Response:
by
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

[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.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Algebra.Group.ClosedSubgroup

Namespace:
InfiniteGalois

Local context:
/-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
/-!

# The Fundamental Theorem of Infinite Galois Theory

In this file, we prove the fundamental theorem of infinite Galois theory and the special case for
open subgroups and normal subgroups. We first verify that `IntermediateField.fixingSubgroup` and
`IntermediateField.fixedField` are inverses of each other between intermediate fields and
closed subgroups of the Galois group.

## Main definitions and results

In `K/k`, for any intermediate field `L` :

* `fixingSubgroup_isClosed` : the subgroup fixing `L` (`Gal(K/L)`) is closed.

* `fixedField_fixingSubgroup` : the field fixed by the
  subgroup fixing `L` is equal to `L` itself.

For any subgroup `H` of `Gal(K/k)` :

* `restrict_fixedField` : For a Galois intermediate field `M`, the fixed field of the image of `H`
  restricted to `M` is equal to the fixed field of `H` intersected with `M`.
* `fixingSubgroup_fixedField` : If `H` is closed, the fixing subgroup of the fixed field of `H`
  is equal to `H` itself.

The fundamental theorem of infinite Galois theory :

* `IntermediateFieldEquivClosedSubgroup` : The order equivalence is given by mapping any
  intermediate field `L` to the subgroup fixing `L`, and the inverse maps any
  closed subgroup of `Gal(K/k)` `H` to the fixed field of `H`. The composition is equal to
  the identity as described in the lemmas above, and compatibility with the order follows easily.

Special cases :

* `isOpen_iff_finite` : The fixing subgroup of an intermediate field `L` is open if and only if
  `L` is finite-dimensional.

* `normal_iff_isGalois` : The fixing subgroup of an intermediate field `L` is normal if and only if
  `L` is Galois.

-/

@[expose] public section

variable {k K : Type*} [Field k] [Field K] [Algebra k K]

namespace InfiniteGalois

open scoped Pointwise
open FiniteGaloisIntermediateField AlgEquiv
--Note: The `adjoin`s below are `FiniteGaloisIntermediateField.adjoin`

lemma fixingSubgroup_isClosed (L : IntermediateField k K) [IsGalois k K] :
    IsClosed (L.fixingSubgroup : Set Gal(K/k)) where
  isOpen_compl := isOpen_iff_mem_nhds.mpr fun σ h => by
    apply mem_nhds_iff.mpr
    rcases Set.not_subset.mp ((mem_fixingSubgroup_iff Gal(K/k)).not.mp h) with ⟨y, yL, ne⟩
    use σ • ((adjoin k {y}).1.fixingSubgroup : Set Gal(K/k))
    constructor
    · intro f hf
      rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
      simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
      apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
      push Not
      use y
      simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
      have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
        adjoin_simple_le_iff.mp le_rfl
      simpa only [this, ne_eq, AlgEquiv.smul_def] using! ne
    · simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
      use 1
      simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem]

lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
    IntermediateField.fixedField L.fixingSubgroup = L := by
  apply le_antisymm
  · intro x hx
    rw [IntermediateField.mem_fixedField_iff] at hx
    have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
    have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
      (IsGalois.tfae.out 0 1).mp (by infer_instance)
    have : ⟨x, mem⟩ ∈ (⊥ : IntermediateField L (adjoin L {x})) := by
      rw [← this, IntermediateField.mem_fixedField_iff]
      intro f _
      rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
      apply Subtype.val_injective
      rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
      have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
      simpa only [SetLike.coe_mem, true_implies]
    rcases IntermediateField.mem_bot.mp this with ⟨y, hy⟩
    obtain ⟨rfl⟩ : y = x := congrArg Subtype.val hy
    exact y.2
  · exact (IntermediateField.le_iff_le L.fixingSubgroup L).mpr le_rfl

Target:
lemma fixedField_bot [IsGalois k K] :
    IntermediateField.fixedField (⊤ : Subgroup Gal(K/k)) = ⊥ :=

Proof body:

[Assistant]
by
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

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.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Algebra.Group.ClosedSubgroup

Namespace:
InfiniteGalois

Local context:
/-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
/-!

# The Fundamental Theorem of Infinite Galois Theory

In this file, we prove the fundamental theorem of infinite Galois theory and the special case for
open subgroups and normal subgroups. We first verify that `IntermediateField.fixingSubgroup` and
`IntermediateField.fixedField` are inverses of each other between intermediate fields and
closed subgroups of the Galois group.

## Main definitions and results

In `K/k`, for any intermediate field `L` :

* `fixingSubgroup_isClosed` : the subgroup fixing `L` (`Gal(K/L)`) is closed.

* `fixedField_fixingSubgroup` : the field fixed by the
  subgroup fixing `L` is equal to `L` itself.

For any subgroup `H` of `Gal(K/k)` :

* `restrict_fixedField` : For a Galois intermediate field `M`, the fixed field of the image of `H`
  restricted to `M` is equal to the fixed field of `H` intersected with `M`.
* `fixingSubgroup_fixedField` : If `H` is closed, the fixing subgroup of the fixed field of `H`
  is equal to `H` itself.

The fundamental theorem of infinite Galois theory :

* `IntermediateFieldEquivClosedSubgroup` : The order equivalence is given by mapping any
  intermediate field `L` to the subgroup fixing `L`, and the inverse maps any
  closed subgroup of `Gal(K/k)` `H` to the fixed field of `H`. The composition is equal to
  the identity as described in the lemmas above, and compatibility with the order follows easily.

Special cases :

* `isOpen_iff_finite` : The fixing subgroup of an intermediate field `L` is open if and only if
  `L` is finite-dimensional.

* `normal_iff_isGalois` : The fixing subgroup of an intermediate field `L` is normal if and only if
  `L` is Galois.

-/

@[expose] public section

variable {k K : Type*} [Field k] [Field K] [Algebra k K]

namespace InfiniteGalois

open scoped Pointwise
open FiniteGaloisIntermediateField AlgEquiv
--Note: The `adjoin`s below are `FiniteGaloisIntermediateField.adjoin`

lemma fixingSubgroup_isClosed (L : IntermediateField k K) [IsGalois k K] :
    IsClosed (L.fixingSubgroup : Set Gal(K/k)) where
  isOpen_compl := isOpen_iff_mem_nhds.mpr fun σ h => by
    apply mem_nhds_iff.mpr
    rcases Set.not_subset.mp ((mem_fixingSubgroup_iff Gal(K/k)).not.mp h) with ⟨y, yL, ne⟩
    use σ • ((adjoin k {y}).1.fixingSubgroup : Set Gal(K/k))
    constructor
    · intro f hf
      rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
      simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
      apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
      push Not
      use y
      simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
      have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
        adjoin_simple_le_iff.mp le_rfl
      simpa only [this, ne_eq, AlgEquiv.smul_def] using! ne
    · simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
      use 1
      simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem]

lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
    IntermediateField.fixedField L.fixingSubgroup = L := by
  apply le_antisymm
  · intro x hx
    rw [IntermediateField.mem_fixedField_iff] at hx
    have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
    have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
      (IsGalois.tfae.out 0 1).mp (by infer_instance)
    have : ⟨x, mem⟩ ∈ (⊥ : IntermediateField L (adjoin L {x})) := by
      rw [← this, IntermediateField.mem_fixedField_iff]
      intro f _
      rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
      apply Subtype.val_injective
      rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
      have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
      simpa only [SetLike.coe_mem, true_implies]
    rcases IntermediateField.mem_bot.mp this with ⟨y, hy⟩
    obtain ⟨rfl⟩ : y = x := congrArg Subtype.val hy
    exact y.2
  · exact (IntermediateField.le_iff_le L.fixingSubgroup L).mpr le_rfl

Target:
lemma fixedField_bot [IsGalois k K] :
    IntermediateField.fixedField (⊤ : Subgroup Gal(K/k)) = ⊥ :=

Proof body:


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

Imports:
public import Mathlib.Analysis.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 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 :=

Proof body:

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

Imports:
public import Mathlib.Analysis.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 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 :=

Proof body:


<LEAN_FILE>:190:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  obtain ⟨n, ι, inst, x, p, e, equiv⟩ := AddCommGroup.equiv_free_prod_directSum_zmod (Additive G)
  exact ⟨ι, Fin n, inst, inferInstance, x, p, e, ⟨MulEquiv.toAdditive.symm <| equiv.some.trans <|
    ((Finsupp.addEquivFunOnFinite.trans <| ((AddEquiv.piAdditive _).trans <|
        (AddEquiv.additiveMultiplicative ℤ).arrowCongr (Equiv.refl _)).symm).prodCongr
          (DirectSum.addEquivProd _ )).trans <| (AddEquiv.prodAdditive _ _).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.PID
public import Mathlib.Algebra.Group.TypeTags.Finite
public import Mathlib.Data.ZMod.QuotientRing

Namespace:
CommGroup

Local safe 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)

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

end Module

variable (G : Type u)

namespace AddCommGroup

variable [AddCommGroup G]

/-- **Structure theorem of finitely generated abelian groups** : 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`. -/
theorem equiv_free_prod_directSum_zmod [hG : AddGroup.FG G] :
    ∃ (n : ℕ) (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃+ (Fin n →₀ ℤ) × ⨁ i : ι, ZMod (p i ^ e i) := by
  obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ :=
    @Module.equiv_free_prod_directSum _ _ _ _ _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG)
  refine ⟨n, ι, fι, fun i => (p i).natAbs, fun i => ?_, e, ⟨?_⟩⟩
  · rw [← Int.prime_iff_natAbs_prime, ← irreducible_iff_prime]; exact hp i
  exact
    f.toAddEquiv.trans
      ((AddEquiv.refl _).prodCongr <|
        DFinsupp.mapRange.addEquiv fun i =>
          ((Int.quotientSpanEquivZMod _).trans <|
              ZMod.ringEquivCongr <| (p i).natAbs_pow _).toAddEquiv)

/-- **Structure theorem of finite abelian groups** : Any finite abelian group is a direct sum of
some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`. -/
theorem equiv_directSum_zmod_of_finite [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃+ ⨁ i : ι, ZMod (p i ^ e i) := by
  cases nonempty_fintype G
  obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ := equiv_free_prod_directSum_zmod G
  rcases n with - | n
  · have : Unique (Fin Nat.zero →₀ ℤ) :=
      { uniq := by subsingleton }
    exact ⟨ι, fι, p, hp, e, ⟨f.trans AddEquiv.uniqueProd⟩⟩
  · haveI := @Fintype.prodLeft _ _ _ (Fintype.ofEquiv G f.toEquiv) _
    exact
      (Fintype.ofSurjective (fun f : Fin n.succ →₀ ℤ => f 0) fun a =>
            ⟨Finsupp.single 0 a, Finsupp.single_eq_same⟩).false.elim

/-- **Structure theorem of finite abelian groups** : Any finite abelian group is a direct sum of
some `ZMod (n i)` for some natural numbers `n i > 1`. -/
lemma equiv_directSum_zmod_of_finite' (G : Type*) [AddCommGroup G] [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (n : ι → ℕ),
      (∀ i, 1 < n i) ∧ Nonempty (G ≃+ ⨁ i, ZMod (n i)) := by
  classical
  obtain ⟨ι, hι, p, hp, n, ⟨e⟩⟩ := AddCommGroup.equiv_directSum_zmod_of_finite G
  refine ⟨{i : ι // n i ≠ 0}, inferInstance, fun i ↦ p i ^ n i, ?_,
    ⟨e.trans (directSumNeZeroMulEquiv ι _ _).symm⟩⟩
  rintro ⟨i, hi⟩
  exact one_lt_pow₀ (hp _).one_lt hi

theorem finite_of_fg_torsion [hG' : AddGroup.FG G] (hG : AddMonoid.IsTorsion G) : Finite G :=
  @Module.finite_of_fg_torsion _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG') <|
    AddMonoid.isTorsion_iff_isTorsion_int.mp hG

end AddCommGroup

namespace CommGroup

theorem finite_of_fg_torsion [CommGroup G] [Group.FG G] (hG : Monoid.IsTorsion G) : Finite G :=
  @Finite.of_equiv _ _ (AddCommGroup.finite_of_fg_torsion (Additive G) hG) Multiplicative.ofAdd

/-- The **Structure Theorem For Finite Abelian Groups** in a multiplicative version:
A finite commutative group `G` is isomorphic to a finite product of finite cyclic groups. -/
theorem equiv_prod_multiplicative_zmod_of_finite (G : Type*) [CommGroup G] [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (n : ι → ℕ),
       (∀ (i : ι), 1 < n i) ∧ Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i)))) := by
  obtain ⟨ι, inst, n, h₁, h₂⟩ := AddCommGroup.equiv_directSum_zmod_of_finite' (Additive G)
  exact ⟨ι, inst, n, h₁, ⟨MulEquiv.toAdditive.symm <| h₂.some.trans <|
    (DirectSum.addEquivProd _).trans (MulEquiv.piMultiplicative _).toAdditiveRight⟩⟩

/-- The **Structure theorem of finitely generated abelian groups** in a multiplicative version :
    Any finitely generated abelian group is the product of a power of `ℤ`
    and a direct product of some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`. -/

Target:
theorem equiv_free_prod_prod_multiplicative_zmod (G : Type*) [CommGroup G] [hG : Group.FG G] :
    ∃ (ι j : Type) (_ : Fintype ι) (_ : Fintype j) (p : ι → ℕ)
    (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃* (j → Multiplicative ℤ) × ((i : ι) → Multiplicative (ZMod (p i ^ e i))) :=

Failed candidate:
by
  have __repair_fefde0126f : False := True.intro
  obtain ⟨n, ι, inst, x, p, e, equiv⟩ := AddCommGroup.equiv_free_prod_directSum_zmod (Additive G)
  exact ⟨ι, Fin n, inst, inferInstance, x, p, e, ⟨MulEquiv.toAdditive.symm <| equiv.some.trans <|
    ((Finsupp.addEquivFunOnFinite.trans <| ((AddEquiv.piAdditive _).trans <|
        (AddEquiv.additiveMultiplicative ℤ).arrowCongr (Equiv.refl _)).symm).prodCongr
          (DirectSum.addEquivProd _ )).trans <| (AddEquiv.prodAdditive _ _).symm⟩⟩

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:190:38: 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 ⟨n, ι, inst, x, p, e, equiv⟩ := AddCommGroup.equiv_free_prod_directSum_zmod (Additive G)
  exact ⟨ι, Fin n, inst, inferInstance, x, p, e, ⟨MulEquiv.toAdditive.symm <| equiv.some.trans <|
    ((Finsupp.addEquivFunOnFinite.trans <| ((AddEquiv.piAdditive _).trans <|
        (AddEquiv.additiveMultiplicative ℤ).arrowCongr (Equiv.refl _)).symm).prodCongr
          (DirectSum.addEquivProd _ )).trans <| (AddEquiv.prodAdditive _ _).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.PID
public import Mathlib.Algebra.Group.TypeTags.Finite
public import Mathlib.Data.ZMod.QuotientRing

Namespace:
CommGroup

Local safe 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)

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

end Module

variable (G : Type u)

namespace AddCommGroup

variable [AddCommGroup G]

/-- **Structure theorem of finitely generated abelian groups** : 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`. -/
theorem equiv_free_prod_directSum_zmod [hG : AddGroup.FG G] :
    ∃ (n : ℕ) (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃+ (Fin n →₀ ℤ) × ⨁ i : ι, ZMod (p i ^ e i) := by
  obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ :=
    @Module.equiv_free_prod_directSum _ _ _ _ _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG)
  refine ⟨n, ι, fι, fun i => (p i).natAbs, fun i => ?_, e, ⟨?_⟩⟩
  · rw [← Int.prime_iff_natAbs_prime, ← irreducible_iff_prime]; exact hp i
  exact
    f.toAddEquiv.trans
      ((AddEquiv.refl _).prodCongr <|
        DFinsupp.mapRange.addEquiv fun i =>
          ((Int.quotientSpanEquivZMod _).trans <|
              ZMod.ringEquivCongr <| (p i).natAbs_pow _).toAddEquiv)

/-- **Structure theorem of finite abelian groups** : Any finite abelian group is a direct sum of
some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`. -/
theorem equiv_directSum_zmod_of_finite [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃+ ⨁ i : ι, ZMod (p i ^ e i) := by
  cases nonempty_fintype G
  obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ := equiv_free_prod_directSum_zmod G
  rcases n with - | n
  · have : Unique (Fin Nat.zero →₀ ℤ) :=
      { uniq := by subsingleton }
    exact ⟨ι, fι, p, hp, e, ⟨f.trans AddEquiv.uniqueProd⟩⟩
  · haveI := @Fintype.prodLeft _ _ _ (Fintype.ofEquiv G f.toEquiv) _
    exact
      (Fintype.ofSurjective (fun f : Fin n.succ →₀ ℤ => f 0) fun a =>
            ⟨Finsupp.single 0 a, Finsupp.single_eq_same⟩).false.elim

/-- **Structure theorem of finite abelian groups** : Any finite abelian group is a direct sum of
some `ZMod (n i)` for some natural numbers `n i > 1`. -/
lemma equiv_directSum_zmod_of_finite' (G : Type*) [AddCommGroup G] [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (n : ι → ℕ),
      (∀ i, 1 < n i) ∧ Nonempty (G ≃+ ⨁ i, ZMod (n i)) := by
  classical
  obtain ⟨ι, hι, p, hp, n, ⟨e⟩⟩ := AddCommGroup.equiv_directSum_zmod_of_finite G
  refine ⟨{i : ι // n i ≠ 0}, inferInstance, fun i ↦ p i ^ n i, ?_,
    ⟨e.trans (directSumNeZeroMulEquiv ι _ _).symm⟩⟩
  rintro ⟨i, hi⟩
  exact one_lt_pow₀ (hp _).one_lt hi

theorem finite_of_fg_torsion [hG' : AddGroup.FG G] (hG : AddMonoid.IsTorsion G) : Finite G :=
  @Module.finite_of_fg_torsion _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG') <|
    AddMonoid.isTorsion_iff_isTorsion_int.mp hG

end AddCommGroup

namespace CommGroup

theorem finite_of_fg_torsion [CommGroup G] [Group.FG G] (hG : Monoid.IsTorsion G) : Finite G :=
  @Finite.of_equiv _ _ (AddCommGroup.finite_of_fg_torsion (Additive G) hG) Multiplicative.ofAdd

/-- The **Structure Theorem For Finite Abelian Groups** in a multiplicative version:
A finite commutative group `G` is isomorphic to a finite product of finite cyclic groups. -/
theorem equiv_prod_multiplicative_zmod_of_finite (G : Type*) [CommGroup G] [Finite G] :
    ∃ (ι : Type) (_ : Fintype ι) (n : ι → ℕ),
       (∀ (i : ι), 1 < n i) ∧ Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i)))) := by
  obtain ⟨ι, inst, n, h₁, h₂⟩ := AddCommGroup.equiv_directSum_zmod_of_finite' (Additive G)
  exact ⟨ι, inst, n, h₁, ⟨MulEquiv.toAdditive.symm <| h₂.some.trans <|
    (DirectSum.addEquivProd _).trans (MulEquiv.piMultiplicative _).toAdditiveRight⟩⟩

/-- The **Structure theorem of finitely generated abelian groups** in a multiplicative version :
    Any finitely generated abelian group is the product of a power of `ℤ`
    and a direct product of some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`. -/

Target:
theorem equiv_free_prod_prod_multiplicative_zmod (G : Type*) [CommGroup G] [hG : Group.FG G] :
    ∃ (ι j : Type) (_ : Fintype ι) (_ : Fintype j) (p : ι → ℕ)
    (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
      Nonempty <| G ≃* (j → Multiplicative ℤ) × ((i : ι) → Multiplicative (ZMod (p i ^ e i))) :=

Failed candidate:
by
  have __repair_fefde0126f : False := True.intro
  obtain ⟨n, ι, inst, x, p, e, equiv⟩ := AddCommGroup.equiv_free_prod_directSum_zmod (Additive G)
  exact ⟨ι, Fin n, inst, inferInstance, x, p, e, ⟨MulEquiv.toAdditive.symm <| equiv.some.trans <|
    ((Finsupp.addEquivFunOnFinite.trans <| ((AddEquiv.piAdditive _).trans <|
        (AddEquiv.additiveMultiplicative ℤ).arrowCongr (Equiv.refl _)).symm).prodCongr
          (DirectSum.addEquivProd _ )).trans <| (AddEquiv.prodAdditive _ _).symm⟩⟩

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:190:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:71:2: warning: declaration uses `sorry`
<LEAN_FILE>:79:43: error: unsolved goals
M : Type u_1
α : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Monoid M
inst✝ : MulAction M α
⊢ False

Response:
by
  simp only [dense_iff_inter_open, inter_iUnion, nonempty_iUnion, ← image_inter_nonempty_iff]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h.1 h₃ h₄ h₁ h₂, fun h ↦ ⟨fun h₁ h₂ h₃ h₄ ↦ h h₃ h₄ _ h₁ h₂⟩⟩

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

Imports:
public import Mathlib.Dynamics.Minimal

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Daniel Figueroa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Figueroa
-/
/-!
# Topologically transitive monoid actions

In this file we define an action of a monoid `M` on a topological space `α` to be
*topologically transitive* if for any pair of nonempty open sets `U` and `V` in `α` there exists an
`m : M` such that `(m • U) ∩ V` is nonempty. We also provide an additive version of this definition
and prove basic facts about topologically transitive actions.

## Tags

group action, topologically transitive
-/

public section

open scoped Pointwise

/-- An action of an additive monoid `M` on a topological space `α` is called
*topologically transitive* if for any pair of nonempty open sets `U` and `V` in `α` there exists an
`m : M` such that `(m +ᵥ U) ∩ V` is nonempty. -/
class AddAction.IsTopologicallyTransitive (M α : Type*) [AddMonoid M] [TopologicalSpace α]
    [AddAction M α] : Prop where
  exists_vadd_inter : ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V → V.Nonempty →
    ∃ m : M, ((m +ᵥ U) ∩ V).Nonempty

/-- An action of a monoid `M` on a topological space `α` is called *topologically transitive* if for
any pair of nonempty open sets `U` and `V` in `α` there exists an `m : M` such that `(m • U) ∩ V` is
nonempty. -/
@[to_additive]
class MulAction.IsTopologicallyTransitive (M α : Type*) [Monoid M] [TopologicalSpace α]
    [MulAction M α] : Prop where
  exists_smul_inter : ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V → V.Nonempty →
    ∃ m : M, ((m • U) ∩ V).Nonempty

open MulAction Set

variable (M : Type*) {α : Type*} [TopologicalSpace α] [Monoid M] [MulAction M α]

section IsTopologicallyTransitive

@[to_additive]
theorem MulAction.isTopologicallyTransitive_iff :
    IsTopologicallyTransitive M α ↔ ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V →
    V.Nonempty → ∃ m : M, ((m • U) ∩ V).Nonempty := ⟨fun h ↦ h.1, fun h ↦ ⟨h⟩⟩

/-- An action of a monoid `M` on `α` is topologically transitive if and only if for any nonempty
open subset `U` of `α` the union over the elements of `M` of images of `U` is dense in `α`. -/
@[to_additive /-- An action of an additive monoid `M` on `α` is topologically transitive if and only
if for any nonempty open subset `U` of `α` the union over the elements of `M` of images of `U` is
dense in `α`. -/]
theorem MulAction.isTopologicallyTransitive_iff_dense_iUnion :
    IsTopologicallyTransitive M α ↔
    ∀ {U : Set α}, IsOpen U → U.Nonempty → Dense (⋃ m : M, m • U) := by
  simp only [isTopologicallyTransitive_iff, inter_comm, dense_iff_inter_open, inter_iUnion,
    nonempty_iUnion]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h h₁ h₂ h₃ h₄, fun h _ _ h₁ h₂ h₃ h₄ ↦ h h₁ h₂ _ h₃ h₄⟩

/-- An action of a monoid `M` on `α` is topologically transitive if and only if for any nonempty
open subset `U` of `α` the union of the preimages of `U` over the elements of `M` is dense in `α`.
-/
@[to_additive /-- An action of an additive monoid `M` on `α` is topologically transitive if and only
if for any nonempty open subset `U` of `α` the union of the preimages of `U` over the elements of
`M` is dense in `α`. -/]

Target:
theorem MulAction.isTopologicallyTransitive_iff_dense_iUnion_preimage :
    IsTopologicallyTransitive M α ↔
    ∀ {U : Set α}, IsOpen U → U.Nonempty → Dense (⋃ m : M, (m • ·) ⁻¹' U) :=

Failed candidate:
by
  have __repair_plan_318c4c9b84 : False := by
    skip
  simp only [dense_iff_inter_open, inter_iUnion, nonempty_iUnion, ← image_inter_nonempty_iff]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h.1 h₃ h₄ h₁ h₂, fun h ↦ ⟨fun h₁ h₂ h₃ h₄ ↦ h h₃ h₄ _ h₁ h₂⟩⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:71:2: warning: declaration uses `sorry`\n<LEAN_FILE>:79:43: error: unsolved goals\nM : Type u_1\nα : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp only [dense_iff_inter_open, inter_iUnion, nonempty_iUnion, ← image_inter_nonempty_iff]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h.1 h₃ h₄ h₁ h₂, fun h ↦ ⟨fun h₁ h₂ h₃ h₄ ↦ h h₃ h₄ _ h₁ h₂⟩⟩

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

Imports:
public import Mathlib.Dynamics.Minimal

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Daniel Figueroa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Figueroa
-/
/-!
# Topologically transitive monoid actions

In this file we define an action of a monoid `M` on a topological space `α` to be
*topologically transitive* if for any pair of nonempty open sets `U` and `V` in `α` there exists an
`m : M` such that `(m • U) ∩ V` is nonempty. We also provide an additive version of this definition
and prove basic facts about topologically transitive actions.

## Tags

group action, topologically transitive
-/

public section

open scoped Pointwise

/-- An action of an additive monoid `M` on a topological space `α` is called
*topologically transitive* if for any pair of nonempty open sets `U` and `V` in `α` there exists an
`m : M` such that `(m +ᵥ U) ∩ V` is nonempty. -/
class AddAction.IsTopologicallyTransitive (M α : Type*) [AddMonoid M] [TopologicalSpace α]
    [AddAction M α] : Prop where
  exists_vadd_inter : ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V → V.Nonempty →
    ∃ m : M, ((m +ᵥ U) ∩ V).Nonempty

/-- An action of a monoid `M` on a topological space `α` is called *topologically transitive* if for
any pair of nonempty open sets `U` and `V` in `α` there exists an `m : M` such that `(m • U) ∩ V` is
nonempty. -/
@[to_additive]
class MulAction.IsTopologicallyTransitive (M α : Type*) [Monoid M] [TopologicalSpace α]
    [MulAction M α] : Prop where
  exists_smul_inter : ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V → V.Nonempty →
    ∃ m : M, ((m • U) ∩ V).Nonempty

open MulAction Set

variable (M : Type*) {α : Type*} [TopologicalSpace α] [Monoid M] [MulAction M α]

section IsTopologicallyTransitive

@[to_additive]
theorem MulAction.isTopologicallyTransitive_iff :
    IsTopologicallyTransitive M α ↔ ∀ {U V : Set α}, IsOpen U → U.Nonempty → IsOpen V →
    V.Nonempty → ∃ m : M, ((m • U) ∩ V).Nonempty := ⟨fun h ↦ h.1, fun h ↦ ⟨h⟩⟩

/-- An action of a monoid `M` on `α` is topologically transitive if and only if for any nonempty
open subset `U` of `α` the union over the elements of `M` of images of `U` is dense in `α`. -/
@[to_additive /-- An action of an additive monoid `M` on `α` is topologically transitive if and only
if for any nonempty open subset `U` of `α` the union over the elements of `M` of images of `U` is
dense in `α`. -/]
theorem MulAction.isTopologicallyTransitive_iff_dense_iUnion :
    IsTopologicallyTransitive M α ↔
    ∀ {U : Set α}, IsOpen U → U.Nonempty → Dense (⋃ m : M, m • U) := by
  simp only [isTopologicallyTransitive_iff, inter_comm, dense_iff_inter_open, inter_iUnion,
    nonempty_iUnion]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h h₁ h₂ h₃ h₄, fun h _ _ h₁ h₂ h₃ h₄ ↦ h h₁ h₂ _ h₃ h₄⟩

/-- An action of a monoid `M` on `α` is topologically transitive if and only if for any nonempty
open subset `U` of `α` the union of the preimages of `U` over the elements of `M` is dense in `α`.
-/
@[to_additive /-- An action of an additive monoid `M` on `α` is topologically transitive if and only
if for any nonempty open subset `U` of `α` the union of the preimages of `U` over the elements of
`M` is dense in `α`. -/]

Target:
theorem MulAction.isTopologicallyTransitive_iff_dense_iUnion_preimage :
    IsTopologicallyTransitive M α ↔
    ∀ {U : Set α}, IsOpen U → U.Nonempty → Dense (⋃ m : M, (m • ·) ⁻¹' U) :=

Failed candidate:
by
  have __repair_plan_318c4c9b84 : False := by
    skip
  simp only [dense_iff_inter_open, inter_iUnion, nonempty_iUnion, ← image_inter_nonempty_iff]
  exact ⟨fun h _ h₁ h₂ _ h₃ h₄ ↦ h.1 h₃ h₄ h₁ h₂, fun h ↦ ⟨fun h₁ h₂ h₃ h₄ ↦ h h₃ h₄ _ h₁ h₂⟩⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:71:2: warning: declaration uses `sorry`\n<LEAN_FILE>:79:43: error: unsolved goals\nM : Type u_1\nα : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\n⊢ False", "returncode": 1}

Corrected proof:


Response:
by
  simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
  tauto

[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.Data.Fintype.Powerset
public import Mathlib.Topology.Constructions
public import Mathlib.Topology.GDelta.Basic
public import Mathlib.Topology.Maps.OpenQuotient
public import Mathlib.Tactic.CrossRefAttribute

Namespace:
(root)

Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
/-!
# Baire spaces

A topological space is called a *Baire space*
if a countable intersection of dense open subsets is dense.
Baire theorems say that all completely metrizable spaces
and all locally compact regular spaces are Baire spaces.
We prove the theorems in `Mathlib/Topology/Baire/CompleteMetrizable`
and `Mathlib/Topology/Baire/LocallyCompactRegular`.

In this file we prove some lemmas about Baire spaces.

The good concept underlying the theorems is that of a Gδ set, i.e., a countable intersection
of open sets. Then Baire theorem can also be formulated as the fact that a countable
intersection of dense Gδ sets is a dense Gδ set. We deduce this version from Baire property.
We also prove the important consequence that, if the space is
covered by a countable union of closed sets, then the union of their interiors is dense.

We also prove that in Baire spaces, the `residual` sets are exactly those containing a dense Gδ set.
-/

public section

noncomputable section

open scoped Topology
open Filter Set TopologicalSpace

variable {X α : Type*} {ι : Sort*}

section BaireTheorem

variable [TopologicalSpace X]

/-- The intersection of finitely many open dense sets is dense. -/
theorem Set.Finite.dense_sInter {s : Set (Set X)} (hs : s.Finite)
    (ho : ∀ t ∈ s, IsOpen t) (hd : ∀ t ∈ s, Dense t) : Dense (⋂₀ s) := by
  induction s, hs using Set.Finite.induction_on with
  | empty => simp [sInter_empty]
  | insert ha hsf ih =>
    simp only [sInter_insert, forall_mem_insert] at hd ⊢
    refine hd.1.inter_of_isOpen_right ?_ (hsf.isOpen_sInter (fun y hy => ho y (Or.inr hy)))
    exact ih ((fun y hy => ho y (Or.inr hy))) (fun y hy => hd.2 y hy)

/-- A finite set is Baire. -/
theorem baire_of_finite [Finite X] : BaireSpace X where
  baire_property f _ _ := sInter_range f ▸ (toFinite (range f)).dense_sInter (by grind) (by grind)

variable [BaireSpace X]

/-- Definition of a Baire space. -/
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
    (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
  BaireSpace.baire_property f ho hd

/-- A dense Gδ subset of a Baire space is Baire. -/
theorem IsGδ.baireSpace_of_dense {s : Set X} (hG : IsGδ s) (hd : Dense s) : BaireSpace s := by
  constructor
  intro f hof hdf
  obtain ⟨V, hV⟩ : ∃ V : ℕ → Set X, (∀ n, IsOpen (V n)) ∧ s = ⋂ n, V n := eq_iInter_nat hG
  choose g hg1 hg2 hg3 using fun n => exists_open_dense_of_open_dense_subtype hd (hof n) (hdf n)
  have h_inter_dense : Dense (⋂ n, g n ∩ V n) := BaireSpace.baire_property (fun n ↦ g n ∩ V n)
    (fun n => (hg1 n).inter (hV.1 n))
    (fun n => (hg2 n).inter_of_isOpen_left (hd.mono (by simp [hV.2, iInter_subset])) (hg1 n))
  have h_inter_eq : ⋂ n, g n ∩ V n = ⋂ n, f n := by ext; simp_all; grind
  exact Subtype.dense_iff.mpr fun a _ ↦ h_inter_eq ▸ h_inter_dense a

/-- If `p : Y → X` is an open embedding and `X` is a Baire space, then `Y` is a Baire space. -/
theorem Topology.IsOpenEmbedding.baireSpace {Y : Type*} [TopologicalSpace Y] {p : Y → X}
    (hp : Topology.IsOpenEmbedding p) : BaireSpace Y := by
  constructor
  intro f hof hdf
  let s := range p
  let c := fun n : ℕ => p '' f n ∪ (closure s)ᶜ
  have c_open (n : ℕ) : IsOpen (c n) := IsOpen.union (hp.isOpenMap (f n) (hof n))
    isClosed_closure.isOpen_compl
  have c_dense (n : ℕ) : Dense (c n) := by
    rw [dense_iff_closure_eq, subset_antisymm_iff]
    have : univ ⊆ closure (c n) := calc
      _ ⊆ (interior (closure s)) ∪ (interior (closure s))ᶜ := by grind
      _ ⊆ closure s ∪ (interior (closure s))ᶜ := by gcongr; exact interior_subset
      _ ⊆ closure (p '' f n) ∪ (interior (closure s))ᶜ := union_subset_union
          (closure_minimal (hp.continuous.range_subset_closure_image_dense (hdf n))
          isClosed_closure) (subset_refl (interior (closure s))ᶜ)
      _ ⊆ closure (p '' f n) ∪ closure ((closure s)ᶜ) := union_subset_union (by simp) (by simp)
      _ = closure (c n) := closure_union.symm
    grind
  have c_inter_dense : Dense (⋂ n, c n) := dense_iInter_of_isOpen_nat c_open c_dense
  have c_inter_eq : ⋂ n, f n = p ⁻¹' (⋂ n, c n) := by
    ext x
    simp only [mem_iInter, mem_preimage, mem_union, mem_compl_iff, c]
    refine ⟨fun h i => by grind, fun h i => ?_⟩
    exact hp.injective.mem_set_image.mp (imp_iff_or_not.mpr (h i)
      (subset_closure (mem_range_self x)))
  exact c_inter_eq ▸ Dense.preimage c_inter_dense hp.isOpenMap

/-- An open subset of a Baire space is Baire. -/
theorem IsOpen.baireSpace {s : Set X} (hO : IsOpen s) : BaireSpace s :=
  hO.isOpenEmbedding_subtypeVal.baireSpace

/-- If `f` is an open quotient map and `X` is Baire, then `Y` is Baire. -/
theorem IsOpenQuotientMap.baireSpace {Y : Type*} [TopologicalSpace Y] {f : X → Y}
    (hf : IsOpenQuotientMap f) : BaireSpace Y := by
  constructor
  intro u hou hdu
  have := dense_iInter_of_isOpen_nat (fun n => hf.continuous.isOpen_preimage (u n) (hou n))
    (fun n => (IsOpenQuotientMap.dense_preimage_iff hf).mpr (hdu n))
  simp_all [← preimage_iInter, IsOpenQuotientMap.dense_preimage_iff]

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
    (hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
  rcases S.eq_empty_or_nonempty with h | h
  · simp [h]
  · rcases hS.exists_eq_range h with ⟨f, rfl⟩
    exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
    (hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
  rw [← sInter_image]
  refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable type. -/
@[wikidata Q1052678]
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
    (hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
  dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)

/-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
  constructor
  · rw [mem_residual_iff]
    rintro ⟨S, hSo, hSd, Sct, Ss⟩
    refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
    exact dense_sInter_of_isOpen hSo Sct hSd
  rintro ⟨t, ts, ho, hd⟩
  exact mem_of_superset (residual_of_dense_Gδ ho hd) ts

/-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/

Target:
theorem eventually_residual {p : X → Prop} :
    (∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x :=

Proof body:

[Assistant]
by
  simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
  tauto

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.Data.Fintype.Powerset
public import Mathlib.Topology.Constructions
public import Mathlib.Topology.GDelta.Basic
public import Mathlib.Topology.Maps.OpenQuotient
public import Mathlib.Tactic.CrossRefAttribute

Namespace:
(root)

Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
/-!
# Baire spaces

A topological space is called a *Baire space*
if a countable intersection of dense open subsets is dense.
Baire theorems say that all completely metrizable spaces
and all locally compact regular spaces are Baire spaces.
We prove the theorems in `Mathlib/Topology/Baire/CompleteMetrizable`
and `Mathlib/Topology/Baire/LocallyCompactRegular`.

In this file we prove some lemmas about Baire spaces.

The good concept underlying the theorems is that of a Gδ set, i.e., a countable intersection
of open sets. Then Baire theorem can also be formulated as the fact that a countable
intersection of dense Gδ sets is a dense Gδ set. We deduce this version from Baire property.
We also prove the important consequence that, if the space is
covered by a countable union of closed sets, then the union of their interiors is dense.

We also prove that in Baire spaces, the `residual` sets are exactly those containing a dense Gδ set.
-/

public section

noncomputable section

open scoped Topology
open Filter Set TopologicalSpace

variable {X α : Type*} {ι : Sort*}

section BaireTheorem

variable [TopologicalSpace X]

/-- The intersection of finitely many open dense sets is dense. -/
theorem Set.Finite.dense_sInter {s : Set (Set X)} (hs : s.Finite)
    (ho : ∀ t ∈ s, IsOpen t) (hd : ∀ t ∈ s, Dense t) : Dense (⋂₀ s) := by
  induction s, hs using Set.Finite.induction_on with
  | empty => simp [sInter_empty]
  | insert ha hsf ih =>
    simp only [sInter_insert, forall_mem_insert] at hd ⊢
    refine hd.1.inter_of_isOpen_right ?_ (hsf.isOpen_sInter (fun y hy => ho y (Or.inr hy)))
    exact ih ((fun y hy => ho y (Or.inr hy))) (fun y hy => hd.2 y hy)

/-- A finite set is Baire. -/
theorem baire_of_finite [Finite X] : BaireSpace X where
  baire_property f _ _ := sInter_range f ▸ (toFinite (range f)).dense_sInter (by grind) (by grind)

variable [BaireSpace X]

/-- Definition of a Baire space. -/
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
    (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
  BaireSpace.baire_property f ho hd

/-- A dense Gδ subset of a Baire space is Baire. -/
theorem IsGδ.baireSpace_of_dense {s : Set X} (hG : IsGδ s) (hd : Dense s) : BaireSpace s := by
  constructor
  intro f hof hdf
  obtain ⟨V, hV⟩ : ∃ V : ℕ → Set X, (∀ n, IsOpen (V n)) ∧ s = ⋂ n, V n := eq_iInter_nat hG
  choose g hg1 hg2 hg3 using fun n => exists_open_dense_of_open_dense_subtype hd (hof n) (hdf n)
  have h_inter_dense : Dense (⋂ n, g n ∩ V n) := BaireSpace.baire_property (fun n ↦ g n ∩ V n)
    (fun n => (hg1 n).inter (hV.1 n))
    (fun n => (hg2 n).inter_of_isOpen_left (hd.mono (by simp [hV.2, iInter_subset])) (hg1 n))
  have h_inter_eq : ⋂ n, g n ∩ V n = ⋂ n, f n := by ext; simp_all; grind
  exact Subtype.dense_iff.mpr fun a _ ↦ h_inter_eq ▸ h_inter_dense a

/-- If `p : Y → X` is an open embedding and `X` is a Baire space, then `Y` is a Baire space. -/
theorem Topology.IsOpenEmbedding.baireSpace {Y : Type*} [TopologicalSpace Y] {p : Y → X}
    (hp : Topology.IsOpenEmbedding p) : BaireSpace Y := by
  constructor
  intro f hof hdf
  let s := range p
  let c := fun n : ℕ => p '' f n ∪ (closure s)ᶜ
  have c_open (n : ℕ) : IsOpen (c n) := IsOpen.union (hp.isOpenMap (f n) (hof n))
    isClosed_closure.isOpen_compl
  have c_dense (n : ℕ) : Dense (c n) := by
    rw [dense_iff_closure_eq, subset_antisymm_iff]
    have : univ ⊆ closure (c n) := calc
      _ ⊆ (interior (closure s)) ∪ (interior (closure s))ᶜ := by grind
      _ ⊆ closure s ∪ (interior (closure s))ᶜ := by gcongr; exact interior_subset
      _ ⊆ closure (p '' f n) ∪ (interior (closure s))ᶜ := union_subset_union
          (closure_minimal (hp.continuous.range_subset_closure_image_dense (hdf n))
          isClosed_closure) (subset_refl (interior (closure s))ᶜ)
      _ ⊆ closure (p '' f n) ∪ closure ((closure s)ᶜ) := union_subset_union (by simp) (by simp)
      _ = closure (c n) := closure_union.symm
    grind
  have c_inter_dense : Dense (⋂ n, c n) := dense_iInter_of_isOpen_nat c_open c_dense
  have c_inter_eq : ⋂ n, f n = p ⁻¹' (⋂ n, c n) := by
    ext x
    simp only [mem_iInter, mem_preimage, mem_union, mem_compl_iff, c]
    refine ⟨fun h i => by grind, fun h i => ?_⟩
    exact hp.injective.mem_set_image.mp (imp_iff_or_not.mpr (h i)
      (subset_closure (mem_range_self x)))
  exact c_inter_eq ▸ Dense.preimage c_inter_dense hp.isOpenMap

/-- An open subset of a Baire space is Baire. -/
theorem IsOpen.baireSpace {s : Set X} (hO : IsOpen s) : BaireSpace s :=
  hO.isOpenEmbedding_subtypeVal.baireSpace

/-- If `f` is an open quotient map and `X` is Baire, then `Y` is Baire. -/
theorem IsOpenQuotientMap.baireSpace {Y : Type*} [TopologicalSpace Y] {f : X → Y}
    (hf : IsOpenQuotientMap f) : BaireSpace Y := by
  constructor
  intro u hou hdu
  have := dense_iInter_of_isOpen_nat (fun n => hf.continuous.isOpen_preimage (u n) (hou n))
    (fun n => (IsOpenQuotientMap.dense_preimage_iff hf).mpr (hdu n))
  simp_all [← preimage_iInter, IsOpenQuotientMap.dense_preimage_iff]

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
    (hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
  rcases S.eq_empty_or_nonempty with h | h
  · simp [h]
  · rcases hS.exists_eq_range h with ⟨f, rfl⟩
    exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
    (hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
  rw [← sInter_image]
  refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]

/-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable type. -/
@[wikidata Q1052678]
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
    (hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
  dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)

/-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
  constructor
  · rw [mem_residual_iff]
    rintro ⟨S, hSo, hSd, Sct, Ss⟩
    refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
    exact dense_sInter_of_isOpen hSo Sct hSd
  rintro ⟨t, ts, ho, hd⟩
  exact mem_of_superset (residual_of_dense_Gδ ho hd) ts

/-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/

Target:
theorem eventually_residual {p : X → Prop} :
    (∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x :=

Proof body:


<LEAN_FILE>:125:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_5001599314`
<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_pad_0_5001599314 : True := by trivial
  have __repair_pad_1_5001599314 : True := by trivial
  have __repair_pad_2_5001599314 : True := by trivial
  have __repair_pad_3_5001599314 : True := by trivial
  have __repair_pad_4_5001599314 : True := by trivial
  have __repair_pad_5_5001599314 : True := by trivial
  have __repair_pad_6_5001599314 : True := by trivial
  have __repair_pad_7_5001599314 : True := by trivial
  have __repair_pad_8_5001599314 : True := by trivial
  have __repair_pad_9_5001599314 : True := by trivial
  have __repair_pad_10_5001599314 : True := by trivial
  have __repair_pad_11_5001599314 : True := by trivial
  have __repair_pad_12_5001599314 : True := by trivial
  have __repair_pad_13_5001599314 : True := by trivial
  have __repair_pad_14_5001599314 : True := by trivial
  have __repair_pad_15_5001599314 : True := by trivial
  have __repair_pad_16_5001599314 : True := by trivial
  have __repair_pad_17_5001599314 : True := by trivial
  have __repair_pad_18_5001599314 : True := by trivial
  have __repair_pad_19_5001599314 : True := by trivial
  have __repair_pad_20_5001599314 : True := by trivial
  have __repair_pad_21_5001599314 : True := by trivial
  have __repair_pad_22_5001599314 : True := by trivial
  have __repair_pad_23_5001599314 : True := by trivial
  have __repair_fail_5001599314 := __oprover_overlong_5001599314
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:125:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_5001599314`\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_pad_0_5001599314 : True := by trivial
  have __repair_pad_1_5001599314 : True := by trivial
  have __repair_pad_2_5001599314 : True := by trivial
  have __repair_pad_3_5001599314 : True := by trivial
  have __repair_pad_4_5001599314 : True := by trivial
  have __repair_pad_5_5001599314 : True := by trivial
  have __repair_pad_6_5001599314 : True := by trivial
  have __repair_pad_7_5001599314 : True := by trivial
  have __repair_pad_8_5001599314 : True := by trivial
  have __repair_pad_9_5001599314 : True := by trivial
  have __repair_pad_10_5001599314 : True := by trivial
  have __repair_pad_11_5001599314 : True := by trivial
  have __repair_pad_12_5001599314 : True := by trivial
  have __repair_pad_13_5001599314 : True := by trivial
  have __repair_pad_14_5001599314 : True := by trivial
  have __repair_pad_15_5001599314 : True := by trivial
  have __repair_pad_16_5001599314 : True := by trivial
  have __repair_pad_17_5001599314 : True := by trivial
  have __repair_pad_18_5001599314 : True := by trivial
  have __repair_pad_19_5001599314 : True := by trivial
  have __repair_pad_20_5001599314 : True := by trivial
  have __repair_pad_21_5001599314 : True := by trivial
  have __repair_pad_22_5001599314 : True := by trivial
  have __repair_pad_23_5001599314 : True := by trivial
  have __repair_fail_5001599314 := __oprover_overlong_5001599314
  simpa [Algebra.TensorProduct.one_def] using mul_antipode_rTensor_comul_apply (R := R) (1 : A)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:125:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_5001599314`\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:


<LEAN_FILE>:203:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 4019206) 0
in the target expression
  IsNilpotent (x - y)

R : Type u_1
x y : R
inst✝ : Ring R
h_comm : Commute x y
hx : IsNilpotent x
hy : IsNilpotent y
⊢ IsNilpotent (x - y)

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

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

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

Namespace:
Commute

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

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

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

## Main definitions

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

-/

public section

universe u v

open Function Set

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

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

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

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

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

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

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

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

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

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

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

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

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

alias IsNilpotent.eq_zero_of_isIdempotentElem := IsIdempotentElem.eq_zero_of_isNilpotent

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

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

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

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

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

namespace Commute

section Semiring

variable [Semiring R]

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

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

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

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

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

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

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

end Semiring

section Ring

variable [Ring R]

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4019206) 0) = ((fun x : Nat => x + 4019206) 0) by rfl]
  rw [← neg_right_iff] at h_comm
  rw [← isNilpotent_neg_iff] at hy
  rw [sub_eq_add_neg]
  exact h_comm.isNilpotent_add hx hy

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:203:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4019206) 0\nin the target expression\n  IsNilpotent (x - y)\n\nR : Type u_1\nx y : R\ninst✝ : Ring R\nh_comm : Commute x y\nhx : IsNilpotent x\nhy : IsNilpotent y\n⊢ IsNilpotent (x - y)", "returncode": 1}

Corrected proof:

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

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

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

Namespace:
Commute

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

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

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

## Main definitions

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

-/

public section

universe u v

open Function Set

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

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

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

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

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

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

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

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

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

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

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

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

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

alias IsNilpotent.eq_zero_of_isIdempotentElem := IsIdempotentElem.eq_zero_of_isNilpotent

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

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

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

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

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

namespace Commute

section Semiring

variable [Semiring R]

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

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

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

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

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

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

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

end Semiring

section Ring

variable [Ring R]

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 4019206) 0) = ((fun x : Nat => x + 4019206) 0) by rfl]
  rw [← neg_right_iff] at h_comm
  rw [← isNilpotent_neg_iff] at hy
  rw [sub_eq_add_neg]
  exact h_comm.isNilpotent_add hx hy

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:203:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 4019206) 0\nin the target expression\n  IsNilpotent (x - y)\n\nR : Type u_1\nx y : R\ninst✝ : Ring R\nh_comm : Commute x y\nhx : IsNilpotent x\nhy : IsNilpotent y\n⊢ IsNilpotent (x - y)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:88:43: 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
⊢ False

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_plan_487c519b7a : False := by
    skip
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:88:43: 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⊢ False", "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_plan_487c519b7a : False := by
    skip
  rw [← comp_bitraverse]
  simp only [Function.comp_def, map_pure]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:88:43: 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⊢ False", "returncode": 1}

Corrected proof: