<LEAN_FILE>:90:43: error: unsolved goals
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
P : MorphismProperty C
S : C
inst✝⁴ : P.IsStableUnderComposition
K : Precoverage C
inst✝³ : K.HasIsos
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasPullbacks
H : K ≤ P.precoverage
⊢ False

Response:
by
  have := MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [toGrothendieck_comap_forget_eq_restrictedTopology _ H]
  exact Functor.coverPreserving_restrictedTopology (Over.forget P ⊤ S) (K.toGrothendieck.over S)

variable [HasFiniteWidePullbacks C] [P.HasOfPostcompProperty P] [P.IsStableUnderBaseChange]
  [P.ContainsIdentities]

[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.MorphismProperty.Comma
public import Mathlib.CategoryTheory.Limits.MorphismProperty
public import Mathlib.CategoryTheory.Sites.MorphismProperty
public import Mathlib.CategoryTheory.Sites.Over

Namespace:
CategoryTheory.MorphismProperty

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

# Sites on `P.Over ⊤ X`

We provide some API for proving properties of `P.Over ⊤ X` in relation to precoverages. Consider
a precoverage `K` on `C`.

## Main results

- `CategoryTheory.MorphismProperty.locallyCoverDense_forget_of_le`:
  The forgetful functor `P.Over ⊤ X ⥤ Over X` is locally cover dense if morphisms in `K`
  satisfy `P`.
- `CategoryTheory.MorphismProperty.toGrothendieck_comap_forget_eq_restrictedTopology`: The topology
  generated by the precoverage induced from `K` via the composition `P.Over ⊤ X ⥤ Over X ⥤ C` is
  the induced topology from the forgetful functor `P.Over ⊤ X ⥤ Over X`.

-/

public section

universe v u

open CategoryTheory MorphismProperty Limits

namespace CategoryTheory.MorphismProperty

variable {C : Type*} [Category* C]

variable {P : MorphismProperty C} {S : C} [P.IsStableUnderComposition]

set_option backward.defeqAttrib.useBackward true in
lemma exists_map_eq_of_presieve (K : Precoverage C) (H : K ≤ P.precoverage)
    {X : P.Over ⊤ S} {R : Presieve ((MorphismProperty.Over.forget P ⊤ S).obj X)}
    (hR : R ∈ (K.comap <| CategoryTheory.Over.forget S) _) :
    ∃ T : Presieve X, T.map (MorphismProperty.Over.forget P ⊤ S) = R := by
  rw [Precoverage.mem_iff_exists_zeroHypercover] at hR
  obtain ⟨𝒰, rfl⟩ := hR
  let 𝒱 : PreZeroHypercover X :=
    ⟨𝒰.I₀, fun i ↦ Over.mk _ (𝒰.X i).hom ?_, fun i ↦ Over.homMk (𝒰.f i).left (by simp) trivial⟩
  · use 𝒱.presieve₀
    rw [Presieve.map_ofArrows]
    rfl
  · rw [← CategoryTheory.Over.w (𝒰.f i)]
    exact P.comp_mem _ _ (H _ 𝒰.mem₀ ⟨⟨i⟩⟩) X.prop

variable (K : Precoverage C) [K.HasIsos] [K.IsStableUnderBaseChange] [K.IsStableUnderComposition]
  [K.HasPullbacks]

lemma locallyCoverDense_forget_of_le (H : K ≤ P.precoverage) :
    (MorphismProperty.Over.forget P ⊤ S).LocallyCoverDense (K.toGrothendieck.over S) := by
  rw [over_toGrothendieck_eq_toGrothendieck_comap_forget]
  apply Precoverage.locallyCoverDense_of_map_functorPullback_mem
  intro X R hR
  obtain ⟨R, rfl⟩ := MorphismProperty.exists_map_eq_of_presieve _ H hR
  simpa

lemma toGrothendieck_comap_forget_eq_restrictedTopology
    (H : K ≤ P.precoverage) :
    (K.comap (MorphismProperty.Over.forget P ⊤ _ ⋙ CategoryTheory.Over.forget S)).toGrothendieck =
      (MorphismProperty.Over.forget P ⊤ _).restrictedTopology (K.toGrothendieck.over S) := by
  have : (Over.forget P ⊤ S).LocallyCoverDense
      (K.comap (CategoryTheory.Over.forget S)).toGrothendieck := by
    rw [← over_toGrothendieck_eq_toGrothendieck_comap_forget]
    exact MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [Precoverage.comap_comp]
  simp_rw [over_toGrothendieck_eq_toGrothendieck_comap_forget]
  apply Precoverage.toGrothendieck_comap_eq_restrictedTopology
  intro X R hR
  obtain ⟨T, rfl⟩ := MorphismProperty.exists_map_eq_of_presieve K H hR
  simpa

Target:
lemma coverPreserving_comap_forget (H : K ≤ P.precoverage) :
    CoverPreserving
      (K.comap (MorphismProperty.Over.forget P ⊤ _ ⋙ CategoryTheory.Over.forget S)).toGrothendieck
      (K.toGrothendieck.over S)
      (MorphismProperty.Over.forget P ⊤ S) :=

Failed candidate:
by
  have __repair_plan_e3091ebaa2 : False := by
    skip
  have := MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [toGrothendieck_comap_forget_eq_restrictedTopology _ H]
  exact Functor.coverPreserving_restrictedTopology (Over.forget P ⊤ S) (K.toGrothendieck.over S)

variable [HasFiniteWidePullbacks C] [P.HasOfPostcompProperty P] [P.IsStableUnderBaseChange]
  [P.ContainsIdentities]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:90:43: error: unsolved goals\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nP : MorphismProperty C\nS : C\ninst✝⁴ : P.IsStableUnderComposition\nK : Precoverage C\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nH : K ≤ P.precoverage\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  have := MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [toGrothendieck_comap_forget_eq_restrictedTopology _ H]
  exact Functor.coverPreserving_restrictedTopology (Over.forget P ⊤ S) (K.toGrothendieck.over S)

variable [HasFiniteWidePullbacks C] [P.HasOfPostcompProperty P] [P.IsStableUnderBaseChange]
  [P.ContainsIdentities]

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.MorphismProperty.Comma
public import Mathlib.CategoryTheory.Limits.MorphismProperty
public import Mathlib.CategoryTheory.Sites.MorphismProperty
public import Mathlib.CategoryTheory.Sites.Over

Namespace:
CategoryTheory.MorphismProperty

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

# Sites on `P.Over ⊤ X`

We provide some API for proving properties of `P.Over ⊤ X` in relation to precoverages. Consider
a precoverage `K` on `C`.

## Main results

- `CategoryTheory.MorphismProperty.locallyCoverDense_forget_of_le`:
  The forgetful functor `P.Over ⊤ X ⥤ Over X` is locally cover dense if morphisms in `K`
  satisfy `P`.
- `CategoryTheory.MorphismProperty.toGrothendieck_comap_forget_eq_restrictedTopology`: The topology
  generated by the precoverage induced from `K` via the composition `P.Over ⊤ X ⥤ Over X ⥤ C` is
  the induced topology from the forgetful functor `P.Over ⊤ X ⥤ Over X`.

-/

public section

universe v u

open CategoryTheory MorphismProperty Limits

namespace CategoryTheory.MorphismProperty

variable {C : Type*} [Category* C]

variable {P : MorphismProperty C} {S : C} [P.IsStableUnderComposition]

set_option backward.defeqAttrib.useBackward true in
lemma exists_map_eq_of_presieve (K : Precoverage C) (H : K ≤ P.precoverage)
    {X : P.Over ⊤ S} {R : Presieve ((MorphismProperty.Over.forget P ⊤ S).obj X)}
    (hR : R ∈ (K.comap <| CategoryTheory.Over.forget S) _) :
    ∃ T : Presieve X, T.map (MorphismProperty.Over.forget P ⊤ S) = R := by
  rw [Precoverage.mem_iff_exists_zeroHypercover] at hR
  obtain ⟨𝒰, rfl⟩ := hR
  let 𝒱 : PreZeroHypercover X :=
    ⟨𝒰.I₀, fun i ↦ Over.mk _ (𝒰.X i).hom ?_, fun i ↦ Over.homMk (𝒰.f i).left (by simp) trivial⟩
  · use 𝒱.presieve₀
    rw [Presieve.map_ofArrows]
    rfl
  · rw [← CategoryTheory.Over.w (𝒰.f i)]
    exact P.comp_mem _ _ (H _ 𝒰.mem₀ ⟨⟨i⟩⟩) X.prop

variable (K : Precoverage C) [K.HasIsos] [K.IsStableUnderBaseChange] [K.IsStableUnderComposition]
  [K.HasPullbacks]

lemma locallyCoverDense_forget_of_le (H : K ≤ P.precoverage) :
    (MorphismProperty.Over.forget P ⊤ S).LocallyCoverDense (K.toGrothendieck.over S) := by
  rw [over_toGrothendieck_eq_toGrothendieck_comap_forget]
  apply Precoverage.locallyCoverDense_of_map_functorPullback_mem
  intro X R hR
  obtain ⟨R, rfl⟩ := MorphismProperty.exists_map_eq_of_presieve _ H hR
  simpa

lemma toGrothendieck_comap_forget_eq_restrictedTopology
    (H : K ≤ P.precoverage) :
    (K.comap (MorphismProperty.Over.forget P ⊤ _ ⋙ CategoryTheory.Over.forget S)).toGrothendieck =
      (MorphismProperty.Over.forget P ⊤ _).restrictedTopology (K.toGrothendieck.over S) := by
  have : (Over.forget P ⊤ S).LocallyCoverDense
      (K.comap (CategoryTheory.Over.forget S)).toGrothendieck := by
    rw [← over_toGrothendieck_eq_toGrothendieck_comap_forget]
    exact MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [Precoverage.comap_comp]
  simp_rw [over_toGrothendieck_eq_toGrothendieck_comap_forget]
  apply Precoverage.toGrothendieck_comap_eq_restrictedTopology
  intro X R hR
  obtain ⟨T, rfl⟩ := MorphismProperty.exists_map_eq_of_presieve K H hR
  simpa

Target:
lemma coverPreserving_comap_forget (H : K ≤ P.precoverage) :
    CoverPreserving
      (K.comap (MorphismProperty.Over.forget P ⊤ _ ⋙ CategoryTheory.Over.forget S)).toGrothendieck
      (K.toGrothendieck.over S)
      (MorphismProperty.Over.forget P ⊤ S) :=

Failed candidate:
by
  have __repair_plan_e3091ebaa2 : False := by
    skip
  have := MorphismProperty.locallyCoverDense_forget_of_le (S := S) K H
  rw [toGrothendieck_comap_forget_eq_restrictedTopology _ H]
  exact Functor.coverPreserving_restrictedTopology (Over.forget P ⊤ S) (K.toGrothendieck.over S)

variable [HasFiniteWidePullbacks C] [P.HasOfPostcompProperty P] [P.IsStableUnderBaseChange]
  [P.ContainsIdentities]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:90:43: error: unsolved goals\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nP : MorphismProperty C\nS : C\ninst✝⁴ : P.IsStableUnderComposition\nK : Precoverage C\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nH : K ≤ P.precoverage\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:128:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing795b0886a1.value`
<LEAN_FILE>:127:0: error: unsolved goals
ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'

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

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

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

Namespace:
BoxIntegral.Prepartition

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

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

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

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

namespace TaggedPrepartition

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

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

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

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

Failed candidate:
by
  have __repair_795b0886a1 := OProverMissing795b0886a1.value
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:128:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing795b0886a1.value`\n<LEAN_FILE>:127:0: error: unsolved goals\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nhJ : J ∈ π\nJ' : Box ι\nhJ' : J' ∈ πi J\n⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'", "returncode": 1}

Corrected proof:

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

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

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

Namespace:
BoxIntegral.Prepartition

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

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

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

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

namespace TaggedPrepartition

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

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

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

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

Failed candidate:
by
  have __repair_795b0886a1 := OProverMissing795b0886a1.value
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:128:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing795b0886a1.value`\n<LEAN_FILE>:127:0: error: unsolved goals\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → TaggedPrepartition J\nhJ : J ∈ π\nJ' : Box ι\nhJ' : J' ∈ πi J\n⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'", "returncode": 1}

Corrected proof:


<LEAN_FILE>:100:43: error: unsolved goals
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
s : Set X
⊢ False

Response:
by
  rw [UniformSpace.mem_nhds_iff]
  constructor
  · rintro ⟨V, V_in, V_sub⟩
    rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
    obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
    refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
  · rintro ⟨V, V_in, _, _, V_sub⟩
    exact ⟨V, V_in, V_sub⟩

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

Imports:
public import Mathlib.Topology.UniformSpace.Defs
public import Mathlib.Topology.Bases

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
/-!
# Ultrametric (nonarchimedean) uniform spaces

Ultrametric (nonarchimedean) uniform spaces are ones that generalize ultrametric spaces by
having a uniformity based on equivalence relations.

## Main definitions

In this file we define `IsUltraUniformity`, a Prop mixin typeclass.

## Main results

* `TopologicalSpace.isTopologicalBasis_clopens`: a uniform space with a nonarchimedean uniformity
  has a topological basis of clopen sets in the topology, meaning that it is topologically
  zero-dimensional.

## Implementation notes

As in the `Mathlib/Topology/UniformSpace/Defs.lean` file, we do not reuse `Mathlib/Data/Rel.lean`
but rather extend the relation properties as needed.

## TODOs

* Prove that `IsUltraUniformity` iff metrizable by `IsUltrametricDist` on a `PseudoMetricSpace`
  under a countable system/basis condition
* Generalize `IsUltrametricDist` to `IsUltrametricUniformity`
* Provide `IsUltraUniformity` for the uniformity in a `Valued` ring
* Generalize results about open/closed balls and spheres in `IsUltraUniformity` to
  combine applications for `MetricSpace.ball` and valued "balls"
* Use `IsUltraUniformity` to work with profinite/totally separated spaces

## References

* [D. Windisch, *Equivalent characterizations of non-Archimedean uniform spaces*][windisch2021]
* [A. C. M. van Rooij, *Non-Archimedean uniformities*][vanrooij1970]

-/

@[expose] public section

open Set Filter Topology
open scoped SetRel Uniformity

variable {X : Type*}

lemma IsTransitiveRel.prod_subset_trans {s : SetRel X X} {t u v : Set X} [s.IsTrans]
    (htu : t ×ˢ u ⊆ s) (huv : u ×ˢ v ⊆ s) (hu : u.Nonempty) :
    t ×ˢ v ⊆ s := by
  rintro ⟨a, b⟩ hab
  simp only [mem_prod] at hab
  obtain ⟨x, hx⟩ := hu
  exact s.trans (@htu ⟨a, x⟩ ⟨hab.left, hx⟩) (@huv ⟨x, b⟩ ⟨hx, hab.right⟩)

lemma IsTransitiveRel.mem_filter_prod_trans {s : SetRel X X} {f g h : Filter X} [g.NeBot]
    [s.IsTrans] (hfg : s ∈ f ×ˢ g) (hgh : s ∈ g ×ˢ h) :
    s ∈ f ×ˢ h :=
  Eventually.trans_prod (p := (fun x y ↦ (x, y) ∈ s)) (q := (fun x y ↦ (x, y) ∈ s))
    (r := (fun x y ↦ (x, y) ∈ s)) hfg hgh fun _ _ _ ↦ s.trans

open UniformSpace

lemma ball_subset_of_mem {V : SetRel X X} [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
    ball y V ⊆ ball x V :=
  ball_subset_of_comp_subset hy SetRel.comp_subset_self

lemma ball_eq_of_mem {V : SetRel X X} [V.IsSymm] [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
    ball x V = ball y V := by
  refine le_antisymm (ball_subset_of_mem ?_) (ball_subset_of_mem hy)
  rwa [← mem_ball_symmetry]

variable [UniformSpace X]

variable (X) in
/-- A uniform space is ultrametric if the uniformity `𝓤 X` has a basis of equivalence relations. -/
class IsUltraUniformity : Prop where
  hasBasis : (𝓤 X).HasBasis
    (fun s : SetRel X X => s ∈ 𝓤 X ∧ SetRel.IsSymm s ∧ SetRel.IsTrans s) id

lemma IsUltraUniformity.mk_of_hasBasis {ι : Type*} {p : ι → Prop} {s : ι → SetRel X X}
    (h_basis : (𝓤 X).HasBasis p s) (h_symm : ∀ i, p i → SetRel.IsSymm (s i))
    (h_trans : ∀ i, p i → SetRel.IsTrans (s i)) :
    IsUltraUniformity X where
  hasBasis := h_basis.to_hasBasis'
    (fun i hi ↦ ⟨s i, ⟨h_basis.mem_of_mem hi, h_symm i hi, h_trans i hi⟩, subset_rfl⟩)
    (fun _ hs ↦ hs.1)

Target:
lemma IsUltraUniformity.mem_nhds_iff_symm_trans [IsUltraUniformity X] {x : X} {s : Set X} :
    s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ UniformSpace.ball x V ⊆ s :=

Failed candidate:
by
  have __repair_plan_fc63f51a4a : False := by
    skip
  rw [UniformSpace.mem_nhds_iff]
  constructor
  · rintro ⟨V, V_in, V_sub⟩
    rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
    obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
    refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
  · rintro ⟨V, V_in, _, _, V_sub⟩
    exact ⟨V, V_in, V_sub⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:100:43: error: unsolved goals\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsUltraUniformity X\nx : X\ns : Set X\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [UniformSpace.mem_nhds_iff]
  constructor
  · rintro ⟨V, V_in, V_sub⟩
    rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
    obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
    refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
  · rintro ⟨V, V_in, _, _, V_sub⟩
    exact ⟨V, V_in, V_sub⟩

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

Imports:
public import Mathlib.Topology.UniformSpace.Defs
public import Mathlib.Topology.Bases

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
/-!
# Ultrametric (nonarchimedean) uniform spaces

Ultrametric (nonarchimedean) uniform spaces are ones that generalize ultrametric spaces by
having a uniformity based on equivalence relations.

## Main definitions

In this file we define `IsUltraUniformity`, a Prop mixin typeclass.

## Main results

* `TopologicalSpace.isTopologicalBasis_clopens`: a uniform space with a nonarchimedean uniformity
  has a topological basis of clopen sets in the topology, meaning that it is topologically
  zero-dimensional.

## Implementation notes

As in the `Mathlib/Topology/UniformSpace/Defs.lean` file, we do not reuse `Mathlib/Data/Rel.lean`
but rather extend the relation properties as needed.

## TODOs

* Prove that `IsUltraUniformity` iff metrizable by `IsUltrametricDist` on a `PseudoMetricSpace`
  under a countable system/basis condition
* Generalize `IsUltrametricDist` to `IsUltrametricUniformity`
* Provide `IsUltraUniformity` for the uniformity in a `Valued` ring
* Generalize results about open/closed balls and spheres in `IsUltraUniformity` to
  combine applications for `MetricSpace.ball` and valued "balls"
* Use `IsUltraUniformity` to work with profinite/totally separated spaces

## References

* [D. Windisch, *Equivalent characterizations of non-Archimedean uniform spaces*][windisch2021]
* [A. C. M. van Rooij, *Non-Archimedean uniformities*][vanrooij1970]

-/

@[expose] public section

open Set Filter Topology
open scoped SetRel Uniformity

variable {X : Type*}

lemma IsTransitiveRel.prod_subset_trans {s : SetRel X X} {t u v : Set X} [s.IsTrans]
    (htu : t ×ˢ u ⊆ s) (huv : u ×ˢ v ⊆ s) (hu : u.Nonempty) :
    t ×ˢ v ⊆ s := by
  rintro ⟨a, b⟩ hab
  simp only [mem_prod] at hab
  obtain ⟨x, hx⟩ := hu
  exact s.trans (@htu ⟨a, x⟩ ⟨hab.left, hx⟩) (@huv ⟨x, b⟩ ⟨hx, hab.right⟩)

lemma IsTransitiveRel.mem_filter_prod_trans {s : SetRel X X} {f g h : Filter X} [g.NeBot]
    [s.IsTrans] (hfg : s ∈ f ×ˢ g) (hgh : s ∈ g ×ˢ h) :
    s ∈ f ×ˢ h :=
  Eventually.trans_prod (p := (fun x y ↦ (x, y) ∈ s)) (q := (fun x y ↦ (x, y) ∈ s))
    (r := (fun x y ↦ (x, y) ∈ s)) hfg hgh fun _ _ _ ↦ s.trans

open UniformSpace

lemma ball_subset_of_mem {V : SetRel X X} [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
    ball y V ⊆ ball x V :=
  ball_subset_of_comp_subset hy SetRel.comp_subset_self

lemma ball_eq_of_mem {V : SetRel X X} [V.IsSymm] [V.IsTrans] {x y : X} (hy : y ∈ ball x V) :
    ball x V = ball y V := by
  refine le_antisymm (ball_subset_of_mem ?_) (ball_subset_of_mem hy)
  rwa [← mem_ball_symmetry]

variable [UniformSpace X]

variable (X) in
/-- A uniform space is ultrametric if the uniformity `𝓤 X` has a basis of equivalence relations. -/
class IsUltraUniformity : Prop where
  hasBasis : (𝓤 X).HasBasis
    (fun s : SetRel X X => s ∈ 𝓤 X ∧ SetRel.IsSymm s ∧ SetRel.IsTrans s) id

lemma IsUltraUniformity.mk_of_hasBasis {ι : Type*} {p : ι → Prop} {s : ι → SetRel X X}
    (h_basis : (𝓤 X).HasBasis p s) (h_symm : ∀ i, p i → SetRel.IsSymm (s i))
    (h_trans : ∀ i, p i → SetRel.IsTrans (s i)) :
    IsUltraUniformity X where
  hasBasis := h_basis.to_hasBasis'
    (fun i hi ↦ ⟨s i, ⟨h_basis.mem_of_mem hi, h_symm i hi, h_trans i hi⟩, subset_rfl⟩)
    (fun _ hs ↦ hs.1)

Target:
lemma IsUltraUniformity.mem_nhds_iff_symm_trans [IsUltraUniformity X] {x : X} {s : Set X} :
    s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 X, SetRel.IsSymm V ∧ SetRel.IsTrans V ∧ UniformSpace.ball x V ⊆ s :=

Failed candidate:
by
  have __repair_plan_fc63f51a4a : False := by
    skip
  rw [UniformSpace.mem_nhds_iff]
  constructor
  · rintro ⟨V, V_in, V_sub⟩
    rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
    obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
    refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
  · rintro ⟨V, V_in, _, _, V_sub⟩
    exact ⟨V, V_in, V_sub⟩

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:100:43: error: unsolved goals\nX : Type u_1\ninst✝¹ : UniformSpace X\ninst✝ : IsUltraUniformity X\nx : X\ns : Set X\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:102:43: error: unsolved goals
R : Type u
inst✝⁹ : CommSemiring R
S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
ι₁ : Type v₁
inst✝⁶ : DecidableEq ι₁
M₁ : ι₁ → Type w₁
M₂' : Type w₂'
inst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)
inst✝⁴ : AddCommMonoid M₂'
inst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)
inst✝² : Module R M₂'
inst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)
inst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)
i : ι₁
x : M₁ i
y : M₂'
⊢ False

Response:
by
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ 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.LinearAlgebra.TensorProduct.Tower
public import Mathlib.Algebra.DirectSum.Module

Namespace:
TensorProduct

Local safe context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Eric Wieser
-/
/-!
# Tensor products of direct sums

This file shows that taking `TensorProduct`s commutes with taking `DirectSum`s in both arguments.

## Main results

* `TensorProduct.directSum`
* `TensorProduct.directSumLeft`
* `TensorProduct.directSumRight`
-/

@[expose] public section

universe u v₁ v₂ w₁ w₁' w₂ w₂'

section Ring

namespace TensorProduct

open TensorProduct

open DirectSum

open LinearMap

attribute [local ext] TensorProduct.ext

variable (R : Type u) [CommSemiring R] (S) [Semiring S] [Algebra R S]
variable {ι₁ : Type v₁} {ι₂ : Type v₂}
variable [DecidableEq ι₁] [DecidableEq ι₂]
variable (M₁ : ι₁ → Type w₁) (M₁' : Type w₁') (M₂ : ι₂ → Type w₂) (M₂' : Type w₂')
variable [∀ i₁, AddCommMonoid (M₁ i₁)] [AddCommMonoid M₁']
variable [∀ i₂, AddCommMonoid (M₂ i₂)] [AddCommMonoid M₂']
variable [∀ i₁, Module R (M₁ i₁)] [Module R M₁'] [∀ i₂, Module R (M₂ i₂)] [Module R M₂']
variable [∀ i₁, Module S (M₁ i₁)] [∀ i₁, IsScalarTower R S (M₁ i₁)]
variable [Module S M₁'] [IsScalarTower R S M₁']

/-- The linear equivalence `(⨁ i₁, M₁ i₁) ⊗ (⨁ i₂, M₂ i₂) ≃ (⨁ i₁, ⨁ i₂, M₁ i₁ ⊗ M₂ i₂)`, i.e.
"tensor product distributes over direct sum". -/
protected def directSum :
    ((⨁ i₁, M₁ i₁) ⊗[R] ⨁ i₂, M₂ i₂) ≃ₗ[S] ⨁ i : ι₁ × ι₂, M₁ i.1 ⊗[R] M₂ i.2 := by
  refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
  · exact AlgebraTensorModule.lift <|
      toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
  · exact toModule S _ _ fun i => AlgebraTensorModule.map (lof S _ M₁ i.1) (lof R _ M₂ i.2)
  · ext ⟨i₁, i₂⟩ x₁ x₂ : 4
    simp only [coe_comp, Function.comp_apply, toModule_lof, AlgebraTensorModule.map_tmul,
      AlgebraTensorModule.lift_apply, lift.tmul, coe_restrictScalars, flip_apply,
      AlgebraTensorModule.curry_apply, curry_apply, id_comp]
  · ext i₁ i₂ x₁ x₂ : 5
    simp only [coe_comp, Function.comp_apply, AlgebraTensorModule.curry_apply, curry_apply,
      coe_restrictScalars, AlgebraTensorModule.lift_apply, lift.tmul, toModule_lof, flip_apply,
      AlgebraTensorModule.map_tmul, id_coe, id_eq]

/-- Tensor products distribute over a direct sum on the left . -/
def directSumLeft : (⨁ i₁, M₁ i₁) ⊗[R] M₂' ≃ₗ[S] ⨁ i, M₁ i ⊗[R] M₂' :=
  TensorProduct.AlgebraTensorModule.congr 1 (DirectSum.lid _ _).symm ≪≫ₗ
  TensorProduct.directSum R S M₁ (fun _ : Unit ↦ M₂') ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.prodUnique _ _)

/-- Tensor products distribute over a direct sum on the right. -/
def directSumRight : (M₁' ⊗[R] ⨁ i, M₂ i) ≃ₗ[S] ⨁ i, M₁' ⊗[R] M₂ i :=
  TensorProduct.AlgebraTensorModule.congr (DirectSum.lid _ _).symm 1 ≪≫ₗ
  TensorProduct.directSum R S (fun _ : Unit ↦ M₁') M₂ ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.uniqueProd _ _)

@[deprecated (since := "2026-03-04")] alias directSumRight' := directSumRight

variable {M₁ M₁' M₂ M₂'}

@[simp]
theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) =
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by
  simp [TensorProduct.directSum]

@[simp]
theorem directSum_symm_lof_tmul (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    (TensorProduct.directSum R S M₁ M₂).symm
      (DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂)) =
      (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) := by
  rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof]

@[simp]

Target:
theorem directSumLeft_tmul_lof (i : ι₁) (x : M₁ i) (y : M₂') :
    directSumLeft R S M₁ M₂' (DirectSum.lof S _ _ i x ⊗ₜ[R] y) =
    DirectSum.lof S _ _ i (x ⊗ₜ[R] y) :=

Failed candidate:
by
  have __repair_plan_31a9f3b99d : False := by
    skip
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ rfl

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:102:43: error: unsolved goals\nR : Type u\ninst✝⁹ : CommSemiring R\nS : Type u_1\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nι₁ : Type v₁\ninst✝⁶ : DecidableEq ι₁\nM₁ : ι₁ → Type w₁\nM₂' : Type w₂'\ninst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝⁴ : AddCommMonoid M₂'\ninst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)\ninst✝² : Module R M₂'\ninst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)\ninst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)\ni : ι₁\nx : M₁ i\ny : M₂'\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ 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.LinearAlgebra.TensorProduct.Tower
public import Mathlib.Algebra.DirectSum.Module

Namespace:
TensorProduct

Local safe context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Eric Wieser
-/
/-!
# Tensor products of direct sums

This file shows that taking `TensorProduct`s commutes with taking `DirectSum`s in both arguments.

## Main results

* `TensorProduct.directSum`
* `TensorProduct.directSumLeft`
* `TensorProduct.directSumRight`
-/

@[expose] public section

universe u v₁ v₂ w₁ w₁' w₂ w₂'

section Ring

namespace TensorProduct

open TensorProduct

open DirectSum

open LinearMap

attribute [local ext] TensorProduct.ext

variable (R : Type u) [CommSemiring R] (S) [Semiring S] [Algebra R S]
variable {ι₁ : Type v₁} {ι₂ : Type v₂}
variable [DecidableEq ι₁] [DecidableEq ι₂]
variable (M₁ : ι₁ → Type w₁) (M₁' : Type w₁') (M₂ : ι₂ → Type w₂) (M₂' : Type w₂')
variable [∀ i₁, AddCommMonoid (M₁ i₁)] [AddCommMonoid M₁']
variable [∀ i₂, AddCommMonoid (M₂ i₂)] [AddCommMonoid M₂']
variable [∀ i₁, Module R (M₁ i₁)] [Module R M₁'] [∀ i₂, Module R (M₂ i₂)] [Module R M₂']
variable [∀ i₁, Module S (M₁ i₁)] [∀ i₁, IsScalarTower R S (M₁ i₁)]
variable [Module S M₁'] [IsScalarTower R S M₁']

/-- The linear equivalence `(⨁ i₁, M₁ i₁) ⊗ (⨁ i₂, M₂ i₂) ≃ (⨁ i₁, ⨁ i₂, M₁ i₁ ⊗ M₂ i₂)`, i.e.
"tensor product distributes over direct sum". -/
protected def directSum :
    ((⨁ i₁, M₁ i₁) ⊗[R] ⨁ i₂, M₂ i₂) ≃ₗ[S] ⨁ i : ι₁ × ι₂, M₁ i.1 ⊗[R] M₂ i.2 := by
  refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
  · exact AlgebraTensorModule.lift <|
      toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
  · exact toModule S _ _ fun i => AlgebraTensorModule.map (lof S _ M₁ i.1) (lof R _ M₂ i.2)
  · ext ⟨i₁, i₂⟩ x₁ x₂ : 4
    simp only [coe_comp, Function.comp_apply, toModule_lof, AlgebraTensorModule.map_tmul,
      AlgebraTensorModule.lift_apply, lift.tmul, coe_restrictScalars, flip_apply,
      AlgebraTensorModule.curry_apply, curry_apply, id_comp]
  · ext i₁ i₂ x₁ x₂ : 5
    simp only [coe_comp, Function.comp_apply, AlgebraTensorModule.curry_apply, curry_apply,
      coe_restrictScalars, AlgebraTensorModule.lift_apply, lift.tmul, toModule_lof, flip_apply,
      AlgebraTensorModule.map_tmul, id_coe, id_eq]

/-- Tensor products distribute over a direct sum on the left . -/
def directSumLeft : (⨁ i₁, M₁ i₁) ⊗[R] M₂' ≃ₗ[S] ⨁ i, M₁ i ⊗[R] M₂' :=
  TensorProduct.AlgebraTensorModule.congr 1 (DirectSum.lid _ _).symm ≪≫ₗ
  TensorProduct.directSum R S M₁ (fun _ : Unit ↦ M₂') ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.prodUnique _ _)

/-- Tensor products distribute over a direct sum on the right. -/
def directSumRight : (M₁' ⊗[R] ⨁ i, M₂ i) ≃ₗ[S] ⨁ i, M₁' ⊗[R] M₂ i :=
  TensorProduct.AlgebraTensorModule.congr (DirectSum.lid _ _).symm 1 ≪≫ₗ
  TensorProduct.directSum R S (fun _ : Unit ↦ M₁') M₂ ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.uniqueProd _ _)

@[deprecated (since := "2026-03-04")] alias directSumRight' := directSumRight

variable {M₁ M₁' M₂ M₂'}

@[simp]
theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) =
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by
  simp [TensorProduct.directSum]

@[simp]
theorem directSum_symm_lof_tmul (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    (TensorProduct.directSum R S M₁ M₂).symm
      (DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂)) =
      (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) := by
  rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof]

@[simp]

Target:
theorem directSumLeft_tmul_lof (i : ι₁) (x : M₁ i) (y : M₂') :
    directSumLeft R S M₁ M₂' (DirectSum.lof S _ _ i x ⊗ₜ[R] y) =
    DirectSum.lof S _ _ i (x ⊗ₜ[R] y) :=

Failed candidate:
by
  have __repair_plan_31a9f3b99d : False := by
    skip
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ rfl

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:102:43: error: unsolved goals\nR : Type u\ninst✝⁹ : CommSemiring R\nS : Type u_1\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nι₁ : Type v₁\ninst✝⁶ : DecidableEq ι₁\nM₁ : ι₁ → Type w₁\nM₂' : Type w₂'\ninst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝⁴ : AddCommMonoid M₂'\ninst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)\ninst✝² : Module R M₂'\ninst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)\ninst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)\ni : ι₁\nx : M₁ i\ny : M₂'\n⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd17af10508.value`
<LEAN_FILE>:101:0: error: unsolved goals
R : Type u
inst✝⁹ : CommSemiring R
S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
ι₁ : Type v₁
inst✝⁶ : DecidableEq ι₁
M₁ : ι₁ → Type w₁
M₂' : Type w₂'
inst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)
inst✝⁴ : AddCommMonoid M₂'
inst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)
inst✝² : Module R M₂'
inst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)
inst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)
i : ι₁
x : M₁ i
y : M₂'
⊢ (directSumLeft R S M₁ M₂') ((lof S ι₁ M₁ i) x ⊗ₜ[R] y) = (lof S ι₁ (fun i => M₁ i ⊗[R] M₂') i) (x ⊗ₜ[R] y)

Response:
by
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ 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.LinearAlgebra.TensorProduct.Tower
public import Mathlib.Algebra.DirectSum.Module

Namespace:
TensorProduct

Local safe context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Eric Wieser
-/
/-!
# Tensor products of direct sums

This file shows that taking `TensorProduct`s commutes with taking `DirectSum`s in both arguments.

## Main results

* `TensorProduct.directSum`
* `TensorProduct.directSumLeft`
* `TensorProduct.directSumRight`
-/

@[expose] public section

universe u v₁ v₂ w₁ w₁' w₂ w₂'

section Ring

namespace TensorProduct

open TensorProduct

open DirectSum

open LinearMap

attribute [local ext] TensorProduct.ext

variable (R : Type u) [CommSemiring R] (S) [Semiring S] [Algebra R S]
variable {ι₁ : Type v₁} {ι₂ : Type v₂}
variable [DecidableEq ι₁] [DecidableEq ι₂]
variable (M₁ : ι₁ → Type w₁) (M₁' : Type w₁') (M₂ : ι₂ → Type w₂) (M₂' : Type w₂')
variable [∀ i₁, AddCommMonoid (M₁ i₁)] [AddCommMonoid M₁']
variable [∀ i₂, AddCommMonoid (M₂ i₂)] [AddCommMonoid M₂']
variable [∀ i₁, Module R (M₁ i₁)] [Module R M₁'] [∀ i₂, Module R (M₂ i₂)] [Module R M₂']
variable [∀ i₁, Module S (M₁ i₁)] [∀ i₁, IsScalarTower R S (M₁ i₁)]
variable [Module S M₁'] [IsScalarTower R S M₁']

/-- The linear equivalence `(⨁ i₁, M₁ i₁) ⊗ (⨁ i₂, M₂ i₂) ≃ (⨁ i₁, ⨁ i₂, M₁ i₁ ⊗ M₂ i₂)`, i.e.
"tensor product distributes over direct sum". -/
protected def directSum :
    ((⨁ i₁, M₁ i₁) ⊗[R] ⨁ i₂, M₂ i₂) ≃ₗ[S] ⨁ i : ι₁ × ι₂, M₁ i.1 ⊗[R] M₂ i.2 := by
  refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
  · exact AlgebraTensorModule.lift <|
      toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
  · exact toModule S _ _ fun i => AlgebraTensorModule.map (lof S _ M₁ i.1) (lof R _ M₂ i.2)
  · ext ⟨i₁, i₂⟩ x₁ x₂ : 4
    simp only [coe_comp, Function.comp_apply, toModule_lof, AlgebraTensorModule.map_tmul,
      AlgebraTensorModule.lift_apply, lift.tmul, coe_restrictScalars, flip_apply,
      AlgebraTensorModule.curry_apply, curry_apply, id_comp]
  · ext i₁ i₂ x₁ x₂ : 5
    simp only [coe_comp, Function.comp_apply, AlgebraTensorModule.curry_apply, curry_apply,
      coe_restrictScalars, AlgebraTensorModule.lift_apply, lift.tmul, toModule_lof, flip_apply,
      AlgebraTensorModule.map_tmul, id_coe, id_eq]

/-- Tensor products distribute over a direct sum on the left . -/
def directSumLeft : (⨁ i₁, M₁ i₁) ⊗[R] M₂' ≃ₗ[S] ⨁ i, M₁ i ⊗[R] M₂' :=
  TensorProduct.AlgebraTensorModule.congr 1 (DirectSum.lid _ _).symm ≪≫ₗ
  TensorProduct.directSum R S M₁ (fun _ : Unit ↦ M₂') ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.prodUnique _ _)

/-- Tensor products distribute over a direct sum on the right. -/
def directSumRight : (M₁' ⊗[R] ⨁ i, M₂ i) ≃ₗ[S] ⨁ i, M₁' ⊗[R] M₂ i :=
  TensorProduct.AlgebraTensorModule.congr (DirectSum.lid _ _).symm 1 ≪≫ₗ
  TensorProduct.directSum R S (fun _ : Unit ↦ M₁') M₂ ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.uniqueProd _ _)

@[deprecated (since := "2026-03-04")] alias directSumRight' := directSumRight

variable {M₁ M₁' M₂ M₂'}

@[simp]
theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) =
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by
  simp [TensorProduct.directSum]

@[simp]
theorem directSum_symm_lof_tmul (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    (TensorProduct.directSum R S M₁ M₂).symm
      (DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂)) =
      (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) := by
  rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof]

@[simp]

Target:
theorem directSumLeft_tmul_lof (i : ι₁) (x : M₁ i) (y : M₂') :
    directSumLeft R S M₁ M₂' (DirectSum.lof S _ _ i x ⊗ₜ[R] y) =
    DirectSum.lof S _ _ i (x ⊗ₜ[R] y) :=

Failed candidate:
by
  have __repair_d17af10508 := OProverMissingd17af10508.value
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd17af10508.value`\n<LEAN_FILE>:101:0: error: unsolved goals\nR : Type u\ninst✝⁹ : CommSemiring R\nS : Type u_1\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nι₁ : Type v₁\ninst✝⁶ : DecidableEq ι₁\nM₁ : ι₁ → Type w₁\nM₂' : Type w₂'\ninst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝⁴ : AddCommMonoid M₂'\ninst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)\ninst✝² : Module R M₂'\ninst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)\ninst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)\ni : ι₁\nx : M₁ i\ny : M₂'\n⊢ (directSumLeft R S M₁ M₂') ((lof S ι₁ M₁ i) x ⊗ₜ[R] y) = (lof S ι₁ (fun i => M₁ i ⊗[R] M₂') i) (x ⊗ₜ[R] y)", "returncode": 1}

Corrected proof:

[Assistant]
by
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ 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.LinearAlgebra.TensorProduct.Tower
public import Mathlib.Algebra.DirectSum.Module

Namespace:
TensorProduct

Local safe context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Eric Wieser
-/
/-!
# Tensor products of direct sums

This file shows that taking `TensorProduct`s commutes with taking `DirectSum`s in both arguments.

## Main results

* `TensorProduct.directSum`
* `TensorProduct.directSumLeft`
* `TensorProduct.directSumRight`
-/

@[expose] public section

universe u v₁ v₂ w₁ w₁' w₂ w₂'

section Ring

namespace TensorProduct

open TensorProduct

open DirectSum

open LinearMap

attribute [local ext] TensorProduct.ext

variable (R : Type u) [CommSemiring R] (S) [Semiring S] [Algebra R S]
variable {ι₁ : Type v₁} {ι₂ : Type v₂}
variable [DecidableEq ι₁] [DecidableEq ι₂]
variable (M₁ : ι₁ → Type w₁) (M₁' : Type w₁') (M₂ : ι₂ → Type w₂) (M₂' : Type w₂')
variable [∀ i₁, AddCommMonoid (M₁ i₁)] [AddCommMonoid M₁']
variable [∀ i₂, AddCommMonoid (M₂ i₂)] [AddCommMonoid M₂']
variable [∀ i₁, Module R (M₁ i₁)] [Module R M₁'] [∀ i₂, Module R (M₂ i₂)] [Module R M₂']
variable [∀ i₁, Module S (M₁ i₁)] [∀ i₁, IsScalarTower R S (M₁ i₁)]
variable [Module S M₁'] [IsScalarTower R S M₁']

/-- The linear equivalence `(⨁ i₁, M₁ i₁) ⊗ (⨁ i₂, M₂ i₂) ≃ (⨁ i₁, ⨁ i₂, M₁ i₁ ⊗ M₂ i₂)`, i.e.
"tensor product distributes over direct sum". -/
protected def directSum :
    ((⨁ i₁, M₁ i₁) ⊗[R] ⨁ i₂, M₂ i₂) ≃ₗ[S] ⨁ i : ι₁ × ι₂, M₁ i.1 ⊗[R] M₂ i.2 := by
  refine LinearEquiv.ofLinear ?toFun ?invFun ?left ?right
  · exact AlgebraTensorModule.lift <|
      toModule S _ _ fun i₁ => flip <| toModule R _ _ fun i₂ => flip <| AlgebraTensorModule.curry <|
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂)
  · exact toModule S _ _ fun i => AlgebraTensorModule.map (lof S _ M₁ i.1) (lof R _ M₂ i.2)
  · ext ⟨i₁, i₂⟩ x₁ x₂ : 4
    simp only [coe_comp, Function.comp_apply, toModule_lof, AlgebraTensorModule.map_tmul,
      AlgebraTensorModule.lift_apply, lift.tmul, coe_restrictScalars, flip_apply,
      AlgebraTensorModule.curry_apply, curry_apply, id_comp]
  · ext i₁ i₂ x₁ x₂ : 5
    simp only [coe_comp, Function.comp_apply, AlgebraTensorModule.curry_apply, curry_apply,
      coe_restrictScalars, AlgebraTensorModule.lift_apply, lift.tmul, toModule_lof, flip_apply,
      AlgebraTensorModule.map_tmul, id_coe, id_eq]

/-- Tensor products distribute over a direct sum on the left . -/
def directSumLeft : (⨁ i₁, M₁ i₁) ⊗[R] M₂' ≃ₗ[S] ⨁ i, M₁ i ⊗[R] M₂' :=
  TensorProduct.AlgebraTensorModule.congr 1 (DirectSum.lid _ _).symm ≪≫ₗ
  TensorProduct.directSum R S M₁ (fun _ : Unit ↦ M₂') ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.prodUnique _ _)

/-- Tensor products distribute over a direct sum on the right. -/
def directSumRight : (M₁' ⊗[R] ⨁ i, M₂ i) ≃ₗ[S] ⨁ i, M₁' ⊗[R] M₂ i :=
  TensorProduct.AlgebraTensorModule.congr (DirectSum.lid _ _).symm 1 ≪≫ₗ
  TensorProduct.directSum R S (fun _ : Unit ↦ M₁') M₂ ≪≫ₗ
  DirectSum.lequivCongrLeft S (Equiv.uniqueProd _ _)

@[deprecated (since := "2026-03-04")] alias directSumRight' := directSumRight

variable {M₁ M₁' M₂ M₂'}

@[simp]
theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) =
      DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by
  simp [TensorProduct.directSum]

@[simp]
theorem directSum_symm_lof_tmul (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
    (TensorProduct.directSum R S M₁ M₂).symm
      (DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂)) =
      (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) := by
  rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof]

@[simp]

Target:
theorem directSumLeft_tmul_lof (i : ι₁) (x : M₁ i) (y : M₂') :
    directSumLeft R S M₁ M₂' (DirectSum.lof S _ _ i x ⊗ₜ[R] y) =
    DirectSum.lof S _ _ i (x ⊗ₜ[R] y) :=

Failed candidate:
by
  have __repair_d17af10508 := OProverMissingd17af10508.value
  simpa [directSumLeft] using! lequivCongrLeft_lof S (by simp) _ _ rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd17af10508.value`\n<LEAN_FILE>:101:0: error: unsolved goals\nR : Type u\ninst✝⁹ : CommSemiring R\nS : Type u_1\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nι₁ : Type v₁\ninst✝⁶ : DecidableEq ι₁\nM₁ : ι₁ → Type w₁\nM₂' : Type w₂'\ninst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)\ninst✝⁴ : AddCommMonoid M₂'\ninst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)\ninst✝² : Module R M₂'\ninst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)\ninst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)\ni : ι₁\nx : M₁ i\ny : M₂'\n⊢ (directSumLeft R S M₁ M₂') ((lof S ι₁ M₁ i) x ⊗ₜ[R] y) = (lof S ι₁ (fun i => M₁ i ⊗[R] M₂') i) (x ⊗ₜ[R] y)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:226:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_4c58e26e07`
<LEAN_FILE>:201:0: error: unsolved goals
X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
⊢ SeparableSpace ↑s

Response:
by
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

In this file we define metrizable topological spaces, i.e., topological spaces for which there
exists a metric space structure that generates the same topology.
We define it without any reference to metric spaces in order to avoid importing the real numbers.
For the proof that metrizable spaces admit a compatible metric,
see `Mathlib/Topology/Metrizable/Uniformity.lean`.
-/

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

variable {ι X Y : Type*} {A : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
  [∀ i, TopologicalSpace (A i)]

/-- A topological space is *pseudometrizable* if there exists a pseudometric space structure
compatible with the topology. To minimize imports, we implement this class in terms of the
existence of a countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_countably_generated :
    ∃ u : UniformSpace X, u.toTopologicalSpace = t ∧ (uniformity X).IsCountablyGenerated

/-- A uniform space with countably generated `𝓤 X` is pseudometrizable. -/
instance (priority := 100) _root_.UniformSpace.pseudoMetrizableSpace {X : Type*}
    [u : UniformSpace X] [hu : IsCountablyGenerated (uniformity X)] : PseudoMetrizableSpace X :=
  ⟨⟨u, rfl, hu⟩⟩

/-- Construct on a pseudometrizable space a countably generated uniformity
compatible with the topology. Use `pseudoMetrizableSpaceUniformity_countably_generated` for a proof
that this uniformity is countably generated. -/
-- see note [reducible non-instances]
noncomputable abbrev pseudoMetrizableSpaceUniformity (X : Type*) [TopologicalSpace X]
    [h : PseudoMetrizableSpace X] : UniformSpace X :=
  h.exists_countably_generated.choose.replaceTopology
    h.exists_countably_generated.choose_spec.1.symm

example {X : Type*} [t : TopologicalSpace X] [PseudoMetrizableSpace X] :
    (pseudoMetrizableSpaceUniformity X).toTopologicalSpace = t := by
  with_reducible_and_instances rfl

/-- The uniformity coming from `pseudoMetrizableSpaceUniformity` is countably generated.. -/
theorem pseudoMetrizableSpaceUniformity_countably_generated
    (X : Type*) [TopologicalSpace X] [h : PseudoMetrizableSpace X] :
    𝓤[pseudoMetrizableSpaceUniformity X].IsCountablyGenerated :=
  h.exists_countably_generated.choose_spec.2

instance pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
    PseudoMetrizableSpace (X × Y) :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  let : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  inferInstance

/-- Given an inducing map of a topological space into a pseudometrizable space, the source space
is also pseudometrizable. -/
theorem _root_.Topology.IsInducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : IsInducing f) : PseudoMetrizableSpace X :=
  let u : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  ⟨⟨u.comap f, u.toTopologicalSpace_comap.trans hf.eq_induced.symm,
    Filter.comap.isCountablyGenerated (uniformity Y) (Prod.map f f)⟩⟩

/-- Every pseudo-metrizable space is first countable. -/
instance (priority := 100) PseudoMetrizableSpace.firstCountableTopology
    [h : PseudoMetrizableSpace X] : FirstCountableTopology X :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  inferInstance

instance PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
    PseudoMetrizableSpace s :=
  IsInducing.subtypeVal.pseudoMetrizableSpace

instance pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (A i)] :
    PseudoMetrizableSpace (∀ i, A i) :=
  let := fun i => pseudoMetrizableSpaceUniformity (A i)
  have := fun i => pseudoMetrizableSpaceUniformity_countably_generated (A i)
  inferInstance

instance PseudoMetrizableSpace.regularSpace [PseudoMetrizableSpace X] : RegularSpace X :=
  let := pseudoMetrizableSpaceUniformity X
  inferInstance

instance (priority := 100) IndiscreteTopology.pseudoMetrizableSpace [IndiscreteTopology X] :
    PseudoMetrizableSpace X where
  exists_countably_generated :=
    ⟨⊤, (IndiscreteTopology.eq_top X).symm, isCountablyGenerated_top⟩

/-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To minimize imports, we implement this class in terms of the existence of a
countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop extends
    PseudoMetrizableSpace X, T0Space X

-- See note [lower instance priority]
attribute [instance 100] MetrizableSpace.toT0Space
attribute [instance 100] MetrizableSpace.toPseudoMetrizableSpace

instance (priority := 100) PseudoMetrizableSpace.toMetrizableSpace
    [T0Space X] [h : PseudoMetrizableSpace X] : MetrizableSpace X where

instance (priority := 100) t2Space_of_metrizableSpace [MetrizableSpace X] : T2Space X :=
  letI : UniformSpace X := pseudoMetrizableSpaceUniformity X
  inferInstance

instance metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] :
    MetrizableSpace (X × Y) where

/-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
theorem _root_.Topology.IsEmbedding.metrizableSpace [MetrizableSpace Y] {f : X → Y}
    (hf : IsEmbedding f) : MetrizableSpace X where
  toPseudoMetrizableSpace := hf.toIsInducing.pseudoMetrizableSpace
  toT0Space := hf.t0Space

instance MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s :=
  IsEmbedding.subtypeVal.metrizableSpace

instance metrizableSpace_pi [∀ i, MetrizableSpace (A i)] : MetrizableSpace (∀ i, A i) where

theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s :=
  let ⟨u, hu, hs⟩ := hs
  have := hu.to_subtype
  have : SeparableSpace (closure u) :=
    ⟨Set.range (u.inclusion subset_closure), Set.countable_range (u.inclusion subset_closure),
      Subtype.dense_iff.2 <| by rw [← Set.range_comp, Set.val_comp_inclusion, Subtype.range_coe]⟩
  let := pseudoMetrizableSpaceUniformity (closure u)
  have := pseudoMetrizableSpaceUniformity_countably_generated (closure u)
  have := secondCountable_of_separable (closure u)
  (Topology.IsEmbedding.inclusion hs).secondCountableTopology

instance (X : Type*) [TopologicalSpace X] [LindelofSpace X] [PseudoMetrizableSpace X] :
    SecondCountableTopology X := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  suffices _ : SeparableSpace X from secondCountable_of_separable X
  obtain ⟨V, hVb, hVs⟩ := has_seq_basis X
  choose U hUc hUu using fun n =>
    LindelofSpace.elim_nhds_subcover (fun x => ball x (V n))
      (fun x => ball_mem_nhds x (hVb.mem n))
  refine ⟨Set.iUnion U, Set.countable_iUnion hUc, fun x => ?_⟩
  rw [mem_closure_iff_frequently, nhds_eq_comap_uniformity, frequently_comap, hVb.frequently_iff]
  intro n _
  obtain ⟨i, hi, hx⟩ := Set.mem_iUnion₂.1 (Set.eq_univ_iff_forall.1 (hUu n) x)
  rw [ball_eq_of_symmetry] at hx
  exact ⟨(x, i), hx, i, rfl, Set.mem_iUnion_of_mem n hi⟩

/-- If a set `s` is separable in a pseudo metrizable space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. -/
theorem IsSeparable.exists_countable_dense_subset [PseudoMetrizableSpace X]
    {s : Set X} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  apply subset_countable_closure_of_almost_dense_set
  intro U hU
  obtain ⟨t, htc, hst⟩ := hs
  refine ⟨t, htc, fun x hx => ?_⟩
  obtain ⟨y, hyx, hyt⟩ := mem_closure_iff_ball.1 (hst hx) (symmetrize_mem_uniformity hU)
  exact mem_biUnion hyt (ball_mono SetRel.symmetrize_subset_inv x hyx)

/-- If a set `s` is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. -/

Target:
theorem IsSeparable.separableSpace [PseudoMetrizableSpace X] {s : Set X} (hs : IsSeparable s) :
    SeparableSpace s :=

Failed candidate:
by
  have __repair_pad_0_4c58e26e07 : True := by trivial
  have __repair_pad_1_4c58e26e07 : True := by trivial
  have __repair_pad_2_4c58e26e07 : True := by trivial
  have __repair_pad_3_4c58e26e07 : True := by trivial
  have __repair_pad_4_4c58e26e07 : True := by trivial
  have __repair_pad_5_4c58e26e07 : True := by trivial
  have __repair_pad_6_4c58e26e07 : True := by trivial
  have __repair_pad_7_4c58e26e07 : True := by trivial
  have __repair_pad_8_4c58e26e07 : True := by trivial
  have __repair_pad_9_4c58e26e07 : True := by trivial
  have __repair_pad_10_4c58e26e07 : True := by trivial
  have __repair_pad_11_4c58e26e07 : True := by trivial
  have __repair_pad_12_4c58e26e07 : True := by trivial
  have __repair_pad_13_4c58e26e07 : True := by trivial
  have __repair_pad_14_4c58e26e07 : True := by trivial
  have __repair_pad_15_4c58e26e07 : True := by trivial
  have __repair_pad_16_4c58e26e07 : True := by trivial
  have __repair_pad_17_4c58e26e07 : True := by trivial
  have __repair_pad_18_4c58e26e07 : True := by trivial
  have __repair_pad_19_4c58e26e07 : True := by trivial
  have __repair_pad_20_4c58e26e07 : True := by trivial
  have __repair_pad_21_4c58e26e07 : True := by trivial
  have __repair_pad_22_4c58e26e07 : True := by trivial
  have __repair_pad_23_4c58e26e07 : True := by trivial
  have __repair_fail_4c58e26e07 := __oprover_overlong_4c58e26e07
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:226:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_4c58e26e07`\n<LEAN_FILE>:201:0: error: unsolved goals\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof:

[Assistant]
by
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

In this file we define metrizable topological spaces, i.e., topological spaces for which there
exists a metric space structure that generates the same topology.
We define it without any reference to metric spaces in order to avoid importing the real numbers.
For the proof that metrizable spaces admit a compatible metric,
see `Mathlib/Topology/Metrizable/Uniformity.lean`.
-/

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

variable {ι X Y : Type*} {A : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
  [∀ i, TopologicalSpace (A i)]

/-- A topological space is *pseudometrizable* if there exists a pseudometric space structure
compatible with the topology. To minimize imports, we implement this class in terms of the
existence of a countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_countably_generated :
    ∃ u : UniformSpace X, u.toTopologicalSpace = t ∧ (uniformity X).IsCountablyGenerated

/-- A uniform space with countably generated `𝓤 X` is pseudometrizable. -/
instance (priority := 100) _root_.UniformSpace.pseudoMetrizableSpace {X : Type*}
    [u : UniformSpace X] [hu : IsCountablyGenerated (uniformity X)] : PseudoMetrizableSpace X :=
  ⟨⟨u, rfl, hu⟩⟩

/-- Construct on a pseudometrizable space a countably generated uniformity
compatible with the topology. Use `pseudoMetrizableSpaceUniformity_countably_generated` for a proof
that this uniformity is countably generated. -/
-- see note [reducible non-instances]
noncomputable abbrev pseudoMetrizableSpaceUniformity (X : Type*) [TopologicalSpace X]
    [h : PseudoMetrizableSpace X] : UniformSpace X :=
  h.exists_countably_generated.choose.replaceTopology
    h.exists_countably_generated.choose_spec.1.symm

example {X : Type*} [t : TopologicalSpace X] [PseudoMetrizableSpace X] :
    (pseudoMetrizableSpaceUniformity X).toTopologicalSpace = t := by
  with_reducible_and_instances rfl

/-- The uniformity coming from `pseudoMetrizableSpaceUniformity` is countably generated.. -/
theorem pseudoMetrizableSpaceUniformity_countably_generated
    (X : Type*) [TopologicalSpace X] [h : PseudoMetrizableSpace X] :
    𝓤[pseudoMetrizableSpaceUniformity X].IsCountablyGenerated :=
  h.exists_countably_generated.choose_spec.2

instance pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
    PseudoMetrizableSpace (X × Y) :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  let : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  inferInstance

/-- Given an inducing map of a topological space into a pseudometrizable space, the source space
is also pseudometrizable. -/
theorem _root_.Topology.IsInducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : IsInducing f) : PseudoMetrizableSpace X :=
  let u : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  ⟨⟨u.comap f, u.toTopologicalSpace_comap.trans hf.eq_induced.symm,
    Filter.comap.isCountablyGenerated (uniformity Y) (Prod.map f f)⟩⟩

/-- Every pseudo-metrizable space is first countable. -/
instance (priority := 100) PseudoMetrizableSpace.firstCountableTopology
    [h : PseudoMetrizableSpace X] : FirstCountableTopology X :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  inferInstance

instance PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
    PseudoMetrizableSpace s :=
  IsInducing.subtypeVal.pseudoMetrizableSpace

instance pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (A i)] :
    PseudoMetrizableSpace (∀ i, A i) :=
  let := fun i => pseudoMetrizableSpaceUniformity (A i)
  have := fun i => pseudoMetrizableSpaceUniformity_countably_generated (A i)
  inferInstance

instance PseudoMetrizableSpace.regularSpace [PseudoMetrizableSpace X] : RegularSpace X :=
  let := pseudoMetrizableSpaceUniformity X
  inferInstance

instance (priority := 100) IndiscreteTopology.pseudoMetrizableSpace [IndiscreteTopology X] :
    PseudoMetrizableSpace X where
  exists_countably_generated :=
    ⟨⊤, (IndiscreteTopology.eq_top X).symm, isCountablyGenerated_top⟩

/-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To minimize imports, we implement this class in terms of the existence of a
countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop extends
    PseudoMetrizableSpace X, T0Space X

-- See note [lower instance priority]
attribute [instance 100] MetrizableSpace.toT0Space
attribute [instance 100] MetrizableSpace.toPseudoMetrizableSpace

instance (priority := 100) PseudoMetrizableSpace.toMetrizableSpace
    [T0Space X] [h : PseudoMetrizableSpace X] : MetrizableSpace X where

instance (priority := 100) t2Space_of_metrizableSpace [MetrizableSpace X] : T2Space X :=
  letI : UniformSpace X := pseudoMetrizableSpaceUniformity X
  inferInstance

instance metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] :
    MetrizableSpace (X × Y) where

/-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
theorem _root_.Topology.IsEmbedding.metrizableSpace [MetrizableSpace Y] {f : X → Y}
    (hf : IsEmbedding f) : MetrizableSpace X where
  toPseudoMetrizableSpace := hf.toIsInducing.pseudoMetrizableSpace
  toT0Space := hf.t0Space

instance MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s :=
  IsEmbedding.subtypeVal.metrizableSpace

instance metrizableSpace_pi [∀ i, MetrizableSpace (A i)] : MetrizableSpace (∀ i, A i) where

theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s :=
  let ⟨u, hu, hs⟩ := hs
  have := hu.to_subtype
  have : SeparableSpace (closure u) :=
    ⟨Set.range (u.inclusion subset_closure), Set.countable_range (u.inclusion subset_closure),
      Subtype.dense_iff.2 <| by rw [← Set.range_comp, Set.val_comp_inclusion, Subtype.range_coe]⟩
  let := pseudoMetrizableSpaceUniformity (closure u)
  have := pseudoMetrizableSpaceUniformity_countably_generated (closure u)
  have := secondCountable_of_separable (closure u)
  (Topology.IsEmbedding.inclusion hs).secondCountableTopology

instance (X : Type*) [TopologicalSpace X] [LindelofSpace X] [PseudoMetrizableSpace X] :
    SecondCountableTopology X := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  suffices _ : SeparableSpace X from secondCountable_of_separable X
  obtain ⟨V, hVb, hVs⟩ := has_seq_basis X
  choose U hUc hUu using fun n =>
    LindelofSpace.elim_nhds_subcover (fun x => ball x (V n))
      (fun x => ball_mem_nhds x (hVb.mem n))
  refine ⟨Set.iUnion U, Set.countable_iUnion hUc, fun x => ?_⟩
  rw [mem_closure_iff_frequently, nhds_eq_comap_uniformity, frequently_comap, hVb.frequently_iff]
  intro n _
  obtain ⟨i, hi, hx⟩ := Set.mem_iUnion₂.1 (Set.eq_univ_iff_forall.1 (hUu n) x)
  rw [ball_eq_of_symmetry] at hx
  exact ⟨(x, i), hx, i, rfl, Set.mem_iUnion_of_mem n hi⟩

/-- If a set `s` is separable in a pseudo metrizable space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. -/
theorem IsSeparable.exists_countable_dense_subset [PseudoMetrizableSpace X]
    {s : Set X} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  apply subset_countable_closure_of_almost_dense_set
  intro U hU
  obtain ⟨t, htc, hst⟩ := hs
  refine ⟨t, htc, fun x hx => ?_⟩
  obtain ⟨y, hyx, hyt⟩ := mem_closure_iff_ball.1 (hst hx) (symmetrize_mem_uniformity hU)
  exact mem_biUnion hyt (ball_mono SetRel.symmetrize_subset_inv x hyx)

/-- If a set `s` is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. -/

Target:
theorem IsSeparable.separableSpace [PseudoMetrizableSpace X] {s : Set X} (hs : IsSeparable s) :
    SeparableSpace s :=

Failed candidate:
by
  have __repair_pad_0_4c58e26e07 : True := by trivial
  have __repair_pad_1_4c58e26e07 : True := by trivial
  have __repair_pad_2_4c58e26e07 : True := by trivial
  have __repair_pad_3_4c58e26e07 : True := by trivial
  have __repair_pad_4_4c58e26e07 : True := by trivial
  have __repair_pad_5_4c58e26e07 : True := by trivial
  have __repair_pad_6_4c58e26e07 : True := by trivial
  have __repair_pad_7_4c58e26e07 : True := by trivial
  have __repair_pad_8_4c58e26e07 : True := by trivial
  have __repair_pad_9_4c58e26e07 : True := by trivial
  have __repair_pad_10_4c58e26e07 : True := by trivial
  have __repair_pad_11_4c58e26e07 : True := by trivial
  have __repair_pad_12_4c58e26e07 : True := by trivial
  have __repair_pad_13_4c58e26e07 : True := by trivial
  have __repair_pad_14_4c58e26e07 : True := by trivial
  have __repair_pad_15_4c58e26e07 : True := by trivial
  have __repair_pad_16_4c58e26e07 : True := by trivial
  have __repair_pad_17_4c58e26e07 : True := by trivial
  have __repair_pad_18_4c58e26e07 : True := by trivial
  have __repair_pad_19_4c58e26e07 : True := by trivial
  have __repair_pad_20_4c58e26e07 : True := by trivial
  have __repair_pad_21_4c58e26e07 : True := by trivial
  have __repair_pad_22_4c58e26e07 : True := by trivial
  have __repair_pad_23_4c58e26e07 : True := by trivial
  have __repair_fail_4c58e26e07 := __oprover_overlong_4c58e26e07
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:226:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_4c58e26e07`\n<LEAN_FILE>:201:0: error: unsolved goals\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof:


<LEAN_FILE>:202:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing6aa264b136.value`
<LEAN_FILE>:201:0: error: unsolved goals
X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
⊢ SeparableSpace ↑s

Response:
by
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

In this file we define metrizable topological spaces, i.e., topological spaces for which there
exists a metric space structure that generates the same topology.
We define it without any reference to metric spaces in order to avoid importing the real numbers.
For the proof that metrizable spaces admit a compatible metric,
see `Mathlib/Topology/Metrizable/Uniformity.lean`.
-/

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

variable {ι X Y : Type*} {A : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
  [∀ i, TopologicalSpace (A i)]

/-- A topological space is *pseudometrizable* if there exists a pseudometric space structure
compatible with the topology. To minimize imports, we implement this class in terms of the
existence of a countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_countably_generated :
    ∃ u : UniformSpace X, u.toTopologicalSpace = t ∧ (uniformity X).IsCountablyGenerated

/-- A uniform space with countably generated `𝓤 X` is pseudometrizable. -/
instance (priority := 100) _root_.UniformSpace.pseudoMetrizableSpace {X : Type*}
    [u : UniformSpace X] [hu : IsCountablyGenerated (uniformity X)] : PseudoMetrizableSpace X :=
  ⟨⟨u, rfl, hu⟩⟩

/-- Construct on a pseudometrizable space a countably generated uniformity
compatible with the topology. Use `pseudoMetrizableSpaceUniformity_countably_generated` for a proof
that this uniformity is countably generated. -/
-- see note [reducible non-instances]
noncomputable abbrev pseudoMetrizableSpaceUniformity (X : Type*) [TopologicalSpace X]
    [h : PseudoMetrizableSpace X] : UniformSpace X :=
  h.exists_countably_generated.choose.replaceTopology
    h.exists_countably_generated.choose_spec.1.symm

example {X : Type*} [t : TopologicalSpace X] [PseudoMetrizableSpace X] :
    (pseudoMetrizableSpaceUniformity X).toTopologicalSpace = t := by
  with_reducible_and_instances rfl

/-- The uniformity coming from `pseudoMetrizableSpaceUniformity` is countably generated.. -/
theorem pseudoMetrizableSpaceUniformity_countably_generated
    (X : Type*) [TopologicalSpace X] [h : PseudoMetrizableSpace X] :
    𝓤[pseudoMetrizableSpaceUniformity X].IsCountablyGenerated :=
  h.exists_countably_generated.choose_spec.2

instance pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
    PseudoMetrizableSpace (X × Y) :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  let : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  inferInstance

/-- Given an inducing map of a topological space into a pseudometrizable space, the source space
is also pseudometrizable. -/
theorem _root_.Topology.IsInducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : IsInducing f) : PseudoMetrizableSpace X :=
  let u : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  ⟨⟨u.comap f, u.toTopologicalSpace_comap.trans hf.eq_induced.symm,
    Filter.comap.isCountablyGenerated (uniformity Y) (Prod.map f f)⟩⟩

/-- Every pseudo-metrizable space is first countable. -/
instance (priority := 100) PseudoMetrizableSpace.firstCountableTopology
    [h : PseudoMetrizableSpace X] : FirstCountableTopology X :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  inferInstance

instance PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
    PseudoMetrizableSpace s :=
  IsInducing.subtypeVal.pseudoMetrizableSpace

instance pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (A i)] :
    PseudoMetrizableSpace (∀ i, A i) :=
  let := fun i => pseudoMetrizableSpaceUniformity (A i)
  have := fun i => pseudoMetrizableSpaceUniformity_countably_generated (A i)
  inferInstance

instance PseudoMetrizableSpace.regularSpace [PseudoMetrizableSpace X] : RegularSpace X :=
  let := pseudoMetrizableSpaceUniformity X
  inferInstance

instance (priority := 100) IndiscreteTopology.pseudoMetrizableSpace [IndiscreteTopology X] :
    PseudoMetrizableSpace X where
  exists_countably_generated :=
    ⟨⊤, (IndiscreteTopology.eq_top X).symm, isCountablyGenerated_top⟩

/-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To minimize imports, we implement this class in terms of the existence of a
countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop extends
    PseudoMetrizableSpace X, T0Space X

-- See note [lower instance priority]
attribute [instance 100] MetrizableSpace.toT0Space
attribute [instance 100] MetrizableSpace.toPseudoMetrizableSpace

instance (priority := 100) PseudoMetrizableSpace.toMetrizableSpace
    [T0Space X] [h : PseudoMetrizableSpace X] : MetrizableSpace X where

instance (priority := 100) t2Space_of_metrizableSpace [MetrizableSpace X] : T2Space X :=
  letI : UniformSpace X := pseudoMetrizableSpaceUniformity X
  inferInstance

instance metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] :
    MetrizableSpace (X × Y) where

/-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
theorem _root_.Topology.IsEmbedding.metrizableSpace [MetrizableSpace Y] {f : X → Y}
    (hf : IsEmbedding f) : MetrizableSpace X where
  toPseudoMetrizableSpace := hf.toIsInducing.pseudoMetrizableSpace
  toT0Space := hf.t0Space

instance MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s :=
  IsEmbedding.subtypeVal.metrizableSpace

instance metrizableSpace_pi [∀ i, MetrizableSpace (A i)] : MetrizableSpace (∀ i, A i) where

theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s :=
  let ⟨u, hu, hs⟩ := hs
  have := hu.to_subtype
  have : SeparableSpace (closure u) :=
    ⟨Set.range (u.inclusion subset_closure), Set.countable_range (u.inclusion subset_closure),
      Subtype.dense_iff.2 <| by rw [← Set.range_comp, Set.val_comp_inclusion, Subtype.range_coe]⟩
  let := pseudoMetrizableSpaceUniformity (closure u)
  have := pseudoMetrizableSpaceUniformity_countably_generated (closure u)
  have := secondCountable_of_separable (closure u)
  (Topology.IsEmbedding.inclusion hs).secondCountableTopology

instance (X : Type*) [TopologicalSpace X] [LindelofSpace X] [PseudoMetrizableSpace X] :
    SecondCountableTopology X := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  suffices _ : SeparableSpace X from secondCountable_of_separable X
  obtain ⟨V, hVb, hVs⟩ := has_seq_basis X
  choose U hUc hUu using fun n =>
    LindelofSpace.elim_nhds_subcover (fun x => ball x (V n))
      (fun x => ball_mem_nhds x (hVb.mem n))
  refine ⟨Set.iUnion U, Set.countable_iUnion hUc, fun x => ?_⟩
  rw [mem_closure_iff_frequently, nhds_eq_comap_uniformity, frequently_comap, hVb.frequently_iff]
  intro n _
  obtain ⟨i, hi, hx⟩ := Set.mem_iUnion₂.1 (Set.eq_univ_iff_forall.1 (hUu n) x)
  rw [ball_eq_of_symmetry] at hx
  exact ⟨(x, i), hx, i, rfl, Set.mem_iUnion_of_mem n hi⟩

/-- If a set `s` is separable in a pseudo metrizable space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. -/
theorem IsSeparable.exists_countable_dense_subset [PseudoMetrizableSpace X]
    {s : Set X} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  apply subset_countable_closure_of_almost_dense_set
  intro U hU
  obtain ⟨t, htc, hst⟩ := hs
  refine ⟨t, htc, fun x hx => ?_⟩
  obtain ⟨y, hyx, hyt⟩ := mem_closure_iff_ball.1 (hst hx) (symmetrize_mem_uniformity hU)
  exact mem_biUnion hyt (ball_mono SetRel.symmetrize_subset_inv x hyx)

/-- If a set `s` is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. -/

Target:
theorem IsSeparable.separableSpace [PseudoMetrizableSpace X] {s : Set X} (hs : IsSeparable s) :
    SeparableSpace s :=

Failed candidate:
by
  have __repair_6aa264b136 := OProverMissing6aa264b136.value
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:202:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing6aa264b136.value`\n<LEAN_FILE>:201:0: error: unsolved goals\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof:

[Assistant]
by
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

In this file we define metrizable topological spaces, i.e., topological spaces for which there
exists a metric space structure that generates the same topology.
We define it without any reference to metric spaces in order to avoid importing the real numbers.
For the proof that metrizable spaces admit a compatible metric,
see `Mathlib/Topology/Metrizable/Uniformity.lean`.
-/

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

variable {ι X Y : Type*} {A : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
  [∀ i, TopologicalSpace (A i)]

/-- A topological space is *pseudometrizable* if there exists a pseudometric space structure
compatible with the topology. To minimize imports, we implement this class in terms of the
existence of a countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_countably_generated :
    ∃ u : UniformSpace X, u.toTopologicalSpace = t ∧ (uniformity X).IsCountablyGenerated

/-- A uniform space with countably generated `𝓤 X` is pseudometrizable. -/
instance (priority := 100) _root_.UniformSpace.pseudoMetrizableSpace {X : Type*}
    [u : UniformSpace X] [hu : IsCountablyGenerated (uniformity X)] : PseudoMetrizableSpace X :=
  ⟨⟨u, rfl, hu⟩⟩

/-- Construct on a pseudometrizable space a countably generated uniformity
compatible with the topology. Use `pseudoMetrizableSpaceUniformity_countably_generated` for a proof
that this uniformity is countably generated. -/
-- see note [reducible non-instances]
noncomputable abbrev pseudoMetrizableSpaceUniformity (X : Type*) [TopologicalSpace X]
    [h : PseudoMetrizableSpace X] : UniformSpace X :=
  h.exists_countably_generated.choose.replaceTopology
    h.exists_countably_generated.choose_spec.1.symm

example {X : Type*} [t : TopologicalSpace X] [PseudoMetrizableSpace X] :
    (pseudoMetrizableSpaceUniformity X).toTopologicalSpace = t := by
  with_reducible_and_instances rfl

/-- The uniformity coming from `pseudoMetrizableSpaceUniformity` is countably generated.. -/
theorem pseudoMetrizableSpaceUniformity_countably_generated
    (X : Type*) [TopologicalSpace X] [h : PseudoMetrizableSpace X] :
    𝓤[pseudoMetrizableSpaceUniformity X].IsCountablyGenerated :=
  h.exists_countably_generated.choose_spec.2

instance pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
    PseudoMetrizableSpace (X × Y) :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  let : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  inferInstance

/-- Given an inducing map of a topological space into a pseudometrizable space, the source space
is also pseudometrizable. -/
theorem _root_.Topology.IsInducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : IsInducing f) : PseudoMetrizableSpace X :=
  let u : UniformSpace Y := pseudoMetrizableSpaceUniformity Y
  have : (uniformity Y).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated Y
  ⟨⟨u.comap f, u.toTopologicalSpace_comap.trans hf.eq_induced.symm,
    Filter.comap.isCountablyGenerated (uniformity Y) (Prod.map f f)⟩⟩

/-- Every pseudo-metrizable space is first countable. -/
instance (priority := 100) PseudoMetrizableSpace.firstCountableTopology
    [h : PseudoMetrizableSpace X] : FirstCountableTopology X :=
  let : UniformSpace X := pseudoMetrizableSpaceUniformity X
  have : (uniformity X).IsCountablyGenerated :=
    pseudoMetrizableSpaceUniformity_countably_generated X
  inferInstance

instance PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
    PseudoMetrizableSpace s :=
  IsInducing.subtypeVal.pseudoMetrizableSpace

instance pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (A i)] :
    PseudoMetrizableSpace (∀ i, A i) :=
  let := fun i => pseudoMetrizableSpaceUniformity (A i)
  have := fun i => pseudoMetrizableSpaceUniformity_countably_generated (A i)
  inferInstance

instance PseudoMetrizableSpace.regularSpace [PseudoMetrizableSpace X] : RegularSpace X :=
  let := pseudoMetrizableSpaceUniformity X
  inferInstance

instance (priority := 100) IndiscreteTopology.pseudoMetrizableSpace [IndiscreteTopology X] :
    PseudoMetrizableSpace X where
  exists_countably_generated :=
    ⟨⊤, (IndiscreteTopology.eq_top X).symm, isCountablyGenerated_top⟩

/-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To minimize imports, we implement this class in terms of the existence of a
countably generated uniformity inducing the topology, which is mathematically
equivalent.
To endow such a space with a compatible uniformity, use
`letI : UniformSpace X := TopologicalSpace.pseudoMetrizableSpaceUniformity X`.
To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop extends
    PseudoMetrizableSpace X, T0Space X

-- See note [lower instance priority]
attribute [instance 100] MetrizableSpace.toT0Space
attribute [instance 100] MetrizableSpace.toPseudoMetrizableSpace

instance (priority := 100) PseudoMetrizableSpace.toMetrizableSpace
    [T0Space X] [h : PseudoMetrizableSpace X] : MetrizableSpace X where

instance (priority := 100) t2Space_of_metrizableSpace [MetrizableSpace X] : T2Space X :=
  letI : UniformSpace X := pseudoMetrizableSpaceUniformity X
  inferInstance

instance metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] :
    MetrizableSpace (X × Y) where

/-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
theorem _root_.Topology.IsEmbedding.metrizableSpace [MetrizableSpace Y] {f : X → Y}
    (hf : IsEmbedding f) : MetrizableSpace X where
  toPseudoMetrizableSpace := hf.toIsInducing.pseudoMetrizableSpace
  toT0Space := hf.t0Space

instance MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s :=
  IsEmbedding.subtypeVal.metrizableSpace

instance metrizableSpace_pi [∀ i, MetrizableSpace (A i)] : MetrizableSpace (∀ i, A i) where

theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s :=
  let ⟨u, hu, hs⟩ := hs
  have := hu.to_subtype
  have : SeparableSpace (closure u) :=
    ⟨Set.range (u.inclusion subset_closure), Set.countable_range (u.inclusion subset_closure),
      Subtype.dense_iff.2 <| by rw [← Set.range_comp, Set.val_comp_inclusion, Subtype.range_coe]⟩
  let := pseudoMetrizableSpaceUniformity (closure u)
  have := pseudoMetrizableSpaceUniformity_countably_generated (closure u)
  have := secondCountable_of_separable (closure u)
  (Topology.IsEmbedding.inclusion hs).secondCountableTopology

instance (X : Type*) [TopologicalSpace X] [LindelofSpace X] [PseudoMetrizableSpace X] :
    SecondCountableTopology X := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  suffices _ : SeparableSpace X from secondCountable_of_separable X
  obtain ⟨V, hVb, hVs⟩ := has_seq_basis X
  choose U hUc hUu using fun n =>
    LindelofSpace.elim_nhds_subcover (fun x => ball x (V n))
      (fun x => ball_mem_nhds x (hVb.mem n))
  refine ⟨Set.iUnion U, Set.countable_iUnion hUc, fun x => ?_⟩
  rw [mem_closure_iff_frequently, nhds_eq_comap_uniformity, frequently_comap, hVb.frequently_iff]
  intro n _
  obtain ⟨i, hi, hx⟩ := Set.mem_iUnion₂.1 (Set.eq_univ_iff_forall.1 (hUu n) x)
  rw [ball_eq_of_symmetry] at hx
  exact ⟨(x, i), hx, i, rfl, Set.mem_iUnion_of_mem n hi⟩

/-- If a set `s` is separable in a pseudo metrizable space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. -/
theorem IsSeparable.exists_countable_dense_subset [PseudoMetrizableSpace X]
    {s : Set X} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
  let := pseudoMetrizableSpaceUniformity X
  have := pseudoMetrizableSpaceUniformity_countably_generated X
  apply subset_countable_closure_of_almost_dense_set
  intro U hU
  obtain ⟨t, htc, hst⟩ := hs
  refine ⟨t, htc, fun x hx => ?_⟩
  obtain ⟨y, hyx, hyt⟩ := mem_closure_iff_ball.1 (hst hx) (symmetrize_mem_uniformity hU)
  exact mem_biUnion hyt (ball_mono SetRel.symmetrize_subset_inv x hyx)

/-- If a set `s` is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. -/

Target:
theorem IsSeparable.separableSpace [PseudoMetrizableSpace X] {s : Set X} (hs : IsSeparable s) :
    SeparableSpace s :=

Failed candidate:
by
  have __repair_6aa264b136 := OProverMissing6aa264b136.value
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:202:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing6aa264b136.value`\n<LEAN_FILE>:201:0: error: unsolved goals\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof: