<LEAN_FILE>:32:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge48dc5ebf3.value`
<LEAN_FILE>:31:0: error: unsolved goals
A : Type u_1
V : Type u_2
k : Type u_3
inst✝⁹ : Field k
inst✝⁸ : Ring A
inst✝⁷ : Algebra k A
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module k V
inst✝⁴ : Module A V
inst✝³ : IsScalarTower k A V
inst✝² : IsSimpleModule A V
inst✝¹ : FiniteDimensional k V
inst✝ : IsAlgClosed k
⊢ Function.Bijective ⇑(algebraMap k (Module.End A V))

Response:
by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

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

Imports:
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
/-!
# Basic facts about algebra representations

This file collects basic general facts about algebra representations. The purpose of this file is
to have general results so that when we prove a corresponding fact about group representations
(or Lie algebra representations etc), we can deduce them as special cases of facts from this file.

-/

public section

variable {A V : Type*} (k : Type*) [Field k] [Ring A] [Algebra k A] [AddCommGroup V] [Module k V]
  [Module A V] [IsScalarTower k A V]
  [IsSimpleModule A V] [FiniteDimensional k V] [IsAlgClosed k]

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/

Target:
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) :=

Failed candidate:
by
  have __repair_e48dc5ebf3 := OProverMissinge48dc5ebf3.value
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:32:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge48dc5ebf3.value`\n<LEAN_FILE>:31:0: error: unsolved goals\nA : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝⁹ : Field k\ninst✝⁸ : Ring A\ninst✝⁷ : Algebra k A\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : Module A V\ninst✝³ : IsScalarTower k A V\ninst✝² : IsSimpleModule A V\ninst✝¹ : FiniteDimensional k V\ninst✝ : IsAlgClosed k\n⊢ Function.Bijective ⇑(algebraMap k (Module.End A V))", "returncode": 1}

Corrected proof:

[Assistant]
by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

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

Imports:
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Stepan Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stepan Nesterov
-/
/-!
# Basic facts about algebra representations

This file collects basic general facts about algebra representations. The purpose of this file is
to have general results so that when we prove a corresponding fact about group representations
(or Lie algebra representations etc), we can deduce them as special cases of facts from this file.

-/

public section

variable {A V : Type*} (k : Type*) [Field k] [Ring A] [Algebra k A] [AddCommGroup V] [Module k V]
  [Module A V] [IsScalarTower k A V]
  [IsSimpleModule A V] [FiniteDimensional k V] [IsAlgClosed k]

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/

Target:
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) :=

Failed candidate:
by
  have __repair_e48dc5ebf3 := OProverMissinge48dc5ebf3.value
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:32:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge48dc5ebf3.value`\n<LEAN_FILE>:31:0: error: unsolved goals\nA : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝⁹ : Field k\ninst✝⁸ : Ring A\ninst✝⁷ : Algebra k A\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : Module A V\ninst✝³ : IsScalarTower k A V\ninst✝² : IsSimpleModule A V\ninst✝¹ : FiniteDimensional k V\ninst✝ : IsAlgClosed k\n⊢ Function.Bijective ⇑(algebraMap k (Module.End A V))", "returncode": 1}

Corrected proof:


Response:
by
  have h_finrank_inf : finrank K ↑(s ⊓ t) = 0 := by
    rw [disjoint_iff_inf_le, le_bot_iff] at hdisjoint
    rw [hdisjoint, finrank_bot]
  apply eq_top_of_finrank_eq
  replace hdim : finrank K V = finrank K s + finrank K t :=
    le_antisymm hdim (finrank_add_finrank_le_of_disjoint hdisjoint)
  rw [hdim]
  convert! s.finrank_sup_add_finrank_inf_eq t
  rw [h_finrank_inf, add_zero]

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

Imports:
public import Mathlib.LinearAlgebra.Dimension.DivisionRing
public import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
public import Mathlib.LinearAlgebra.FiniteDimensional.Basic
public import Mathlib.Tactic.IntervalCases

Namespace:
Submodule

Local context:
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
/-!
# Finite-dimensional vector spaces

This file contains some further development of finite-dimensional vector spaces, their dimensions,
and linear maps on such spaces.

Definitions are in `Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean`
and results that require fewer imports are in `Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean`.
-/

@[expose] public section

assert_not_exists Monoid.exponent Module.IsTorsion

universe u v v'

open Cardinal Submodule Module Function

variable {K : Type u} {V : Type v}

namespace Submodule

open IsNoetherian Module

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V]

/-- The dimension of a strict submodule is strictly bounded by the dimension of the ambient
space.

See also `Submodule.length_lt`. -/
theorem finrank_lt [FiniteDimensional K V] {s : Submodule K V} (h : s ≠ ⊤) :
    finrank K s < finrank K V := by
  rw [← s.finrank_quotient_add_finrank, add_comm]
  rw [← Quotient.nontrivial_iff] at h
  exact Nat.lt_add_of_pos_right finrank_pos

/-- The sum of the dimensions of s + t and s ∩ t is the sum of the dimensions of s and t -/
theorem finrank_sup_add_finrank_inf_eq (s t : Submodule K V) [FiniteDimensional K s]
    [FiniteDimensional K t] :
    finrank K ↑(s ⊔ t) + finrank K ↑(s ⊓ t) = finrank K ↑s + finrank K ↑t := by
  have key : Module.rank K ↑(s ⊔ t) + Module.rank K ↑(s ⊓ t) = Module.rank K s + Module.rank K t :=
    rank_sup_add_rank_inf_eq s t
  repeat rw [← finrank_eq_rank] at key
  norm_cast at key

theorem finrank_add_le_finrank_add_finrank (s t : Submodule K V) [FiniteDimensional K s]
    [FiniteDimensional K t] : finrank K (s ⊔ t : Submodule K V) ≤ finrank K s + finrank K t := by
  rw [← finrank_sup_add_finrank_inf_eq]
  exact self_le_add_right _ _

theorem finrank_add_finrank_le_of_disjoint [FiniteDimensional K V]
    {s t : Submodule K V} (hdisjoint : Disjoint s t) :
    finrank K s + finrank K t ≤ finrank K V := by
  rw [← Submodule.finrank_sup_add_finrank_inf_eq s t, hdisjoint.eq_bot, finrank_bot, add_zero]
  exact Submodule.finrank_le _

Target:
theorem eq_top_of_disjoint [FiniteDimensional K V] (s t : Submodule K V)
    (hdim : finrank K V ≤ finrank K s + finrank K t) (hdisjoint : Disjoint s t) : s ⊔ t = ⊤ :=

Proof body:

[Assistant]
by
  have h_finrank_inf : finrank K ↑(s ⊓ t) = 0 := by
    rw [disjoint_iff_inf_le, le_bot_iff] at hdisjoint
    rw [hdisjoint, finrank_bot]
  apply eq_top_of_finrank_eq
  replace hdim : finrank K V = finrank K s + finrank K t :=
    le_antisymm hdim (finrank_add_finrank_le_of_disjoint hdisjoint)
  rw [hdim]
  convert! s.finrank_sup_add_finrank_inf_eq t
  rw [h_finrank_inf, add_zero]

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

Imports:
public import Mathlib.LinearAlgebra.Dimension.DivisionRing
public import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
public import Mathlib.LinearAlgebra.FiniteDimensional.Basic
public import Mathlib.Tactic.IntervalCases

Namespace:
Submodule

Local context:
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
/-!
# Finite-dimensional vector spaces

This file contains some further development of finite-dimensional vector spaces, their dimensions,
and linear maps on such spaces.

Definitions are in `Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean`
and results that require fewer imports are in `Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean`.
-/

@[expose] public section

assert_not_exists Monoid.exponent Module.IsTorsion

universe u v v'

open Cardinal Submodule Module Function

variable {K : Type u} {V : Type v}

namespace Submodule

open IsNoetherian Module

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V]

/-- The dimension of a strict submodule is strictly bounded by the dimension of the ambient
space.

See also `Submodule.length_lt`. -/
theorem finrank_lt [FiniteDimensional K V] {s : Submodule K V} (h : s ≠ ⊤) :
    finrank K s < finrank K V := by
  rw [← s.finrank_quotient_add_finrank, add_comm]
  rw [← Quotient.nontrivial_iff] at h
  exact Nat.lt_add_of_pos_right finrank_pos

/-- The sum of the dimensions of s + t and s ∩ t is the sum of the dimensions of s and t -/
theorem finrank_sup_add_finrank_inf_eq (s t : Submodule K V) [FiniteDimensional K s]
    [FiniteDimensional K t] :
    finrank K ↑(s ⊔ t) + finrank K ↑(s ⊓ t) = finrank K ↑s + finrank K ↑t := by
  have key : Module.rank K ↑(s ⊔ t) + Module.rank K ↑(s ⊓ t) = Module.rank K s + Module.rank K t :=
    rank_sup_add_rank_inf_eq s t
  repeat rw [← finrank_eq_rank] at key
  norm_cast at key

theorem finrank_add_le_finrank_add_finrank (s t : Submodule K V) [FiniteDimensional K s]
    [FiniteDimensional K t] : finrank K (s ⊔ t : Submodule K V) ≤ finrank K s + finrank K t := by
  rw [← finrank_sup_add_finrank_inf_eq]
  exact self_le_add_right _ _

theorem finrank_add_finrank_le_of_disjoint [FiniteDimensional K V]
    {s t : Submodule K V} (hdisjoint : Disjoint s t) :
    finrank K s + finrank K t ≤ finrank K V := by
  rw [← Submodule.finrank_sup_add_finrank_inf_eq s t, hdisjoint.eq_bot, finrank_bot, add_zero]
  exact Submodule.finrank_le _

Target:
theorem eq_top_of_disjoint [FiniteDimensional K V] (s t : Submodule K V)
    (hdim : finrank K V ≤ finrank K s + finrank K t) (hdisjoint : Disjoint s t) : s ⊔ t = ⊤ :=

Proof body:


<LEAN_FILE>:102:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_e3602e652f`
<LEAN_FILE>:77:0: error: unsolved goals
X : Type u_1
U : SetRel X X
s N : Set X
inst✝¹ : U.IsRefl
inst✝ : U.IsSymm
hN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N
⊢ U.IsCover s N

Response:
by
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Covers in a uniform space

This file defines covers, aka nets, which are a quantitative notion of compactness given an
entourage.

A `U`-cover of a set `s` is a set `N` such that every element of `s` is `U`-close to some element of
`N`.

The concept of uniform covers is used to define two further notions of covering:
* Metric covers: `Metric.IsCover`, defined using the distance entourage.
* Dynamical covers: `Dynamics.IsDynCoverOf`, defined using the dynamical entourage.

## References

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], Section 4.2.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {U V : SetRel X X} {s t N N₁ N₂ : Set X} {x : X}

/-- For an entourage `U`, a set `N` is a *`U`-cover* of a set `s` if every point of `s` is `U`-close
to some point of `N`.

This is also called a *`U`-net* in the literature.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.1. -/
def IsCover (U : SetRel X X) (s N : Set X) : Prop := ∀ ⦃x⦄, x ∈ s → ∃ y ∈ N, x ~[U] y

@[simp] lemma IsCover.empty : IsCover U ∅ N := by simp [IsCover]

@[simp] lemma isCover_empty_right : IsCover U s ∅ ↔ s = ∅ := by
  simp [IsCover, eq_empty_iff_forall_notMem]

protected nonrec lemma IsCover.nonempty (hsN : IsCover U s N) (hs : s.Nonempty) : N.Nonempty :=
  let ⟨_x, hx⟩ := hs; let ⟨y, hy, _⟩ := hsN hx; ⟨y, hy⟩

@[simp] lemma IsCover.refl (U : SetRel X X) [U.IsRefl] (s : Set X) : IsCover U s s :=
  fun a ha ↦ ⟨a, ha, U.rfl⟩

lemma IsCover.rfl {U : SetRel X X} [U.IsRefl] {s : Set X} : IsCover U s s := refl U s

@[simp] protected lemma isCover_univ : IsCover univ s N ↔ (s.Nonempty → N.Nonempty) := by
  simp [IsCover, Set.Nonempty]

lemma IsCover.mono (hN : N₁ ⊆ N₂) (h₁ : IsCover U s N₁) : IsCover U s N₂ :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := h₁ hx; ⟨y, hN hy, hxy⟩

lemma IsCover.anti (hst : s ⊆ t) (ht : IsCover U t N) : IsCover U s N := fun _x hx ↦ ht <| hst hx

lemma IsCover.mono_entourage (hUV : U ⊆ V) (hU : IsCover U s N) : IsCover V s N :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := hU hx; ⟨y, hy, hUV hxy⟩

lemma IsCover.union (hs : IsCover U s N₁) (ht : IsCover U t N₂) : IsCover U (s ∪ t) (N₁ ∪ N₂) := fun
  | _x, .inl hx => let ⟨y, hy, hxy⟩ := hs hx; ⟨y, .inl hy, hxy⟩
  | _x, .inr hx => let ⟨y, hy, hxy⟩ := ht hx; ⟨y, .inr hy, hxy⟩

/-- A maximal `U`-separated subset of a set `s` is a `U`-cover of `s`.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.6. -/

Target:
lemma IsCover.of_maximal_isSeparated [U.IsRefl] [U.IsSymm]
    (hN : Maximal (fun N ↦ N ⊆ s ∧ IsSeparated U N) N) : IsCover U s N :=

Failed candidate:
by
  have __repair_pad_0_e3602e652f : True := by trivial
  have __repair_pad_1_e3602e652f : True := by trivial
  have __repair_pad_2_e3602e652f : True := by trivial
  have __repair_pad_3_e3602e652f : True := by trivial
  have __repair_pad_4_e3602e652f : True := by trivial
  have __repair_pad_5_e3602e652f : True := by trivial
  have __repair_pad_6_e3602e652f : True := by trivial
  have __repair_pad_7_e3602e652f : True := by trivial
  have __repair_pad_8_e3602e652f : True := by trivial
  have __repair_pad_9_e3602e652f : True := by trivial
  have __repair_pad_10_e3602e652f : True := by trivial
  have __repair_pad_11_e3602e652f : True := by trivial
  have __repair_pad_12_e3602e652f : True := by trivial
  have __repair_pad_13_e3602e652f : True := by trivial
  have __repair_pad_14_e3602e652f : True := by trivial
  have __repair_pad_15_e3602e652f : True := by trivial
  have __repair_pad_16_e3602e652f : True := by trivial
  have __repair_pad_17_e3602e652f : True := by trivial
  have __repair_pad_18_e3602e652f : True := by trivial
  have __repair_pad_19_e3602e652f : True := by trivial
  have __repair_pad_20_e3602e652f : True := by trivial
  have __repair_pad_21_e3602e652f : True := by trivial
  have __repair_pad_22_e3602e652f : True := by trivial
  have __repair_pad_23_e3602e652f : True := by trivial
  have __repair_fail_e3602e652f := __oprover_overlong_e3602e652f
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_e3602e652f`\n<LEAN_FILE>:77:0: error: unsolved goals\nX : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N\n⊢ U.IsCover s N", "returncode": 1}

Corrected proof:

[Assistant]
by
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Covers in a uniform space

This file defines covers, aka nets, which are a quantitative notion of compactness given an
entourage.

A `U`-cover of a set `s` is a set `N` such that every element of `s` is `U`-close to some element of
`N`.

The concept of uniform covers is used to define two further notions of covering:
* Metric covers: `Metric.IsCover`, defined using the distance entourage.
* Dynamical covers: `Dynamics.IsDynCoverOf`, defined using the dynamical entourage.

## References

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], Section 4.2.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {U V : SetRel X X} {s t N N₁ N₂ : Set X} {x : X}

/-- For an entourage `U`, a set `N` is a *`U`-cover* of a set `s` if every point of `s` is `U`-close
to some point of `N`.

This is also called a *`U`-net* in the literature.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.1. -/
def IsCover (U : SetRel X X) (s N : Set X) : Prop := ∀ ⦃x⦄, x ∈ s → ∃ y ∈ N, x ~[U] y

@[simp] lemma IsCover.empty : IsCover U ∅ N := by simp [IsCover]

@[simp] lemma isCover_empty_right : IsCover U s ∅ ↔ s = ∅ := by
  simp [IsCover, eq_empty_iff_forall_notMem]

protected nonrec lemma IsCover.nonempty (hsN : IsCover U s N) (hs : s.Nonempty) : N.Nonempty :=
  let ⟨_x, hx⟩ := hs; let ⟨y, hy, _⟩ := hsN hx; ⟨y, hy⟩

@[simp] lemma IsCover.refl (U : SetRel X X) [U.IsRefl] (s : Set X) : IsCover U s s :=
  fun a ha ↦ ⟨a, ha, U.rfl⟩

lemma IsCover.rfl {U : SetRel X X} [U.IsRefl] {s : Set X} : IsCover U s s := refl U s

@[simp] protected lemma isCover_univ : IsCover univ s N ↔ (s.Nonempty → N.Nonempty) := by
  simp [IsCover, Set.Nonempty]

lemma IsCover.mono (hN : N₁ ⊆ N₂) (h₁ : IsCover U s N₁) : IsCover U s N₂ :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := h₁ hx; ⟨y, hN hy, hxy⟩

lemma IsCover.anti (hst : s ⊆ t) (ht : IsCover U t N) : IsCover U s N := fun _x hx ↦ ht <| hst hx

lemma IsCover.mono_entourage (hUV : U ⊆ V) (hU : IsCover U s N) : IsCover V s N :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := hU hx; ⟨y, hy, hUV hxy⟩

lemma IsCover.union (hs : IsCover U s N₁) (ht : IsCover U t N₂) : IsCover U (s ∪ t) (N₁ ∪ N₂) := fun
  | _x, .inl hx => let ⟨y, hy, hxy⟩ := hs hx; ⟨y, .inl hy, hxy⟩
  | _x, .inr hx => let ⟨y, hy, hxy⟩ := ht hx; ⟨y, .inr hy, hxy⟩

/-- A maximal `U`-separated subset of a set `s` is a `U`-cover of `s`.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.6. -/

Target:
lemma IsCover.of_maximal_isSeparated [U.IsRefl] [U.IsSymm]
    (hN : Maximal (fun N ↦ N ⊆ s ∧ IsSeparated U N) N) : IsCover U s N :=

Failed candidate:
by
  have __repair_pad_0_e3602e652f : True := by trivial
  have __repair_pad_1_e3602e652f : True := by trivial
  have __repair_pad_2_e3602e652f : True := by trivial
  have __repair_pad_3_e3602e652f : True := by trivial
  have __repair_pad_4_e3602e652f : True := by trivial
  have __repair_pad_5_e3602e652f : True := by trivial
  have __repair_pad_6_e3602e652f : True := by trivial
  have __repair_pad_7_e3602e652f : True := by trivial
  have __repair_pad_8_e3602e652f : True := by trivial
  have __repair_pad_9_e3602e652f : True := by trivial
  have __repair_pad_10_e3602e652f : True := by trivial
  have __repair_pad_11_e3602e652f : True := by trivial
  have __repair_pad_12_e3602e652f : True := by trivial
  have __repair_pad_13_e3602e652f : True := by trivial
  have __repair_pad_14_e3602e652f : True := by trivial
  have __repair_pad_15_e3602e652f : True := by trivial
  have __repair_pad_16_e3602e652f : True := by trivial
  have __repair_pad_17_e3602e652f : True := by trivial
  have __repair_pad_18_e3602e652f : True := by trivial
  have __repair_pad_19_e3602e652f : True := by trivial
  have __repair_pad_20_e3602e652f : True := by trivial
  have __repair_pad_21_e3602e652f : True := by trivial
  have __repair_pad_22_e3602e652f : True := by trivial
  have __repair_pad_23_e3602e652f : True := by trivial
  have __repair_fail_e3602e652f := __oprover_overlong_e3602e652f
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_e3602e652f`\n<LEAN_FILE>:77:0: error: unsolved goals\nX : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N\n⊢ U.IsCover s N", "returncode": 1}

Corrected proof:


<LEAN_FILE>:155:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show ⟨b, y⟩ ≤ a by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

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

Imports:
public import Mathlib.Data.Set.Image
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.Interval.Set.Defs
public import Mathlib.Order.SetNotation

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
/-!
# Properties of relative upper/lower sets

This file proves results on `IsRelUpperSet` and `IsRelLowerSet`.
-/

public section

open Set

variable {α : Type*} {ι : Sort*} {κ : ι → Sort*} {s t : Set α} {a b : α} {P : α → Prop}

section LE

variable [LE α]

@[simp] lemma isRelUpperSet_true_iff_isUpperSet :
    IsRelUpperSet s (fun _ ↦ True) ↔ IsUpperSet s := by
  grind [IsUpperSet, IsRelUpperSet]

@[simp] lemma isRelLowerSet_true_iff_isLowerSet :
    IsRelLowerSet s (fun _ ↦ True) ↔ IsLowerSet s := by
  grind [IsLowerSet, IsRelLowerSet]

variable (P) in
lemma IsUpperSet.isRelUpperSet_sep (hs : IsUpperSet s) : IsRelUpperSet {x ∈ s | P x} P :=
  fun _ h => ⟨h.2, fun _ ht hp => ⟨hs ht h.1, hp⟩⟩
variable (P) in
lemma IsLowerSet.isRelLowerSet_sep (hs : IsLowerSet s) : IsRelLowerSet {x ∈ s | P x} P :=
  fun _ h => ⟨h.2, fun _ ht hp => ⟨hs ht h.1, hp⟩⟩

/-- A subset that is a lower set is additionally a _relative_ lower set. -/
lemma IsRelLowerSet.mono_isLowerSet (ht : IsRelLowerSet t P) (hs : IsLowerSet s) (hst : s ⊆ t) :
    IsRelLowerSet s P :=
  fun _ h => ⟨(ht (hst h)).1, fun _ ht _ => hs ht h⟩

/-- A subset that is an upper set is additionally a _relative_ upper set. -/
lemma IsRelUpperSet.mono_isUpperSet (ht : IsRelUpperSet t P) (hs : IsUpperSet s) (hst : s ⊆ t) :
    IsRelUpperSet s P :=
  fun _ h => ⟨(ht (hst h)).1, fun _ ht _ => hs ht h⟩

lemma IsRelUpperSet.prop_of_mem (hs : IsRelUpperSet s P) (h : a ∈ s) : P a := (hs h).1
lemma IsRelLowerSet.prop_of_mem (hs : IsRelLowerSet s P) (h : a ∈ s) : P a := (hs h).1

lemma IsRelUpperSet.mem_of_le (hs : IsRelUpperSet s P) (h : a ∈ s) (h₁ : a ≤ b) (h₂ : P b) :
    b ∈ s := (hs h).2 h₁ h₂
lemma IsRelLowerSet.mem_of_le (hs : IsRelLowerSet s P) (h : a ∈ s) (h₁ : b ≤ a) (h₂ : P b) :
    b ∈ s := (hs h).2 h₁ h₂

@[simp] lemma isRelUpperSet_empty : IsRelUpperSet (∅ : Set α) P := fun _ ↦ False.elim
@[simp] lemma isRelLowerSet_empty : IsRelLowerSet (∅ : Set α) P := fun _ ↦ False.elim

@[simp] lemma isRelUpperSet_self : IsRelUpperSet s (· ∈ s) := fun _ b ↦ ⟨b, fun _ _ ↦ id⟩
@[simp] lemma isRelLowerSet_self : IsRelLowerSet s (· ∈ s) := fun _ b ↦ ⟨b, fun _ _ ↦ id⟩

lemma IsRelUpperSet.union (hs : IsRelUpperSet s P) (ht : IsRelUpperSet t P) :
    IsRelUpperSet (s ∪ t) P := fun b mb ↦ by
  cases mb with
  | inl h => exact ⟨(hs h).1, fun _ x y ↦ .inl ((hs h).2 x y)⟩
  | inr h => exact ⟨(ht h).1, fun _ x y ↦ .inr ((ht h).2 x y)⟩

lemma IsRelLowerSet.union (hs : IsRelLowerSet s P) (ht : IsRelLowerSet t P) :
    IsRelLowerSet (s ∪ t) P := fun b mb ↦ by
  cases mb with
  | inl h => exact ⟨(hs h).1, fun _ x y ↦ .inl ((hs h).2 x y)⟩
  | inr h => exact ⟨(ht h).1, fun _ x y ↦ .inr ((ht h).2 x y)⟩

lemma IsRelUpperSet.inter (hs : IsRelUpperSet s P) (ht : IsRelUpperSet t P) :
    IsRelUpperSet (s ∩ t) P := fun b ⟨bs, bt⟩ ↦ by
  simp_all only [IsRelUpperSet, true_and]
  exact fun _ x y ↦ ⟨(hs bs).2 x y, (ht bt).2 x y⟩

lemma IsRelLowerSet.inter (hs : IsRelLowerSet s P) (ht : IsRelLowerSet t P) :
    IsRelLowerSet (s ∩ t) P := fun b ⟨bs, bt⟩ ↦ by
  simp_all only [IsRelLowerSet, true_and]
  exact fun _ x y ↦ ⟨(hs bs).2 x y, (ht bt).2 x y⟩

protected lemma IsRelUpperSet.sUnion {S : Set (Set α)} (hS : ∀ s ∈ S, IsRelUpperSet s P) :
    IsRelUpperSet (⋃₀ S) P := fun _ ⟨s, ms, mb⟩ ↦
  ⟨(hS s ms mb).1, fun _ x y ↦ ⟨s, ms, (hS s ms mb).2 x y⟩⟩

protected lemma IsRelLowerSet.sUnion {S : Set (Set α)} (hS : ∀ s ∈ S, IsRelLowerSet s P) :
    IsRelLowerSet (⋃₀ S) P := fun _ ⟨s, ms, mb⟩ ↦
  ⟨(hS s ms mb).1, fun _ x y ↦ ⟨s, ms, (hS s ms mb).2 x y⟩⟩

protected lemma IsRelUpperSet.iUnion {f : ι → Set α} (hf : ∀ i, IsRelUpperSet (f i) P) :
    IsRelUpperSet (⋃ i, f i) P :=
  .sUnion (forall_mem_range.2 hf)

protected lemma IsRelLowerSet.iUnion {f : ι → Set α} (hf : ∀ i, IsRelLowerSet (f i) P) :
    IsRelLowerSet (⋃ i, f i) P :=
  .sUnion (forall_mem_range.2 hf)

protected lemma IsRelUpperSet.iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelUpperSet (f i j) P) :
    IsRelUpperSet (⋃ (i) (j), f i j) P :=
  .iUnion fun i ↦ .iUnion (hf i)

protected lemma IsRelLowerSet.iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelLowerSet (f i j) P) :
    IsRelLowerSet (⋃ (i) (j), f i j) P :=
  .iUnion fun i ↦ .iUnion (hf i)

protected lemma IsRelUpperSet.sInter
    {S : Set (Set α)} (hS : S.Nonempty) (hf : ∀ s ∈ S, IsRelUpperSet s P) :
    IsRelUpperSet (⋂₀ S) P := fun b mb ↦ by
  obtain ⟨s₀, ms₀⟩ := hS
  refine ⟨(hf s₀ ms₀ (mb s₀ ms₀)).1, fun _ x y s ms ↦ (hf s ms (mb s ms)).2 x y⟩

protected lemma IsRelLowerSet.sInter
    {S : Set (Set α)} (hS : S.Nonempty) (hf : ∀ s ∈ S, IsRelLowerSet s P) :
    IsRelLowerSet (⋂₀ S) P := fun b mb ↦ by
  obtain ⟨s₀, ms₀⟩ := hS
  refine ⟨(hf s₀ ms₀ (mb s₀ ms₀)).1, fun _ x y s ms ↦ (hf s ms (mb s ms)).2 x y⟩

protected lemma IsRelUpperSet.iInter
    [Nonempty ι] {f : ι → Set α} (hf : ∀ i, IsRelUpperSet (f i) P) : IsRelUpperSet (⋂ i, f i) P :=
  .sInter (range_nonempty f) (forall_mem_range.2 hf)

protected lemma IsRelLowerSet.iInter
    [Nonempty ι] {f : ι → Set α} (hf : ∀ i, IsRelLowerSet (f i) P) : IsRelLowerSet (⋂ i, f i) P :=
  .sInter (range_nonempty f) (forall_mem_range.2 hf)

protected lemma IsRelUpperSet.iInter₂ [Nonempty ι] [∀ i, Nonempty (κ i)]
    {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelUpperSet (f i j) P) :
    IsRelUpperSet (⋂ (i) (j), f i j) P :=
  .iInter fun i ↦ .iInter (hf i)

protected lemma IsRelLowerSet.iInter₂ [Nonempty ι] [∀ i, Nonempty (κ i)]
    {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelLowerSet (f i j) P) :
    IsRelLowerSet (⋂ (i) (j), f i j) P :=
  .iInter fun i ↦ .iInter (hf i)

lemma isUpperSet_subtype_iff_isRelUpperSet {s : Set { x // P x }} :
    IsUpperSet s ↔ IsRelUpperSet (Subtype.val '' s) P := by
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show a ≤ ⟨b, y⟩ by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

Target:
lemma isLowerSet_subtype_iff_isRelLowerSet {s : Set { x // P x }} :
    IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P :=

Failed candidate:
by
  let __repair_f16d81750e : Fact False := inferInstance
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show ⟨b, y⟩ ≤ a by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:155:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show ⟨b, y⟩ ≤ a by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

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

Imports:
public import Mathlib.Data.Set.Image
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.Interval.Set.Defs
public import Mathlib.Order.SetNotation

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2025 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
/-!
# Properties of relative upper/lower sets

This file proves results on `IsRelUpperSet` and `IsRelLowerSet`.
-/

public section

open Set

variable {α : Type*} {ι : Sort*} {κ : ι → Sort*} {s t : Set α} {a b : α} {P : α → Prop}

section LE

variable [LE α]

@[simp] lemma isRelUpperSet_true_iff_isUpperSet :
    IsRelUpperSet s (fun _ ↦ True) ↔ IsUpperSet s := by
  grind [IsUpperSet, IsRelUpperSet]

@[simp] lemma isRelLowerSet_true_iff_isLowerSet :
    IsRelLowerSet s (fun _ ↦ True) ↔ IsLowerSet s := by
  grind [IsLowerSet, IsRelLowerSet]

variable (P) in
lemma IsUpperSet.isRelUpperSet_sep (hs : IsUpperSet s) : IsRelUpperSet {x ∈ s | P x} P :=
  fun _ h => ⟨h.2, fun _ ht hp => ⟨hs ht h.1, hp⟩⟩
variable (P) in
lemma IsLowerSet.isRelLowerSet_sep (hs : IsLowerSet s) : IsRelLowerSet {x ∈ s | P x} P :=
  fun _ h => ⟨h.2, fun _ ht hp => ⟨hs ht h.1, hp⟩⟩

/-- A subset that is a lower set is additionally a _relative_ lower set. -/
lemma IsRelLowerSet.mono_isLowerSet (ht : IsRelLowerSet t P) (hs : IsLowerSet s) (hst : s ⊆ t) :
    IsRelLowerSet s P :=
  fun _ h => ⟨(ht (hst h)).1, fun _ ht _ => hs ht h⟩

/-- A subset that is an upper set is additionally a _relative_ upper set. -/
lemma IsRelUpperSet.mono_isUpperSet (ht : IsRelUpperSet t P) (hs : IsUpperSet s) (hst : s ⊆ t) :
    IsRelUpperSet s P :=
  fun _ h => ⟨(ht (hst h)).1, fun _ ht _ => hs ht h⟩

lemma IsRelUpperSet.prop_of_mem (hs : IsRelUpperSet s P) (h : a ∈ s) : P a := (hs h).1
lemma IsRelLowerSet.prop_of_mem (hs : IsRelLowerSet s P) (h : a ∈ s) : P a := (hs h).1

lemma IsRelUpperSet.mem_of_le (hs : IsRelUpperSet s P) (h : a ∈ s) (h₁ : a ≤ b) (h₂ : P b) :
    b ∈ s := (hs h).2 h₁ h₂
lemma IsRelLowerSet.mem_of_le (hs : IsRelLowerSet s P) (h : a ∈ s) (h₁ : b ≤ a) (h₂ : P b) :
    b ∈ s := (hs h).2 h₁ h₂

@[simp] lemma isRelUpperSet_empty : IsRelUpperSet (∅ : Set α) P := fun _ ↦ False.elim
@[simp] lemma isRelLowerSet_empty : IsRelLowerSet (∅ : Set α) P := fun _ ↦ False.elim

@[simp] lemma isRelUpperSet_self : IsRelUpperSet s (· ∈ s) := fun _ b ↦ ⟨b, fun _ _ ↦ id⟩
@[simp] lemma isRelLowerSet_self : IsRelLowerSet s (· ∈ s) := fun _ b ↦ ⟨b, fun _ _ ↦ id⟩

lemma IsRelUpperSet.union (hs : IsRelUpperSet s P) (ht : IsRelUpperSet t P) :
    IsRelUpperSet (s ∪ t) P := fun b mb ↦ by
  cases mb with
  | inl h => exact ⟨(hs h).1, fun _ x y ↦ .inl ((hs h).2 x y)⟩
  | inr h => exact ⟨(ht h).1, fun _ x y ↦ .inr ((ht h).2 x y)⟩

lemma IsRelLowerSet.union (hs : IsRelLowerSet s P) (ht : IsRelLowerSet t P) :
    IsRelLowerSet (s ∪ t) P := fun b mb ↦ by
  cases mb with
  | inl h => exact ⟨(hs h).1, fun _ x y ↦ .inl ((hs h).2 x y)⟩
  | inr h => exact ⟨(ht h).1, fun _ x y ↦ .inr ((ht h).2 x y)⟩

lemma IsRelUpperSet.inter (hs : IsRelUpperSet s P) (ht : IsRelUpperSet t P) :
    IsRelUpperSet (s ∩ t) P := fun b ⟨bs, bt⟩ ↦ by
  simp_all only [IsRelUpperSet, true_and]
  exact fun _ x y ↦ ⟨(hs bs).2 x y, (ht bt).2 x y⟩

lemma IsRelLowerSet.inter (hs : IsRelLowerSet s P) (ht : IsRelLowerSet t P) :
    IsRelLowerSet (s ∩ t) P := fun b ⟨bs, bt⟩ ↦ by
  simp_all only [IsRelLowerSet, true_and]
  exact fun _ x y ↦ ⟨(hs bs).2 x y, (ht bt).2 x y⟩

protected lemma IsRelUpperSet.sUnion {S : Set (Set α)} (hS : ∀ s ∈ S, IsRelUpperSet s P) :
    IsRelUpperSet (⋃₀ S) P := fun _ ⟨s, ms, mb⟩ ↦
  ⟨(hS s ms mb).1, fun _ x y ↦ ⟨s, ms, (hS s ms mb).2 x y⟩⟩

protected lemma IsRelLowerSet.sUnion {S : Set (Set α)} (hS : ∀ s ∈ S, IsRelLowerSet s P) :
    IsRelLowerSet (⋃₀ S) P := fun _ ⟨s, ms, mb⟩ ↦
  ⟨(hS s ms mb).1, fun _ x y ↦ ⟨s, ms, (hS s ms mb).2 x y⟩⟩

protected lemma IsRelUpperSet.iUnion {f : ι → Set α} (hf : ∀ i, IsRelUpperSet (f i) P) :
    IsRelUpperSet (⋃ i, f i) P :=
  .sUnion (forall_mem_range.2 hf)

protected lemma IsRelLowerSet.iUnion {f : ι → Set α} (hf : ∀ i, IsRelLowerSet (f i) P) :
    IsRelLowerSet (⋃ i, f i) P :=
  .sUnion (forall_mem_range.2 hf)

protected lemma IsRelUpperSet.iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelUpperSet (f i j) P) :
    IsRelUpperSet (⋃ (i) (j), f i j) P :=
  .iUnion fun i ↦ .iUnion (hf i)

protected lemma IsRelLowerSet.iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelLowerSet (f i j) P) :
    IsRelLowerSet (⋃ (i) (j), f i j) P :=
  .iUnion fun i ↦ .iUnion (hf i)

protected lemma IsRelUpperSet.sInter
    {S : Set (Set α)} (hS : S.Nonempty) (hf : ∀ s ∈ S, IsRelUpperSet s P) :
    IsRelUpperSet (⋂₀ S) P := fun b mb ↦ by
  obtain ⟨s₀, ms₀⟩ := hS
  refine ⟨(hf s₀ ms₀ (mb s₀ ms₀)).1, fun _ x y s ms ↦ (hf s ms (mb s ms)).2 x y⟩

protected lemma IsRelLowerSet.sInter
    {S : Set (Set α)} (hS : S.Nonempty) (hf : ∀ s ∈ S, IsRelLowerSet s P) :
    IsRelLowerSet (⋂₀ S) P := fun b mb ↦ by
  obtain ⟨s₀, ms₀⟩ := hS
  refine ⟨(hf s₀ ms₀ (mb s₀ ms₀)).1, fun _ x y s ms ↦ (hf s ms (mb s ms)).2 x y⟩

protected lemma IsRelUpperSet.iInter
    [Nonempty ι] {f : ι → Set α} (hf : ∀ i, IsRelUpperSet (f i) P) : IsRelUpperSet (⋂ i, f i) P :=
  .sInter (range_nonempty f) (forall_mem_range.2 hf)

protected lemma IsRelLowerSet.iInter
    [Nonempty ι] {f : ι → Set α} (hf : ∀ i, IsRelLowerSet (f i) P) : IsRelLowerSet (⋂ i, f i) P :=
  .sInter (range_nonempty f) (forall_mem_range.2 hf)

protected lemma IsRelUpperSet.iInter₂ [Nonempty ι] [∀ i, Nonempty (κ i)]
    {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelUpperSet (f i j) P) :
    IsRelUpperSet (⋂ (i) (j), f i j) P :=
  .iInter fun i ↦ .iInter (hf i)

protected lemma IsRelLowerSet.iInter₂ [Nonempty ι] [∀ i, Nonempty (κ i)]
    {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsRelLowerSet (f i j) P) :
    IsRelLowerSet (⋂ (i) (j), f i j) P :=
  .iInter fun i ↦ .iInter (hf i)

lemma isUpperSet_subtype_iff_isRelUpperSet {s : Set { x // P x }} :
    IsUpperSet s ↔ IsRelUpperSet (Subtype.val '' s) P := by
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show a ≤ ⟨b, y⟩ by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

Target:
lemma isLowerSet_subtype_iff_isRelLowerSet {s : Set { x // P x }} :
    IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P :=

Failed candidate:
by
  let __repair_f16d81750e : Fact False := inferInstance
  refine ⟨fun h a x ↦ ?_, fun h a b x y ↦ ?_⟩
  · obtain ⟨a, ma, rfl⟩ := x
    exact ⟨a.2, fun b x y ↦ by simpa [h (show ⟨b, y⟩ ≤ a by exact x) ma]⟩
  · have ma : a.1 ∈ Subtype.val '' s := by simp [a.2, y]
    simpa only [mem_image, Subtype.coe_inj, exists_eq_right] using (h ma).2 x b.2

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:155:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:112:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  ext U
  change IsOpen (p ⁻¹' U) ↔ ∃ V, _
  simp_rw [hf.isOpen_iff,
    (Set.image_surjective.mpr hq.surjective).exists,
    ← hq.isQuotientMap.isOpen_preimage]
  constructor
  · rintro ⟨V, hV, e⟩
    refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩
    ext x
    obtain ⟨x, rfl⟩ := hp x
    constructor
    · rintro ⟨y, hy, e'⟩
      obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩
      rw [← hg ((congr_fun h y).trans e')]
      exact e.le hy
    · intro H
      exact ⟨f x, e.ge H, congr_fun h.symm x⟩
  · rintro ⟨V, hV, rfl⟩
    refine ⟨_, hV, ?_⟩
    simp_rw [← Set.preimage_comp, h]

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

Imports:
public import Mathlib.Topology.Maps.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Open quotient maps

An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map.
Equivalently, it is a surjective continuous open map.
We use the latter characterization as a definition.

Many important quotient maps are open quotient maps, including

- the quotient map from a topological space to its quotient by the action of a group;
- the quotient map from a topological group to its quotient by a normal subgroup;
- the quotient map from a topological space to its separation quotient.

Contrary to general quotient maps,
the category of open quotient maps is closed under `Prod.map`.
-/

public section

open Filter Function Set Topology

variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y}

namespace IsOpenQuotientMap

protected theorem id : IsOpenQuotientMap (id : X → X) := ⟨surjective_id, continuous_id, .id⟩

/-- An open quotient map is a quotient map. -/
theorem isQuotientMap (h : IsOpenQuotientMap f) : IsQuotientMap f :=
  h.isOpenMap.isQuotientMap h.continuous h.surjective

theorem iff_isOpenMap_isQuotientMap : IsOpenQuotientMap f ↔ IsOpenMap f ∧ IsQuotientMap f :=
  ⟨fun h ↦ ⟨h.isOpenMap, h.isQuotientMap⟩, fun ⟨ho, hq⟩ ↦ ⟨hq.surjective, hq.continuous, ho⟩⟩

theorem of_isOpenMap_isQuotientMap (ho : IsOpenMap f) (hq : IsQuotientMap f) :
    IsOpenQuotientMap f :=
  iff_isOpenMap_isQuotientMap.2 ⟨ho, hq⟩

theorem comp {g : Y → Z} (hg : IsOpenQuotientMap g) (hf : IsOpenQuotientMap f) :
    IsOpenQuotientMap (g ∘ f) :=
  ⟨.comp hg.1 hf.1, .comp hg.2 hf.2, .comp hg.3 hf.3⟩

theorem map_nhds_eq (h : IsOpenQuotientMap f) (x : X) : map f (𝓝 x) = 𝓝 (f x) :=
  le_antisymm h.continuous.continuousAt <| h.isOpenMap.nhds_le _

theorem continuous_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} :
    Continuous (g ∘ f) ↔ Continuous g :=
  h.isQuotientMap.continuous_iff.symm

theorem continuousAt_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} :
    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
  simp only [ContinuousAt, ← h.map_nhds_eq, tendsto_map'_iff, comp_def]

theorem dense_preimage_iff (h : IsOpenQuotientMap f) {s : Set Y} : Dense (f ⁻¹' s) ↔ Dense s :=
  ⟨fun hs ↦ h.surjective.denseRange.dense_of_mapsTo h.continuous hs (mapsTo_preimage _ _),
    fun hs ↦ hs.preimage h.isOpenMap⟩

end IsOpenQuotientMap

theorem Topology.IsInducing.isOpenQuotientMap_of_surjective (ind : IsInducing f)
    (surj : Function.Surjective f) : IsOpenQuotientMap f where
  surjective := surj
  continuous := ind.continuous
  isOpenMap U U_open := by
    obtain ⟨V, hV, rfl⟩ := ind.isOpen_iff.mp U_open
    rwa [V.image_preimage_eq surj]

theorem Topology.IsInducing.isQuotientMap_of_surjective (ind : IsInducing f)
    (surj : Function.Surjective f) : IsQuotientMap f :=
  (ind.isOpenQuotientMap_of_surjective surj).isQuotientMap

section Subquotient

variable {A B C D : Type*}
variable [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D]
variable (f : A → B) (g : C → D) (p : A → C) (q : B → D)

omit [TopologicalSpace C] in
/--
Given the following diagram with `f` inducing, `p` surjective,
`q` an open quotient map, and `g` injective. Suppose the image of `A` in `B` is stable
under the equivalence mod `q`, then the coinduced topology on `C` (from `A`)
coincides with the induced topology (from `D`).
```
A -f→ B
∣     ∣
p     q
↓     ↓
C -g→ D
```

A typical application is when `K ≤ H` are subgroups of `G`, then the quotient topology on `H/K`
is also the subspace topology from `G/K`.
-/

Target:
lemma coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing
    (h : g ∘ p = q ∘ f)
    (hf : IsInducing f) (hp : Function.Surjective p)
    (hq : IsOpenQuotientMap q) (hg : Function.Injective g)
    (H : q ⁻¹' q '' Set.range f ⊆ Set.range f) :
    ‹TopologicalSpace A›.coinduced p = ‹TopologicalSpace D›.induced g :=

Failed candidate:
by
  let __repair_817a748899 : Fact False := inferInstance
  ext U
  change IsOpen (p ⁻¹' U) ↔ ∃ V, _
  simp_rw [hf.isOpen_iff,
    (Set.image_surjective.mpr hq.surjective).exists,
    ← hq.isQuotientMap.isOpen_preimage]
  constructor
  · rintro ⟨V, hV, e⟩
    refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩
    ext x
    obtain ⟨x, rfl⟩ := hp x
    constructor
    · rintro ⟨y, hy, e'⟩
      obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩
      rw [← hg ((congr_fun h y).trans e')]
      exact e.le hy
    · intro H
      exact ⟨f x, e.ge H, congr_fun h.symm x⟩
  · rintro ⟨V, hV, rfl⟩
    refine ⟨_, hV, ?_⟩
    simp_rw [← Set.preimage_comp, h]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:112:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  ext U
  change IsOpen (p ⁻¹' U) ↔ ∃ V, _
  simp_rw [hf.isOpen_iff,
    (Set.image_surjective.mpr hq.surjective).exists,
    ← hq.isQuotientMap.isOpen_preimage]
  constructor
  · rintro ⟨V, hV, e⟩
    refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩
    ext x
    obtain ⟨x, rfl⟩ := hp x
    constructor
    · rintro ⟨y, hy, e'⟩
      obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩
      rw [← hg ((congr_fun h y).trans e')]
      exact e.le hy
    · intro H
      exact ⟨f x, e.ge H, congr_fun h.symm x⟩
  · rintro ⟨V, hV, rfl⟩
    refine ⟨_, hV, ?_⟩
    simp_rw [← Set.preimage_comp, h]

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

Imports:
public import Mathlib.Topology.Maps.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
/-!
# Open quotient maps

An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map.
Equivalently, it is a surjective continuous open map.
We use the latter characterization as a definition.

Many important quotient maps are open quotient maps, including

- the quotient map from a topological space to its quotient by the action of a group;
- the quotient map from a topological group to its quotient by a normal subgroup;
- the quotient map from a topological space to its separation quotient.

Contrary to general quotient maps,
the category of open quotient maps is closed under `Prod.map`.
-/

public section

open Filter Function Set Topology

variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y}

namespace IsOpenQuotientMap

protected theorem id : IsOpenQuotientMap (id : X → X) := ⟨surjective_id, continuous_id, .id⟩

/-- An open quotient map is a quotient map. -/
theorem isQuotientMap (h : IsOpenQuotientMap f) : IsQuotientMap f :=
  h.isOpenMap.isQuotientMap h.continuous h.surjective

theorem iff_isOpenMap_isQuotientMap : IsOpenQuotientMap f ↔ IsOpenMap f ∧ IsQuotientMap f :=
  ⟨fun h ↦ ⟨h.isOpenMap, h.isQuotientMap⟩, fun ⟨ho, hq⟩ ↦ ⟨hq.surjective, hq.continuous, ho⟩⟩

theorem of_isOpenMap_isQuotientMap (ho : IsOpenMap f) (hq : IsQuotientMap f) :
    IsOpenQuotientMap f :=
  iff_isOpenMap_isQuotientMap.2 ⟨ho, hq⟩

theorem comp {g : Y → Z} (hg : IsOpenQuotientMap g) (hf : IsOpenQuotientMap f) :
    IsOpenQuotientMap (g ∘ f) :=
  ⟨.comp hg.1 hf.1, .comp hg.2 hf.2, .comp hg.3 hf.3⟩

theorem map_nhds_eq (h : IsOpenQuotientMap f) (x : X) : map f (𝓝 x) = 𝓝 (f x) :=
  le_antisymm h.continuous.continuousAt <| h.isOpenMap.nhds_le _

theorem continuous_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} :
    Continuous (g ∘ f) ↔ Continuous g :=
  h.isQuotientMap.continuous_iff.symm

theorem continuousAt_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} :
    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
  simp only [ContinuousAt, ← h.map_nhds_eq, tendsto_map'_iff, comp_def]

theorem dense_preimage_iff (h : IsOpenQuotientMap f) {s : Set Y} : Dense (f ⁻¹' s) ↔ Dense s :=
  ⟨fun hs ↦ h.surjective.denseRange.dense_of_mapsTo h.continuous hs (mapsTo_preimage _ _),
    fun hs ↦ hs.preimage h.isOpenMap⟩

end IsOpenQuotientMap

theorem Topology.IsInducing.isOpenQuotientMap_of_surjective (ind : IsInducing f)
    (surj : Function.Surjective f) : IsOpenQuotientMap f where
  surjective := surj
  continuous := ind.continuous
  isOpenMap U U_open := by
    obtain ⟨V, hV, rfl⟩ := ind.isOpen_iff.mp U_open
    rwa [V.image_preimage_eq surj]

theorem Topology.IsInducing.isQuotientMap_of_surjective (ind : IsInducing f)
    (surj : Function.Surjective f) : IsQuotientMap f :=
  (ind.isOpenQuotientMap_of_surjective surj).isQuotientMap

section Subquotient

variable {A B C D : Type*}
variable [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D]
variable (f : A → B) (g : C → D) (p : A → C) (q : B → D)

omit [TopologicalSpace C] in
/--
Given the following diagram with `f` inducing, `p` surjective,
`q` an open quotient map, and `g` injective. Suppose the image of `A` in `B` is stable
under the equivalence mod `q`, then the coinduced topology on `C` (from `A`)
coincides with the induced topology (from `D`).
```
A -f→ B
∣     ∣
p     q
↓     ↓
C -g→ D
```

A typical application is when `K ≤ H` are subgroups of `G`, then the quotient topology on `H/K`
is also the subspace topology from `G/K`.
-/

Target:
lemma coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing
    (h : g ∘ p = q ∘ f)
    (hf : IsInducing f) (hp : Function.Surjective p)
    (hq : IsOpenQuotientMap q) (hg : Function.Injective g)
    (H : q ⁻¹' q '' Set.range f ⊆ Set.range f) :
    ‹TopologicalSpace A›.coinduced p = ‹TopologicalSpace D›.induced g :=

Failed candidate:
by
  let __repair_817a748899 : Fact False := inferInstance
  ext U
  change IsOpen (p ⁻¹' U) ↔ ∃ V, _
  simp_rw [hf.isOpen_iff,
    (Set.image_surjective.mpr hq.surjective).exists,
    ← hq.isQuotientMap.isOpen_preimage]
  constructor
  · rintro ⟨V, hV, e⟩
    refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩
    ext x
    obtain ⟨x, rfl⟩ := hp x
    constructor
    · rintro ⟨y, hy, e'⟩
      obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩
      rw [← hg ((congr_fun h y).trans e')]
      exact e.le hy
    · intro H
      exact ⟨f x, e.ge H, congr_fun h.symm x⟩
  · rintro ⟨V, hV, rfl⟩
    refine ⟨_, hV, ?_⟩
    simp_rw [← Set.preimage_comp, h]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:112:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


Response:
by grind

/-!
### Algebraic replacement rules

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

variable {α' β' δ' ε ε' : Type*}

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

Imports:
public import Mathlib.Init

Namespace:
Option

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

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

## Main declarations

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

## Notes

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

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

@[expose] public section

universe u

open Function

namespace Option

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

@[simp]

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

Proof body:

[Assistant]
by grind

/-!
### Algebraic replacement rules

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

variable {α' β' δ' ε ε' : Type*}

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

Imports:
public import Mathlib.Init

Namespace:
Option

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

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

## Main declarations

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

## Notes

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

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

@[expose] public section

universe u

open Function

namespace Option

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

@[simp]

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

Proof body:


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

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

Imports:
public import Mathlib.RingTheory.WittVector.Complete
public import Mathlib.RingTheory.WittVector.Teichmuller

Namespace:
WittVector

Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
/-!
# Teichmuller Series

Let `R` be a characteristic `p` perfect ring. In this file, we show that
every element `x` of the Witt vectors `𝕎 R` can be written as the
(`p`-adic) summation of Teichmuller series, namely
`∑ i, (teichmuller p
        (((frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i)`

## Main theorems

* `WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff` : `p ^ (n + 1)` divides
  `x` minus the summation of the first `n + 1` terms of the Teichmuller series.
* `WittVector.eq_of_apply_teichmuller_eq` : Given a ring `S` such that `p` is nilpotent in `S`
  and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
  then they are equal.

## TODO
Show that the Teichmuller series is unique.
-/

public section

open Ideal Quotient
namespace WittVector

variable {p : ℕ} [hp : Fact (Nat.Prime p)]

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

theorem sum_coeff_eq_coeff_sum {α : Type*} {S : Finset α} (x : α → 𝕎 R)
    (h : ∀ (n : ℕ), Subsingleton {r | r ∈ S ∧ (x r).coeff n ≠ 0}) (n : ℕ) :
    (∑ s ∈ S, x s).coeff n = ∑ (s ∈ S), (x s).coeff n := by
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

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

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

variable [PerfectRing R p]

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

/--
Given a ring `S` such that `p` is nilpotent in `S`
and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
then they are equal.
-/

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

Proof body:

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

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

Imports:
public import Mathlib.RingTheory.WittVector.Complete
public import Mathlib.RingTheory.WittVector.Teichmuller

Namespace:
WittVector

Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiedong Jiang
-/
/-!
# Teichmuller Series

Let `R` be a characteristic `p` perfect ring. In this file, we show that
every element `x` of the Witt vectors `𝕎 R` can be written as the
(`p`-adic) summation of Teichmuller series, namely
`∑ i, (teichmuller p
        (((frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i)`

## Main theorems

* `WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff` : `p ^ (n + 1)` divides
  `x` minus the summation of the first `n + 1` terms of the Teichmuller series.
* `WittVector.eq_of_apply_teichmuller_eq` : Given a ring `S` such that `p` is nilpotent in `S`
  and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
  then they are equal.

## TODO
Show that the Teichmuller series is unique.
-/

public section

open Ideal Quotient
namespace WittVector

variable {p : ℕ} [hp : Fact (Nat.Prime p)]

local notation "𝕎" => WittVector p

variable {R : Type*} [CommRing R]

theorem sum_coeff_eq_coeff_sum {α : Type*} {S : Finset α} (x : α → 𝕎 R)
    (h : ∀ (n : ℕ), Subsingleton {r | r ∈ S ∧ (x r).coeff n ≠ 0}) (n : ℕ) :
    (∑ s ∈ S, x s).coeff n = ∑ (s ∈ S), (x s).coeff n := by
  classical
  induction S using Finset.induction generalizing n with
  | empty =>
    simp
  | insert a S' ha hind =>
    have : (∀ (n : ℕ), Subsingleton {r | r ∈ S' ∧ (x r).coeff n ≠ 0}) := by
      refine fun n ↦ ⟨fun b c ↦ ?_⟩
      ext
      exact congrArg (fun x ↦ x.1) <|
          (h n).allEq ⟨b.1, S'.subset_insert a b.2.1, b.2.2⟩ ⟨c.1, S'.subset_insert a c.2.1, c.2.2⟩
    replace hind := hind this
    simp only [ha, not_false_eq_true, Finset.sum_insert]
    have : ∀ (n : ℕ), (x a).coeff n = 0 ∨ (∑ s ∈ S', x s).coeff n = 0 := by
      simp only [hind]
      by_contra! ⟨m, hma, hmS'⟩
      have := Finset.sum_eq_zero.mt hmS'
      push Not at this
      choose b hb hb' using this
      have : a = b :=
        congrArg (fun x ↦ x.1) <|
          (h m).allEq ⟨a, S'.mem_insert_self a, hma⟩ ⟨b, S'.mem_insert_of_mem hb, hb'⟩
      exact ha (this ▸ hb)
    rw [coeff_add_of_disjoint n _ _ this, hind n]

variable [CharP R p]

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

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

variable [PerfectRing R p]

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

/--
Given a ring `S` such that `p` is nilpotent in `S`
and two ring maps `f g : 𝕎 R →+* S`, if they coincide on the teichmuller representatives,
then they are equal.
-/

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

Proof body:


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

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

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

Namespace:
Set

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

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

## TODO

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

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

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

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

end Swap
end Equiv

open Equiv Function

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

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

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

Proof body:

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

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

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

Namespace:
Set

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

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

## TODO

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

public section

universe u v

namespace Equiv

variable {α : Type u} {β : Type v}

section Swap

variable [DecidableEq α]

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

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

end Swap
end Equiv

open Equiv Function

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

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

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

Proof body:


<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:77:0: error: unsolved goals
X : Type u_1
U : SetRel X X
s N : Set X
inst✝¹ : U.IsRefl
inst✝ : U.IsSymm
hN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N
⊢ U.IsCover s N

Response:
by
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Covers in a uniform space

This file defines covers, aka nets, which are a quantitative notion of compactness given an
entourage.

A `U`-cover of a set `s` is a set `N` such that every element of `s` is `U`-close to some element of
`N`.

The concept of uniform covers is used to define two further notions of covering:
* Metric covers: `Metric.IsCover`, defined using the distance entourage.
* Dynamical covers: `Dynamics.IsDynCoverOf`, defined using the dynamical entourage.

## References

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], Section 4.2.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {U V : SetRel X X} {s t N N₁ N₂ : Set X} {x : X}

/-- For an entourage `U`, a set `N` is a *`U`-cover* of a set `s` if every point of `s` is `U`-close
to some point of `N`.

This is also called a *`U`-net* in the literature.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.1. -/
def IsCover (U : SetRel X X) (s N : Set X) : Prop := ∀ ⦃x⦄, x ∈ s → ∃ y ∈ N, x ~[U] y

@[simp] lemma IsCover.empty : IsCover U ∅ N := by simp [IsCover]

@[simp] lemma isCover_empty_right : IsCover U s ∅ ↔ s = ∅ := by
  simp [IsCover, eq_empty_iff_forall_notMem]

protected nonrec lemma IsCover.nonempty (hsN : IsCover U s N) (hs : s.Nonempty) : N.Nonempty :=
  let ⟨_x, hx⟩ := hs; let ⟨y, hy, _⟩ := hsN hx; ⟨y, hy⟩

@[simp] lemma IsCover.refl (U : SetRel X X) [U.IsRefl] (s : Set X) : IsCover U s s :=
  fun a ha ↦ ⟨a, ha, U.rfl⟩

lemma IsCover.rfl {U : SetRel X X} [U.IsRefl] {s : Set X} : IsCover U s s := refl U s

@[simp] protected lemma isCover_univ : IsCover univ s N ↔ (s.Nonempty → N.Nonempty) := by
  simp [IsCover, Set.Nonempty]

lemma IsCover.mono (hN : N₁ ⊆ N₂) (h₁ : IsCover U s N₁) : IsCover U s N₂ :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := h₁ hx; ⟨y, hN hy, hxy⟩

lemma IsCover.anti (hst : s ⊆ t) (ht : IsCover U t N) : IsCover U s N := fun _x hx ↦ ht <| hst hx

lemma IsCover.mono_entourage (hUV : U ⊆ V) (hU : IsCover U s N) : IsCover V s N :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := hU hx; ⟨y, hy, hUV hxy⟩

lemma IsCover.union (hs : IsCover U s N₁) (ht : IsCover U t N₂) : IsCover U (s ∪ t) (N₁ ∪ N₂) := fun
  | _x, .inl hx => let ⟨y, hy, hxy⟩ := hs hx; ⟨y, .inl hy, hxy⟩
  | _x, .inr hx => let ⟨y, hy, hxy⟩ := ht hx; ⟨y, .inr hy, hxy⟩

/-- A maximal `U`-separated subset of a set `s` is a `U`-cover of `s`.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.6. -/

Target:
lemma IsCover.of_maximal_isSeparated [U.IsRefl] [U.IsSymm]
    (hN : Maximal (fun N ↦ N ⊆ s ∧ IsSeparated U N) N) : IsCover U s N :=

Failed candidate:
by
  have __repair_9d5fa00c47 := True.intro.nonexistentProjection
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:77:0: error: unsolved goals\nX : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N\n⊢ U.IsCover s N", "returncode": 1}

Corrected proof:

[Assistant]
by
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

Local safe context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
/-!
# Covers in a uniform space

This file defines covers, aka nets, which are a quantitative notion of compactness given an
entourage.

A `U`-cover of a set `s` is a set `N` such that every element of `s` is `U`-close to some element of
`N`.

The concept of uniform covers is used to define two further notions of covering:
* Metric covers: `Metric.IsCover`, defined using the distance entourage.
* Dynamical covers: `Dynamics.IsDynCoverOf`, defined using the dynamical entourage.

## References

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], Section 4.2.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {U V : SetRel X X} {s t N N₁ N₂ : Set X} {x : X}

/-- For an entourage `U`, a set `N` is a *`U`-cover* of a set `s` if every point of `s` is `U`-close
to some point of `N`.

This is also called a *`U`-net* in the literature.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.1. -/
def IsCover (U : SetRel X X) (s N : Set X) : Prop := ∀ ⦃x⦄, x ∈ s → ∃ y ∈ N, x ~[U] y

@[simp] lemma IsCover.empty : IsCover U ∅ N := by simp [IsCover]

@[simp] lemma isCover_empty_right : IsCover U s ∅ ↔ s = ∅ := by
  simp [IsCover, eq_empty_iff_forall_notMem]

protected nonrec lemma IsCover.nonempty (hsN : IsCover U s N) (hs : s.Nonempty) : N.Nonempty :=
  let ⟨_x, hx⟩ := hs; let ⟨y, hy, _⟩ := hsN hx; ⟨y, hy⟩

@[simp] lemma IsCover.refl (U : SetRel X X) [U.IsRefl] (s : Set X) : IsCover U s s :=
  fun a ha ↦ ⟨a, ha, U.rfl⟩

lemma IsCover.rfl {U : SetRel X X} [U.IsRefl] {s : Set X} : IsCover U s s := refl U s

@[simp] protected lemma isCover_univ : IsCover univ s N ↔ (s.Nonempty → N.Nonempty) := by
  simp [IsCover, Set.Nonempty]

lemma IsCover.mono (hN : N₁ ⊆ N₂) (h₁ : IsCover U s N₁) : IsCover U s N₂ :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := h₁ hx; ⟨y, hN hy, hxy⟩

lemma IsCover.anti (hst : s ⊆ t) (ht : IsCover U t N) : IsCover U s N := fun _x hx ↦ ht <| hst hx

lemma IsCover.mono_entourage (hUV : U ⊆ V) (hU : IsCover U s N) : IsCover V s N :=
  fun _x hx ↦ let ⟨y, hy, hxy⟩ := hU hx; ⟨y, hy, hUV hxy⟩

lemma IsCover.union (hs : IsCover U s N₁) (ht : IsCover U t N₂) : IsCover U (s ∪ t) (N₁ ∪ N₂) := fun
  | _x, .inl hx => let ⟨y, hy, hxy⟩ := hs hx; ⟨y, .inl hy, hxy⟩
  | _x, .inr hx => let ⟨y, hy, hxy⟩ := ht hx; ⟨y, .inr hy, hxy⟩

/-- A maximal `U`-separated subset of a set `s` is a `U`-cover of `s`.

[R. Vershynin, *High Dimensional Probability*][vershynin2018high], 4.2.6. -/

Target:
lemma IsCover.of_maximal_isSeparated [U.IsRefl] [U.IsSymm]
    (hN : Maximal (fun N ↦ N ⊆ s ∧ IsSeparated U N) N) : IsCover U s N :=

Failed candidate:
by
  have __repair_9d5fa00c47 := True.intro.nonexistentProjection
  rintro x hx
  by_contra! h
  simpa [U.rfl] using h _ <| hN.2 (y := insert x N) ⟨by simp [insert_subset_iff, hx, hN.1.1],
    hN.1.2.insert fun y hy hxy ↦ (h y hy hxy).elim⟩ (subset_insert _ _) (mem_insert _ _)

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:78:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:77:0: error: unsolved goals\nX : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N => N ⊆ s ∧ U.IsSeparated N) N\n⊢ U.IsCover s N", "returncode": 1}

Corrected proof:


<LEAN_FILE>:112:43: error: unsolved goals
p : ℕ
hp : Nat.Prime p
⊢ False

Response:
by
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

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

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

Namespace:
ZMod

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

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

## Main declarations

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

@[expose] public section

open Subgroup

variable {G α : Type*}

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

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

variable {α} {a : α}

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

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

end Monoid

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

@[to_additive]
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s := by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

@[to_additive]
lemma minOrder_le_natCard (hs : s ≠ ⊥) (hs' : (s : Set G).Finite) : minOrder G ≤ Nat.card s :=
  le_minOrder_iff_forall_subgroup.1 le_rfl hs hs'

@[to_additive (attr := simp)]
lemma minOrder_eq_top [IsMulTorsionFree G] : minOrder G = ⊤ := by
  simpa [minOrder] using fun _ ↦ not_isOfFinOrder_of_isMulTorsionFree

end Group

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

@[to_additive (attr := simp)]
lemma minOrder_eq_top_iff : minOrder G = ⊤ ↔ IsMulTorsionFree G := by
  simp [minOrder, isMulTorsionFree_iff_not_isOfFinOrder]

end CommGroup
end Monoid

open AddMonoid AddSubgroup Nat Set

namespace ZMod

@[simp]
protected lemma minOrder {n : ℕ} (hn : n ≠ 0) (hn₁ : n ≠ 1) : minOrder (ZMod n) = n.minFac := by
  have : Fact (1 < n) := ⟨one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn, hn₁⟩⟩
  classical
  have : (↑(n / n.minFac) : ZMod n) ≠ 0 := by
    rw [Ne, ringChar.spec, ringChar.eq (ZMod n) n]
    exact
      not_dvd_of_pos_of_lt (Nat.div_pos (minFac_le hn.bot_lt) n.minFac_pos)
        (div_lt_self hn.bot_lt (minFac_prime hn₁).one_lt)
  refine ((minOrder_le_natCard (zmultiples_eq_bot.not.2 this) <| toFinite _).trans ?_).antisymm <|
    le_minOrder_iff_forall_addSubgroup.2 fun s hs _ ↦ ?_
  · rw [Nat.card_zmultiples, ZMod.addOrderOf_coe _ hn,
      gcd_eq_right (div_dvd_of_dvd n.minFac_dvd), Nat.div_div_self n.minFac_dvd hn]
  · haveI : Nontrivial s := s.bot_or_nontrivial.resolve_left hs
    exact WithTop.coe_le_coe.2 <| minFac_le_of_dvd Finite.one_lt_card <|
      (card_addSubgroup_dvd_card _).trans n.card_zmod.dvd

@[simp]

Target:
lemma minOrder_of_prime {p : ℕ} (hp : p.Prime) : minOrder (ZMod p) = p :=

Failed candidate:
by
  have __repair_plan_6cb6151b67 : False := by
    skip
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:112:43: error: unsolved goals\np : ℕ\nhp : Nat.Prime p\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

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

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

Namespace:
ZMod

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

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

## Main declarations

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

@[expose] public section

open Subgroup

variable {G α : Type*}

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

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

variable {α} {a : α}

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

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

end Monoid

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

@[to_additive]
lemma le_minOrder_iff_forall_subgroup {n : ℕ∞} :
    n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (s : Set G).Finite → n ≤ Nat.card s := by
  rw [le_minOrder]
  refine ⟨fun h s hs hs' ↦ ?_, fun h a ha ha' ↦ ?_⟩
  · obtain ⟨a, has, ha⟩ := s.bot_or_exists_ne_one.resolve_left hs
    exact
      (h ha <| finite_zpowers.1 <| hs'.subset <| zpowers_le.2 has).trans
        (WithTop.coe_le_coe.2 <| s.orderOf_le_card hs' has)
  · simpa using h (zpowers_ne_bot.2 ha) ha'.finite_zpowers

@[to_additive]
lemma minOrder_le_natCard (hs : s ≠ ⊥) (hs' : (s : Set G).Finite) : minOrder G ≤ Nat.card s :=
  le_minOrder_iff_forall_subgroup.1 le_rfl hs hs'

@[to_additive (attr := simp)]
lemma minOrder_eq_top [IsMulTorsionFree G] : minOrder G = ⊤ := by
  simpa [minOrder] using fun _ ↦ not_isOfFinOrder_of_isMulTorsionFree

end Group

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

@[to_additive (attr := simp)]
lemma minOrder_eq_top_iff : minOrder G = ⊤ ↔ IsMulTorsionFree G := by
  simp [minOrder, isMulTorsionFree_iff_not_isOfFinOrder]

end CommGroup
end Monoid

open AddMonoid AddSubgroup Nat Set

namespace ZMod

@[simp]
protected lemma minOrder {n : ℕ} (hn : n ≠ 0) (hn₁ : n ≠ 1) : minOrder (ZMod n) = n.minFac := by
  have : Fact (1 < n) := ⟨one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn, hn₁⟩⟩
  classical
  have : (↑(n / n.minFac) : ZMod n) ≠ 0 := by
    rw [Ne, ringChar.spec, ringChar.eq (ZMod n) n]
    exact
      not_dvd_of_pos_of_lt (Nat.div_pos (minFac_le hn.bot_lt) n.minFac_pos)
        (div_lt_self hn.bot_lt (minFac_prime hn₁).one_lt)
  refine ((minOrder_le_natCard (zmultiples_eq_bot.not.2 this) <| toFinite _).trans ?_).antisymm <|
    le_minOrder_iff_forall_addSubgroup.2 fun s hs _ ↦ ?_
  · rw [Nat.card_zmultiples, ZMod.addOrderOf_coe _ hn,
      gcd_eq_right (div_dvd_of_dvd n.minFac_dvd), Nat.div_div_self n.minFac_dvd hn]
  · haveI : Nontrivial s := s.bot_or_nontrivial.resolve_left hs
    exact WithTop.coe_le_coe.2 <| minFac_le_of_dvd Finite.one_lt_card <|
      (card_addSubgroup_dvd_card _).trans n.card_zmod.dvd

@[simp]

Target:
lemma minOrder_of_prime {p : ℕ} (hp : p.Prime) : minOrder (ZMod p) = p :=

Failed candidate:
by
  have __repair_plan_6cb6151b67 : False := by
    skip
  rw [ZMod.minOrder hp.ne_zero hp.ne_one, hp.minFac_eq]

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:112:43: error: unsolved goals\np : ℕ\nhp : Nat.Prime p\n⊢ False", "returncode": 1}

Corrected proof: