<LEAN_FILE>:133:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_3c3d95964d`
<LEAN_FILE>:108:0: error: unsolved goals
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
⊢ IntermediateField.fixedField ⊤ = ⊥

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

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

Namespace:
InfiniteGalois

Local safe 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)) = ⊥ :=

Failed candidate:
by
  have __repair_pad_0_3c3d95964d : True := by trivial
  have __repair_pad_1_3c3d95964d : True := by trivial
  have __repair_pad_2_3c3d95964d : True := by trivial
  have __repair_pad_3_3c3d95964d : True := by trivial
  have __repair_pad_4_3c3d95964d : True := by trivial
  have __repair_pad_5_3c3d95964d : True := by trivial
  have __repair_pad_6_3c3d95964d : True := by trivial
  have __repair_pad_7_3c3d95964d : True := by trivial
  have __repair_pad_8_3c3d95964d : True := by trivial
  have __repair_pad_9_3c3d95964d : True := by trivial
  have __repair_pad_10_3c3d95964d : True := by trivial
  have __repair_pad_11_3c3d95964d : True := by trivial
  have __repair_pad_12_3c3d95964d : True := by trivial
  have __repair_pad_13_3c3d95964d : True := by trivial
  have __repair_pad_14_3c3d95964d : True := by trivial
  have __repair_pad_15_3c3d95964d : True := by trivial
  have __repair_pad_16_3c3d95964d : True := by trivial
  have __repair_pad_17_3c3d95964d : True := by trivial
  have __repair_pad_18_3c3d95964d : True := by trivial
  have __repair_pad_19_3c3d95964d : True := by trivial
  have __repair_pad_20_3c3d95964d : True := by trivial
  have __repair_pad_21_3c3d95964d : True := by trivial
  have __repair_pad_22_3c3d95964d : True := by trivial
  have __repair_pad_23_3c3d95964d : True := by trivial
  have __repair_fail_3c3d95964d := __oprover_overlong_3c3d95964d
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:133:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_3c3d95964d`\n<LEAN_FILE>:108:0: error: unsolved goals\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\n⊢ IntermediateField.fixedField ⊤ = ⊥", "returncode": 1}

Corrected proof:

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

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

Namespace:
InfiniteGalois

Local safe 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)) = ⊥ :=

Failed candidate:
by
  have __repair_pad_0_3c3d95964d : True := by trivial
  have __repair_pad_1_3c3d95964d : True := by trivial
  have __repair_pad_2_3c3d95964d : True := by trivial
  have __repair_pad_3_3c3d95964d : True := by trivial
  have __repair_pad_4_3c3d95964d : True := by trivial
  have __repair_pad_5_3c3d95964d : True := by trivial
  have __repair_pad_6_3c3d95964d : True := by trivial
  have __repair_pad_7_3c3d95964d : True := by trivial
  have __repair_pad_8_3c3d95964d : True := by trivial
  have __repair_pad_9_3c3d95964d : True := by trivial
  have __repair_pad_10_3c3d95964d : True := by trivial
  have __repair_pad_11_3c3d95964d : True := by trivial
  have __repair_pad_12_3c3d95964d : True := by trivial
  have __repair_pad_13_3c3d95964d : True := by trivial
  have __repair_pad_14_3c3d95964d : True := by trivial
  have __repair_pad_15_3c3d95964d : True := by trivial
  have __repair_pad_16_3c3d95964d : True := by trivial
  have __repair_pad_17_3c3d95964d : True := by trivial
  have __repair_pad_18_3c3d95964d : True := by trivial
  have __repair_pad_19_3c3d95964d : True := by trivial
  have __repair_pad_20_3c3d95964d : True := by trivial
  have __repair_pad_21_3c3d95964d : True := by trivial
  have __repair_pad_22_3c3d95964d : True := by trivial
  have __repair_pad_23_3c3d95964d : True := by trivial
  have __repair_fail_3c3d95964d := __oprover_overlong_3c3d95964d
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:133:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_3c3d95964d`\n<LEAN_FILE>:108:0: error: unsolved goals\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\n⊢ IntermediateField.fixedField ⊤ = ⊥", "returncode": 1}

Corrected proof:


Response:
by
  rw [Topology.IsInducing.continuous_iff (autEmbedding_isClosedEmbedding _).isInducing,
    continuous_pi_iff]
  intro X
  change Continuous fun a ↦ G.mapAut (F.obj X) (autEmbedding F a X)
  fun_prop

/-- If `G` is a fully faithful functor of categories finite types, this is the automorphism of
topological groups `Aut F ≃ Aut (F ⋙ G)`. -/
noncomputable def autEquivAutWhiskerRight {G : FintypeCat.{w} ⥤ FintypeCat.{v}}
    (h : G.FullyFaithful) :
    Aut F ≃ₜ* Aut (F ⋙ G) where
  __ := (h.whiskeringRight C).autMulEquivOfFullyFaithful F
  continuous_toFun := continuous_mapAut_whiskeringRight F G
  continuous_invFun := Continuous.continuous_symm_of_equiv_compact_to_t2
    (f := ((h.whiskeringRight C).autMulEquivOfFullyFaithful F).toEquiv)
    (continuous_mapAut_whiskeringRight F G)

variable [GaloisCategory C] [FiberFunctor F]

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

Imports:
public import Mathlib.CategoryTheory.Galois.Prorepresentability
public import Mathlib.Topology.Algebra.ContinuousMonoidHom
public import Mathlib.Topology.Algebra.Group.Basic

Namespace:
CategoryTheory.PreGaloisCategory

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

# Topology of fundamental group

In this file we define a natural topology on the automorphism group of a functor
`F : C ⥤ FintypeCat`: It is defined as the subspace topology induced by the natural
embedding of `Aut F` into `∀ X, Aut (F.obj X)` where
`Aut (F.obj X)` carries the discrete topology.

## References

- [Stacks 0BMQ](https://stacks.math.columbia.edu/tag/0BMQ)

-/

@[expose] public section

open Topology

universe u₁ u₂ v₁ v₂ v w

namespace CategoryTheory

namespace PreGaloisCategory

open Functor

variable {C : Type u₁} [Category.{u₂} C] (F : C ⥤ FintypeCat.{w})

/-- For a functor `F : C ⥤ FintypeCat`, the canonical embedding of `Aut F` into
the product over `Aut (F.obj X)` for all objects `X`. -/
def autEmbedding : Aut F →* ∀ X, Aut (F.obj X) :=
  MonoidHom.mk' (fun σ X ↦ σ.app X) (fun _ _ ↦ rfl)

@[simp]
lemma autEmbedding_apply (σ : Aut F) (X : C) : autEmbedding F σ X = σ.app X :=
  rfl

lemma autEmbedding_injective : Function.Injective (autEmbedding F) := by
  intro σ τ h
  ext X x
  have : σ.app X = τ.app X := congr_fun h X
  rw [← Iso.app_hom, ← Iso.app_hom, this]

/-- We put the discrete topology on `F.obj X`. -/
scoped instance (X : C) : TopologicalSpace (F.obj X) := ⊥

@[scoped instance]
lemma obj_discreteTopology (X : C) : DiscreteTopology (F.obj X) := ⟨rfl⟩

/-- We put the discrete topology on `Aut (F.obj X)`. -/
scoped instance (X : C) : TopologicalSpace (Aut (F.obj X)) := ⊥

/-- We give `F.obj X  ⟶ F.obj Y` the product topology. -/
@[local simp]
scoped instance {X Y : C} : TopologicalSpace (F.obj X ⟶ F.obj Y) :=
  .coinduced (fun f ↦ ObjectProperty.homMk (↾f)) inferInstance

scoped instance {X Y : C} : DiscreteTopology (F.obj X ⟶ F.obj Y) :=
  ⟨by simp [DiscreteTopology.eq_bot]⟩

@[scoped instance]
lemma aut_discreteTopology (X : C) : DiscreteTopology (Aut (F.obj X)) := ⟨rfl⟩

/-- `Aut F` is equipped with the by the embedding into `∀ X, Aut (F.obj X)` induced embedding. -/
instance : TopologicalSpace (Aut F) :=
  TopologicalSpace.induced (autEmbedding F) inferInstance

/-lemma autEmbedding_range :
    Set.range (autEmbedding F) =
      ⋂ (f : Arrow C), { a | F.map f.hom ≫ (a f.right).hom = (a f.left).hom ≫ F.map f.hom } := by
  ext a
  simp only [Set.mem_range, id_obj, Set.mem_iInter, Set.mem_setOf_eq]
  refine ⟨fun ⟨σ, h⟩ i ↦ h.symm ▸ σ.hom.naturality i.hom, fun h ↦ ?_⟩
  · use NatIso.ofComponents a (fun {X Y} f ↦ h ⟨X, Y, f⟩)
    rfl-/

/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` are precisely the compatible families of
automorphisms. -/
lemma autEmbedding_range :
    Set.range (autEmbedding F) = ⋂ (f : Arrow C), { a | F.map f.hom ≫ (a f.right).hom =
      (a f.left).hom ≫ F.map f.hom } := by
  ext a
  simp +instances only [Set.mem_range, Set.mem_iInter, Set.mem_setOf_eq]
  refine ⟨fun ⟨σ, h⟩ i ↦ by cat_disch, fun h ↦ ?_⟩
  exact ⟨NatIso.ofComponents a (fun {X Y} f ↦ by
    ext; simpa using ConcreteCategory.congr_hom (h ⟨X, Y, f⟩) _), rfl⟩

/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` is closed. -/
lemma autEmbedding_range_isClosed : IsClosed (Set.range (autEmbedding F)) := by
  rw [autEmbedding_range]
  exact isClosed_iInter (fun f ↦ isClosed_eq (by fun_prop) (by fun_prop))

lemma autEmbedding_isClosedEmbedding : IsClosedEmbedding (autEmbedding F) where
  eq_induced := rfl
  injective := autEmbedding_injective F
  isClosed_range := autEmbedding_range_isClosed F

instance : CompactSpace (Aut F) := (autEmbedding_isClosedEmbedding F).compactSpace

instance : T2Space (Aut F) :=
  T2Space.of_injective_continuous (autEmbedding_injective F) continuous_induced_dom

instance : TotallyDisconnectedSpace (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isEmbedding.isTotallyDisconnected_range.mp
    (isTotallyDisconnected_of_totallyDisconnectedSpace _)

instance : ContinuousMul (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isInducing.continuousMul (autEmbedding F)

instance : ContinuousInv (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isInducing.continuousInv fun _ ↦ rfl

instance : IsTopologicalGroup (Aut F) := ⟨⟩

instance (X : C) : SMul (Aut (F.obj X)) (F.obj X) := ⟨fun σ a => σ.hom a⟩

instance (X : C) : ContinuousSMul (Aut (F.obj X)) (F.obj X) := by
  constructor
  fun_prop

instance continuousSMul_aut_fiber (X : C) : ContinuousSMul (Aut F) (F.obj X) where
  continuous_smul := by
    let g : Aut (F.obj X) × F.obj X → F.obj X := fun ⟨σ, x⟩ ↦ σ.hom x
    let h (q : Aut F × F.obj X) : Aut (F.obj X) × F.obj X :=
      ⟨((fun p ↦ p X) ∘ autEmbedding F) q.1, q.2⟩
    change Continuous (g ∘ h)
    fun_prop

/-- If `G` is a functor of categories of finite types, the induced map `Aut F → Aut (F ⋙ G)` is
continuous. -/

Target:
lemma continuous_mapAut_whiskeringRight (G : FintypeCat.{w} ⥤ FintypeCat.{v}) :
    Continuous (((whiskeringRight _ _ _).obj G).mapAut F) :=

Proof body:

[Assistant]
by
  rw [Topology.IsInducing.continuous_iff (autEmbedding_isClosedEmbedding _).isInducing,
    continuous_pi_iff]
  intro X
  change Continuous fun a ↦ G.mapAut (F.obj X) (autEmbedding F a X)
  fun_prop

/-- If `G` is a fully faithful functor of categories finite types, this is the automorphism of
topological groups `Aut F ≃ Aut (F ⋙ G)`. -/
noncomputable def autEquivAutWhiskerRight {G : FintypeCat.{w} ⥤ FintypeCat.{v}}
    (h : G.FullyFaithful) :
    Aut F ≃ₜ* Aut (F ⋙ G) where
  __ := (h.whiskeringRight C).autMulEquivOfFullyFaithful F
  continuous_toFun := continuous_mapAut_whiskeringRight F G
  continuous_invFun := Continuous.continuous_symm_of_equiv_compact_to_t2
    (f := ((h.whiskeringRight C).autMulEquivOfFullyFaithful F).toEquiv)
    (continuous_mapAut_whiskeringRight F G)

variable [GaloisCategory C] [FiberFunctor F]

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

Imports:
public import Mathlib.CategoryTheory.Galois.Prorepresentability
public import Mathlib.Topology.Algebra.ContinuousMonoidHom
public import Mathlib.Topology.Algebra.Group.Basic

Namespace:
CategoryTheory.PreGaloisCategory

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

# Topology of fundamental group

In this file we define a natural topology on the automorphism group of a functor
`F : C ⥤ FintypeCat`: It is defined as the subspace topology induced by the natural
embedding of `Aut F` into `∀ X, Aut (F.obj X)` where
`Aut (F.obj X)` carries the discrete topology.

## References

- [Stacks 0BMQ](https://stacks.math.columbia.edu/tag/0BMQ)

-/

@[expose] public section

open Topology

universe u₁ u₂ v₁ v₂ v w

namespace CategoryTheory

namespace PreGaloisCategory

open Functor

variable {C : Type u₁} [Category.{u₂} C] (F : C ⥤ FintypeCat.{w})

/-- For a functor `F : C ⥤ FintypeCat`, the canonical embedding of `Aut F` into
the product over `Aut (F.obj X)` for all objects `X`. -/
def autEmbedding : Aut F →* ∀ X, Aut (F.obj X) :=
  MonoidHom.mk' (fun σ X ↦ σ.app X) (fun _ _ ↦ rfl)

@[simp]
lemma autEmbedding_apply (σ : Aut F) (X : C) : autEmbedding F σ X = σ.app X :=
  rfl

lemma autEmbedding_injective : Function.Injective (autEmbedding F) := by
  intro σ τ h
  ext X x
  have : σ.app X = τ.app X := congr_fun h X
  rw [← Iso.app_hom, ← Iso.app_hom, this]

/-- We put the discrete topology on `F.obj X`. -/
scoped instance (X : C) : TopologicalSpace (F.obj X) := ⊥

@[scoped instance]
lemma obj_discreteTopology (X : C) : DiscreteTopology (F.obj X) := ⟨rfl⟩

/-- We put the discrete topology on `Aut (F.obj X)`. -/
scoped instance (X : C) : TopologicalSpace (Aut (F.obj X)) := ⊥

/-- We give `F.obj X  ⟶ F.obj Y` the product topology. -/
@[local simp]
scoped instance {X Y : C} : TopologicalSpace (F.obj X ⟶ F.obj Y) :=
  .coinduced (fun f ↦ ObjectProperty.homMk (↾f)) inferInstance

scoped instance {X Y : C} : DiscreteTopology (F.obj X ⟶ F.obj Y) :=
  ⟨by simp [DiscreteTopology.eq_bot]⟩

@[scoped instance]
lemma aut_discreteTopology (X : C) : DiscreteTopology (Aut (F.obj X)) := ⟨rfl⟩

/-- `Aut F` is equipped with the by the embedding into `∀ X, Aut (F.obj X)` induced embedding. -/
instance : TopologicalSpace (Aut F) :=
  TopologicalSpace.induced (autEmbedding F) inferInstance

/-lemma autEmbedding_range :
    Set.range (autEmbedding F) =
      ⋂ (f : Arrow C), { a | F.map f.hom ≫ (a f.right).hom = (a f.left).hom ≫ F.map f.hom } := by
  ext a
  simp only [Set.mem_range, id_obj, Set.mem_iInter, Set.mem_setOf_eq]
  refine ⟨fun ⟨σ, h⟩ i ↦ h.symm ▸ σ.hom.naturality i.hom, fun h ↦ ?_⟩
  · use NatIso.ofComponents a (fun {X Y} f ↦ h ⟨X, Y, f⟩)
    rfl-/

/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` are precisely the compatible families of
automorphisms. -/
lemma autEmbedding_range :
    Set.range (autEmbedding F) = ⋂ (f : Arrow C), { a | F.map f.hom ≫ (a f.right).hom =
      (a f.left).hom ≫ F.map f.hom } := by
  ext a
  simp +instances only [Set.mem_range, Set.mem_iInter, Set.mem_setOf_eq]
  refine ⟨fun ⟨σ, h⟩ i ↦ by cat_disch, fun h ↦ ?_⟩
  exact ⟨NatIso.ofComponents a (fun {X Y} f ↦ by
    ext; simpa using ConcreteCategory.congr_hom (h ⟨X, Y, f⟩) _), rfl⟩

/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` is closed. -/
lemma autEmbedding_range_isClosed : IsClosed (Set.range (autEmbedding F)) := by
  rw [autEmbedding_range]
  exact isClosed_iInter (fun f ↦ isClosed_eq (by fun_prop) (by fun_prop))

lemma autEmbedding_isClosedEmbedding : IsClosedEmbedding (autEmbedding F) where
  eq_induced := rfl
  injective := autEmbedding_injective F
  isClosed_range := autEmbedding_range_isClosed F

instance : CompactSpace (Aut F) := (autEmbedding_isClosedEmbedding F).compactSpace

instance : T2Space (Aut F) :=
  T2Space.of_injective_continuous (autEmbedding_injective F) continuous_induced_dom

instance : TotallyDisconnectedSpace (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isEmbedding.isTotallyDisconnected_range.mp
    (isTotallyDisconnected_of_totallyDisconnectedSpace _)

instance : ContinuousMul (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isInducing.continuousMul (autEmbedding F)

instance : ContinuousInv (Aut F) :=
  (autEmbedding_isClosedEmbedding F).isInducing.continuousInv fun _ ↦ rfl

instance : IsTopologicalGroup (Aut F) := ⟨⟩

instance (X : C) : SMul (Aut (F.obj X)) (F.obj X) := ⟨fun σ a => σ.hom a⟩

instance (X : C) : ContinuousSMul (Aut (F.obj X)) (F.obj X) := by
  constructor
  fun_prop

instance continuousSMul_aut_fiber (X : C) : ContinuousSMul (Aut F) (F.obj X) where
  continuous_smul := by
    let g : Aut (F.obj X) × F.obj X → F.obj X := fun ⟨σ, x⟩ ↦ σ.hom x
    let h (q : Aut F × F.obj X) : Aut (F.obj X) × F.obj X :=
      ⟨((fun p ↦ p X) ∘ autEmbedding F) q.1, q.2⟩
    change Continuous (g ∘ h)
    fun_prop

/-- If `G` is a functor of categories of finite types, the induced map `Aut F → Aut (F ⋙ G)` is
continuous. -/

Target:
lemma continuous_mapAut_whiskeringRight (G : FintypeCat.{w} ⥤ FintypeCat.{v}) :
    Continuous (((whiskeringRight _ _ _).obj G).mapAut F) :=

Proof body:


<LEAN_FILE>:271:43: error: unsolved goals
X : Compactum
A : Set X.A
⊢ False

Response:
by
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

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

Imports:
public import Mathlib.CategoryTheory.Monad.Types
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.Topology.Category.CompHaus.Basic
public import Mathlib.Topology.Category.Profinite.Basic
public import Mathlib.Data.Set.Constructions

Namespace:
Compactum

Local safe context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
/-!

# Compacta and Compact Hausdorff Spaces

Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data:
- A type `X : Type*`
- A "structure" map `M X → X`.

This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category
in an evident way.

See the file `Mathlib/CategoryTheory/Monad/Algebra.lean` for a general version, as well as the
following link.
https://ncatlab.org/nlab/show/monad

This file proves the equivalence between the category of *compact Hausdorff topological spaces*
and the category of algebras for the *ultrafilter monad*.

## Notation:

Here are the main objects introduced in this file.
- `Compactum` is the type of compacta, which we define as algebras for the ultrafilter monad.
- `compactumToCompHaus` is the functor `Compactum ⥤ CompHaus`. Here `CompHaus` is the usual
  category of compact Hausdorff spaces.
- `compactumToCompHaus.isEquivalence` is a term of type `IsEquivalence compactumToCompHaus`.

The proof of this equivalence is a bit technical. But the idea is quite simply that the structure
map `Ultrafilter X → X` for an algebra `X` of the ultrafilter monad should be considered as the map
sending an ultrafilter to its limit in `X`. The topology on `X` is then defined by mimicking the
characterization of open sets in terms of ultrafilters.

Any `X : Compactum` is endowed with a coercion to `Type*`, as well as the following instances:
- `TopologicalSpace X`.
- `CompactSpace X`.
- `T2Space X`.

Any morphism `f : X ⟶ Y` of is endowed with a coercion to a function `X → Y`, which is shown to
be continuous in `continuous_of_hom`.

The function `Compactum.ofTopologicalSpace` can be used to construct a `Compactum` from a
topological space which satisfies `CompactSpace` and `T2Space`.

We also add wrappers around structures which already exist. Here are the main ones, all in the
`Compactum` namespace:

- `forget : Compactum ⥤ Type*` is the forgetful functor, which induces a `ConcreteCategory`
  instance for `Compactum`.
- `free : Type* ⥤ Compactum` is the left adjoint to `forget`, and the adjunction is in `adj`.
- `str : Ultrafilter X → X` is the structure map for `X : Compactum`.
  The notation `X.str` is preferred.
- `join : Ultrafilter (Ultrafilter X) → Ultrafilter X` is the monadic join for `X : Compactum`.
  Again, the notation `X.join` is preferred.
- `incl : X → Ultrafilter X` is the unit for `X : Compactum`. The notation `X.incl` is preferred.

## References

- E. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, 1976.
- https://ncatlab.org/nlab/show/ultrafilter

-/

@[expose] public section

universe u

open CategoryTheory Filter Ultrafilter TopologicalSpace CategoryTheory.Limits FiniteInter
open scoped Topology

local notation "β" => ofTypeMonad Ultrafilter

/-- The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. -/
def Compactum :=
  Monad.Algebra β deriving Category, Inhabited

namespace Compactum

/-- The forgetful functor to TypeCat -/
def forget : Compactum ⥤ Type _ :=
  Monad.forget _

instance : forget.Faithful :=
  show (Monad.forget _).Faithful from inferInstance

noncomputable instance : CreatesLimits forget :=
  show CreatesLimits <| Monad.forget _ from inferInstance

/-- The "free" Compactum functor. -/
def free : Type _ ⥤ Compactum :=
  Monad.free _

/-- The adjunction between `free` and `forget`. -/
def adj : free ⊣ forget :=
  Monad.adj _

instance : CoeSort Compactum Type* :=
  ⟨fun X => X.A⟩

instance {X Y : Compactum} : FunLike (X ⟶ Y) X Y where
  coe f := f.f
  coe_injective _ _ h := (Monad.forget_faithful β).map_injective (by aesop)

-- Basic instances
instance : ConcreteCategory Compactum (· ⟶ ·) where
  hom f := f
  ofHom f := f

instance : HasLimits Compactum :=
  hasLimits_of_hasLimits_createsLimits forget

/-- The structure map for a compactum, essentially sending an ultrafilter to its limit. -/
def str (X : Compactum) : Ultrafilter X → X :=
  X.a

/-- The monadic join. -/
def join (X : Compactum) : Ultrafilter (Ultrafilter X) → Ultrafilter X :=
  (β).μ.app _

/-- The inclusion of `X` into `Ultrafilter X`. -/
def incl (X : Compactum) : X → Ultrafilter X :=
  (β).η.app _

@[simp]
theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by
  change ((β).η.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.unit]
  rfl

@[simp]
theorem str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X) :
    f (X.str xs) = Y.str (map f xs) := by
  change (X.a ≫ f.f) _ = _
  rw [← f.h]
  rfl

@[simp]
theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
    X.str (X.join uux) = X.str (map X.str uux) := by
  change ((β).μ.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.assoc]
  rfl

instance {X : Compactum} : TopologicalSpace X where
  IsOpen U := ∀ F : Ultrafilter X, X.str F ∈ U → U ∈ F
  isOpen_univ _ _ := Filter.univ_sets _
  isOpen_inter _ _ h3 h4 _ h6 := Filter.inter_sets _ (h3 _ h6.1) (h4 _ h6.2)
  isOpen_sUnion := fun _ h1 _ ⟨T, hT, h2⟩ =>
    mem_of_superset (h1 T hT _ h2) (Set.subset_sUnion_of_mem hT)

theorem isClosed_iff {X : Compactum} (S : Set X) :
    IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
  rw [← isOpen_compl_iff]
  constructor
  · intro cond F h
    by_contra c
    specialize cond F c
    rw [compl_mem_iff_notMem] at cond
    contradiction
  · intro h1 F h2
    specialize h1 F
    rcases F.mem_or_compl_mem S with h | h
    exacts [absurd (h1 h) h2, h]

instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine ⟨X.str F, by tauto, ?_⟩
  rw [le_nhds_iff]
  intro S h1 h2
  exact h2 F h1

/-- A local definition used only in the proofs. -/
private def basic {X : Compactum} (A : Set X) : Set (Ultrafilter X) :=
  { F | A ∈ F }

set_option backward.privateInPublic true in
/-- A local definition used only in the proofs. -/
private def cl {X : Compactum} (A : Set X) : Set X :=
  X.str '' basic A

private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · rintro ⟨h1, h2⟩
    exact inter_mem h1 h2

private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha =>
  ⟨X.incl a, ha, by simp⟩

private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine ⟨P, hq2, ?_⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      rcases this P hP with h | h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
    intro P hP
    exact claim5.finiteInterClosure_insert _ hP
  intro T hT
  -- Suffices to show that the intersection of the T's is contained in C2.
  suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this
  -- Finish
  apply claim4.finiteInter_mem T
  intro t ht
  exact finiteInterClosure.basic (@hT t ht)

set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in

Target:
theorem isClosed_cl {X : Compactum} (A : Set X) : IsClosed (cl A) :=

Failed candidate:
by
  have __repair_plan_115e9696b2 : False := by
    skip
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:271:43: error: unsolved goals\nX : Compactum\nA : Set X.A\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

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

Imports:
public import Mathlib.CategoryTheory.Monad.Types
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.Topology.Category.CompHaus.Basic
public import Mathlib.Topology.Category.Profinite.Basic
public import Mathlib.Data.Set.Constructions

Namespace:
Compactum

Local safe context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
/-!

# Compacta and Compact Hausdorff Spaces

Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data:
- A type `X : Type*`
- A "structure" map `M X → X`.

This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category
in an evident way.

See the file `Mathlib/CategoryTheory/Monad/Algebra.lean` for a general version, as well as the
following link.
https://ncatlab.org/nlab/show/monad

This file proves the equivalence between the category of *compact Hausdorff topological spaces*
and the category of algebras for the *ultrafilter monad*.

## Notation:

Here are the main objects introduced in this file.
- `Compactum` is the type of compacta, which we define as algebras for the ultrafilter monad.
- `compactumToCompHaus` is the functor `Compactum ⥤ CompHaus`. Here `CompHaus` is the usual
  category of compact Hausdorff spaces.
- `compactumToCompHaus.isEquivalence` is a term of type `IsEquivalence compactumToCompHaus`.

The proof of this equivalence is a bit technical. But the idea is quite simply that the structure
map `Ultrafilter X → X` for an algebra `X` of the ultrafilter monad should be considered as the map
sending an ultrafilter to its limit in `X`. The topology on `X` is then defined by mimicking the
characterization of open sets in terms of ultrafilters.

Any `X : Compactum` is endowed with a coercion to `Type*`, as well as the following instances:
- `TopologicalSpace X`.
- `CompactSpace X`.
- `T2Space X`.

Any morphism `f : X ⟶ Y` of is endowed with a coercion to a function `X → Y`, which is shown to
be continuous in `continuous_of_hom`.

The function `Compactum.ofTopologicalSpace` can be used to construct a `Compactum` from a
topological space which satisfies `CompactSpace` and `T2Space`.

We also add wrappers around structures which already exist. Here are the main ones, all in the
`Compactum` namespace:

- `forget : Compactum ⥤ Type*` is the forgetful functor, which induces a `ConcreteCategory`
  instance for `Compactum`.
- `free : Type* ⥤ Compactum` is the left adjoint to `forget`, and the adjunction is in `adj`.
- `str : Ultrafilter X → X` is the structure map for `X : Compactum`.
  The notation `X.str` is preferred.
- `join : Ultrafilter (Ultrafilter X) → Ultrafilter X` is the monadic join for `X : Compactum`.
  Again, the notation `X.join` is preferred.
- `incl : X → Ultrafilter X` is the unit for `X : Compactum`. The notation `X.incl` is preferred.

## References

- E. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, 1976.
- https://ncatlab.org/nlab/show/ultrafilter

-/

@[expose] public section

universe u

open CategoryTheory Filter Ultrafilter TopologicalSpace CategoryTheory.Limits FiniteInter
open scoped Topology

local notation "β" => ofTypeMonad Ultrafilter

/-- The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. -/
def Compactum :=
  Monad.Algebra β deriving Category, Inhabited

namespace Compactum

/-- The forgetful functor to TypeCat -/
def forget : Compactum ⥤ Type _ :=
  Monad.forget _

instance : forget.Faithful :=
  show (Monad.forget _).Faithful from inferInstance

noncomputable instance : CreatesLimits forget :=
  show CreatesLimits <| Monad.forget _ from inferInstance

/-- The "free" Compactum functor. -/
def free : Type _ ⥤ Compactum :=
  Monad.free _

/-- The adjunction between `free` and `forget`. -/
def adj : free ⊣ forget :=
  Monad.adj _

instance : CoeSort Compactum Type* :=
  ⟨fun X => X.A⟩

instance {X Y : Compactum} : FunLike (X ⟶ Y) X Y where
  coe f := f.f
  coe_injective _ _ h := (Monad.forget_faithful β).map_injective (by aesop)

-- Basic instances
instance : ConcreteCategory Compactum (· ⟶ ·) where
  hom f := f
  ofHom f := f

instance : HasLimits Compactum :=
  hasLimits_of_hasLimits_createsLimits forget

/-- The structure map for a compactum, essentially sending an ultrafilter to its limit. -/
def str (X : Compactum) : Ultrafilter X → X :=
  X.a

/-- The monadic join. -/
def join (X : Compactum) : Ultrafilter (Ultrafilter X) → Ultrafilter X :=
  (β).μ.app _

/-- The inclusion of `X` into `Ultrafilter X`. -/
def incl (X : Compactum) : X → Ultrafilter X :=
  (β).η.app _

@[simp]
theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by
  change ((β).η.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.unit]
  rfl

@[simp]
theorem str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X) :
    f (X.str xs) = Y.str (map f xs) := by
  change (X.a ≫ f.f) _ = _
  rw [← f.h]
  rfl

@[simp]
theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
    X.str (X.join uux) = X.str (map X.str uux) := by
  change ((β).μ.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.assoc]
  rfl

instance {X : Compactum} : TopologicalSpace X where
  IsOpen U := ∀ F : Ultrafilter X, X.str F ∈ U → U ∈ F
  isOpen_univ _ _ := Filter.univ_sets _
  isOpen_inter _ _ h3 h4 _ h6 := Filter.inter_sets _ (h3 _ h6.1) (h4 _ h6.2)
  isOpen_sUnion := fun _ h1 _ ⟨T, hT, h2⟩ =>
    mem_of_superset (h1 T hT _ h2) (Set.subset_sUnion_of_mem hT)

theorem isClosed_iff {X : Compactum} (S : Set X) :
    IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
  rw [← isOpen_compl_iff]
  constructor
  · intro cond F h
    by_contra c
    specialize cond F c
    rw [compl_mem_iff_notMem] at cond
    contradiction
  · intro h1 F h2
    specialize h1 F
    rcases F.mem_or_compl_mem S with h | h
    exacts [absurd (h1 h) h2, h]

instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine ⟨X.str F, by tauto, ?_⟩
  rw [le_nhds_iff]
  intro S h1 h2
  exact h2 F h1

/-- A local definition used only in the proofs. -/
private def basic {X : Compactum} (A : Set X) : Set (Ultrafilter X) :=
  { F | A ∈ F }

set_option backward.privateInPublic true in
/-- A local definition used only in the proofs. -/
private def cl {X : Compactum} (A : Set X) : Set X :=
  X.str '' basic A

private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · rintro ⟨h1, h2⟩
    exact inter_mem h1 h2

private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha =>
  ⟨X.incl a, ha, by simp⟩

private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine ⟨P, hq2, ?_⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      rcases this P hP with h | h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
    intro P hP
    exact claim5.finiteInterClosure_insert _ hP
  intro T hT
  -- Suffices to show that the intersection of the T's is contained in C2.
  suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this
  -- Finish
  apply claim4.finiteInter_mem T
  intro t ht
  exact finiteInterClosure.basic (@hT t ht)

set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in

Target:
theorem isClosed_cl {X : Compactum} (A : Set X) : IsClosed (cl A) :=

Failed candidate:
by
  have __repair_plan_115e9696b2 : False := by
    skip
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:271:43: error: unsolved goals\nX : Compactum\nA : Set X.A\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:106:43: error: unsolved goals
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
⊢ False

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

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

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

Namespace:
TopCat.Sheaf

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

## Main Results

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

-/

public section

universe u v v₁ v₂

open TopologicalSpace CategoryTheory Limits

namespace TopCat

variable {X : TopCat.{u}}

section

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

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

namespace Sheaf

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

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

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

end Sheaf

end

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

namespace Sheaf

open Presheaf

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

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

open ZeroObject

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

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

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

Failed candidate:
by
  have __repair_plan_fd80ecbe85 : False := by
    skip
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:106:43: error: unsolved goals\nC : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLimits (CategoryTheory.forget C)\ninst✝ : Abelian C\nX : TopCat\nS : ShortComplex (Sheaf C X)\n⊢ False", "returncode": 1}

Corrected proof:

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

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

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

Namespace:
TopCat.Sheaf

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

## Main Results

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

-/

public section

universe u v v₁ v₂

open TopologicalSpace CategoryTheory Limits

namespace TopCat

variable {X : TopCat.{u}}

section

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

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

namespace Sheaf

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

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

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

end Sheaf

end

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

namespace Sheaf

open Presheaf

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

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

open ZeroObject

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

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

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

Failed candidate:
by
  have __repair_plan_fd80ecbe85 : False := by
    skip
  constructor
  · intro h x
    have := (forget C X ⋙ stalkFunctor C x).exact_tfae.out 2 1
    exact this.mp inferInstance S h
  intro h
  simp_rw [ShortComplex.exact_iff_isZero_homology] at h
  rw [ShortComplex.exact_iff_isZero_homology, isZero_iff_stalkFunctor_obj_isZero S.homology]
  exact fun x => (h x).of_iso
    (ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:106:43: error: unsolved goals\nC : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLimits (CategoryTheory.forget C)\ninst✝ : Abelian C\nX : TopCat\nS : ShortComplex (Sheaf C X)\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:77:0: error: unsolved goals
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : AddCommMonoid F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : ContinuousAdd E
inst✝² : ContinuousAdd F
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ Continuous[inst✝⁵, inst✝⁴] (p.partialSum n)

Response:
by
  unfold partialSum
  fun_prop

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

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

Namespace:
FormalMultilinearSeries

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

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

## Main definitions

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

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

## Implementation details

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

@[expose] public section

noncomputable section

variable {𝕜 E F G : Type*}

open Topology NNReal Filter ENNReal Set Asymptotics
open scoped Pointwise

namespace FormalMultilinearSeries

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

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

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

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

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

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

Failed candidate:
by
  have __repair_0c468d26d2 := True.intro.nonexistentProjection
  unfold partialSum
  fun_prop

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:77:0: error: unsolved goals\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : AddCommMonoid F\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : ContinuousAdd E\ninst✝² : ContinuousAdd F\ninst✝¹ : ContinuousConstSMul 𝕜 E\ninst✝ : ContinuousConstSMul 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ Continuous[inst✝⁵, inst✝⁴] (p.partialSum n)", "returncode": 1}

Corrected proof:

[Assistant]
by
  unfold partialSum
  fun_prop

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

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

Namespace:
FormalMultilinearSeries

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

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

## Main definitions

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

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

## Implementation details

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

@[expose] public section

noncomputable section

variable {𝕜 E F G : Type*}

open Topology NNReal Filter ENNReal Set Asymptotics
open scoped Pointwise

namespace FormalMultilinearSeries

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

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

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

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

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

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

Failed candidate:
by
  have __repair_0c468d26d2 := True.intro.nonexistentProjection
  unfold partialSum
  fun_prop

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:77:0: error: unsolved goals\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : AddCommMonoid F\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : ContinuousAdd E\ninst✝² : ContinuousAdd F\ninst✝¹ : ContinuousConstSMul 𝕜 E\ninst✝ : ContinuousConstSMul 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ Continuous[inst✝⁵, inst✝⁴] (p.partialSum n)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:60:43: error: unsolved goals
M : Type u_1
N : Type u_2
inst✝³ : CommMonoidWithZero M
inst✝² : CommMonoidWithZero N
p : M
E : Type u_5
inst✝¹ : EquivLike E M N
inst✝ : MulEquivClass E M N
e : E
⊢ False

Response:
by
  let e := MulEquivClass.toMulEquiv e
  exact ⟨comap_prime e e.symm fun a => by simp,
    fun h => (comap_prime e.symm e fun a => by simp) <| (e.symm_apply_apply p).substr h⟩

[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.Divisibility.Hom
public import Mathlib.Algebra.Group.Irreducible.Lemmas
public import Mathlib.Algebra.GroupWithZero.Equiv
public import Mathlib.Algebra.Prime.Defs
public import Mathlib.Order.Monotone.Defs

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
/-!
# Associated, prime, and irreducible elements.

In this file we define the predicate `Prime p`
saying that an element of a commutative monoid with zero is prime.
Namely, `Prime p` means that `p` isn't zero, it isn't a unit,
and `p ∣ a * b → p ∣ a ∨ p ∣ b` for all `a`, `b`;

In decomposition monoids (e.g., `ℕ`, `ℤ`), this predicate is equivalent to `Irreducible`,
however this is not true in general.

We also define an equivalence relation `Associated`
saying that two elements of a monoid differ by a multiplication by a unit.
Then we show that the quotient type `Associates` is a monoid
and prove basic properties of this quotient.
-/

public section

assert_not_exists IsOrderedMonoid Multiset

variable {M N : Type*}

section Prime

variable [CommMonoidWithZero M]

section Map

variable [CommMonoidWithZero N] {F : Type*} {G : Type*} [FunLike F M N]
variable [MonoidWithZeroHomClass F M N] [FunLike G N M] [MulHomClass G N M]
variable (f : F) (g : G) {p : M}

theorem comap_prime (hinv : ∀ a, g (f a : N) = a) (hp : Prime (f p)) : Prime p :=
  ⟨fun h => hp.1 <| by simp [h], fun h => hp.2.1 <| h.map f, fun a b h => by
    refine
        (hp.2.2 (f a) (f b) <| by
              convert! map_dvd f h
              simp).imp
          ?_ ?_ <;>
      · intro h
        convert! ← map_dvd g h <;> apply hinv⟩

Target:
theorem MulEquiv.prime_iff {E : Type*} [EquivLike E M N] [MulEquivClass E M N] (e : E) :
    Prime (e p) ↔ Prime p :=

Failed candidate:
by
  have __repair_plan_4299722063 : False := by
    skip
  let e := MulEquivClass.toMulEquiv e
  exact ⟨comap_prime e e.symm fun a => by simp,
    fun h => (comap_prime e.symm e fun a => by simp) <| (e.symm_apply_apply p).substr h⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:60:43: error: unsolved goals\nM : Type u_1\nN : Type u_2\ninst✝³ : CommMonoidWithZero M\ninst✝² : CommMonoidWithZero N\np : M\nE : Type u_5\ninst✝¹ : EquivLike E M N\ninst✝ : MulEquivClass E M N\ne : E\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  let e := MulEquivClass.toMulEquiv e
  exact ⟨comap_prime e e.symm fun a => by simp,
    fun h => (comap_prime e.symm e fun a => by simp) <| (e.symm_apply_apply p).substr h⟩

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.Divisibility.Hom
public import Mathlib.Algebra.Group.Irreducible.Lemmas
public import Mathlib.Algebra.GroupWithZero.Equiv
public import Mathlib.Algebra.Prime.Defs
public import Mathlib.Order.Monotone.Defs

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
/-!
# Associated, prime, and irreducible elements.

In this file we define the predicate `Prime p`
saying that an element of a commutative monoid with zero is prime.
Namely, `Prime p` means that `p` isn't zero, it isn't a unit,
and `p ∣ a * b → p ∣ a ∨ p ∣ b` for all `a`, `b`;

In decomposition monoids (e.g., `ℕ`, `ℤ`), this predicate is equivalent to `Irreducible`,
however this is not true in general.

We also define an equivalence relation `Associated`
saying that two elements of a monoid differ by a multiplication by a unit.
Then we show that the quotient type `Associates` is a monoid
and prove basic properties of this quotient.
-/

public section

assert_not_exists IsOrderedMonoid Multiset

variable {M N : Type*}

section Prime

variable [CommMonoidWithZero M]

section Map

variable [CommMonoidWithZero N] {F : Type*} {G : Type*} [FunLike F M N]
variable [MonoidWithZeroHomClass F M N] [FunLike G N M] [MulHomClass G N M]
variable (f : F) (g : G) {p : M}

theorem comap_prime (hinv : ∀ a, g (f a : N) = a) (hp : Prime (f p)) : Prime p :=
  ⟨fun h => hp.1 <| by simp [h], fun h => hp.2.1 <| h.map f, fun a b h => by
    refine
        (hp.2.2 (f a) (f b) <| by
              convert! map_dvd f h
              simp).imp
          ?_ ?_ <;>
      · intro h
        convert! ← map_dvd g h <;> apply hinv⟩

Target:
theorem MulEquiv.prime_iff {E : Type*} [EquivLike E M N] [MulEquivClass E M N] (e : E) :
    Prime (e p) ↔ Prime p :=

Failed candidate:
by
  have __repair_plan_4299722063 : False := by
    skip
  let e := MulEquivClass.toMulEquiv e
  exact ⟨comap_prime e e.symm fun a => by simp,
    fun h => (comap_prime e.symm e fun a => by simp) <| (e.symm_apply_apply p).substr h⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:60:43: error: unsolved goals\nM : Type u_1\nN : Type u_2\ninst✝³ : CommMonoidWithZero M\ninst✝² : CommMonoidWithZero N\np : M\nE : Type u_5\ninst✝¹ : EquivLike E M N\ninst✝ : MulEquivClass E M N\ne : E\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:79:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:78:0: error: unsolved goals
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
P : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D
X : C
Y Y' Y'' : D
g : Y ⟶ Y'
g' : Y' ⟶ Y''
⊢ P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g

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

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

Imports:
public import Mathlib.CategoryTheory.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

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

# Profunctors

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

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

## Future work

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

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

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

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

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

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

variable {C D}

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

namespace ProfunctorCore

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

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

@[simp]

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

Failed candidate:
by
  have __repair_37a726e314 := True.intro.nonexistentProjection
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:79:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:78:0: error: unsolved goals\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX : C\nY Y' Y'' : D\ng : Y ⟶ Y'\ng' : Y' ⟶ Y''\n⊢ P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g", "returncode": 1}

Corrected proof:

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

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

Imports:
public import Mathlib.CategoryTheory.Yoneda

Namespace:
CategoryTheory.ProfunctorCore

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

# Profunctors

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

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

## Future work

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

@[expose] public section

namespace CategoryTheory

open Opposite

universe w' w v₁ v₂ u₁ u₂

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

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

attribute [simp] ProfunctorCore.map_id ProfunctorCore.map_comp

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

variable {C D}

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

namespace ProfunctorCore

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

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

@[simp]

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

Failed candidate:
by
  have __repair_37a726e314 := True.intro.nonexistentProjection
  nth_rw 1 [← Category.id_comp (𝟙 X)]
  simp only [P.map_comp]

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:79:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:78:0: error: unsolved goals\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX : C\nY Y' Y'' : D\ng : Y ⟶ Y'\ng' : Y' ⟶ Y''\n⊢ P.map (𝟙 X) (g ≫ g') = P.map (𝟙 X) g' ≫ P.map (𝟙 X) g", "returncode": 1}

Corrected proof:


<LEAN_FILE>:59:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  split_ifs with h
  · simp only [supNorm, gaussNorm, h, ↓reduceDIte, one_pow, mul_one, Function.comp_apply]
    congr
  · simp [supNorm, gaussNorm, h]

[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.Polynomial.GaussNorm
public import Mathlib.Analysis.Normed.Unbundled.RingSeminorm
public import Mathlib.Algebra.Order.Hom.Basic

Namespace:
Polynomial

Local safe context:
/-
Copyright (c) 2025 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
/-!
# Sup Norm of Polynomials

In this file we define the sup norm on `Polynomial`s based on their coefficients as well as several
basic results about this norm. We note that this is often called the _(naive) height_ of the
polynomial in the literature.

The sup norm is related to the Mahler measure of the polynomial. See
`Mathlib/Analysis/Polynomial/MahlerMeasure.lean`.

## Main definitions

- `Polynomial.supNorm p`: the sup norm of the coefficients of the polynomial, equal to the
  maximum of the norm of its coefficients (or zero for the zero polynomial)

## A Note on Naming

In the literature, the sup norm is often called the _(naive) height_ of a polynomial and the
`l^1` norm is often called the _length_ of the polynomial. Unfortunately, these terms are
extremely overloaded and Mathlib defines _height_ differently.

### TODOs

All other `l^p` norms can be defined on Polynomials as well. In the literature, the `l^1` norm is
sometimes called the polynomial's _length_. The `l^2` norm sometimes arises due to Parseval's
theorem implying that the squared `l^2` norm of a complex polynomial is the integral of the norm of
the polynomial's value on the unit circle.
-/

@[expose] public section supnorm_seminorm

variable {A : Type*} [SeminormedRing A] (p : Polynomial A)

namespace Polynomial

/-- The sup norm of a polynomial on a semi-normed ring, defined as the maximum of its coefficients.
Often called the _(naive) height_ of the polynomial.

This is defined in terms of `Polynomial.gaussNorm`. -/
noncomputable def supNorm : ℝ := p.gaussNorm (SeminormedRing.toRingSeminorm A) 1

/-- The direct definition of the supNorm -/

Target:
lemma supNorm_def' : p.supNorm =
    if hp : p.support.Nonempty then p.support.sup' hp (norm ∘ p.coeff) else 0 :=

Failed candidate:
by
  have __repair_e3d68153a4 : False := True.intro
  split_ifs with h
  · simp only [supNorm, gaussNorm, h, ↓reduceDIte, one_pow, mul_one, Function.comp_apply]
    congr
  · simp [supNorm, gaussNorm, h]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:59:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  split_ifs with h
  · simp only [supNorm, gaussNorm, h, ↓reduceDIte, one_pow, mul_one, Function.comp_apply]
    congr
  · simp [supNorm, gaussNorm, h]

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.Polynomial.GaussNorm
public import Mathlib.Analysis.Normed.Unbundled.RingSeminorm
public import Mathlib.Algebra.Order.Hom.Basic

Namespace:
Polynomial

Local safe context:
/-
Copyright (c) 2025 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
/-!
# Sup Norm of Polynomials

In this file we define the sup norm on `Polynomial`s based on their coefficients as well as several
basic results about this norm. We note that this is often called the _(naive) height_ of the
polynomial in the literature.

The sup norm is related to the Mahler measure of the polynomial. See
`Mathlib/Analysis/Polynomial/MahlerMeasure.lean`.

## Main definitions

- `Polynomial.supNorm p`: the sup norm of the coefficients of the polynomial, equal to the
  maximum of the norm of its coefficients (or zero for the zero polynomial)

## A Note on Naming

In the literature, the sup norm is often called the _(naive) height_ of a polynomial and the
`l^1` norm is often called the _length_ of the polynomial. Unfortunately, these terms are
extremely overloaded and Mathlib defines _height_ differently.

### TODOs

All other `l^p` norms can be defined on Polynomials as well. In the literature, the `l^1` norm is
sometimes called the polynomial's _length_. The `l^2` norm sometimes arises due to Parseval's
theorem implying that the squared `l^2` norm of a complex polynomial is the integral of the norm of
the polynomial's value on the unit circle.
-/

@[expose] public section supnorm_seminorm

variable {A : Type*} [SeminormedRing A] (p : Polynomial A)

namespace Polynomial

/-- The sup norm of a polynomial on a semi-normed ring, defined as the maximum of its coefficients.
Often called the _(naive) height_ of the polynomial.

This is defined in terms of `Polynomial.gaussNorm`. -/
noncomputable def supNorm : ℝ := p.gaussNorm (SeminormedRing.toRingSeminorm A) 1

/-- The direct definition of the supNorm -/

Target:
lemma supNorm_def' : p.supNorm =
    if hp : p.support.Nonempty then p.support.sup' hp (norm ∘ p.coeff) else 0 :=

Failed candidate:
by
  have __repair_e3d68153a4 : False := True.intro
  split_ifs with h
  · simp only [supNorm, gaussNorm, h, ↓reduceDIte, one_pow, mul_one, Function.comp_apply]
    congr
  · simp [supNorm, gaussNorm, h]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:59:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:80:2: warning: declaration uses `sorry`
<LEAN_FILE>:84:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing8ae2b130b5.value`
<LEAN_FILE>:83:0: error: unsolved goals
F : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : FunLike F α β
inst✝³ : MulOne F
inst✝² : MulOneClass β
inst✝¹ : IsOneApply F α β
inst✝ : IsMulApply F α β
⊢ Function.Injective ⇑(coeMonoidHom F α β)

Response:
by
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

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

Imports:
public import Mathlib.Data.FunLike.IsApply
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathlib.Algebra.Group.Pi.Basic

Namespace:
FunLike

Local safe context:
/-
Copyright (c) 2026 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-! # Group instances for `FunLike` types
In this file we define various instances related to groups for `FunLike` types.

For example given a `FunLike F α β` with `IsMulApply F α β` and `Semigroup β`, then `F` is naturally
a semigroup. Note that currently, these are not registered as instances, but only `abbrev`s to
avoid long typeclass searches.

Moreover, we define the homomorphism `FunLike.coeMulHom : F →* α → β` that acts by coercion. This
definition is mainly needed to define a module instance on `F`.

-/

@[expose] public section

namespace FunLike

variable {F α β : Type*} [FunLike F α β]

section CoercionHom

section MulHom

variable [Mul F] [Mul β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a multiplicative homomorphism. -/
@[to_additive
/-- Coercion as an additive homomorphism. -/]
def coeMulHom : F →ₙ* α → β where
  toFun f := f
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMulHom_apply (f : F) : coeMulHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMulHom : (coeMulHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive]
theorem coeMulHom_injective : Function.Injective (coeMulHom F α β) := by
  rw [coe_coeMulHom]
  exact DFunLike.coe_injective

end MulHom

section MonoidHom

variable [MulOne F] [MulOneClass β] [IsOneApply F α β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a monoid homomorphism. -/
@[to_additive
/-- Coercion as an additive monoid homomorphism. -/]
def coeMonoidHom : F →* α → β where
  toFun f := f
  map_one' := coe_one
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMonoidHom_apply (f : F) : coeMonoidHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom : (coeMonoidHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom' : (coeMonoidHom F α β : F →ₙ* α → β) = coeMulHom F α β := rfl

@[to_additive]

Target:
theorem coeMonoidHom_injective : Function.Injective (coeMonoidHom F α β) :=

Failed candidate:
by
  have __repair_8ae2b130b5 := OProverMissing8ae2b130b5.value
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:80:2: warning: declaration uses `sorry`\n<LEAN_FILE>:84:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing8ae2b130b5.value`\n<LEAN_FILE>:83:0: error: unsolved goals\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : FunLike F α β\ninst✝³ : MulOne F\ninst✝² : MulOneClass β\ninst✝¹ : IsOneApply F α β\ninst✝ : IsMulApply F α β\n⊢ Function.Injective ⇑(coeMonoidHom F α β)", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

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

Imports:
public import Mathlib.Data.FunLike.IsApply
public import Mathlib.Algebra.Group.InjSurj
public import Mathlib.Algebra.Group.Hom.Defs
public import Mathlib.Algebra.Group.Pi.Basic

Namespace:
FunLike

Local safe context:
/-
Copyright (c) 2026 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
/-! # Group instances for `FunLike` types
In this file we define various instances related to groups for `FunLike` types.

For example given a `FunLike F α β` with `IsMulApply F α β` and `Semigroup β`, then `F` is naturally
a semigroup. Note that currently, these are not registered as instances, but only `abbrev`s to
avoid long typeclass searches.

Moreover, we define the homomorphism `FunLike.coeMulHom : F →* α → β` that acts by coercion. This
definition is mainly needed to define a module instance on `F`.

-/

@[expose] public section

namespace FunLike

variable {F α β : Type*} [FunLike F α β]

section CoercionHom

section MulHom

variable [Mul F] [Mul β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a multiplicative homomorphism. -/
@[to_additive
/-- Coercion as an additive homomorphism. -/]
def coeMulHom : F →ₙ* α → β where
  toFun f := f
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMulHom_apply (f : F) : coeMulHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMulHom : (coeMulHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive]
theorem coeMulHom_injective : Function.Injective (coeMulHom F α β) := by
  rw [coe_coeMulHom]
  exact DFunLike.coe_injective

end MulHom

section MonoidHom

variable [MulOne F] [MulOneClass β] [IsOneApply F α β] [IsMulApply F α β]

variable (F α β) in
/-- Coercion as a monoid homomorphism. -/
@[to_additive
/-- Coercion as an additive monoid homomorphism. -/]
def coeMonoidHom : F →* α → β where
  toFun f := f
  map_one' := coe_one
  map_mul' := coe_mul

@[to_additive (attr := simp)]
theorem coeMonoidHom_apply (f : F) : coeMonoidHom F α β f = f := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom : (coeMonoidHom F α β : F → α → β) = DFunLike.coe := rfl

@[to_additive (attr := norm_cast)]
theorem coe_coeMonoidHom' : (coeMonoidHom F α β : F →ₙ* α → β) = coeMulHom F α β := rfl

@[to_additive]

Target:
theorem coeMonoidHom_injective : Function.Injective (coeMonoidHom F α β) :=

Failed candidate:
by
  have __repair_8ae2b130b5 := OProverMissing8ae2b130b5.value
  rw [coe_coeMonoidHom]
  exact DFunLike.coe_injective

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:80:2: warning: declaration uses `sorry`\n<LEAN_FILE>:84:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing8ae2b130b5.value`\n<LEAN_FILE>:83:0: error: unsolved goals\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : FunLike F α β\ninst✝³ : MulOne F\ninst✝² : MulOneClass β\ninst✝¹ : IsOneApply F α β\ninst✝ : IsMulApply F α β\n⊢ Function.Injective ⇑(coeMonoidHom F α β)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:59:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing70ee9fcc6b.value`
<LEAN_FILE>:58:0: error: unsolved goals
α : Type u_1
f : Perm α
s : Set α
⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s

Response:
by
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

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

Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.End
public import Mathlib.Logic.Equiv.Set
public import Mathlib.Tactic.Common

Namespace:
Set

Local safe context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
/-!
# Extra lemmas about permutations

This file proves miscellaneous lemmas about `Equiv.Perm`.

## TODO

Most of the content of this file was moved to `Mathlib/Algebra/Group/End.lean` in
https://github.com/leanprover-community/mathlib4/pull/22141.
It would be good to merge the remaining lemmas with other files, e.g.
`GroupTheory.Perm.ViaEmbedding` looks like it could benefit from such a treatment (splitting into
the algebra and non-algebra parts).
-/

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

@[simp]
theorem swap_smul_self_smul [MulAction (Perm α) β] (i j : α) (x : β) :
    swap i j • swap i j • x = x := by simp [smul_smul]

theorem swap_smul_involutive [MulAction (Perm α) β] (i j : α) :
    Function.Involutive (swap i j • · : β → β) := swap_smul_self_smul i j

end Swap
end Equiv

open Equiv Function

namespace Set
variable {α : Type*} {f : Perm α} {s : Set α}

lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s := hf.symm f.invOn

Target:
lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s :=

Failed candidate:
by
  have __repair_70ee9fcc6b := OProverMissing70ee9fcc6b.value
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:59:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing70ee9fcc6b.value`\n<LEAN_FILE>:58:0: error: unsolved goals\nα : Type u_1\nf : Perm α\ns : Set α\n⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

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

Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.End
public import Mathlib.Logic.Equiv.Set
public import Mathlib.Tactic.Common

Namespace:
Set

Local safe context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
/-!
# Extra lemmas about permutations

This file proves miscellaneous lemmas about `Equiv.Perm`.

## TODO

Most of the content of this file was moved to `Mathlib/Algebra/Group/End.lean` in
https://github.com/leanprover-community/mathlib4/pull/22141.
It would be good to merge the remaining lemmas with other files, e.g.
`GroupTheory.Perm.ViaEmbedding` looks like it could benefit from such a treatment (splitting into
the algebra and non-algebra parts).
-/

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

@[simp]
theorem swap_smul_self_smul [MulAction (Perm α) β] (i j : α) (x : β) :
    swap i j • swap i j • x = x := by simp [smul_smul]

theorem swap_smul_involutive [MulAction (Perm α) β] (i j : α) :
    Function.Involutive (swap i j • · : β → β) := swap_smul_self_smul i j

end Swap
end Equiv

open Equiv Function

namespace Set
variable {α : Type*} {f : Perm α} {s : Set α}

lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s := hf.symm f.invOn

Target:
lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s :=

Failed candidate:
by
  have __repair_70ee9fcc6b := OProverMissing70ee9fcc6b.value
  simp_rw [Equiv.Perm.coe_pow]; exact MapsTo.iterate

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:59:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing70ee9fcc6b.value`\n<LEAN_FILE>:58:0: error: unsolved goals\nα : Type u_1\nf : Perm α\ns : Set α\n⊢ MapsTo (⇑f) s s → ∀ (n : ℕ), MapsTo (⇑(f ^ n)) s s", "returncode": 1}

Corrected proof: