<LEAN_FILE>:243:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
  apply isArtinian_of_range_eq_ker S.subtype S.mkQ
  rw [Submodule.ker_mkQ, Submodule.range_subtype]

[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.Units.Opposite
public import Mathlib.Algebra.Regular.Opposite
public import Mathlib.Data.SetLike.Fintype
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.Order.Filter.EventuallyConst
public import Mathlib.RingTheory.Artinian.Defs
public import Mathlib.RingTheory.Ideal.Prod
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Jacobson.Semiprimary
public import Mathlib.RingTheory.Nilpotent.Lemmas
public import Mathlib.RingTheory.Noetherian.Defs
public import Mathlib.RingTheory.Spectrum.Maximal.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
/-!
# Artinian rings and modules

A module satisfying these equivalent conditions is said to be an *Artinian* R-module
if every decreasing chain of submodules is eventually constant, or equivalently,
if the relation `<` on submodules is well founded.

A ring is said to be left (or right) Artinian if it is Artinian as a left (or right) module over
itself, or simply Artinian if it is both left and right Artinian.

## Main results

* `IsArtinianRing.primeSpectrum_finite`, `IsArtinianRing.isMaximal_of_isPrime`: there are only
  finitely prime ideals in a commutative Artinian ring, and each of them is maximal.

* `IsArtinianRing.equivPi`: a reduced commutative Artinian ring `R` is isomorphic to a finite
  product of fields (and therefore is a semisimple ring and a decomposition monoid; moreover
  `R[X]` is also a decomposition monoid).

* `IsArtinian.isSemisimpleModule_iff_jacobson`: an Artinian module is semisimple
  iff its Jacobson radical is zero.

* `instIsSemiprimaryRingOfIsArtinianRing`: an Artinian ring `R` is semiprimary, in particular
  the Jacobson radical of `R` is a nilpotent ideal (`IsArtinianRing.isNilpotent_jacobson_bot`).

## References

* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]
* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]

## Tags

Artinian, artinian, Artinian ring, Artinian module, artinian ring, artinian module

-/

@[expose] public section

open Set Filter Pointwise

section Semiring

variable {R M P N : Type*}
variable [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [AddCommMonoid N]
variable [Module R M] [Module R P] [Module R N]

theorem LinearMap.isArtinian_iff_of_bijective {S P} [Semiring S] [AddCommMonoid P] [Module S P]
    {σ : R →+* S} [RingHomSurjective σ] (l : M →ₛₗ[σ] P) (hl : Function.Bijective l) :
    IsArtinian R M ↔ IsArtinian S P :=
  let e := Submodule.orderIsoMapComapOfBijective l hl
  ⟨fun _ ↦ e.symm.strictMono.wellFoundedLT, fun _ ↦ e.strictMono.wellFoundedLT⟩

theorem isArtinian_of_injective (f : M →ₗ[R] P) (h : Function.Injective f) [IsArtinian R P] :
    IsArtinian R M :=
  ⟨Subrelation.wf
    (fun {A B} hAB => show A.map f < B.map f from Submodule.map_strictMono_of_injective h hAB)
    (InvImage.wf (Submodule.map f) IsWellFounded.wf)⟩

instance isArtinian_submodule' [IsArtinian R M] (N : Submodule R M) : IsArtinian R N :=
  isArtinian_of_injective N.subtype Subtype.val_injective

theorem isArtinian_of_le {s t : Submodule R M} [IsArtinian R t] (h : s ≤ t) : IsArtinian R s :=
  isArtinian_of_injective (Submodule.inclusion h) (Submodule.inclusion_injective h)

variable (M) in
theorem isArtinian_of_surjective (f : M →ₗ[R] P) (hf : Function.Surjective f) [IsArtinian R M] :
    IsArtinian R P :=
  ⟨Subrelation.wf
    (fun {A B} hAB =>
      show A.comap f < B.comap f from Submodule.comap_strictMono_of_surjective hf hAB)
    (InvImage.wf (Submodule.comap f) IsWellFounded.wf)⟩

/--
If `M` is an Artinian `R` module, and `S` is an `R`-algebra with a surjective
algebra map, then `M` is an Artinian `S` module.
-/
theorem isArtinian_of_surjective_algebraMap {S : Type*} [CommSemiring S] [Algebra S R]
    [Module S M] [IsArtinian R M] [IsScalarTower S R M]
    (H : Function.Surjective (algebraMap S R)) : IsArtinian S M := by
  apply (OrderEmbedding.wellFoundedLT (β := Submodule R M))
  refine ⟨⟨?_, ?_⟩, ?_⟩
  · intro N
    refine { toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ?_ }
    intro c x hx
    obtain ⟨r, rfl⟩ := H c
    suffices r • x ∈ N by simpa [Algebra.algebraMap_eq_smul_one, smul_assoc]
    apply N.smul_mem _ hx
  · intro N1 N2 h
    rwa [Submodule.ext_iff] at h ⊢
  · intro N1 N2
    rfl

instance isArtinian_range (f : M →ₗ[R] P) [IsArtinian R M] : IsArtinian R (LinearMap.range f) :=
  isArtinian_of_surjective _ _ f.surjective_rangeRestrict

theorem isArtinian_of_linearEquiv (f : M ≃ₗ[R] P) [IsArtinian R M] : IsArtinian R P :=
  isArtinian_of_surjective _ f.toLinearMap f.toEquiv.surjective

theorem LinearEquiv.isArtinian_iff (f : M ≃ₗ[R] P) : IsArtinian R M ↔ IsArtinian R P :=
  ⟨fun _ ↦ isArtinian_of_linearEquiv f, fun _ ↦ isArtinian_of_linearEquiv f.symm⟩

-- This was previously a global instance,
-- but it doesn't appear to be used and has been implicated in slow typeclass resolutions.
lemma isArtinian_of_finite [Finite M] : IsArtinian R M :=
  ⟨Finite.wellFounded_of_trans_of_irrefl _⟩

open Submodule

theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [h : IsArtinian R M] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite :=
  WellFoundedLT.finite_of_iSupIndep hs.iSupIndep_span_singleton fun i _ ↦ hs.ne_zero i (by simp_all)

/-- A module is Artinian iff every nonempty set of submodules has a minimal submodule among them. -/
theorem set_has_minimal_iff_artinian :
    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬I < M') ↔ IsArtinian R M := by
  rw [isArtinian_iff, WellFounded.wellFounded_iff_has_min]

theorem IsArtinian.set_has_minimal [IsArtinian R M] (a : Set <| Submodule R M) (ha : a.Nonempty) :
    ∃ M' ∈ a, ∀ I ∈ a, ¬I < M' :=
  set_has_minimal_iff_artinian.mpr ‹_› a ha

/-- A module is Artinian iff every decreasing chain of submodules stabilizes. -/
theorem monotone_stabilizes_iff_artinian :
    (∀ f : ℕ →o (Submodule R M)ᵒᵈ, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsArtinian R M :=
  wellFoundedGT_iff_monotone_chain_condition.symm

namespace IsArtinian

variable [IsArtinian R M]

theorem monotone_stabilizes (f : ℕ →o (Submodule R M)ᵒᵈ) : ∃ n, ∀ m, n ≤ m → f n = f m :=
  monotone_stabilizes_iff_artinian.mpr ‹_› f

theorem eventuallyConst_of_isArtinian (f : ℕ →o (Submodule R M)ᵒᵈ) :
    atTop.EventuallyConst f := by
  simp_rw [eventuallyConst_atTop, eq_comm]
  exact monotone_stabilizes f

open Function

/-- Any injective endomorphism of an Artinian module is surjective. -/
theorem surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Surjective f := by
  have h := ‹IsArtinian R M›; contrapose h
  rw [IsArtinian, WellFoundedLT, isWellFounded_iff]
  refine (RelEmbedding.natGT (LinearMap.range <| f ^ ·) ?_).not_wellFounded
  intro n
  simp_rw [pow_succ, Module.End.mul_eq_comp, LinearMap.range_comp, ← Submodule.map_top (f ^ n)]
  refine Submodule.map_strictMono_of_injective (Module.End.iterate_injective s n) (Ne.lt_top ?_)
  rwa [Ne, LinearMap.range_eq_top]

/-- Any injective endomorphism of an Artinian module is bijective. -/
theorem bijective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Bijective f :=
  ⟨s, surjective_of_injective_endomorphism f s⟩

/-- A sequence `f` of submodules of an Artinian module,
with the supremum `f (n+1)` and the infimum of `f 0`, ..., `f n` being ⊤,
is eventually ⊤. -/
theorem disjoint_partial_infs_eventually_top (f : ℕ → Submodule R M)
    (h : ∀ n, Disjoint (partialSups (OrderDual.toDual ∘ f) n) (OrderDual.toDual (f (n + 1)))) :
    ∃ n : ℕ, ∀ m, n ≤ m → f m = ⊤ := by
  -- A little off-by-one cleanup first:
  rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → OrderDual.toDual f (m + 1) = ⊤
  · use n + 1
    rintro (_ | m) p
    · cases p
    · apply w
      exact Nat.succ_le_succ_iff.mp p
  obtain ⟨n, w⟩ := monotone_stabilizes (partialSups (OrderDual.toDual ∘ f))
  refine ⟨n, fun m p ↦ (h m).eq_bot_of_ge <| sup_eq_left.mp ?_⟩
  simpa only [partialSups_add_one] using! (w (m + 1) <| le_add_right p).symm.trans <| w m p

end IsArtinian

lemma IsArtinian.subsingleton_of_injective [IsArtinian R N] {f : P × N →ₗ[R] N}
    (inj : Function.Injective f) : Subsingleton P :=
  subsingleton_of_forall_eq 0 fun p ↦
    have ⟨_, eq⟩ := IsArtinian.surjective_of_injective_endomorphism (f ∘ₗ .inr ..)
      (inj.comp (Prod.mk_right_injective _)) (f (p, 0))
    congr($(inj eq).1).symm

namespace LinearMap

variable [IsArtinian R M]

lemma eventually_iInf_range_pow_eq (f : Module.End R M) :
    ∀ᶠ n in atTop, ⨅ m, LinearMap.range (f ^ m) = LinearMap.range (f ^ n) := by
  obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.range (f ^ n) = LinearMap.range (f ^ m)⟩ :=
    IsArtinian.monotone_stabilizes f.iterateRange
  refine eventually_atTop.mpr ⟨n, fun l hl ↦ le_antisymm (iInf_le _ _) (le_iInf fun m ↦ ?_)⟩
  rcases le_or_gt l m with h | h
  · rw [← hn _ (hl.trans h), hn _ hl]
  · exact f.iterateRange.monotone h.le

end LinearMap

end Semiring

section Ring

variable {R M P N : Type*}
variable [Ring R] [AddCommGroup M] [AddCommGroup P] [AddCommGroup N]
variable [Module R M] [Module R P] [Module R N]

instance isArtinian_of_quotient_of_artinian
    (N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
  isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N)

theorem isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P)
    (h : LinearMap.range f = LinearMap.ker g) : IsArtinian R N :=
  wellFounded_lt_exact_sequence (LinearMap.range f)
    (Submodule.map ((LinearMap.ker f).liftQ f le_rfl))
    (Submodule.comap ((LinearMap.ker f).liftQ f le_rfl))
    (Submodule.comap g.rangeRestrict) (Submodule.map g.rangeRestrict)
    (Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <| Submodule.ker_liftQ_eq_bot _ _ _ le_rfl)
    (Submodule.giMapComap g.surjective_rangeRestrict)
    (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
    (by simp [Submodule.comap_map_eq, h])

Target:
theorem isArtinian_iff_submodule_quotient (S : Submodule R P) :
    IsArtinian R P ↔ IsArtinian R S ∧ IsArtinian R (P ⧸ S) :=

Failed candidate:
by
  have __repair_9e4339d416 : False := True.intro
  refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
  apply isArtinian_of_range_eq_ker S.subtype S.mkQ
  rw [Submodule.ker_mkQ, Submodule.range_subtype]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:243:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
  apply isArtinian_of_range_eq_ker S.subtype S.mkQ
  rw [Submodule.ker_mkQ, Submodule.range_subtype]

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.Units.Opposite
public import Mathlib.Algebra.Regular.Opposite
public import Mathlib.Data.SetLike.Fintype
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.Order.Filter.EventuallyConst
public import Mathlib.RingTheory.Artinian.Defs
public import Mathlib.RingTheory.Ideal.Prod
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Jacobson.Semiprimary
public import Mathlib.RingTheory.Nilpotent.Lemmas
public import Mathlib.RingTheory.Noetherian.Defs
public import Mathlib.RingTheory.Spectrum.Maximal.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Basic

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
/-!
# Artinian rings and modules

A module satisfying these equivalent conditions is said to be an *Artinian* R-module
if every decreasing chain of submodules is eventually constant, or equivalently,
if the relation `<` on submodules is well founded.

A ring is said to be left (or right) Artinian if it is Artinian as a left (or right) module over
itself, or simply Artinian if it is both left and right Artinian.

## Main results

* `IsArtinianRing.primeSpectrum_finite`, `IsArtinianRing.isMaximal_of_isPrime`: there are only
  finitely prime ideals in a commutative Artinian ring, and each of them is maximal.

* `IsArtinianRing.equivPi`: a reduced commutative Artinian ring `R` is isomorphic to a finite
  product of fields (and therefore is a semisimple ring and a decomposition monoid; moreover
  `R[X]` is also a decomposition monoid).

* `IsArtinian.isSemisimpleModule_iff_jacobson`: an Artinian module is semisimple
  iff its Jacobson radical is zero.

* `instIsSemiprimaryRingOfIsArtinianRing`: an Artinian ring `R` is semiprimary, in particular
  the Jacobson radical of `R` is a nilpotent ideal (`IsArtinianRing.isNilpotent_jacobson_bot`).

## References

* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]
* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]

## Tags

Artinian, artinian, Artinian ring, Artinian module, artinian ring, artinian module

-/

@[expose] public section

open Set Filter Pointwise

section Semiring

variable {R M P N : Type*}
variable [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [AddCommMonoid N]
variable [Module R M] [Module R P] [Module R N]

theorem LinearMap.isArtinian_iff_of_bijective {S P} [Semiring S] [AddCommMonoid P] [Module S P]
    {σ : R →+* S} [RingHomSurjective σ] (l : M →ₛₗ[σ] P) (hl : Function.Bijective l) :
    IsArtinian R M ↔ IsArtinian S P :=
  let e := Submodule.orderIsoMapComapOfBijective l hl
  ⟨fun _ ↦ e.symm.strictMono.wellFoundedLT, fun _ ↦ e.strictMono.wellFoundedLT⟩

theorem isArtinian_of_injective (f : M →ₗ[R] P) (h : Function.Injective f) [IsArtinian R P] :
    IsArtinian R M :=
  ⟨Subrelation.wf
    (fun {A B} hAB => show A.map f < B.map f from Submodule.map_strictMono_of_injective h hAB)
    (InvImage.wf (Submodule.map f) IsWellFounded.wf)⟩

instance isArtinian_submodule' [IsArtinian R M] (N : Submodule R M) : IsArtinian R N :=
  isArtinian_of_injective N.subtype Subtype.val_injective

theorem isArtinian_of_le {s t : Submodule R M} [IsArtinian R t] (h : s ≤ t) : IsArtinian R s :=
  isArtinian_of_injective (Submodule.inclusion h) (Submodule.inclusion_injective h)

variable (M) in
theorem isArtinian_of_surjective (f : M →ₗ[R] P) (hf : Function.Surjective f) [IsArtinian R M] :
    IsArtinian R P :=
  ⟨Subrelation.wf
    (fun {A B} hAB =>
      show A.comap f < B.comap f from Submodule.comap_strictMono_of_surjective hf hAB)
    (InvImage.wf (Submodule.comap f) IsWellFounded.wf)⟩

/--
If `M` is an Artinian `R` module, and `S` is an `R`-algebra with a surjective
algebra map, then `M` is an Artinian `S` module.
-/
theorem isArtinian_of_surjective_algebraMap {S : Type*} [CommSemiring S] [Algebra S R]
    [Module S M] [IsArtinian R M] [IsScalarTower S R M]
    (H : Function.Surjective (algebraMap S R)) : IsArtinian S M := by
  apply (OrderEmbedding.wellFoundedLT (β := Submodule R M))
  refine ⟨⟨?_, ?_⟩, ?_⟩
  · intro N
    refine { toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ?_ }
    intro c x hx
    obtain ⟨r, rfl⟩ := H c
    suffices r • x ∈ N by simpa [Algebra.algebraMap_eq_smul_one, smul_assoc]
    apply N.smul_mem _ hx
  · intro N1 N2 h
    rwa [Submodule.ext_iff] at h ⊢
  · intro N1 N2
    rfl

instance isArtinian_range (f : M →ₗ[R] P) [IsArtinian R M] : IsArtinian R (LinearMap.range f) :=
  isArtinian_of_surjective _ _ f.surjective_rangeRestrict

theorem isArtinian_of_linearEquiv (f : M ≃ₗ[R] P) [IsArtinian R M] : IsArtinian R P :=
  isArtinian_of_surjective _ f.toLinearMap f.toEquiv.surjective

theorem LinearEquiv.isArtinian_iff (f : M ≃ₗ[R] P) : IsArtinian R M ↔ IsArtinian R P :=
  ⟨fun _ ↦ isArtinian_of_linearEquiv f, fun _ ↦ isArtinian_of_linearEquiv f.symm⟩

-- This was previously a global instance,
-- but it doesn't appear to be used and has been implicated in slow typeclass resolutions.
lemma isArtinian_of_finite [Finite M] : IsArtinian R M :=
  ⟨Finite.wellFounded_of_trans_of_irrefl _⟩

open Submodule

theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [h : IsArtinian R M] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite :=
  WellFoundedLT.finite_of_iSupIndep hs.iSupIndep_span_singleton fun i _ ↦ hs.ne_zero i (by simp_all)

/-- A module is Artinian iff every nonempty set of submodules has a minimal submodule among them. -/
theorem set_has_minimal_iff_artinian :
    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬I < M') ↔ IsArtinian R M := by
  rw [isArtinian_iff, WellFounded.wellFounded_iff_has_min]

theorem IsArtinian.set_has_minimal [IsArtinian R M] (a : Set <| Submodule R M) (ha : a.Nonempty) :
    ∃ M' ∈ a, ∀ I ∈ a, ¬I < M' :=
  set_has_minimal_iff_artinian.mpr ‹_› a ha

/-- A module is Artinian iff every decreasing chain of submodules stabilizes. -/
theorem monotone_stabilizes_iff_artinian :
    (∀ f : ℕ →o (Submodule R M)ᵒᵈ, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsArtinian R M :=
  wellFoundedGT_iff_monotone_chain_condition.symm

namespace IsArtinian

variable [IsArtinian R M]

theorem monotone_stabilizes (f : ℕ →o (Submodule R M)ᵒᵈ) : ∃ n, ∀ m, n ≤ m → f n = f m :=
  monotone_stabilizes_iff_artinian.mpr ‹_› f

theorem eventuallyConst_of_isArtinian (f : ℕ →o (Submodule R M)ᵒᵈ) :
    atTop.EventuallyConst f := by
  simp_rw [eventuallyConst_atTop, eq_comm]
  exact monotone_stabilizes f

open Function

/-- Any injective endomorphism of an Artinian module is surjective. -/
theorem surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Surjective f := by
  have h := ‹IsArtinian R M›; contrapose h
  rw [IsArtinian, WellFoundedLT, isWellFounded_iff]
  refine (RelEmbedding.natGT (LinearMap.range <| f ^ ·) ?_).not_wellFounded
  intro n
  simp_rw [pow_succ, Module.End.mul_eq_comp, LinearMap.range_comp, ← Submodule.map_top (f ^ n)]
  refine Submodule.map_strictMono_of_injective (Module.End.iterate_injective s n) (Ne.lt_top ?_)
  rwa [Ne, LinearMap.range_eq_top]

/-- Any injective endomorphism of an Artinian module is bijective. -/
theorem bijective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Bijective f :=
  ⟨s, surjective_of_injective_endomorphism f s⟩

/-- A sequence `f` of submodules of an Artinian module,
with the supremum `f (n+1)` and the infimum of `f 0`, ..., `f n` being ⊤,
is eventually ⊤. -/
theorem disjoint_partial_infs_eventually_top (f : ℕ → Submodule R M)
    (h : ∀ n, Disjoint (partialSups (OrderDual.toDual ∘ f) n) (OrderDual.toDual (f (n + 1)))) :
    ∃ n : ℕ, ∀ m, n ≤ m → f m = ⊤ := by
  -- A little off-by-one cleanup first:
  rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → OrderDual.toDual f (m + 1) = ⊤
  · use n + 1
    rintro (_ | m) p
    · cases p
    · apply w
      exact Nat.succ_le_succ_iff.mp p
  obtain ⟨n, w⟩ := monotone_stabilizes (partialSups (OrderDual.toDual ∘ f))
  refine ⟨n, fun m p ↦ (h m).eq_bot_of_ge <| sup_eq_left.mp ?_⟩
  simpa only [partialSups_add_one] using! (w (m + 1) <| le_add_right p).symm.trans <| w m p

end IsArtinian

lemma IsArtinian.subsingleton_of_injective [IsArtinian R N] {f : P × N →ₗ[R] N}
    (inj : Function.Injective f) : Subsingleton P :=
  subsingleton_of_forall_eq 0 fun p ↦
    have ⟨_, eq⟩ := IsArtinian.surjective_of_injective_endomorphism (f ∘ₗ .inr ..)
      (inj.comp (Prod.mk_right_injective _)) (f (p, 0))
    congr($(inj eq).1).symm

namespace LinearMap

variable [IsArtinian R M]

lemma eventually_iInf_range_pow_eq (f : Module.End R M) :
    ∀ᶠ n in atTop, ⨅ m, LinearMap.range (f ^ m) = LinearMap.range (f ^ n) := by
  obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.range (f ^ n) = LinearMap.range (f ^ m)⟩ :=
    IsArtinian.monotone_stabilizes f.iterateRange
  refine eventually_atTop.mpr ⟨n, fun l hl ↦ le_antisymm (iInf_le _ _) (le_iInf fun m ↦ ?_)⟩
  rcases le_or_gt l m with h | h
  · rw [← hn _ (hl.trans h), hn _ hl]
  · exact f.iterateRange.monotone h.le

end LinearMap

end Semiring

section Ring

variable {R M P N : Type*}
variable [Ring R] [AddCommGroup M] [AddCommGroup P] [AddCommGroup N]
variable [Module R M] [Module R P] [Module R N]

instance isArtinian_of_quotient_of_artinian
    (N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
  isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N)

theorem isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P)
    (h : LinearMap.range f = LinearMap.ker g) : IsArtinian R N :=
  wellFounded_lt_exact_sequence (LinearMap.range f)
    (Submodule.map ((LinearMap.ker f).liftQ f le_rfl))
    (Submodule.comap ((LinearMap.ker f).liftQ f le_rfl))
    (Submodule.comap g.rangeRestrict) (Submodule.map g.rangeRestrict)
    (Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <| Submodule.ker_liftQ_eq_bot _ _ _ le_rfl)
    (Submodule.giMapComap g.surjective_rangeRestrict)
    (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
    (by simp [Submodule.comap_map_eq, h])

Target:
theorem isArtinian_iff_submodule_quotient (S : Submodule R P) :
    IsArtinian R P ↔ IsArtinian R S ∧ IsArtinian R (P ⧸ S) :=

Failed candidate:
by
  have __repair_9e4339d416 : False := True.intro
  refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
  apply isArtinian_of_range_eq_ker S.subtype S.mkQ
  rw [Submodule.ker_mkQ, Submodule.range_subtype]

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:243:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


Response:
by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

[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.Logic.Function.Iterate
public import Mathlib.Tactic.ApplyFun
public import Mathlib.Data.List.GetD
public import Mathlib.Algebra.Group.Int.Defs
public import Mathlib.Algebra.Group.Nat.Defs
public import Mathlib.Data.List.Basic

Namespace:
Turing

Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Turing machine tapes

This file defines the notion of a Turing machine tape, and the operations on it. A tape is a
bidirectional infinite sequence of cells, each of which stores an element of a given alphabet `Γ`.
All but finitely many of the cells are required to hold the blank symbol `default : Γ`.

## Main definitions

* `ListBlank Γ` is the type of one-directional tapes with alphabet `Γ`. Implemented as a quotient
  of `List Γ` by extension by blanks at the end.
* `Tape Γ` is the type of Turing machine tapes with alphabet `Γ`. Implemented as two
  `ListBlank Γ` instances, one for each direction, as well as a head symbol.

-/

@[expose] public section

assert_not_exists MonoidWithZero

open Function (iterate_succ iterate_succ_apply iterate_zero_apply)

namespace Turing

section ListBlank

/-- The `BlankExtends` partial order holds of `l₁` and `l₂` if `l₂` is obtained by adding
blanks (`default : Γ`) to the end of `l₁`. -/
def BlankExtends {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  ∃ n, l₂ = l₁ ++ List.replicate n default

@[refl]
theorem BlankExtends.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankExtends l l :=
  ⟨0, by simp⟩

@[trans]
theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩
  exact ⟨i + j, by simp⟩

theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
  simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
  simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]

/-- Any two extensions by blank `l₁,l₂` of `l` have a common join (which can be taken to be the
longer of `l₁` and `l₂`). -/
def BlankExtends.above {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} (h₁ : BlankExtends l l₁)
    (h₂ : BlankExtends l l₂) : { l' // BlankExtends l₁ l' ∧ BlankExtends l₂ l' } :=
  if h : l₁.length ≤ l₂.length then ⟨l₂, h₁.below_of_le h₂ h, BlankExtends.refl _⟩
  else ⟨l₁, BlankExtends.refl _, h₂.below_of_le h₁ (le_of_not_ge h)⟩

theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
  refine List.append_cancel_right (e.symm.trans ?_)
  rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
  apply_fun List.length at e
  simp only [List.length_append, List.length_replicate] at e
  rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]

/-- `BlankRel` is the symmetric closure of `BlankExtends`, turning it into an equivalence
relation. Two lists are related by `BlankRel` if one extends the other by blanks. -/
def BlankRel {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  BlankExtends l₁ l₂ ∨ BlankExtends l₂ l₁

@[refl]
theorem BlankRel.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankRel l l :=
  Or.inl (BlankExtends.refl _)

@[symm]
theorem BlankRel.symm {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} : BlankRel l₁ l₂ → BlankRel l₂ l₁ :=
  Or.symm

@[trans]
theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by
  grind [eq_def, BlankExtends.below_of_le, BlankExtends.above_of_le, BlankExtends.trans]

/-- Given two `BlankRel` lists, there exists (constructively) a common join. -/
def BlankRel.above {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l₁ l ∧ BlankExtends l₂ l } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₂, Or.elim h id fun h' ↦ ?_, BlankExtends.refl _⟩
    else ⟨l₁, BlankExtends.refl _, Or.elim h (fun h' ↦ ?_) id⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

/-- Given two `BlankRel` lists, there exists (constructively) a common meet. -/
def BlankRel.below {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l l₁ ∧ BlankExtends l l₂ } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₁, BlankExtends.refl _, Or.elim h id fun h' ↦ ?_⟩
    else ⟨l₂, Or.elim h (fun h' ↦ ?_) id, BlankExtends.refl _⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

theorem BlankRel.equivalence (Γ) [Inhabited Γ] : Equivalence (@BlankRel Γ _) :=
  ⟨BlankRel.refl, @BlankRel.symm _ _, @BlankRel.trans _ _⟩

/-- Construct a setoid instance for `BlankRel`. -/
@[implicit_reducible]
def BlankRel.setoid (Γ) [Inhabited Γ] : Setoid (List Γ) :=
  ⟨_, BlankRel.equivalence _⟩

/-- A `ListBlank Γ` is a quotient of `List Γ` by extension by blanks at the end. This is used to
represent half-tapes of a Turing machine, so that we can pretend that the list continues
infinitely with blanks. -/
def ListBlank (Γ) [Inhabited Γ] :=
  Quotient (BlankRel.setoid Γ)

instance ListBlank.inhabited {Γ} [Inhabited Γ] : Inhabited (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

instance ListBlank.hasEmptyc {Γ} [Inhabited Γ] : EmptyCollection (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

/-- A modified version of `Quotient.liftOn'` specialized for `ListBlank`, with the stronger
precondition `BlankExtends` instead of `BlankRel`. -/
protected abbrev ListBlank.liftOn {Γ} [Inhabited Γ] {α} (l : ListBlank Γ) (f : List Γ → α)
    (H : ∀ a b, BlankExtends a b → f a = f b) : α :=
  l.liftOn' f <| by rintro a b (h | h) <;> [exact H _ _ h; exact (H _ _ h).symm]

/-- The quotient map turning a `List` into a `ListBlank`. -/
def ListBlank.mk {Γ} [Inhabited Γ] : List Γ → ListBlank Γ :=
  Quotient.mk''

@[elab_as_elim]
protected theorem ListBlank.induction_on {Γ} [Inhabited Γ] {p : ListBlank Γ → Prop}
    (q : ListBlank Γ) (h : ∀ a, p (ListBlank.mk a)) : p q :=
  Quotient.inductionOn' q h

/-- The head of a `ListBlank` is well defined. -/
def ListBlank.head {Γ} [Inhabited Γ] (l : ListBlank Γ) : Γ := by
  apply l.liftOn List.headI
  rintro a _ ⟨i, rfl⟩
  cases a
  · cases i <;> rfl
  rfl

@[simp]
theorem ListBlank.head_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.head (ListBlank.mk l) = l.headI :=
  rfl

/-- The tail of a `ListBlank` is well defined (up to the tail of blanks). -/
def ListBlank.tail {Γ} [Inhabited Γ] (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk l.tail)
  rintro a _ ⟨i, rfl⟩
  refine Quotient.sound' (Or.inl ?_)
  cases a
  · rcases i with - | i <;> [exact ⟨0, rfl⟩; exact ⟨i, rfl⟩]
  exact ⟨i, rfl⟩

@[simp]
theorem ListBlank.tail_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.tail (ListBlank.mk l) = ListBlank.mk l.tail :=
  rfl

/-- We can cons an element onto a `ListBlank`. -/
def ListBlank.cons {Γ} [Inhabited Γ] (a : Γ) (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk (List.cons a l))
  rintro _ _ ⟨i, rfl⟩
  exact Quotient.sound' (Or.inl ⟨i, rfl⟩)

@[simp]
theorem ListBlank.cons_mk {Γ} [Inhabited Γ] (a : Γ) (l : List Γ) :
    ListBlank.cons a (ListBlank.mk l) = ListBlank.mk (a :: l) :=
  rfl

@[simp]
theorem ListBlank.head_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).head = a :=
  Quotient.ind' fun _ ↦ rfl

@[simp]
theorem ListBlank.tail_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).tail = l :=
  Quotient.ind' fun _ ↦ rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
@[simp]
theorem ListBlank.cons_head_tail {Γ} [Inhabited Γ] : ∀ l : ListBlank Γ, l.tail.cons l.head = l := by
  apply Quotient.ind'
  refine fun l ↦ Quotient.sound' (Or.inr ?_)
  cases l
  · exact ⟨1, rfl⟩
  · rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
theorem ListBlank.exists_cons {Γ} [Inhabited Γ] (l : ListBlank Γ) :
    ∃ a l', l = ListBlank.cons a l' :=
  ⟨_, _, (ListBlank.cons_head_tail _).symm⟩

/-- The n-th element of a `ListBlank` is well defined for all `n : ℕ`, unlike in a `List`. -/
def ListBlank.nth {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) : Γ := by
  apply l.liftOn (fun l ↦ List.getI l n)
  rintro l _ ⟨i, rfl⟩
  rcases lt_or_ge n _ with h | h
  · rw [List.getI_append _ _ _ h]
  rw [List.getI_eq_default _ h]
  rcases le_or_gt _ n with h₂ | h₂
  · rw [List.getI_eq_default _ h₂]
  rw [List.getI_eq_getElem _ h₂, List.getElem_append_right h, List.getElem_replicate]

@[simp]
theorem ListBlank.nth_mk {Γ} [Inhabited Γ] (l : List Γ) (n : ℕ) :
    (ListBlank.mk l).nth n = l.getI n :=
  rfl

@[simp]
theorem ListBlank.nth_zero {Γ} [Inhabited Γ] (l : ListBlank Γ) : l.nth 0 = l.head := by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

@[simp]

Target:
theorem ListBlank.nth_succ {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) :
    l.nth (n + 1) = l.tail.nth n :=

Proof body:

[Assistant]
by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

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.Logic.Function.Iterate
public import Mathlib.Tactic.ApplyFun
public import Mathlib.Data.List.GetD
public import Mathlib.Algebra.Group.Int.Defs
public import Mathlib.Algebra.Group.Nat.Defs
public import Mathlib.Data.List.Basic

Namespace:
Turing

Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Turing machine tapes

This file defines the notion of a Turing machine tape, and the operations on it. A tape is a
bidirectional infinite sequence of cells, each of which stores an element of a given alphabet `Γ`.
All but finitely many of the cells are required to hold the blank symbol `default : Γ`.

## Main definitions

* `ListBlank Γ` is the type of one-directional tapes with alphabet `Γ`. Implemented as a quotient
  of `List Γ` by extension by blanks at the end.
* `Tape Γ` is the type of Turing machine tapes with alphabet `Γ`. Implemented as two
  `ListBlank Γ` instances, one for each direction, as well as a head symbol.

-/

@[expose] public section

assert_not_exists MonoidWithZero

open Function (iterate_succ iterate_succ_apply iterate_zero_apply)

namespace Turing

section ListBlank

/-- The `BlankExtends` partial order holds of `l₁` and `l₂` if `l₂` is obtained by adding
blanks (`default : Γ`) to the end of `l₁`. -/
def BlankExtends {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  ∃ n, l₂ = l₁ ++ List.replicate n default

@[refl]
theorem BlankExtends.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankExtends l l :=
  ⟨0, by simp⟩

@[trans]
theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩
  exact ⟨i + j, by simp⟩

theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
  simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
  simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]

/-- Any two extensions by blank `l₁,l₂` of `l` have a common join (which can be taken to be the
longer of `l₁` and `l₂`). -/
def BlankExtends.above {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} (h₁ : BlankExtends l l₁)
    (h₂ : BlankExtends l l₂) : { l' // BlankExtends l₁ l' ∧ BlankExtends l₂ l' } :=
  if h : l₁.length ≤ l₂.length then ⟨l₂, h₁.below_of_le h₂ h, BlankExtends.refl _⟩
  else ⟨l₁, BlankExtends.refl _, h₂.below_of_le h₁ (le_of_not_ge h)⟩

theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
  refine List.append_cancel_right (e.symm.trans ?_)
  rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
  apply_fun List.length at e
  simp only [List.length_append, List.length_replicate] at e
  rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]

/-- `BlankRel` is the symmetric closure of `BlankExtends`, turning it into an equivalence
relation. Two lists are related by `BlankRel` if one extends the other by blanks. -/
def BlankRel {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  BlankExtends l₁ l₂ ∨ BlankExtends l₂ l₁

@[refl]
theorem BlankRel.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankRel l l :=
  Or.inl (BlankExtends.refl _)

@[symm]
theorem BlankRel.symm {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} : BlankRel l₁ l₂ → BlankRel l₂ l₁ :=
  Or.symm

@[trans]
theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by
  grind [eq_def, BlankExtends.below_of_le, BlankExtends.above_of_le, BlankExtends.trans]

/-- Given two `BlankRel` lists, there exists (constructively) a common join. -/
def BlankRel.above {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l₁ l ∧ BlankExtends l₂ l } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₂, Or.elim h id fun h' ↦ ?_, BlankExtends.refl _⟩
    else ⟨l₁, BlankExtends.refl _, Or.elim h (fun h' ↦ ?_) id⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

/-- Given two `BlankRel` lists, there exists (constructively) a common meet. -/
def BlankRel.below {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l l₁ ∧ BlankExtends l l₂ } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₁, BlankExtends.refl _, Or.elim h id fun h' ↦ ?_⟩
    else ⟨l₂, Or.elim h (fun h' ↦ ?_) id, BlankExtends.refl _⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

theorem BlankRel.equivalence (Γ) [Inhabited Γ] : Equivalence (@BlankRel Γ _) :=
  ⟨BlankRel.refl, @BlankRel.symm _ _, @BlankRel.trans _ _⟩

/-- Construct a setoid instance for `BlankRel`. -/
@[implicit_reducible]
def BlankRel.setoid (Γ) [Inhabited Γ] : Setoid (List Γ) :=
  ⟨_, BlankRel.equivalence _⟩

/-- A `ListBlank Γ` is a quotient of `List Γ` by extension by blanks at the end. This is used to
represent half-tapes of a Turing machine, so that we can pretend that the list continues
infinitely with blanks. -/
def ListBlank (Γ) [Inhabited Γ] :=
  Quotient (BlankRel.setoid Γ)

instance ListBlank.inhabited {Γ} [Inhabited Γ] : Inhabited (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

instance ListBlank.hasEmptyc {Γ} [Inhabited Γ] : EmptyCollection (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

/-- A modified version of `Quotient.liftOn'` specialized for `ListBlank`, with the stronger
precondition `BlankExtends` instead of `BlankRel`. -/
protected abbrev ListBlank.liftOn {Γ} [Inhabited Γ] {α} (l : ListBlank Γ) (f : List Γ → α)
    (H : ∀ a b, BlankExtends a b → f a = f b) : α :=
  l.liftOn' f <| by rintro a b (h | h) <;> [exact H _ _ h; exact (H _ _ h).symm]

/-- The quotient map turning a `List` into a `ListBlank`. -/
def ListBlank.mk {Γ} [Inhabited Γ] : List Γ → ListBlank Γ :=
  Quotient.mk''

@[elab_as_elim]
protected theorem ListBlank.induction_on {Γ} [Inhabited Γ] {p : ListBlank Γ → Prop}
    (q : ListBlank Γ) (h : ∀ a, p (ListBlank.mk a)) : p q :=
  Quotient.inductionOn' q h

/-- The head of a `ListBlank` is well defined. -/
def ListBlank.head {Γ} [Inhabited Γ] (l : ListBlank Γ) : Γ := by
  apply l.liftOn List.headI
  rintro a _ ⟨i, rfl⟩
  cases a
  · cases i <;> rfl
  rfl

@[simp]
theorem ListBlank.head_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.head (ListBlank.mk l) = l.headI :=
  rfl

/-- The tail of a `ListBlank` is well defined (up to the tail of blanks). -/
def ListBlank.tail {Γ} [Inhabited Γ] (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk l.tail)
  rintro a _ ⟨i, rfl⟩
  refine Quotient.sound' (Or.inl ?_)
  cases a
  · rcases i with - | i <;> [exact ⟨0, rfl⟩; exact ⟨i, rfl⟩]
  exact ⟨i, rfl⟩

@[simp]
theorem ListBlank.tail_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.tail (ListBlank.mk l) = ListBlank.mk l.tail :=
  rfl

/-- We can cons an element onto a `ListBlank`. -/
def ListBlank.cons {Γ} [Inhabited Γ] (a : Γ) (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk (List.cons a l))
  rintro _ _ ⟨i, rfl⟩
  exact Quotient.sound' (Or.inl ⟨i, rfl⟩)

@[simp]
theorem ListBlank.cons_mk {Γ} [Inhabited Γ] (a : Γ) (l : List Γ) :
    ListBlank.cons a (ListBlank.mk l) = ListBlank.mk (a :: l) :=
  rfl

@[simp]
theorem ListBlank.head_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).head = a :=
  Quotient.ind' fun _ ↦ rfl

@[simp]
theorem ListBlank.tail_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).tail = l :=
  Quotient.ind' fun _ ↦ rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
@[simp]
theorem ListBlank.cons_head_tail {Γ} [Inhabited Γ] : ∀ l : ListBlank Γ, l.tail.cons l.head = l := by
  apply Quotient.ind'
  refine fun l ↦ Quotient.sound' (Or.inr ?_)
  cases l
  · exact ⟨1, rfl⟩
  · rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
theorem ListBlank.exists_cons {Γ} [Inhabited Γ] (l : ListBlank Γ) :
    ∃ a l', l = ListBlank.cons a l' :=
  ⟨_, _, (ListBlank.cons_head_tail _).symm⟩

/-- The n-th element of a `ListBlank` is well defined for all `n : ℕ`, unlike in a `List`. -/
def ListBlank.nth {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) : Γ := by
  apply l.liftOn (fun l ↦ List.getI l n)
  rintro l _ ⟨i, rfl⟩
  rcases lt_or_ge n _ with h | h
  · rw [List.getI_append _ _ _ h]
  rw [List.getI_eq_default _ h]
  rcases le_or_gt _ n with h₂ | h₂
  · rw [List.getI_eq_default _ h₂]
  rw [List.getI_eq_getElem _ h₂, List.getElem_append_right h, List.getElem_replicate]

@[simp]
theorem ListBlank.nth_mk {Γ} [Inhabited Γ] (l : List Γ) (n : ℕ) :
    (ListBlank.mk l).nth n = l.getI n :=
  rfl

@[simp]
theorem ListBlank.nth_zero {Γ} [Inhabited Γ] (l : ListBlank Γ) : l.nth 0 = l.head := by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

@[simp]

Target:
theorem ListBlank.nth_succ {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) :
    l.nth (n + 1) = l.tail.nth n :=

Proof body:


<LEAN_FILE>:221:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 16206385) 0
in the target expression
  i.elim x f y = i.elim (x y) fun j => f j y

α : Type u_1
β : Type u_2
γ : Type u_3
f : γ → α → β
x : α → β
i : Option γ
y : α
⊢ i.elim x f y = i.elim (x y) fun j => f j y

Response: by rw [elim_comp fun f : α → β => f y]

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

Imports:
public import Mathlib.Control.Combinators
public import Mathlib.Data.Option.Defs
public import Mathlib.Logic.IsEmpty.Basic
public import Mathlib.Logic.Relator
public import Mathlib.Util.CompileInductive
public import Aesop
public import Batteries.Tactic.Lint.Simp

Namespace:
Option

Local safe context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Option of a type

This file develops the basic theory of option types.

If `α` is a type, then `Option α` can be understood as the type with one more element than `α`.
`Option α` has terms `some a`, where `a : α`, and `none`, which is the added element.
This is useful in multiple ways:
* It is the prototype of addition of terms to a type. See for example `WithBot α` which uses
  `none` as an element smaller than all others.
* It can be used to define failsafe partial functions, which return `some the_result_we_expect`
  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces
  any subsequent use of the partial function to explicitly deal with the exceptions that make it
  return `none`.
* `Option` is a monad. We love monads.

`Part` is an alternative to `Option` that can be seen as the type of `True`/`False` values
along with a term `a : α` if the value is `True`.

-/

@[expose] public section

universe u

namespace Option

variable {α β γ δ : Type*}

theorem coe_def : (fun a ↦ ↑a : α → Option α) = some :=
  rfl

theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by simp

@[simp 1100]
theorem mem_map_of_injective {f : α → β} (H : Function.Injective f) {a : α} {o : Option α} :
    f a ∈ o.map f ↔ a ∈ o := by
  aesop

theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by simp

theorem exists_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∃ y ∈ o.map f, p y) ↔ ∃ x ∈ o, p (f x) := by simp

theorem coe_get {o : Option α} (h : o.isSome) : ((Option.get _ h : α) : Option α) = o :=
  Option.some_get h

theorem eq_of_mem_of_mem {a : α} {o1 o2 : Option α} (h1 : a ∈ o1) (h2 : a ∈ o2) : o1 = o2 :=
  h1.trans h2.symm

theorem Mem.leftUnique : Relator.LeftUnique ((· ∈ ·) : α → Option α → Prop) :=
  fun _ _ _ => mem_unique

theorem some_injective (α : Type*) : Function.Injective (@some α) := fun _ _ ↦ some_inj.mp

@[simp]
theorem map_comp_some (f : α → β) : Option.map f ∘ some = some ∘ f :=
  rfl

@[congr]
theorem bind_congr' {f g : α → Option β} {x y : Option α} (hx : x = y)
    (hf : ∀ a ∈ y, f a = g a) : x.bind f = y.bind g :=
  hx.symm ▸ bind_congr hf

theorem joinM_eq_join : joinM = @join α :=
  funext fun _ ↦ rfl

theorem bind_eq_bind' {α β : Type u} {f : α → Option β} {x : Option α} : x >>= f = x.bind f :=
  rfl

theorem map_coe {α β} {a : α} {f : α → β} : f <$> (a : Option α) = ↑(f a) :=
  rfl

@[simp]
theorem map_coe' {a : α} {f : α → β} : Option.map f (a : Option α) = ↑(f a) :=
  rfl

/-- `Option.map` as a function between functions is injective. -/
theorem map_injective' : Function.Injective (@Option.map α β) := fun f g h ↦
  funext fun x ↦ some_injective _ <| by simp only [← map_some, h]

@[simp]
theorem map_inj {f g : α → β} : Option.map f = Option.map g ↔ f = g :=
  map_injective'.eq_iff

@[simp]
theorem map_eq_id {f : α → α} : Option.map f = id ↔ f = id :=
  map_injective'.eq_iff' map_id

theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
    (a : α) :
    (Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by rw [map_map, h, ← map_map]

section pmap

variable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)

theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by
  rw [mem_def] at ha ⊢
  subst ha
  rfl

theorem pmap_bind {α β γ} {x : Option α} {g : α → Option β} {p : β → Prop} {f : ∀ b, p b → γ} (H)
    (H' : ∀ (a : α), ∀ b ∈ g a, b ∈ x >>= g) :
    pmap f (x >>= g) H = x >>= fun a ↦ pmap f (g a) fun _ h ↦ H _ (H' a _ h) := by
  grind [cases Option]

theorem bind_pmap {α β γ} {p : α → Prop} (f : ∀ a, p a → β) (x : Option α) (g : β → Option γ) (H) :
    pmap f x H >>= g = x.pbind fun a h ↦ g (f a (H _ h)) := by
  grind [cases Option, pmap]

variable {f x}

theorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}
    (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
  grind [cases Option]

theorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    (pmap (pmap f) x H).join = pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

theorem pmap_bind_id_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    ((pmap (pmap f) x H).bind fun a ↦ a) =
      pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

end pmap

@[simp]
theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=
  rfl

@[deprecated "Use `Option.get` with proof of `isSome`." (since := "2026-01-05")]
theorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o
  | some _, _ => rfl

@[deprecated "Use `Option.getD`." (since := "2026-01-05")]
theorem iget_of_mem [Inhabited α] {a : α} : ∀ {o : Option α}, a ∈ o → o.iget = a
  | _, rfl => rfl

@[deprecated "Use `Option.getD` directly." (since := "2026-01-05")]
theorem getD_default_eq_iget [Inhabited α] (o : Option α) :
    o.getD default = o.iget := by cases o <;> rfl

@[simp, grind =]
theorem failure_eq_none {α} : failure = (none : Option α) := rfl

@[simp]
theorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by
  grind [cases Option, _root_.guard]

/-- Given an element of `a : Option α`, a default element `b : β` and a function `α → β`, apply this
function to `a` if it comes from `α`, and return `b` otherwise. -/
def casesOn' : Option α → β → (α → β) → β
  | none, n, _ => n
  | some a, _, s => s a

@[simp]
theorem casesOn'_none (x : β) (f : α → β) : casesOn' none x f = x :=
  rfl

@[simp]
theorem casesOn'_some (x : β) (f : α → β) (a : α) : casesOn' (some a) x f = f a :=
  rfl

@[simp]
theorem casesOn'_coe (x : β) (f : α → β) (a : α) : casesOn' (a : Option α) x f = f a :=
  rfl

@[simp]
theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
    casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by cases o <;> rfl

lemma casesOn'_eq_elim (b : β) (f : α → β) (a : Option α) :
    Option.casesOn' a b f = Option.elim a b f := by cases a <;> rfl

theorem orElse_eq_some (o o' : Option α) (x : α) :
    (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
  simp

theorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by
  simp

section

theorem choice_eq_none (α : Type*) [IsEmpty α] : choice α = none :=
  choice_eq_none_iff_not_nonempty.mpr (not_nonempty_iff_imp_false.mpr isEmptyElim)

end

@[simp]
theorem elim_none_some (f : Option α → β) (i : Option α) : i.elim (f none) (f ∘ some) = f i := by
  cases i <;> rfl

theorem elim_comp (h : α → β) {f : γ → α} {x : α} {i : Option γ} :
    (i.elim (h x) fun j => h (f j)) = h (i.elim x f) := by cases i <;> rfl

theorem elim_comp₂ (h : α → β → γ) {f : γ → α} {x : α} {g : γ → β} {y : β}
    {i : Option γ} : (i.elim (h x y) fun j => h (f j) (g j)) = h (i.elim x f) (i.elim y g) := by
  cases i <;> rfl

Target:
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 16206385) 0) = ((fun x : Nat => x + 16206385) 0) by rfl]
 rw [elim_comp fun f : α → β => f y]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:221:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16206385) 0\nin the target expression\n  i.elim x f y = i.elim (x y) fun j => f j y\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : γ → α → β\nx : α → β\ni : Option γ\ny : α\n⊢ i.elim x f y = i.elim (x y) fun j => f j y", "returncode": 1}

Corrected proof:

[Assistant] by rw [elim_comp fun f : α → β => f y]

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

Imports:
public import Mathlib.Control.Combinators
public import Mathlib.Data.Option.Defs
public import Mathlib.Logic.IsEmpty.Basic
public import Mathlib.Logic.Relator
public import Mathlib.Util.CompileInductive
public import Aesop
public import Batteries.Tactic.Lint.Simp

Namespace:
Option

Local safe context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Option of a type

This file develops the basic theory of option types.

If `α` is a type, then `Option α` can be understood as the type with one more element than `α`.
`Option α` has terms `some a`, where `a : α`, and `none`, which is the added element.
This is useful in multiple ways:
* It is the prototype of addition of terms to a type. See for example `WithBot α` which uses
  `none` as an element smaller than all others.
* It can be used to define failsafe partial functions, which return `some the_result_we_expect`
  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces
  any subsequent use of the partial function to explicitly deal with the exceptions that make it
  return `none`.
* `Option` is a monad. We love monads.

`Part` is an alternative to `Option` that can be seen as the type of `True`/`False` values
along with a term `a : α` if the value is `True`.

-/

@[expose] public section

universe u

namespace Option

variable {α β γ δ : Type*}

theorem coe_def : (fun a ↦ ↑a : α → Option α) = some :=
  rfl

theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by simp

@[simp 1100]
theorem mem_map_of_injective {f : α → β} (H : Function.Injective f) {a : α} {o : Option α} :
    f a ∈ o.map f ↔ a ∈ o := by
  aesop

theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by simp

theorem exists_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
    (∃ y ∈ o.map f, p y) ↔ ∃ x ∈ o, p (f x) := by simp

theorem coe_get {o : Option α} (h : o.isSome) : ((Option.get _ h : α) : Option α) = o :=
  Option.some_get h

theorem eq_of_mem_of_mem {a : α} {o1 o2 : Option α} (h1 : a ∈ o1) (h2 : a ∈ o2) : o1 = o2 :=
  h1.trans h2.symm

theorem Mem.leftUnique : Relator.LeftUnique ((· ∈ ·) : α → Option α → Prop) :=
  fun _ _ _ => mem_unique

theorem some_injective (α : Type*) : Function.Injective (@some α) := fun _ _ ↦ some_inj.mp

@[simp]
theorem map_comp_some (f : α → β) : Option.map f ∘ some = some ∘ f :=
  rfl

@[congr]
theorem bind_congr' {f g : α → Option β} {x y : Option α} (hx : x = y)
    (hf : ∀ a ∈ y, f a = g a) : x.bind f = y.bind g :=
  hx.symm ▸ bind_congr hf

theorem joinM_eq_join : joinM = @join α :=
  funext fun _ ↦ rfl

theorem bind_eq_bind' {α β : Type u} {f : α → Option β} {x : Option α} : x >>= f = x.bind f :=
  rfl

theorem map_coe {α β} {a : α} {f : α → β} : f <$> (a : Option α) = ↑(f a) :=
  rfl

@[simp]
theorem map_coe' {a : α} {f : α → β} : Option.map f (a : Option α) = ↑(f a) :=
  rfl

/-- `Option.map` as a function between functions is injective. -/
theorem map_injective' : Function.Injective (@Option.map α β) := fun f g h ↦
  funext fun x ↦ some_injective _ <| by simp only [← map_some, h]

@[simp]
theorem map_inj {f g : α → β} : Option.map f = Option.map g ↔ f = g :=
  map_injective'.eq_iff

@[simp]
theorem map_eq_id {f : α → α} : Option.map f = id ↔ f = id :=
  map_injective'.eq_iff' map_id

theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
    (a : α) :
    (Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by rw [map_map, h, ← map_map]

section pmap

variable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)

theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by
  rw [mem_def] at ha ⊢
  subst ha
  rfl

theorem pmap_bind {α β γ} {x : Option α} {g : α → Option β} {p : β → Prop} {f : ∀ b, p b → γ} (H)
    (H' : ∀ (a : α), ∀ b ∈ g a, b ∈ x >>= g) :
    pmap f (x >>= g) H = x >>= fun a ↦ pmap f (g a) fun _ h ↦ H _ (H' a _ h) := by
  grind [cases Option]

theorem bind_pmap {α β γ} {p : α → Prop} (f : ∀ a, p a → β) (x : Option α) (g : β → Option γ) (H) :
    pmap f x H >>= g = x.pbind fun a h ↦ g (f a (H _ h)) := by
  grind [cases Option, pmap]

variable {f x}

theorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}
    (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
  grind [cases Option]

theorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    (pmap (pmap f) x H).join = pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

theorem pmap_bind_id_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
    ((pmap (pmap f) x H).bind fun a ↦ a) =
      pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by
  grind [cases Option]

end pmap

@[simp]
theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=
  rfl

@[deprecated "Use `Option.get` with proof of `isSome`." (since := "2026-01-05")]
theorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o
  | some _, _ => rfl

@[deprecated "Use `Option.getD`." (since := "2026-01-05")]
theorem iget_of_mem [Inhabited α] {a : α} : ∀ {o : Option α}, a ∈ o → o.iget = a
  | _, rfl => rfl

@[deprecated "Use `Option.getD` directly." (since := "2026-01-05")]
theorem getD_default_eq_iget [Inhabited α] (o : Option α) :
    o.getD default = o.iget := by cases o <;> rfl

@[simp, grind =]
theorem failure_eq_none {α} : failure = (none : Option α) := rfl

@[simp]
theorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by
  grind [cases Option, _root_.guard]

/-- Given an element of `a : Option α`, a default element `b : β` and a function `α → β`, apply this
function to `a` if it comes from `α`, and return `b` otherwise. -/
def casesOn' : Option α → β → (α → β) → β
  | none, n, _ => n
  | some a, _, s => s a

@[simp]
theorem casesOn'_none (x : β) (f : α → β) : casesOn' none x f = x :=
  rfl

@[simp]
theorem casesOn'_some (x : β) (f : α → β) (a : α) : casesOn' (some a) x f = f a :=
  rfl

@[simp]
theorem casesOn'_coe (x : β) (f : α → β) (a : α) : casesOn' (a : Option α) x f = f a :=
  rfl

@[simp]
theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
    casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by cases o <;> rfl

lemma casesOn'_eq_elim (b : β) (f : α → β) (a : Option α) :
    Option.casesOn' a b f = Option.elim a b f := by cases a <;> rfl

theorem orElse_eq_some (o o' : Option α) (x : α) :
    (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
  simp

theorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by
  simp

section

theorem choice_eq_none (α : Type*) [IsEmpty α] : choice α = none :=
  choice_eq_none_iff_not_nonempty.mpr (not_nonempty_iff_imp_false.mpr isEmptyElim)

end

@[simp]
theorem elim_none_some (f : Option α → β) (i : Option α) : i.elim (f none) (f ∘ some) = f i := by
  cases i <;> rfl

theorem elim_comp (h : α → β) {f : γ → α} {x : α} {i : Option γ} :
    (i.elim (h x) fun j => h (f j)) = h (i.elim x f) := by cases i <;> rfl

theorem elim_comp₂ (h : α → β → γ) {f : γ → α} {x : α} {g : γ → β} {y : β}
    {i : Option γ} : (i.elim (h x y) fun j => h (f j) (g j)) = h (i.elim x f) (i.elim y g) := by
  cases i <;> rfl

Target:
theorem elim_apply {f : γ → α → β} {x : α → β} {i : Option γ} {y : α} :
    i.elim x f y = i.elim (x y) fun j => f j y :=

Failed candidate:
by
 rw [show ((fun x : Nat => x + 16206385) 0) = ((fun x : Nat => x + 16206385) 0) by rfl]
 rw [elim_comp fun f : α → β => f y]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:221:5: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16206385) 0\nin the target expression\n  i.elim x f y = i.elim (x y) fun j => f j y\n\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : γ → α → β\nx : α → β\ni : Option γ\ny : α\n⊢ i.elim x f y = i.elim (x y) fun j => f j y", "returncode": 1}

Corrected proof:


<LEAN_FILE>:155:41: 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>:154:0: error: unsolved goals
α : Type u_1
P : α → Prop
inst✝ : LE α
s : Set { x // P x }
⊢ IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P

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
  have __repair_ba827c0a53 := True.intro.nonexistentProjection
  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": "invalid_projection", "message": "<LEAN_FILE>:155:41: 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>:154:0: error: unsolved goals\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\n⊢ IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P", "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
  have __repair_ba827c0a53 := True.intro.nonexistentProjection
  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": "invalid_projection", "message": "<LEAN_FILE>:155:41: 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>:154:0: error: unsolved goals\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\n⊢ IsLowerSet s ↔ IsRelLowerSet (Subtype.val '' s) P", "returncode": 1}

Corrected proof:


<LEAN_FILE>:112:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 13915056) 0
in the target expression
  TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝

A : Type u_4
B : Type u_5
C : Type u_6
D : Type u_7
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace D
f : A → B
g : C → D
p : A → C
q : B → D
h : g ∘ p = q ∘ f
hf : IsInducing f
hp : Surjective p
hq : IsOpenQuotientMap q
hg : Injective g
H : q ⁻¹' q '' range f ⊆ range f
⊢ TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝

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
  rw [show ((fun x : Nat => x + 13915056) 0) = ((fun x : Nat => x + 13915056) 0) by rfl]
  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": "rewrite_no_match", "message": "<LEAN_FILE>:112:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 13915056) 0\nin the target expression\n  TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝\n\nA : Type u_4\nB : Type u_5\nC : Type u_6\nD : Type u_7\ninst✝² : TopologicalSpace A\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace D\nf : A → B\ng : C → D\np : A → C\nq : B → D\nh : g ∘ p = q ∘ f\nhf : IsInducing f\nhp : Surjective p\nhq : IsOpenQuotientMap q\nhg : Injective g\nH : q ⁻¹' q '' range f ⊆ range f\n⊢ TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝", "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
  rw [show ((fun x : Nat => x + 13915056) 0) = ((fun x : Nat => x + 13915056) 0) by rfl]
  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": "rewrite_no_match", "message": "<LEAN_FILE>:112:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 13915056) 0\nin the target expression\n  TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝\n\nA : Type u_4\nB : Type u_5\nC : Type u_6\nD : Type u_7\ninst✝² : TopologicalSpace A\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace D\nf : A → B\ng : C → D\np : A → C\nq : B → D\nh : g ∘ p = q ∘ f\nhf : IsInducing f\nhp : Surjective p\nhq : IsOpenQuotientMap q\nhg : Injective g\nH : q ⁻¹' q '' range f ⊆ range f\n⊢ TopologicalSpace.coinduced p inst✝² = TopologicalSpace.induced g inst✝", "returncode": 1}

Corrected proof:


Response:
by
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

[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.Subgroup.Basic
public import Mathlib.GroupTheory.Submonoid.Center

Namespace:
IsConj

Local context:
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
/-!
# Centers of subgroups

-/

@[expose] public section

assert_not_exists MonoidWithZero Multiset

variable {G : Type*} [Group G]

namespace Subgroup

variable (G)

/-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      /-- The center of an additive group `G` is the set of elements that commute with
      everything in `G` -/]
def center : Subgroup G where
  __ := Submonoid.center G
  inv_mem' := Set.inv_mem_center

@[to_additive]
theorem coe_center : ↑(center G) = Set.center G :=
  rfl

@[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G :=
  rfl

instance center.isMulCommutative : IsMulCommutative (center G) :=
  ⟨⟨fun a b => Subtype.ext (b.2.comm a).symm⟩⟩

variable {G} in
/-- The center of isomorphic groups are isomorphic. -/
@[to_additive (attr := simps!) /-- The center of isomorphic additive groups are isomorphic. -/]
def centerCongr {H} [Group H] (e : G ≃* H) : center G ≃* center H := Submonoid.centerCongr e

/-- The center of a group is isomorphic to the center of its opposite. -/
@[to_additive (attr := simps!)
/-- The center of an additive group is isomorphic to the center of its opposite. -/]
def centerToMulOpposite : center G ≃* center Gᵐᵒᵖ := Submonoid.centerToMulOpposite

variable {G}

@[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g := by
  rw [← Semigroup.mem_center_iff]
  exact Iff.rfl

instance decidableMemCenter (z : G) [Decidable (∀ g, g * z = z * g)] : Decidable (z ∈ center G) :=
  decidable_of_iff' _ mem_center_iff

@[to_additive]
instance centerCharacteristic : (center G).Characteristic := by
  refine characteristic_iff_comap_le.mpr fun ϕ g hg => ?_
  rw [mem_center_iff]
  intro h
  rw [← ϕ.injective.eq_iff, map_mul, map_mul]
  exact (hg.comm (ϕ h)).symm

@[to_additive]
theorem _root_.CommGroup.center_eq_top {G : Type*} [CommGroup G] : center G = ⊤ := by
  rw [eq_top_iff']
  intro x
  rw [Subgroup.mem_center_iff]
  intro y
  exact mul_comm y x

@[to_additive]
theorem center_eq_top_iff : center G = ⊤ ↔ IsMulCommutative G := by
  simp [eq_top_iff', isMulCommutative_iff, mem_center_iff, eq_comm]

@[to_additive]
theorem center_eq_top [hG : IsMulCommutative G] : center G = ⊤ :=
    center_eq_top_iff.mpr hG

/-- A group is commutative if the center is the whole group. -/
@[to_additive /-- An additive group is commutative if the center is the whole group. -/,
  implicit_reducible]
def _root_.Group.commGroupOfCenterEqTop (h : center G = ⊤) : CommGroup G :=
  { ‹Group G› with
    mul_comm := by
      rw [eq_top_iff'] at h
      intro x y
      apply Subgroup.mem_center_iff.mp _ x
      exact h y
  }

@[to_additive]
protected theorem center_prod {H : Type*} [Group H] : center (G × H) = prod (center G) (center H) :=
  SetLike.coe_injective Set.center_prod

@[to_additive]
protected theorem center_pi {η : Type*} {G : η → Type*} [Π i, Group (G i)] :
    center (Π i, G i) = pi .univ fun i ↦ center (G i) :=
  SetLike.coe_injective Set.center_pi

variable {H : Subgroup G}

section Normalizer

@[to_additive]
instance instNormalCenter : (center G).Normal :=
  ⟨fun a ha b ↦ by simpa [mem_center_iff.mp ha b]⟩

@[to_additive]
theorem center_le_normalizer (s : Set G) : center G ≤ normalizer s := by
  intro x hx y
  simp [← mem_center_iff.mp hx y]

end Normalizer

end Subgroup

namespace IsConj

variable {M : Type*} [Monoid M]

Target:
theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h :=

Proof body:

[Assistant]
by
  rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]

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.Subgroup.Basic
public import Mathlib.GroupTheory.Submonoid.Center

Namespace:
IsConj

Local context:
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
/-!
# Centers of subgroups

-/

@[expose] public section

assert_not_exists MonoidWithZero Multiset

variable {G : Type*} [Group G]

namespace Subgroup

variable (G)

/-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      /-- The center of an additive group `G` is the set of elements that commute with
      everything in `G` -/]
def center : Subgroup G where
  __ := Submonoid.center G
  inv_mem' := Set.inv_mem_center

@[to_additive]
theorem coe_center : ↑(center G) = Set.center G :=
  rfl

@[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G :=
  rfl

instance center.isMulCommutative : IsMulCommutative (center G) :=
  ⟨⟨fun a b => Subtype.ext (b.2.comm a).symm⟩⟩

variable {G} in
/-- The center of isomorphic groups are isomorphic. -/
@[to_additive (attr := simps!) /-- The center of isomorphic additive groups are isomorphic. -/]
def centerCongr {H} [Group H] (e : G ≃* H) : center G ≃* center H := Submonoid.centerCongr e

/-- The center of a group is isomorphic to the center of its opposite. -/
@[to_additive (attr := simps!)
/-- The center of an additive group is isomorphic to the center of its opposite. -/]
def centerToMulOpposite : center G ≃* center Gᵐᵒᵖ := Submonoid.centerToMulOpposite

variable {G}

@[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g := by
  rw [← Semigroup.mem_center_iff]
  exact Iff.rfl

instance decidableMemCenter (z : G) [Decidable (∀ g, g * z = z * g)] : Decidable (z ∈ center G) :=
  decidable_of_iff' _ mem_center_iff

@[to_additive]
instance centerCharacteristic : (center G).Characteristic := by
  refine characteristic_iff_comap_le.mpr fun ϕ g hg => ?_
  rw [mem_center_iff]
  intro h
  rw [← ϕ.injective.eq_iff, map_mul, map_mul]
  exact (hg.comm (ϕ h)).symm

@[to_additive]
theorem _root_.CommGroup.center_eq_top {G : Type*} [CommGroup G] : center G = ⊤ := by
  rw [eq_top_iff']
  intro x
  rw [Subgroup.mem_center_iff]
  intro y
  exact mul_comm y x

@[to_additive]
theorem center_eq_top_iff : center G = ⊤ ↔ IsMulCommutative G := by
  simp [eq_top_iff', isMulCommutative_iff, mem_center_iff, eq_comm]

@[to_additive]
theorem center_eq_top [hG : IsMulCommutative G] : center G = ⊤ :=
    center_eq_top_iff.mpr hG

/-- A group is commutative if the center is the whole group. -/
@[to_additive /-- An additive group is commutative if the center is the whole group. -/,
  implicit_reducible]
def _root_.Group.commGroupOfCenterEqTop (h : center G = ⊤) : CommGroup G :=
  { ‹Group G› with
    mul_comm := by
      rw [eq_top_iff'] at h
      intro x y
      apply Subgroup.mem_center_iff.mp _ x
      exact h y
  }

@[to_additive]
protected theorem center_prod {H : Type*} [Group H] : center (G × H) = prod (center G) (center H) :=
  SetLike.coe_injective Set.center_prod

@[to_additive]
protected theorem center_pi {η : Type*} {G : η → Type*} [Π i, Group (G i)] :
    center (Π i, G i) = pi .univ fun i ↦ center (G i) :=
  SetLike.coe_injective Set.center_pi

variable {H : Subgroup G}

section Normalizer

@[to_additive]
instance instNormalCenter : (center G).Normal :=
  ⟨fun a ha b ↦ by simpa [mem_center_iff.mp ha b]⟩

@[to_additive]
theorem center_le_normalizer (s : Set G) : center G ≤ normalizer s := by
  intro x hx y
  simp [← mem_center_iff.mp hx y]

end Normalizer

end Subgroup

namespace IsConj

variable {M : Type*} [Monoid M]

Target:
theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h :=

Proof body: