[to_dual existing mk]
abbrev CommaMorphism.mk' {X Y : Comma R L}
    (right : Y.right ⟶ X.right) (left : Y.left ⟶ X.left)
    (w : Y.hom ≫ L.map right = R.map left ≫ X.hom) :
    CommaMorphism Y X where
  left; right; w := w.symm

-- Satisfying the inhabited linter
instance CommaMorphism.inhabited [Inhabited (Comma L R)] :
    Inhabited (CommaMorphism (default : Comma L R) default) :=
    ⟨{ left := 𝟙 _, right := 𝟙 _}⟩

attribute [reassoc (attr := simp)] CommaMorphism.w

@[to_dual self]
instance commaCategory : Category (Comma L R) where
  Hom X Y := CommaMorphism X Y
  id X :=
    { left := 𝟙 X.left
      right := 𝟙 X.right }
  comp f g :=
    { left := f.left ≫ g.left
      right := f.right ≫ g.right }

namespace Comma

section

variable {X Y Z : Comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}

@[ext, to_dual self (reorder := A B, 2 4, L R, X Y, h₁ h₂)]
lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=
  CommaMorphism.ext h₁ h₂

@[to_dual (attr := simp)]
theorem id_left : (𝟙 X : CommaMorphism X X).left = 𝟙 X.left :=
  rfl

@[to_dual (attr := simp)]
theorem comp_left : (f ≫ g).left = f.left ≫ g.left :=
  rfl

end

variable (L) (R)

/-- The functor sending an object `X` in the comma category to `X.left`. -/
@[to_dual (reorder := L R) (attr := simps)
/-- The functor sending an object `X` in the comma category to `X.right`. -/]
def fst : Comma L R ⥤ A where
  obj X := X.left
  map f := f.left

set_option linter.existingAttributeWarning false in
attribute [to_dual existing] fst_map

set_option backward.defeqAttrib.useBackward true in
/-- We can interpret the commutative square constituting a morphism in the comma category as a
natural transformation between the functors `fst ⋙ L` and `snd ⋙ R` from the comma category
to `T`, where the components are given by the morphism that constitutes an object of the comma
category. -/
@[simps, to_dual self]
def natTrans : fst L R ⋙ L ⟶ snd L R ⋙ R where app X := X.hom

@[simp]
theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by
  cases H
  rfl

@[simp]

Target:
theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
    CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) :=

Failed candidate:
by
  have __repair_plan_89f366c48c : False := by
    skip
  cases H
  rfl

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:187:43: error: unsolved goals\nA : Type u₁\ninst✝² : Category.{v₁, u₁} A\nB : Type u₂\ninst✝¹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nX Y : Comma L R\nH : X = Y\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:56:2: warning: declaration uses `sorry`
<LEAN_FILE>:85:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_7e187ed6f6`
<LEAN_FILE>:60:0: error: unsolved goals
G : Type u_1
inst✝ : Group G
n : ℕ∞
⊢ n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)

Response:
by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

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

Imports:
public import Mathlib.Algebra.Group.Torsion
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.ZMod.QuotientGroup

Namespace:
Monoid

Local safe context:
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Minimum order of an element

This file defines the minimum order of an element of a monoid.

## Main declarations

* `Monoid.minOrder`: The minimum order of an element of a given monoid.
* `Monoid.minOrder_eq_top`: The minimum order is infinite iff the monoid is torsion-free.
* `ZMod.minOrder`: The minimum order of $ℤ/nℤ$ is the smallest factor of `n`, unless `n = 0, 1`.
-/

@[expose] public section

open Subgroup

variable {G α : Type*}

namespace Monoid
section Monoid
variable (α) [Monoid α]

/-- The minimum order of a non-identity element. Also the minimum size of a nontrivial subgroup, see
`Monoid.le_minOrder_iff_forall_subgroup`. Returns `∞` if the monoid is torsion-free. -/
@[to_additive /-- The minimum order of a non-identity element. Also the minimum size of a nontrivial
subgroup, see `AddMonoid.le_minOrder_iff_forall_addSubgroup`. Returns `∞` if the monoid is
torsion-free. -/]
noncomputable def minOrder : ℕ∞ := ⨅ (a : α) (_ha : a ≠ 1) (_ha' : IsOfFinOrder a), orderOf a

variable {α} {a : α}

@[to_additive (attr := simp)]
lemma le_minOrder {n : ℕ∞} :
    n ≤ minOrder α ↔ ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ orderOf a := by simp [minOrder]

@[to_additive]
lemma minOrder_le_orderOf (ha : a ≠ 1) (ha' : IsOfFinOrder a) : minOrder α ≤ orderOf a :=
  le_minOrder.1 le_rfl ha ha'

end Monoid

section Group
variable [Group G] {s : Subgroup G}

@[to_additive]

Target:
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s :=

Failed candidate:
by
  have __repair_pad_0_7e187ed6f6 : True := by trivial
  have __repair_pad_1_7e187ed6f6 : True := by trivial
  have __repair_pad_2_7e187ed6f6 : True := by trivial
  have __repair_pad_3_7e187ed6f6 : True := by trivial
  have __repair_pad_4_7e187ed6f6 : True := by trivial
  have __repair_pad_5_7e187ed6f6 : True := by trivial
  have __repair_pad_6_7e187ed6f6 : True := by trivial
  have __repair_pad_7_7e187ed6f6 : True := by trivial
  have __repair_pad_8_7e187ed6f6 : True := by trivial
  have __repair_pad_9_7e187ed6f6 : True := by trivial
  have __repair_pad_10_7e187ed6f6 : True := by trivial
  have __repair_pad_11_7e187ed6f6 : True := by trivial
  have __repair_pad_12_7e187ed6f6 : True := by trivial
  have __repair_pad_13_7e187ed6f6 : True := by trivial
  have __repair_pad_14_7e187ed6f6 : True := by trivial
  have __repair_pad_15_7e187ed6f6 : True := by trivial
  have __repair_pad_16_7e187ed6f6 : True := by trivial
  have __repair_pad_17_7e187ed6f6 : True := by trivial
  have __repair_pad_18_7e187ed6f6 : True := by trivial
  have __repair_pad_19_7e187ed6f6 : True := by trivial
  have __repair_pad_20_7e187ed6f6 : True := by trivial
  have __repair_pad_21_7e187ed6f6 : True := by trivial
  have __repair_pad_22_7e187ed6f6 : True := by trivial
  have __repair_pad_23_7e187ed6f6 : True := by trivial
  have __repair_fail_7e187ed6f6 := __oprover_overlong_7e187ed6f6
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:56:2: warning: declaration uses `sorry`\n<LEAN_FILE>:85:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_7e187ed6f6`\n<LEAN_FILE>:60:0: error: unsolved goals\nG : Type u_1\ninst✝ : Group G\nn : ℕ∞\n⊢ n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

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

Imports:
public import Mathlib.Algebra.Group.Torsion
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.ZMod.QuotientGroup

Namespace:
Monoid

Local safe context:
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Minimum order of an element

This file defines the minimum order of an element of a monoid.

## Main declarations

* `Monoid.minOrder`: The minimum order of an element of a given monoid.
* `Monoid.minOrder_eq_top`: The minimum order is infinite iff the monoid is torsion-free.
* `ZMod.minOrder`: The minimum order of $ℤ/nℤ$ is the smallest factor of `n`, unless `n = 0, 1`.
-/

@[expose] public section

open Subgroup

variable {G α : Type*}

namespace Monoid
section Monoid
variable (α) [Monoid α]

/-- The minimum order of a non-identity element. Also the minimum size of a nontrivial subgroup, see
`Monoid.le_minOrder_iff_forall_subgroup`. Returns `∞` if the monoid is torsion-free. -/
@[to_additive /-- The minimum order of a non-identity element. Also the minimum size of a nontrivial
subgroup, see `AddMonoid.le_minOrder_iff_forall_addSubgroup`. Returns `∞` if the monoid is
torsion-free. -/]
noncomputable def minOrder : ℕ∞ := ⨅ (a : α) (_ha : a ≠ 1) (_ha' : IsOfFinOrder a), orderOf a

variable {α} {a : α}

@[to_additive (attr := simp)]
lemma le_minOrder {n : ℕ∞} :
    n ≤ minOrder α ↔ ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ orderOf a := by simp [minOrder]

@[to_additive]
lemma minOrder_le_orderOf (ha : a ≠ 1) (ha' : IsOfFinOrder a) : minOrder α ≤ orderOf a :=
  le_minOrder.1 le_rfl ha ha'

end Monoid

section Group
variable [Group G] {s : Subgroup G}

@[to_additive]

Target:
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s :=

Failed candidate:
by
  have __repair_pad_0_7e187ed6f6 : True := by trivial
  have __repair_pad_1_7e187ed6f6 : True := by trivial
  have __repair_pad_2_7e187ed6f6 : True := by trivial
  have __repair_pad_3_7e187ed6f6 : True := by trivial
  have __repair_pad_4_7e187ed6f6 : True := by trivial
  have __repair_pad_5_7e187ed6f6 : True := by trivial
  have __repair_pad_6_7e187ed6f6 : True := by trivial
  have __repair_pad_7_7e187ed6f6 : True := by trivial
  have __repair_pad_8_7e187ed6f6 : True := by trivial
  have __repair_pad_9_7e187ed6f6 : True := by trivial
  have __repair_pad_10_7e187ed6f6 : True := by trivial
  have __repair_pad_11_7e187ed6f6 : True := by trivial
  have __repair_pad_12_7e187ed6f6 : True := by trivial
  have __repair_pad_13_7e187ed6f6 : True := by trivial
  have __repair_pad_14_7e187ed6f6 : True := by trivial
  have __repair_pad_15_7e187ed6f6 : True := by trivial
  have __repair_pad_16_7e187ed6f6 : True := by trivial
  have __repair_pad_17_7e187ed6f6 : True := by trivial
  have __repair_pad_18_7e187ed6f6 : True := by trivial
  have __repair_pad_19_7e187ed6f6 : True := by trivial
  have __repair_pad_20_7e187ed6f6 : True := by trivial
  have __repair_pad_21_7e187ed6f6 : True := by trivial
  have __repair_pad_22_7e187ed6f6 : True := by trivial
  have __repair_pad_23_7e187ed6f6 : True := by trivial
  have __repair_fail_7e187ed6f6 := __oprover_overlong_7e187ed6f6
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:56:2: warning: declaration uses `sorry`\n<LEAN_FILE>:85:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_7e187ed6f6`\n<LEAN_FILE>:60:0: error: unsolved goals\nG : Type u_1\ninst✝ : Group G\nn : ℕ∞\n⊢ n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:86:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

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

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

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

Namespace:
MeasureTheory.MemLp

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

# Density of smooth compactly supported functions in `Lp`

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

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

public section

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

open scoped Nat NNReal ContDiff
open MeasureTheory Pointwise ENNReal

namespace HasCompactSupport

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

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

end HasCompactSupport

namespace MeasureTheory.MemLp

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

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

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

Failed candidate:
by
  have __repair_b9a51224d5 : False := True.intro
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:86:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

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

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

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

Namespace:
MeasureTheory.MemLp

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

# Density of smooth compactly supported functions in `Lp`

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

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

public section

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

open scoped Nat NNReal ContDiff
open MeasureTheory Pointwise ENNReal

namespace HasCompactSupport

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

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

end HasCompactSupport

namespace MeasureTheory.MemLp

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

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

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

Failed candidate:
by
  have __repair_b9a51224d5 : False := True.intro
  -- We use a standard ε / 2 argument to deduce the result from the approximation for
  -- continuous compactly supported functions.
  have hε₂ : 0 < ε / 2 := by positivity
  have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
  obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
  obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
  refine ⟨g', hg'₁, hg'₂, ?_⟩
  have : f - g' = (f - g) - (g' - g) := by simp
  grw [this, eLpNorm_sub_le (hf.aestronglyMeasurable.sub hg₄.aestronglyMeasurable)
    (hg'₂.continuous.aestronglyMeasurable.sub hg₄.aestronglyMeasurable) hp₂, hg₂,
    eLpNorm_sub_comm, hg'₃, ← ENNReal.ofReal_add hε₂.le hε₂.le, add_halves]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:86:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:78:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  rw [continuousGeneratedBy_iff]
  exact (hg.continuousMap.comp hf.continuousMap).continuous

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

Imports:
public import Mathlib.Topology.Convenient.GeneratedBy

Namespace:
Topology

Local safe context:
/-
Copyright (c) 2026 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# `X`-continuous maps

Given a family `X i` of topological spaces, we introduce a predicate
`ContinuousGeneratedBy X` on maps `g : Y ⟶ Z` saying that
`g` is `X`-continuous, i.e. for any continuous map `f : X i → Y`,
the composition `g ∘ f` is continuous.

## References
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

universe v v' t u

@[expose] public section

open Topology

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

namespace Topology

variable (X) in
/-- Given a family `X i` of topological space, this is a predicate
on maps `g : Y → Z` between topological spaces saying that for any
continuous map `f : X i → Y`, the composition `g ∘ f` is continuous.
We say that `g` is `X`-continuous. -/
def ContinuousGeneratedBy (g : Y → Z) : Prop :=
  ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ f)

lemma continuousGeneratedBy_def (g : Y → Z) :
    ContinuousGeneratedBy X g ↔
      ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ f) := Iff.rfl

lemma continuousGeneratedBy_iff (g : Y → Z) :
    ContinuousGeneratedBy X g ↔
      Continuous ((WithGeneratedByTopology.equiv (X := X)).symm ∘ g ∘
        WithGeneratedByTopology.equiv (X := X)) := by
  rw [IsGeneratedBy.equiv_symm_comp_continuous_iff,
    WithGeneratedByTopology.continuous_from_iff]
  rfl

/-- A `X`-continuous map `g : Y → Z` induces a continuous map
between the types `Y` and `Z` equipped with the `X`-generated topology. -/
def ContinuousGeneratedBy.continuousMap {g : Y → Z}
    (hg : ContinuousGeneratedBy X g) :
    C(WithGeneratedByTopology X Y, WithGeneratedByTopology X Z) :=
  ⟨WithGeneratedByTopology.equiv.symm ∘ g ∘ WithGeneratedByTopology.equiv, by
    rwa [← continuousGeneratedBy_iff]⟩

@[simp]
lemma ContinuousGeneratedBy.continuousMap_coe {g : Y → Z}
    (hg : ContinuousGeneratedBy X g) :
    ⇑hg.continuousMap = WithGeneratedByTopology.equiv.symm ∘ g ∘ WithGeneratedByTopology.equiv :=
  rfl

@[simp]
lemma ContinuousGeneratedBy.id :
    ContinuousGeneratedBy X (id : Y → Y) := by
  simpa [continuousGeneratedBy_iff] using continuous_id

Target:
lemma ContinuousGeneratedBy.comp {g : Y → Z} (hg : ContinuousGeneratedBy X g)
    {T : Type*} [TopologicalSpace T] {f : T → Y} (hf : ContinuousGeneratedBy X f) :
    ContinuousGeneratedBy X (g ∘ f) :=

Failed candidate:
by
  have __repair_7d13cb4505 : False := True.intro
  rw [continuousGeneratedBy_iff]
  exact (hg.continuousMap.comp hf.continuousMap).continuous

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:78:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [continuousGeneratedBy_iff]
  exact (hg.continuousMap.comp hf.continuousMap).continuous

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

Imports:
public import Mathlib.Topology.Convenient.GeneratedBy

Namespace:
Topology

Local safe context:
/-
Copyright (c) 2026 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
/-!
# `X`-continuous maps

Given a family `X i` of topological spaces, we introduce a predicate
`ContinuousGeneratedBy X` on maps `g : Y ⟶ Z` saying that
`g` is `X`-continuous, i.e. for any continuous map `f : X i → Y`,
the composition `g ∘ f` is continuous.

## References
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

universe v v' t u

@[expose] public section

open Topology

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

namespace Topology

variable (X) in
/-- Given a family `X i` of topological space, this is a predicate
on maps `g : Y → Z` between topological spaces saying that for any
continuous map `f : X i → Y`, the composition `g ∘ f` is continuous.
We say that `g` is `X`-continuous. -/
def ContinuousGeneratedBy (g : Y → Z) : Prop :=
  ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ f)

lemma continuousGeneratedBy_def (g : Y → Z) :
    ContinuousGeneratedBy X g ↔
      ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ f) := Iff.rfl

lemma continuousGeneratedBy_iff (g : Y → Z) :
    ContinuousGeneratedBy X g ↔
      Continuous ((WithGeneratedByTopology.equiv (X := X)).symm ∘ g ∘
        WithGeneratedByTopology.equiv (X := X)) := by
  rw [IsGeneratedBy.equiv_symm_comp_continuous_iff,
    WithGeneratedByTopology.continuous_from_iff]
  rfl

/-- A `X`-continuous map `g : Y → Z` induces a continuous map
between the types `Y` and `Z` equipped with the `X`-generated topology. -/
def ContinuousGeneratedBy.continuousMap {g : Y → Z}
    (hg : ContinuousGeneratedBy X g) :
    C(WithGeneratedByTopology X Y, WithGeneratedByTopology X Z) :=
  ⟨WithGeneratedByTopology.equiv.symm ∘ g ∘ WithGeneratedByTopology.equiv, by
    rwa [← continuousGeneratedBy_iff]⟩

@[simp]
lemma ContinuousGeneratedBy.continuousMap_coe {g : Y → Z}
    (hg : ContinuousGeneratedBy X g) :
    ⇑hg.continuousMap = WithGeneratedByTopology.equiv.symm ∘ g ∘ WithGeneratedByTopology.equiv :=
  rfl

@[simp]
lemma ContinuousGeneratedBy.id :
    ContinuousGeneratedBy X (id : Y → Y) := by
  simpa [continuousGeneratedBy_iff] using continuous_id

Target:
lemma ContinuousGeneratedBy.comp {g : Y → Z} (hg : ContinuousGeneratedBy X g)
    {T : Type*} [TopologicalSpace T] {f : T → Y} (hf : ContinuousGeneratedBy X f) :
    ContinuousGeneratedBy X (g ∘ f) :=

Failed candidate:
by
  have __repair_7d13cb4505 : False := True.intro
  rw [continuousGeneratedBy_iff]
  exact (hg.continuousMap.comp hf.continuousMap).continuous

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:78:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:53:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

[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.NumberTheory.RamificationInertia.Inertia
public import Mathlib.RingTheory.QuasiFinite.Basic

Namespace:
Ideal

Local safe context:
/-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
/-!
# Inertia degree

Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

## Main definitions

* `Ideal.inertiaDeg' q R`: The inertia degree of `q` over `R`.

## Main statements

* `inertiaDeg_eq_inertiaDeg'`: The inertia degree agrees with the usual definition in the case of
  maximal ideals.
* `inertiaDeg'_tower`: Inertia degree is multiplicative in towers.
-/

@[expose] public section

namespace Ideal

section

variable {S : Type*} [CommRing S] (q : Ideal S) (R : Type*) [CommRing R] [Algebra R S]

open Classical in
/-- Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

When `q` is not prime, we use a junk value of `0`.

This will eventually replace the existing definition of `Ideal.inertiaDeg`. -/
noncomputable def inertiaDeg' : ℕ :=
  if _ : q.IsPrime then
    letI := Localization.AtPrime.algebraOfLiesOver (q.under R) q
    Module.finrank (q.under R).ResidueField q.ResidueField else 0

Target:
theorem inertiaDeg'_def [hq : q.IsPrime]
    [Algebra (Localization.AtPrime (q.under R)) (Localization.AtPrime q)]
    [Localization.AtPrime.IsLiesOverAlgebra (q.under R) q] :
    q.inertiaDeg' R = Module.finrank (q.under R).ResidueField q.ResidueField :=

Failed candidate:
by
  have __repair_183d4bb216 : False := True.intro
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:53:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

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.NumberTheory.RamificationInertia.Inertia
public import Mathlib.RingTheory.QuasiFinite.Basic

Namespace:
Ideal

Local safe context:
/-
Copyright (c) 2026 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
/-!
# Inertia degree

Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

## Main definitions

* `Ideal.inertiaDeg' q R`: The inertia degree of `q` over `R`.

## Main statements

* `inertiaDeg_eq_inertiaDeg'`: The inertia degree agrees with the usual definition in the case of
  maximal ideals.
* `inertiaDeg'_tower`: Inertia degree is multiplicative in towers.
-/

@[expose] public section

namespace Ideal

section

variable {S : Type*} [CommRing S] (q : Ideal S) (R : Type*) [CommRing R] [Algebra R S]

open Classical in
/-- Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

When `q` is not prime, we use a junk value of `0`.

This will eventually replace the existing definition of `Ideal.inertiaDeg`. -/
noncomputable def inertiaDeg' : ℕ :=
  if _ : q.IsPrime then
    letI := Localization.AtPrime.algebraOfLiesOver (q.under R) q
    Module.finrank (q.under R).ResidueField q.ResidueField else 0

Target:
theorem inertiaDeg'_def [hq : q.IsPrime]
    [Algebra (Localization.AtPrime (q.under R)) (Localization.AtPrime q)]
    [Localization.AtPrime.IsLiesOverAlgebra (q.under R) q] :
    q.inertiaDeg' R = Module.finrank (q.under R).ResidueField q.ResidueField :=

Failed candidate:
by
  have __repair_183d4bb216 : False := True.intro
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:53:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:101:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:100:0: error: unsolved goals
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
⊢ (antipode R) 1 = 1

Response:
by
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

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

Imports:
public import Mathlib.RingTheory.Bialgebra.Basic
public import Mathlib.RingTheory.Coalgebra.Convolution

Namespace:
HopfAlgebra

Local safe context:
/-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey
-/
/-!
# Hopf algebras

In this file we define `HopfAlgebra`, and provide instances for:

* Commutative semirings: `CommSemiring.toHopfAlgebra`

## Main definitions

* `HopfAlgebra R A` : the Hopf algebra structure on an `R`-bialgebra `A`.
* `HopfAlgebra.antipode` : the `R`-linear map `A →ₗ[R] A`.
* `HopfAlgebra.ofConvInverse` : construct a Hopf algebra from a two-sided convolution inverse
  of the identity.
* `HopfAlgebra.ofAlgHom` : the same for commutative `A`, with `AlgHom` hypotheses.

## Main results

* `HopfAlgebra.antipode_one` : the antipode of the unit is the unit.
* `HopfAlgebra.antipode_mul` : the antipode is an antihomomorphism: `S(ab) = S(b)S(a)`.

## TODO

* Uniqueness of Hopf algebra structure on a bialgebra (i.e. if the algebra and coalgebra structures
  agree then the antipodes must also agree).

* If `A` is commutative then `antipode` is an algebra homomorphism.

* If `A` is commutative then `antipode` is necessarily a bijection and its square is
  the identity.

(Note that all three facts have been proved for Hopf bimonoids in an arbitrary braided category,
so we could deduce the facts here from an equivalence `HopfAlgCat R ≌ Hopf (ModuleCat R)`.)

## References

* <https://en.wikipedia.org/wiki/Hopf_algebra>

* [C. Kassel, *Quantum Groups* (§III.3)][Kassel1995]

-/

public section

open Bialgebra

universe u v w

/-- Isolates the antipode of a Hopf algebra, to allow API to be constructed before proving the
Hopf algebra axioms. See `HopfAlgebra` for documentation. -/
class HopfAlgebraStruct (R : Type u) (A : Type v) [CommSemiring R] [Semiring A]
    extends Bialgebra R A where
  /-- The antipode of the Hopf algebra. -/
  antipode (R) : A →ₗ[R] A

/-- A Hopf algebra over a commutative (semi)ring `R` is a bialgebra over `R` equipped with an
`R`-linear endomorphism `antipode` satisfying the antipode axioms. -/
class HopfAlgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    HopfAlgebraStruct R A where
  /-- One of the antipode axioms for a Hopf algebra. -/
  mul_antipode_rTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.rTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
  /-- One of the antipode axioms for a Hopf algebra. -/
  mul_antipode_lTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.lTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit

namespace HopfAlgebra

export HopfAlgebraStruct (antipode)

variable {R : Type u} {A : Type v} {ι : Type*} [CommSemiring R] [Semiring A] [HopfAlgebra R A]
  {a : A}

@[simp]
theorem mul_antipode_rTensor_comul_apply (a : A) :
    LinearMap.mul' R A ((antipode R).rTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) :=
  LinearMap.congr_fun mul_antipode_rTensor_comul a

@[simp]
theorem mul_antipode_lTensor_comul_apply (a : A) :
    LinearMap.mul' R A ((antipode R).lTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) :=
  LinearMap.congr_fun mul_antipode_lTensor_comul a

@[simp]

Target:
theorem antipode_one :
    HopfAlgebra.antipode R (1 : A) = 1 :=

Failed candidate:
by
  have __repair_f89d921def := True.intro.nonexistentProjection
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:101:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:100:0: error: unsolved goals\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\n⊢ (antipode R) 1 = 1", "returncode": 1}

Corrected proof:

[Assistant]
by
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

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

Imports:
public import Mathlib.RingTheory.Bialgebra.Basic
public import Mathlib.RingTheory.Coalgebra.Convolution

Namespace:
HopfAlgebra

Local safe context:
/-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey
-/
/-!
# Hopf algebras

In this file we define `HopfAlgebra`, and provide instances for:

* Commutative semirings: `CommSemiring.toHopfAlgebra`

## Main definitions

* `HopfAlgebra R A` : the Hopf algebra structure on an `R`-bialgebra `A`.
* `HopfAlgebra.antipode` : the `R`-linear map `A →ₗ[R] A`.
* `HopfAlgebra.ofConvInverse` : construct a Hopf algebra from a two-sided convolution inverse
  of the identity.
* `HopfAlgebra.ofAlgHom` : the same for commutative `A`, with `AlgHom` hypotheses.

## Main results

* `HopfAlgebra.antipode_one` : the antipode of the unit is the unit.
* `HopfAlgebra.antipode_mul` : the antipode is an antihomomorphism: `S(ab) = S(b)S(a)`.

## TODO

* Uniqueness of Hopf algebra structure on a bialgebra (i.e. if the algebra and coalgebra structures
  agree then the antipodes must also agree).

* If `A` is commutative then `antipode` is an algebra homomorphism.

* If `A` is commutative then `antipode` is necessarily a bijection and its square is
  the identity.

(Note that all three facts have been proved for Hopf bimonoids in an arbitrary braided category,
so we could deduce the facts here from an equivalence `HopfAlgCat R ≌ Hopf (ModuleCat R)`.)

## References

* <https://en.wikipedia.org/wiki/Hopf_algebra>

* [C. Kassel, *Quantum Groups* (§III.3)][Kassel1995]

-/

public section

open Bialgebra

universe u v w

/-- Isolates the antipode of a Hopf algebra, to allow API to be constructed before proving the
Hopf algebra axioms. See `HopfAlgebra` for documentation. -/
class HopfAlgebraStruct (R : Type u) (A : Type v) [CommSemiring R] [Semiring A]
    extends Bialgebra R A where
  /-- The antipode of the Hopf algebra. -/
  antipode (R) : A →ₗ[R] A

/-- A Hopf algebra over a commutative (semi)ring `R` is a bialgebra over `R` equipped with an
`R`-linear endomorphism `antipode` satisfying the antipode axioms. -/
class HopfAlgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    HopfAlgebraStruct R A where
  /-- One of the antipode axioms for a Hopf algebra. -/
  mul_antipode_rTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.rTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit
  /-- One of the antipode axioms for a Hopf algebra. -/
  mul_antipode_lTensor_comul :
    LinearMap.mul' R A ∘ₗ antipode.lTensor A ∘ₗ comul = (Algebra.linearMap R A) ∘ₗ counit

namespace HopfAlgebra

export HopfAlgebraStruct (antipode)

variable {R : Type u} {A : Type v} {ι : Type*} [CommSemiring R] [Semiring A] [HopfAlgebra R A]
  {a : A}

@[simp]
theorem mul_antipode_rTensor_comul_apply (a : A) :
    LinearMap.mul' R A ((antipode R).rTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) :=
  LinearMap.congr_fun mul_antipode_rTensor_comul a

@[simp]
theorem mul_antipode_lTensor_comul_apply (a : A) :
    LinearMap.mul' R A ((antipode R).lTensor A (Coalgebra.comul a)) =
    algebraMap R A (Coalgebra.counit a) :=
  LinearMap.congr_fun mul_antipode_lTensor_comul a

@[simp]

Target:
theorem antipode_one :
    HopfAlgebra.antipode R (1 : A) = 1 :=

Failed candidate:
by
  have __repair_f89d921def := True.intro.nonexistentProjection
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:101:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:100:0: error: unsolved goals\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\n⊢ (antipode R) 1 = 1", "returncode": 1}

Corrected proof:


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

[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.CategoryTheory.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

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

# Profunctors

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

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

## Future work

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

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

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

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

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

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

variable {C D}

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

namespace ProfunctorCore

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

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

@[simp]

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

Proof body:

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

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.CategoryTheory.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

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

# Profunctors

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

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

## Future work

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

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

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

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

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

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

variable {C D}

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

namespace ProfunctorCore

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

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

@[simp]

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

Proof body:


<LEAN_FILE>:88:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:87:0: error: unsolved goals
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
F G : Type u → Type u
inst✝⁴ : Applicative F
inst✝³ : Applicative G
inst✝² : LawfulBitraversable t
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
α₀ α₁ β₀ β₁ : Type u
f : α₀ → F α₁
f' : β₀ → G β₁
x : t α₀ β₀
⊢ Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x

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

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

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

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

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

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

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

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

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

namespace Bitraversable

open Functor LawfulApplicative

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

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

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

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

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

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

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

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

@[higher_order tsnd_comp_tfst]

Target:
theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tsnd f' <$> tfst f x)
      = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x :=

Failed candidate:
by
  have __repair_9588bcb829 := True.intro.nonexistentProjection
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:88:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:87:0: error: unsolved goals\nt : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : LawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ β₀ β₁ : Type u\nf : α₀ → F α₁\nf' : β₀ → G β₁\nx : t α₀ β₀\n⊢ Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

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

Imports:
public import Mathlib.Control.Bitraversable.Basic

Namespace:
Bitraversable

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

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

## Lemmas

Combination of
  * bitraverse
  * tfst
  * tsnd

with the applicatives `id` and `comp`

## References

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

## Tags

traversable bitraversable functor bifunctor applicative

-/

public section

universe u

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

namespace Bitraversable

open Functor LawfulApplicative

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

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

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

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

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

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

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

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

@[higher_order tsnd_comp_tfst]

Target:
theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
    Comp.mk (tsnd f' <$> tfst f x)
      = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x :=

Failed candidate:
by
  have __repair_9588bcb829 := True.intro.nonexistentProjection
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:88:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:87:0: error: unsolved goals\nt : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : LawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ β₀ β₁ : Type u\nf : α₀ → F α₁\nf' : β₀ → G β₁\nx : t α₀ β₀\n⊢ Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x", "returncode": 1}

Corrected proof: