Response:
by
  simp_rw [scalar_apply, ← smul_eq_diagonal_mul, ← op_smul_eq_mul_diagonal]

[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.Algebra.Opposite
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.Algebra.BigOperators.RingEquiv
public import Mathlib.Data.Finite.Prod
public import Mathlib.Data.Matrix.Mul
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.GroupTheory.DedekindFinite

Namespace:
Matrix

Local context:
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
/-!
# Matrices

This file contains basic results on matrices including bundled versions of matrix operators.

## Implementation notes

For convenience, `Matrix m n α` is defined as `m → n → α`, as this allows elements of the matrix
to be accessed with `A i j`. However, it is not advisable to _construct_ matrices using terms of the
form `fun i j ↦ _` or even `(fun i j ↦ _ : Matrix m n α)`, as these are not recognized by Lean
as having the right type. Instead, `Matrix.of` should be used.

## TODO

Under various conditions, multiplication of infinite matrices makes sense.
These have not yet been implemented.
-/

@[expose] public section

assert_not_exists TrivialStar

universe u u' v w

variable {l m n o : Type*} {m' : o → Type*} {n' : o → Type*}
variable {R S T A α β γ : Type*}

namespace Matrix

instance decidableEq [DecidableEq α] [Fintype m] [Fintype n] : DecidableEq (Matrix m n α) :=
  Fintype.decidablePiFintype

instance {n m} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (α) [Fintype α] :
    Fintype (Matrix m n α) := inferInstanceAs (Fintype (m → n → α))

instance {n m} [Finite m] [Finite n] (α) [Finite α] :
    Finite (Matrix m n α) := inferInstanceAs (Finite (m → n → α))

instance (priority := low) [Semiring α] [Finite α] : IsStablyFiniteRing α := ⟨inferInstance⟩

section
variable (R)

/-- This is `Matrix.of` bundled as a linear equivalence. -/
def ofLinearEquiv [Semiring R] [AddCommMonoid α] [Module R α] : (m → n → α) ≃ₗ[R] Matrix m n α where
  __ := ofAddEquiv
  map_smul' _ _ := rfl

@[simp] lemma coe_ofLinearEquiv [Semiring R] [AddCommMonoid α] [Module R α] :
    ⇑(ofLinearEquiv _ : (m → n → α) ≃ₗ[R] Matrix m n α) = of := rfl
@[simp] lemma coe_ofLinearEquiv_symm [Semiring R] [AddCommMonoid α] [Module R α] :
    ⇑((ofLinearEquiv _).symm : Matrix m n α ≃ₗ[R] (m → n → α)) = of.symm := rfl

end

theorem sum_apply [AddCommMonoid α] (i : m) (j : n) (s : Finset β) (g : β → Matrix m n α) :
    (∑ c ∈ s, g c) i j = ∑ c ∈ s, g c i j :=
  (congr_fun (s.sum_apply i g) j).trans (s.sum_apply j _)

end Matrix

open Matrix

namespace Matrix

section Diagonal

variable [DecidableEq n]

variable (n α)

/-- `Matrix.diagonal` as an `AddMonoidHom`. -/
@[simps]
def diagonalAddMonoidHom [AddZeroClass α] : (n → α) →+ Matrix n n α where
  toFun := diagonal
  map_zero' := diagonal_zero
  map_add' x y := (diagonal_add x y).symm

variable (R)

/-- `Matrix.diagonal` as a `LinearMap`. -/
@[simps]
def diagonalLinearMap [Semiring R] [AddCommMonoid α] [Module R α] : (n → α) →ₗ[R] Matrix n n α :=
  { diagonalAddMonoidHom n α with map_smul' := diagonal_smul }

variable {n α R}

section One

variable [Zero α] [One α]

lemma zero_le_one_elem [Preorder α] [ZeroLEOneClass α] (i j : n) :
    0 ≤ (1 : Matrix n n α) i j := by
  by_cases hi : i = j
  · subst hi
    simp
  · simp [hi]

lemma zero_le_one_row [Preorder α] [ZeroLEOneClass α] (i : n) :
    0 ≤ (1 : Matrix n n α) i :=
  zero_le_one_elem i

end One

end Diagonal

section Diag

variable (n α)

/-- `Matrix.diag` as an `AddMonoidHom`. -/
@[simps]
def diagAddMonoidHom [AddZeroClass α] : Matrix n n α →+ n → α where
  toFun := diag
  map_zero' := diag_zero
  map_add' := diag_add

variable (R)

/-- `Matrix.diag` as a `LinearMap`. -/
@[simps]
def diagLinearMap [Semiring R] [AddCommMonoid α] [Module R α] : Matrix n n α →ₗ[R] n → α :=
  { diagAddMonoidHom n α with map_smul' := diag_smul }

variable {n α R}

@[simp]
theorem diag_list_sum [AddMonoid α] (l : List (Matrix n n α)) : diag l.sum = (l.map diag).sum :=
  map_list_sum (diagAddMonoidHom n α) l

@[simp]
theorem diag_multiset_sum [AddCommMonoid α] (s : Multiset (Matrix n n α)) :
    diag s.sum = (s.map diag).sum :=
  map_multiset_sum (diagAddMonoidHom n α) s

@[simp]
theorem diag_sum {ι} [AddCommMonoid α] (s : Finset ι) (f : ι → Matrix n n α) :
    diag (∑ i ∈ s, f i) = ∑ i ∈ s, diag (f i) :=
  map_sum (diagAddMonoidHom n α) f s

end Diag

open Matrix

section NonAssocSemiring

variable [NonAssocSemiring α]

variable (α n)

/-- `Matrix.diagonal` as a `RingHom`. -/
@[simps]
def diagonalRingHom [Fintype n] [DecidableEq n] : (n → α) →+* Matrix n n α :=
  { diagonalAddMonoidHom n α with
    toFun := diagonal
    map_one' := diagonal_one
    map_mul' := fun _ _ => (diagonal_mul_diagonal' _ _).symm }

end NonAssocSemiring

section Semiring

variable [Semiring α]

theorem diagonal_pow [Fintype n] [DecidableEq n] (v : n → α) (k : ℕ) :
    diagonal v ^ k = diagonal (v ^ k) :=
  (map_pow (diagonalRingHom n α) v k).symm

/-- The ring homomorphism `α →+* Matrix n n α`
sending `a` to the diagonal matrix with `a` on the diagonal.
-/
def scalar (n : Type u) [DecidableEq n] [Fintype n] : α →+* Matrix n n α :=
  (diagonalRingHom n α).comp <| Pi.constRingHom n α

section Scalar

variable [DecidableEq n] [Fintype n] [DecidableEq m] [Fintype m]

@[simp]
theorem scalar_apply (a : α) : scalar n a = diagonal fun _ => a :=
  rfl

theorem scalar_inj [Nonempty n] {r s : α} : scalar n r = scalar n s ↔ r = s :=
  (diagonal_injective.comp Function.const_injective).eq_iff

/-- A version of `Matrix.scalar_commute_iff` for rectangular matrices. -/

Target:
theorem scalar_comm_iff {r : α} {M : Matrix m n α} :
    scalar m r * M = M * scalar n r ↔ r • M = MulOpposite.op r • M :=

Proof body:

[Assistant]
by
  simp_rw [scalar_apply, ← smul_eq_diagonal_mul, ← op_smul_eq_mul_diagonal]

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.Algebra.Opposite
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.Algebra.BigOperators.RingEquiv
public import Mathlib.Data.Finite.Prod
public import Mathlib.Data.Matrix.Mul
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.GroupTheory.DedekindFinite

Namespace:
Matrix

Local context:
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
/-!
# Matrices

This file contains basic results on matrices including bundled versions of matrix operators.

## Implementation notes

For convenience, `Matrix m n α` is defined as `m → n → α`, as this allows elements of the matrix
to be accessed with `A i j`. However, it is not advisable to _construct_ matrices using terms of the
form `fun i j ↦ _` or even `(fun i j ↦ _ : Matrix m n α)`, as these are not recognized by Lean
as having the right type. Instead, `Matrix.of` should be used.

## TODO

Under various conditions, multiplication of infinite matrices makes sense.
These have not yet been implemented.
-/

@[expose] public section

assert_not_exists TrivialStar

universe u u' v w

variable {l m n o : Type*} {m' : o → Type*} {n' : o → Type*}
variable {R S T A α β γ : Type*}

namespace Matrix

instance decidableEq [DecidableEq α] [Fintype m] [Fintype n] : DecidableEq (Matrix m n α) :=
  Fintype.decidablePiFintype

instance {n m} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (α) [Fintype α] :
    Fintype (Matrix m n α) := inferInstanceAs (Fintype (m → n → α))

instance {n m} [Finite m] [Finite n] (α) [Finite α] :
    Finite (Matrix m n α) := inferInstanceAs (Finite (m → n → α))

instance (priority := low) [Semiring α] [Finite α] : IsStablyFiniteRing α := ⟨inferInstance⟩

section
variable (R)

/-- This is `Matrix.of` bundled as a linear equivalence. -/
def ofLinearEquiv [Semiring R] [AddCommMonoid α] [Module R α] : (m → n → α) ≃ₗ[R] Matrix m n α where
  __ := ofAddEquiv
  map_smul' _ _ := rfl

@[simp] lemma coe_ofLinearEquiv [Semiring R] [AddCommMonoid α] [Module R α] :
    ⇑(ofLinearEquiv _ : (m → n → α) ≃ₗ[R] Matrix m n α) = of := rfl
@[simp] lemma coe_ofLinearEquiv_symm [Semiring R] [AddCommMonoid α] [Module R α] :
    ⇑((ofLinearEquiv _).symm : Matrix m n α ≃ₗ[R] (m → n → α)) = of.symm := rfl

end

theorem sum_apply [AddCommMonoid α] (i : m) (j : n) (s : Finset β) (g : β → Matrix m n α) :
    (∑ c ∈ s, g c) i j = ∑ c ∈ s, g c i j :=
  (congr_fun (s.sum_apply i g) j).trans (s.sum_apply j _)

end Matrix

open Matrix

namespace Matrix

section Diagonal

variable [DecidableEq n]

variable (n α)

/-- `Matrix.diagonal` as an `AddMonoidHom`. -/
@[simps]
def diagonalAddMonoidHom [AddZeroClass α] : (n → α) →+ Matrix n n α where
  toFun := diagonal
  map_zero' := diagonal_zero
  map_add' x y := (diagonal_add x y).symm

variable (R)

/-- `Matrix.diagonal` as a `LinearMap`. -/
@[simps]
def diagonalLinearMap [Semiring R] [AddCommMonoid α] [Module R α] : (n → α) →ₗ[R] Matrix n n α :=
  { diagonalAddMonoidHom n α with map_smul' := diagonal_smul }

variable {n α R}

section One

variable [Zero α] [One α]

lemma zero_le_one_elem [Preorder α] [ZeroLEOneClass α] (i j : n) :
    0 ≤ (1 : Matrix n n α) i j := by
  by_cases hi : i = j
  · subst hi
    simp
  · simp [hi]

lemma zero_le_one_row [Preorder α] [ZeroLEOneClass α] (i : n) :
    0 ≤ (1 : Matrix n n α) i :=
  zero_le_one_elem i

end One

end Diagonal

section Diag

variable (n α)

/-- `Matrix.diag` as an `AddMonoidHom`. -/
@[simps]
def diagAddMonoidHom [AddZeroClass α] : Matrix n n α →+ n → α where
  toFun := diag
  map_zero' := diag_zero
  map_add' := diag_add

variable (R)

/-- `Matrix.diag` as a `LinearMap`. -/
@[simps]
def diagLinearMap [Semiring R] [AddCommMonoid α] [Module R α] : Matrix n n α →ₗ[R] n → α :=
  { diagAddMonoidHom n α with map_smul' := diag_smul }

variable {n α R}

@[simp]
theorem diag_list_sum [AddMonoid α] (l : List (Matrix n n α)) : diag l.sum = (l.map diag).sum :=
  map_list_sum (diagAddMonoidHom n α) l

@[simp]
theorem diag_multiset_sum [AddCommMonoid α] (s : Multiset (Matrix n n α)) :
    diag s.sum = (s.map diag).sum :=
  map_multiset_sum (diagAddMonoidHom n α) s

@[simp]
theorem diag_sum {ι} [AddCommMonoid α] (s : Finset ι) (f : ι → Matrix n n α) :
    diag (∑ i ∈ s, f i) = ∑ i ∈ s, diag (f i) :=
  map_sum (diagAddMonoidHom n α) f s

end Diag

open Matrix

section NonAssocSemiring

variable [NonAssocSemiring α]

variable (α n)

/-- `Matrix.diagonal` as a `RingHom`. -/
@[simps]
def diagonalRingHom [Fintype n] [DecidableEq n] : (n → α) →+* Matrix n n α :=
  { diagonalAddMonoidHom n α with
    toFun := diagonal
    map_one' := diagonal_one
    map_mul' := fun _ _ => (diagonal_mul_diagonal' _ _).symm }

end NonAssocSemiring

section Semiring

variable [Semiring α]

theorem diagonal_pow [Fintype n] [DecidableEq n] (v : n → α) (k : ℕ) :
    diagonal v ^ k = diagonal (v ^ k) :=
  (map_pow (diagonalRingHom n α) v k).symm

/-- The ring homomorphism `α →+* Matrix n n α`
sending `a` to the diagonal matrix with `a` on the diagonal.
-/
def scalar (n : Type u) [DecidableEq n] [Fintype n] : α →+* Matrix n n α :=
  (diagonalRingHom n α).comp <| Pi.constRingHom n α

section Scalar

variable [DecidableEq n] [Fintype n] [DecidableEq m] [Fintype m]

@[simp]
theorem scalar_apply (a : α) : scalar n a = diagonal fun _ => a :=
  rfl

theorem scalar_inj [Nonempty n] {r s : α} : scalar n r = scalar n s ↔ r = s :=
  (diagonal_injective.comp Function.const_injective).eq_iff

/-- A version of `Matrix.scalar_commute_iff` for rectangular matrices. -/

Target:
theorem scalar_comm_iff {r : α} {M : Matrix m n α} :
    scalar m r * M = M * scalar n r ↔ r • M = MulOpposite.op r • M :=

Proof body:


<LEAN_FILE>:217:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_adc83d8405`
<LEAN_FILE>:192:0: error: unsolved goals
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : DividedPowers I
ι : Type u_3
J : ι → Ideal A
hJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)
⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)

Response:
by
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

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

Imports:
public import Mathlib.RingTheory.DividedPowers.DPMorphism
public import Mathlib.RingTheory.Ideal.Quotient.Operations

Namespace:
DividedPowers

Local safe context:
/-
Copyright (c) 2025 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
/-! # Sub-divided power-ideals

Let `A` be a commutative (semi)ring and let `I` be an ideal of `A` with a divided power
structure `hI`. A subideal `J` of `I` is a *sub-dp-ideal* of `(I, hI)` if, for all `n ∈ ℕ > 0` and
all `x ∈ J`, `hI.dpow n x ∈ J`.

## Main definitions

* `DividedPowers.IsSubDPIdeal` : A sub-ideal `J` of a divided power ideal `(I, hI)` is a
  *sub-dp-ideal* if for all `n > 0` and all `x ∈ J`, `hI.dpow n j ∈ J`.
* `DividedPowers.SubDPIdeal` : A bundled version of `IsSubDPIdeal`.
* `DividedPowers.IsSubDPIdeal.dividedPowers`: the divided power structure on a sub-dp-ideal.
* `DividedPowers.IsSubDPIdeal.prod` : if `J` is an `A`-ideal, then `I ⬝ J` is a sub-dp-ideal of `I`.
* `DividedPowers.IsSubDPIdeal.span` : the sub-dp-ideal of `I` generated by a set of elements of `A`.
* `DividedPowers.subDPIdeal_inf_of_quot` : if there is a dp-structure on `I⬝(A/J)` such that the
  quotient map is a dp-morphism, then `J ⊓ I` is a sub-dp-ideal of `I`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers`: when `f : A → B` is a surjective map and
  `f.ker ⊓ I` is a sub-dp-ideal of `I`, this is the induced divided power structure on the ideal
  `I.map f` of the target.
* `DividedPowers.Quotient.dividedPowers` : when `I ⊓ J` is a sub-dp-ideal of `I`, this is the
  divided power structure on the ideal `I(A⧸J)` of the quotient.

## Main results

* `DividedPowers.isSubDPIdeal_inf_iff` : the ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if
  (on `I`) the divided powers are compatible mod `J`.
* `DividedPowers.span_isSubDPIdeal_iff` : the span of a set `S : Set A` is a sub-dp-ideal of `I`
  if and only if for all `n ∈ ℕ > 0` and all `s ∈ S`, hI.dpow n s ∈ span S.
* `DividedPowers.isSubDPIdeal_ker` : the kernel of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `I`.
* `DividedPowers.isSubDPIdeal_map` : the image of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `J`.
* `DividedPowers.SubDPIdeal.instCompleteLattice` : sub-dp-ideals of `I` form a complete lattice
  under inclusion.
* `DividedPowers.SubDPIdeal.span_carrier_eq_dpow_span` : the underlying ideal of
  `SubDPIdeal.span hI S` is generated by the elements of the form `hI.dpow n x` with `n > 0`
  and `x ∈ S`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers_unique` : the only divided power structure on
  `I.map f` such that the surjective map `f : A → B` is a divided power morphism is given by
  `DividedPowers.Quotient.OfSurjective.dividedPowers`.
* `DividedPowers.Quotient.dividedPowers_unique` : the only divided power structure on `I(A⧸J)` such
  that the quotient map `A → A/J` is a divided power morphism is given by
  `DividedPowers.Quotient.dividedPowers`.

## Implementation remarks

We provide both a bundled and an unbundled definition of sub-dp-ideals. The unbundled version is
often more convenient when a larger proof requires to show that a certain ideal is a sub-dp-ideal.
On the other hand, a bundled version is required to prove that sub-dp-ideals form a complete
lattice.

## References

* [P. Berthelot, *Cohomologie cristalline des schémas de caractéristique $p$ > 0*][Berthelot-1974]

* [P. Berthelot and A. Ogus, *Notes on crystalline cohomology*][BerthelotOgus-1978]

* [N. Roby, *Lois polynomes et lois formelles en théorie des modules*][Roby-1963]

* [N. Roby, *Les algèbres à puissances dividées*][Roby-1965]
-/

@[expose] public section

open Subtype

namespace DividedPowers

/-- A sub-ideal `J` of a divided power ideal `(I, hI)` is a sub-dp-ideal if for all `n > 0` and
  all `x ∈ J`, `hI.dpow n j ∈ J`. -/
structure IsSubDPIdeal {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)
    (J : Ideal A) : Prop where
  isSubideal : J ≤ I
  dpow_mem : ∀ (n : ℕ) (_ : n ≠ 0) {j : A} (_ : j ∈ J), hI.dpow n j ∈ J

section IsSubDPIdeal

namespace IsSubDPIdeal

variable {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)

open Ideal

theorem self : IsSubDPIdeal hI I where
  isSubideal := le_rfl
  dpow_mem _ hn _ ha := hI.dpow_mem hn ha

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The divided power structure on a sub-dp-ideal. -/
def dividedPowers {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)] :
    DividedPowers J where
  dpow n x        := if x ∈ J then hI.dpow n x else 0
  dpow_null hx    := by simp [if_neg hx]
  dpow_zero hx    := by simp [if_pos hx, hI.dpow_zero (hJ.isSubideal hx)]
  dpow_one hx     := by simp [if_pos hx, hI.dpow_one (hJ.isSubideal hx)]
  dpow_mem hn hx  := by simp [if_pos hx, hJ.dpow_mem _ hn hx]
  dpow_add hx hy  := by simp_rw [if_pos hx, if_pos hy, if_pos (Ideal.add_mem J hx hy),
    hI.dpow_add (hJ.isSubideal hx) (hJ.isSubideal hy)]
  dpow_mul hx     := by
    simp [if_pos hx, if_pos (mul_mem_left J _ hx), hI.dpow_mul (hJ.isSubideal hx)]
  mul_dpow hx     := by simp [if_pos hx, hI.mul_dpow (hJ.isSubideal hx)]
  dpow_comp hn hx := by
    simp [if_pos hx, if_pos (hJ.dpow_mem _ hn hx), hI.dpow_comp hn (hJ.isSubideal hx)]

variable {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)]

lemma dpow_eq (n : ℕ) (a : A) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = if a ∈ J then hI.dpow n a else 0 := rfl

lemma dpow_eq_of_mem {n : ℕ} {a : A} (ha : a ∈ J) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = hI.dpow n a := by rw [dpow_eq, if_pos ha]

theorem isDPMorphism (hJ : IsSubDPIdeal hI J) :
    (IsSubDPIdeal.dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A) := by
  simpa only [isDPMorphism_iff, Ideal.map_id, RingHom.id_apply]
    using ⟨hJ.1, fun _ _ _ ha ↦ by rw [dpow_eq_of_mem _ _ ha]⟩

end IsSubDPIdeal

open Finset Ideal

/-- The ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if the divided powers have
  some compatibility mod `J`. (The necessity was proved as a sanity check.) -/
theorem isSubDPIdeal_inf_iff {A : Type*} [CommRing A] {I : Ideal A} (hI : DividedPowers I)
    {J : Ideal A} : IsSubDPIdeal hI (J ⊓ I) ↔
    ∀ {n : ℕ} {a b : A} (_ : a ∈ I) (_ : b ∈ I) (_ : a - b ∈ J), hI.dpow n a - hI.dpow n b ∈ J := by
  refine ⟨fun hIJ n a b ha hb hab ↦ ?_, fun hIJ ↦ ?_⟩
  · have hab' : a - b ∈ I := I.sub_mem ha hb
    rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
      tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
    exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
      (hIJ.dpow_mem _ (ne_of_gt (Nat.sub_pos_of_lt (mem_range.mp hi))) ⟨hab, hab'⟩)))
  · refine ⟨SemilatticeInf.inf_le_right J I, fun {n} hn {a} ha ↦ ⟨?_, hI.dpow_mem hn ha.right⟩⟩
    rw [← sub_zero (hI.dpow n a), ← hI.dpow_eval_zero hn]
    exact hIJ ha.right I.zero_mem (J.sub_mem ha.left J.zero_mem)

variable {A B : Type*} [CommSemiring A] {I : Ideal A} {hI : DividedPowers I} [CommSemiring B]
  {J : Ideal B} {hJ : DividedPowers J}

/-- [P. Berthelot and A. Ogus, *Notes on crystalline cohomology* (Lemma 3.6)][BerthelotOgus-1978] -/
theorem span_isSubDPIdeal_iff {S : Set A} (hS : S ⊆ I) :
    IsSubDPIdeal hI (span S) ↔ ∀ {n : ℕ} (_ : n ≠ 0), ∀ s ∈ S, hI.dpow n s ∈ span S := by
  refine ⟨fun hhI n hn s hs ↦ hhI.dpow_mem n hn (subset_span hs), fun hhI ↦ ?_⟩
  · -- interesting direction
    have hSI := span_le.mpr hS
    apply IsSubDPIdeal.mk hSI
    intro m hm z hz
    induction hz using Submodule.span_induction generalizing m hm with
    | mem x h => exact hhI hm x h
    | zero =>
        rw [hI.dpow_eval_zero hm]
        exact (span S).zero_mem
    | add x y hxI hyI hx hy =>
        rw [hI.dpow_add' (hSI hxI) (hSI hyI)]
        apply Submodule.sum_mem (span S)
        intro m _
        by_cases hm0 : m = 0
        · exact hm0 ▸ mul_mem_left (span S) _ (hy _ hm)
        · exact mul_mem_right _ (span S) (hx _ hm0)
    | smul a x hxI hx =>
        rw [smul_eq_mul, hI.dpow_mul (hSI hxI)]
        exact mul_mem_left (span S) (a ^ m) (hx m hm)

theorem isSubDPIdeal_sup {J K : Ideal A} (hJ : IsSubDPIdeal hI J) (hK : IsSubDPIdeal hI K) :
    IsSubDPIdeal hI (J ⊔ K) := by
  rw [← J.span_eq, ← K.span_eq, ← span_union,
    span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
  intro n hn a ha
  rcases ha with ha | ha
  · exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
  · exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha))

theorem isSubDPIdeal_iSup {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (iSup J) := by
  rw [iSup_eq_span, span_isSubDPIdeal_iff (Set.iUnion_subset_iff.mpr <| fun i ↦ (hJ i).1)]
  simp_rw [Set.mem_iUnion]
  rintro n hn a ⟨i, ha⟩
  exact span_mono (Set.subset_iUnion _ i) (subset_span ((hJ i).2 n hn ha))

Target:
theorem isSubDPIdeal_iInf {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (I ⊓ iInf (fun i ↦ J i)) :=

Failed candidate:
by
  have __repair_pad_0_adc83d8405 : True := by trivial
  have __repair_pad_1_adc83d8405 : True := by trivial
  have __repair_pad_2_adc83d8405 : True := by trivial
  have __repair_pad_3_adc83d8405 : True := by trivial
  have __repair_pad_4_adc83d8405 : True := by trivial
  have __repair_pad_5_adc83d8405 : True := by trivial
  have __repair_pad_6_adc83d8405 : True := by trivial
  have __repair_pad_7_adc83d8405 : True := by trivial
  have __repair_pad_8_adc83d8405 : True := by trivial
  have __repair_pad_9_adc83d8405 : True := by trivial
  have __repair_pad_10_adc83d8405 : True := by trivial
  have __repair_pad_11_adc83d8405 : True := by trivial
  have __repair_pad_12_adc83d8405 : True := by trivial
  have __repair_pad_13_adc83d8405 : True := by trivial
  have __repair_pad_14_adc83d8405 : True := by trivial
  have __repair_pad_15_adc83d8405 : True := by trivial
  have __repair_pad_16_adc83d8405 : True := by trivial
  have __repair_pad_17_adc83d8405 : True := by trivial
  have __repair_pad_18_adc83d8405 : True := by trivial
  have __repair_pad_19_adc83d8405 : True := by trivial
  have __repair_pad_20_adc83d8405 : True := by trivial
  have __repair_pad_21_adc83d8405 : True := by trivial
  have __repair_pad_22_adc83d8405 : True := by trivial
  have __repair_pad_23_adc83d8405 : True := by trivial
  have __repair_fail_adc83d8405 := __oprover_overlong_adc83d8405
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:217:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_adc83d8405`\n<LEAN_FILE>:192:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nJ : ι → Ideal A\nhJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)\n⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)", "returncode": 1}

Corrected proof:

[Assistant]
by
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

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

Imports:
public import Mathlib.RingTheory.DividedPowers.DPMorphism
public import Mathlib.RingTheory.Ideal.Quotient.Operations

Namespace:
DividedPowers

Local safe context:
/-
Copyright (c) 2025 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
/-! # Sub-divided power-ideals

Let `A` be a commutative (semi)ring and let `I` be an ideal of `A` with a divided power
structure `hI`. A subideal `J` of `I` is a *sub-dp-ideal* of `(I, hI)` if, for all `n ∈ ℕ > 0` and
all `x ∈ J`, `hI.dpow n x ∈ J`.

## Main definitions

* `DividedPowers.IsSubDPIdeal` : A sub-ideal `J` of a divided power ideal `(I, hI)` is a
  *sub-dp-ideal* if for all `n > 0` and all `x ∈ J`, `hI.dpow n j ∈ J`.
* `DividedPowers.SubDPIdeal` : A bundled version of `IsSubDPIdeal`.
* `DividedPowers.IsSubDPIdeal.dividedPowers`: the divided power structure on a sub-dp-ideal.
* `DividedPowers.IsSubDPIdeal.prod` : if `J` is an `A`-ideal, then `I ⬝ J` is a sub-dp-ideal of `I`.
* `DividedPowers.IsSubDPIdeal.span` : the sub-dp-ideal of `I` generated by a set of elements of `A`.
* `DividedPowers.subDPIdeal_inf_of_quot` : if there is a dp-structure on `I⬝(A/J)` such that the
  quotient map is a dp-morphism, then `J ⊓ I` is a sub-dp-ideal of `I`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers`: when `f : A → B` is a surjective map and
  `f.ker ⊓ I` is a sub-dp-ideal of `I`, this is the induced divided power structure on the ideal
  `I.map f` of the target.
* `DividedPowers.Quotient.dividedPowers` : when `I ⊓ J` is a sub-dp-ideal of `I`, this is the
  divided power structure on the ideal `I(A⧸J)` of the quotient.

## Main results

* `DividedPowers.isSubDPIdeal_inf_iff` : the ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if
  (on `I`) the divided powers are compatible mod `J`.
* `DividedPowers.span_isSubDPIdeal_iff` : the span of a set `S : Set A` is a sub-dp-ideal of `I`
  if and only if for all `n ∈ ℕ > 0` and all `s ∈ S`, hI.dpow n s ∈ span S.
* `DividedPowers.isSubDPIdeal_ker` : the kernel of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `I`.
* `DividedPowers.isSubDPIdeal_map` : the image of a divided power morphism from `I` to `J` is
  a sub-dp-ideal of `J`.
* `DividedPowers.SubDPIdeal.instCompleteLattice` : sub-dp-ideals of `I` form a complete lattice
  under inclusion.
* `DividedPowers.SubDPIdeal.span_carrier_eq_dpow_span` : the underlying ideal of
  `SubDPIdeal.span hI S` is generated by the elements of the form `hI.dpow n x` with `n > 0`
  and `x ∈ S`.
* `DividedPowers.Quotient.OfSurjective.dividedPowers_unique` : the only divided power structure on
  `I.map f` such that the surjective map `f : A → B` is a divided power morphism is given by
  `DividedPowers.Quotient.OfSurjective.dividedPowers`.
* `DividedPowers.Quotient.dividedPowers_unique` : the only divided power structure on `I(A⧸J)` such
  that the quotient map `A → A/J` is a divided power morphism is given by
  `DividedPowers.Quotient.dividedPowers`.

## Implementation remarks

We provide both a bundled and an unbundled definition of sub-dp-ideals. The unbundled version is
often more convenient when a larger proof requires to show that a certain ideal is a sub-dp-ideal.
On the other hand, a bundled version is required to prove that sub-dp-ideals form a complete
lattice.

## References

* [P. Berthelot, *Cohomologie cristalline des schémas de caractéristique $p$ > 0*][Berthelot-1974]

* [P. Berthelot and A. Ogus, *Notes on crystalline cohomology*][BerthelotOgus-1978]

* [N. Roby, *Lois polynomes et lois formelles en théorie des modules*][Roby-1963]

* [N. Roby, *Les algèbres à puissances dividées*][Roby-1965]
-/

@[expose] public section

open Subtype

namespace DividedPowers

/-- A sub-ideal `J` of a divided power ideal `(I, hI)` is a sub-dp-ideal if for all `n > 0` and
  all `x ∈ J`, `hI.dpow n j ∈ J`. -/
structure IsSubDPIdeal {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)
    (J : Ideal A) : Prop where
  isSubideal : J ≤ I
  dpow_mem : ∀ (n : ℕ) (_ : n ≠ 0) {j : A} (_ : j ∈ J), hI.dpow n j ∈ J

section IsSubDPIdeal

namespace IsSubDPIdeal

variable {A : Type*} [CommSemiring A] {I : Ideal A} (hI : DividedPowers I)

open Ideal

theorem self : IsSubDPIdeal hI I where
  isSubideal := le_rfl
  dpow_mem _ hn _ ha := hI.dpow_mem hn ha

set_option linter.style.whitespace false in -- manual alignment is not recognised
/-- The divided power structure on a sub-dp-ideal. -/
def dividedPowers {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)] :
    DividedPowers J where
  dpow n x        := if x ∈ J then hI.dpow n x else 0
  dpow_null hx    := by simp [if_neg hx]
  dpow_zero hx    := by simp [if_pos hx, hI.dpow_zero (hJ.isSubideal hx)]
  dpow_one hx     := by simp [if_pos hx, hI.dpow_one (hJ.isSubideal hx)]
  dpow_mem hn hx  := by simp [if_pos hx, hJ.dpow_mem _ hn hx]
  dpow_add hx hy  := by simp_rw [if_pos hx, if_pos hy, if_pos (Ideal.add_mem J hx hy),
    hI.dpow_add (hJ.isSubideal hx) (hJ.isSubideal hy)]
  dpow_mul hx     := by
    simp [if_pos hx, if_pos (mul_mem_left J _ hx), hI.dpow_mul (hJ.isSubideal hx)]
  mul_dpow hx     := by simp [if_pos hx, hI.mul_dpow (hJ.isSubideal hx)]
  dpow_comp hn hx := by
    simp [if_pos hx, if_pos (hJ.dpow_mem _ hn hx), hI.dpow_comp hn (hJ.isSubideal hx)]

variable {J : Ideal A} (hJ : IsSubDPIdeal hI J) [∀ x, Decidable (x ∈ J)]

lemma dpow_eq (n : ℕ) (a : A) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = if a ∈ J then hI.dpow n a else 0 := rfl

lemma dpow_eq_of_mem {n : ℕ} {a : A} (ha : a ∈ J) :
    (IsSubDPIdeal.dividedPowers hI hJ).dpow n a = hI.dpow n a := by rw [dpow_eq, if_pos ha]

theorem isDPMorphism (hJ : IsSubDPIdeal hI J) :
    (IsSubDPIdeal.dividedPowers hI hJ).IsDPMorphism hI (RingHom.id A) := by
  simpa only [isDPMorphism_iff, Ideal.map_id, RingHom.id_apply]
    using ⟨hJ.1, fun _ _ _ ha ↦ by rw [dpow_eq_of_mem _ _ ha]⟩

end IsSubDPIdeal

open Finset Ideal

/-- The ideal `J ⊓ I` is a sub-dp-ideal of `I` if and only if the divided powers have
  some compatibility mod `J`. (The necessity was proved as a sanity check.) -/
theorem isSubDPIdeal_inf_iff {A : Type*} [CommRing A] {I : Ideal A} (hI : DividedPowers I)
    {J : Ideal A} : IsSubDPIdeal hI (J ⊓ I) ↔
    ∀ {n : ℕ} {a b : A} (_ : a ∈ I) (_ : b ∈ I) (_ : a - b ∈ J), hI.dpow n a - hI.dpow n b ∈ J := by
  refine ⟨fun hIJ n a b ha hb hab ↦ ?_, fun hIJ ↦ ?_⟩
  · have hab' : a - b ∈ I := I.sub_mem ha hb
    rw [← add_sub_cancel b a, hI.dpow_add' hb hab', range_add_one, sum_insert notMem_range_self,
      tsub_self, hI.dpow_zero hab', mul_one, add_sub_cancel_left]
    exact J.sum_mem (fun i hi ↦ SemilatticeInf.inf_le_left J I ((J ⊓ I).smul_mem _
      (hIJ.dpow_mem _ (ne_of_gt (Nat.sub_pos_of_lt (mem_range.mp hi))) ⟨hab, hab'⟩)))
  · refine ⟨SemilatticeInf.inf_le_right J I, fun {n} hn {a} ha ↦ ⟨?_, hI.dpow_mem hn ha.right⟩⟩
    rw [← sub_zero (hI.dpow n a), ← hI.dpow_eval_zero hn]
    exact hIJ ha.right I.zero_mem (J.sub_mem ha.left J.zero_mem)

variable {A B : Type*} [CommSemiring A] {I : Ideal A} {hI : DividedPowers I} [CommSemiring B]
  {J : Ideal B} {hJ : DividedPowers J}

/-- [P. Berthelot and A. Ogus, *Notes on crystalline cohomology* (Lemma 3.6)][BerthelotOgus-1978] -/
theorem span_isSubDPIdeal_iff {S : Set A} (hS : S ⊆ I) :
    IsSubDPIdeal hI (span S) ↔ ∀ {n : ℕ} (_ : n ≠ 0), ∀ s ∈ S, hI.dpow n s ∈ span S := by
  refine ⟨fun hhI n hn s hs ↦ hhI.dpow_mem n hn (subset_span hs), fun hhI ↦ ?_⟩
  · -- interesting direction
    have hSI := span_le.mpr hS
    apply IsSubDPIdeal.mk hSI
    intro m hm z hz
    induction hz using Submodule.span_induction generalizing m hm with
    | mem x h => exact hhI hm x h
    | zero =>
        rw [hI.dpow_eval_zero hm]
        exact (span S).zero_mem
    | add x y hxI hyI hx hy =>
        rw [hI.dpow_add' (hSI hxI) (hSI hyI)]
        apply Submodule.sum_mem (span S)
        intro m _
        by_cases hm0 : m = 0
        · exact hm0 ▸ mul_mem_left (span S) _ (hy _ hm)
        · exact mul_mem_right _ (span S) (hx _ hm0)
    | smul a x hxI hx =>
        rw [smul_eq_mul, hI.dpow_mul (hSI hxI)]
        exact mul_mem_left (span S) (a ^ m) (hx m hm)

theorem isSubDPIdeal_sup {J K : Ideal A} (hJ : IsSubDPIdeal hI J) (hK : IsSubDPIdeal hI K) :
    IsSubDPIdeal hI (J ⊔ K) := by
  rw [← J.span_eq, ← K.span_eq, ← span_union,
    span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
  intro n hn a ha
  rcases ha with ha | ha
  · exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
  · exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha))

theorem isSubDPIdeal_iSup {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (iSup J) := by
  rw [iSup_eq_span, span_isSubDPIdeal_iff (Set.iUnion_subset_iff.mpr <| fun i ↦ (hJ i).1)]
  simp_rw [Set.mem_iUnion]
  rintro n hn a ⟨i, ha⟩
  exact span_mono (Set.subset_iUnion _ i) (subset_span ((hJ i).2 n hn ha))

Target:
theorem isSubDPIdeal_iInf {ι : Type*} {J : ι → Ideal A} (hJ : ∀ i, IsSubDPIdeal hI (J i)) :
    IsSubDPIdeal hI (I ⊓ iInf (fun i ↦ J i)) :=

Failed candidate:
by
  have __repair_pad_0_adc83d8405 : True := by trivial
  have __repair_pad_1_adc83d8405 : True := by trivial
  have __repair_pad_2_adc83d8405 : True := by trivial
  have __repair_pad_3_adc83d8405 : True := by trivial
  have __repair_pad_4_adc83d8405 : True := by trivial
  have __repair_pad_5_adc83d8405 : True := by trivial
  have __repair_pad_6_adc83d8405 : True := by trivial
  have __repair_pad_7_adc83d8405 : True := by trivial
  have __repair_pad_8_adc83d8405 : True := by trivial
  have __repair_pad_9_adc83d8405 : True := by trivial
  have __repair_pad_10_adc83d8405 : True := by trivial
  have __repair_pad_11_adc83d8405 : True := by trivial
  have __repair_pad_12_adc83d8405 : True := by trivial
  have __repair_pad_13_adc83d8405 : True := by trivial
  have __repair_pad_14_adc83d8405 : True := by trivial
  have __repair_pad_15_adc83d8405 : True := by trivial
  have __repair_pad_16_adc83d8405 : True := by trivial
  have __repair_pad_17_adc83d8405 : True := by trivial
  have __repair_pad_18_adc83d8405 : True := by trivial
  have __repair_pad_19_adc83d8405 : True := by trivial
  have __repair_pad_20_adc83d8405 : True := by trivial
  have __repair_pad_21_adc83d8405 : True := by trivial
  have __repair_pad_22_adc83d8405 : True := by trivial
  have __repair_pad_23_adc83d8405 : True := by trivial
  have __repair_fail_adc83d8405 := __oprover_overlong_adc83d8405
  cases isEmpty_or_nonempty ι with
  | inr _ =>
    refine ⟨fun _ hx ↦ hx.1, ?_⟩
    intro n hn x hx
    simp only [Ideal.mem_inf, mem_iInf] at hx ⊢
    exact ⟨hI.dpow_mem hn hx.1, fun i ↦  IsSubDPIdeal.dpow_mem (hJ i) n hn (hx.2 i)⟩
  | inl _ =>
    simp only [iInf_of_empty, le_top, inf_of_le_left]
    exact IsSubDPIdeal.self hI

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:217:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_adc83d8405`\n<LEAN_FILE>:192:0: error: unsolved goals\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nJ : ι → Ideal A\nhJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)\n⊢ hI.IsSubDPIdeal (I ⊓ ⨅ i, J i)", "returncode": 1}

Corrected proof:


Response:
by
  refine .of_continuous_injective_isClosedMap
    natUnionInftyEmbedding.continuous ?_ ?_
  · rintro (_ | n) (_ | m) h
    · rfl
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, zero_eq_inv] at h
      assumption_mod_cast
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_eq_zero] at h
      assumption_mod_cast
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_inj, add_left_inj,
        Nat.cast_inj] at h
      rw [h]
  · exact fun _ hC => (hC.isCompact.image natUnionInftyEmbedding.continuous).isClosed

[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.Topology.Compactification.OnePoint.Basic
public import Mathlib.Topology.Category.LightProfinite.Basic

Namespace:
LightProfinite

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

# The light profinite set classifying convergent sequences

This file defines the light profinite set `ℕ∪{∞}`, defined as the one point compactification of
`ℕ`.
-/

@[expose] public section

open CategoryTheory OnePoint TopologicalSpace Topology

namespace LightProfinite

/-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. -/
noncomputable def natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where
  toFun
    | ∞ => 0
    | OnePoint.some n => 1 / (n + 1 : ℝ)
  continuous_toFun := OnePoint.continuous_iff_from_nat _ |>.mpr
    tendsto_one_div_add_atTop_nhds_zero_nat

/--
The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed
embedding.
-/

Target:
lemma isClosedEmbedding_natUnionInftyEmbedding : IsClosedEmbedding natUnionInftyEmbedding :=

Proof body:

[Assistant]
by
  refine .of_continuous_injective_isClosedMap
    natUnionInftyEmbedding.continuous ?_ ?_
  · rintro (_ | n) (_ | m) h
    · rfl
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, zero_eq_inv] at h
      assumption_mod_cast
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_eq_zero] at h
      assumption_mod_cast
    · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_inj, add_left_inj,
        Nat.cast_inj] at h
      rw [h]
  · exact fun _ hC => (hC.isCompact.image natUnionInftyEmbedding.continuous).isClosed

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.Topology.Compactification.OnePoint.Basic
public import Mathlib.Topology.Category.LightProfinite.Basic

Namespace:
LightProfinite

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

# The light profinite set classifying convergent sequences

This file defines the light profinite set `ℕ∪{∞}`, defined as the one point compactification of
`ℕ`.
-/

@[expose] public section

open CategoryTheory OnePoint TopologicalSpace Topology

namespace LightProfinite

/-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. -/
noncomputable def natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where
  toFun
    | ∞ => 0
    | OnePoint.some n => 1 / (n + 1 : ℝ)
  continuous_toFun := OnePoint.continuous_iff_from_nat _ |>.mpr
    tendsto_one_div_add_atTop_nhds_zero_nat

/--
The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed
embedding.
-/

Target:
lemma isClosedEmbedding_natUnionInftyEmbedding : IsClosedEmbedding natUnionInftyEmbedding :=

Proof body:


Response:
by
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

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

Imports:
public import Mathlib.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
public import Mathlib.LinearAlgebra.Semisimple

Namespace:
Module.End

Local context:
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Eigenspaces of semisimple linear endomorphisms

This file contains basic results relevant to the study of eigenspaces of semisimple linear
endomorphisms.

## Main definitions / results

* `Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspace`: for a semisimple endomorphism,
  a generalized eigenspace is an eigenspace.
* `Module.End.IsSemisimple.iSup_eigenspace_eq_top`: over an algebraically closed field,
  the eigenspaces of a semisimple endomorphism span the whole space.
* `Module.End.IsSemisimple.eq_zero_iff_forall_eigenvalue`: a semisimple endomorphism over
  an algebraically closed field is zero iff all eigenvalues are zero.

-/

public section

open Function Set

namespace Module.End

variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] {f g : End R M}

lemma apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil
    {μ : R} {k : ℕ∞} {m : M} (hm : m ∈ f.genEigenspace μ k)
    (hfg : Commute f g) (hss : g.IsFinitelySemisimple) (hnil : IsNilpotent (f - g)) :
    g m = μ • m := by
  rw [f.mem_genEigenspace] at hm
  obtain ⟨l, -, hm⟩ := hm
  rw [← f.mem_genEigenspace_nat] at hm
  set p := f.genEigenspace μ l
  have h₁ : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg μ l
  have h₂ : MapsTo (g - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right μ f) μ l
  have h₃ : MapsTo (f - g) p p :=
    mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) μ l
  have h₄ : MapsTo (f - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ l
  replace hfg : Commute (f - algebraMap R (End R M) μ) (f - g) :=
    (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left μ (f - g)
  suffices IsNilpotent ((g - algebraMap R (End R M) μ).restrict h₂) by
    replace this : g.restrict h₁ - algebraMap R (End R p) μ = 0 :=
      eq_zero_of_isNilpotent_of_isFinitelySemisimple this (by simpa using hss.restrict _)
    simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this ⟨m, hm⟩
  simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub
    (f.isNilpotent_restrict_sub_algebraMap μ l) (Module.End.isNilpotent.restrict h₃ hnil)

Target:
lemma IsFinitelySemisimple.genEigenspace_eq_eigenspace
    (hf : f.IsFinitelySemisimple) (μ : R) {k : ℕ∞} (hk : 0 < k) :
    f.genEigenspace μ k = f.eigenspace μ :=

Proof body:

[Assistant]
by
  refine le_antisymm (fun m hm ↦ mem_eigenspace_iff.mpr ?_) (f.genEigenspace μ |>.mono ?_)
  · apply apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hm rfl hf
    simp
  · exact Order.one_le_iff_pos.mpr hk

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

Imports:
public import Mathlib.LinearAlgebra.Eigenspace.Basic
public import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
public import Mathlib.LinearAlgebra.Semisimple

Namespace:
Module.End

Local context:
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
/-!
# Eigenspaces of semisimple linear endomorphisms

This file contains basic results relevant to the study of eigenspaces of semisimple linear
endomorphisms.

## Main definitions / results

* `Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspace`: for a semisimple endomorphism,
  a generalized eigenspace is an eigenspace.
* `Module.End.IsSemisimple.iSup_eigenspace_eq_top`: over an algebraically closed field,
  the eigenspaces of a semisimple endomorphism span the whole space.
* `Module.End.IsSemisimple.eq_zero_iff_forall_eigenvalue`: a semisimple endomorphism over
  an algebraically closed field is zero iff all eigenvalues are zero.

-/

public section

open Function Set

namespace Module.End

variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] {f g : End R M}

lemma apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil
    {μ : R} {k : ℕ∞} {m : M} (hm : m ∈ f.genEigenspace μ k)
    (hfg : Commute f g) (hss : g.IsFinitelySemisimple) (hnil : IsNilpotent (f - g)) :
    g m = μ • m := by
  rw [f.mem_genEigenspace] at hm
  obtain ⟨l, -, hm⟩ := hm
  rw [← f.mem_genEigenspace_nat] at hm
  set p := f.genEigenspace μ l
  have h₁ : MapsTo g p p := mapsTo_genEigenspace_of_comm hfg μ l
  have h₂ : MapsTo (g - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (hfg.sub_right <| Algebra.commute_algebraMap_right μ f) μ l
  have h₃ : MapsTo (f - g) p p :=
    mapsTo_genEigenspace_of_comm (Commute.sub_right rfl hfg) μ l
  have h₄ : MapsTo (f - algebraMap R (End R M) μ) p p :=
    mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ l
  replace hfg : Commute (f - algebraMap R (End R M) μ) (f - g) :=
    (Commute.sub_right rfl hfg).sub_left <| Algebra.commute_algebraMap_left μ (f - g)
  suffices IsNilpotent ((g - algebraMap R (End R M) μ).restrict h₂) by
    replace this : g.restrict h₁ - algebraMap R (End R p) μ = 0 :=
      eq_zero_of_isNilpotent_of_isFinitelySemisimple this (by simpa using hss.restrict _)
    simpa [LinearMap.restrict_apply, sub_eq_zero] using LinearMap.congr_fun this ⟨m, hm⟩
  simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub
    (f.isNilpotent_restrict_sub_algebraMap μ l) (Module.End.isNilpotent.restrict h₃ hnil)

Target:
lemma IsFinitelySemisimple.genEigenspace_eq_eigenspace
    (hf : f.IsFinitelySemisimple) (μ : R) {k : ℕ∞} (hk : 0 < k) :
    f.genEigenspace μ k = f.eigenspace μ :=

Proof body:


Response:
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp hc

[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.Constructions

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
-/
/-!
# Bundled morphisms with continuous evaluation at a point

In this file we define a typeclass
saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`.

## Implementation Notes

For now, we define the typeclass for non-dependent bundled functions only.
Whenever we add a type of bundled dependent functions with a topology having this property,
we may decide to generalize from `FunLike` to `DFunLike`.
-/

public section

open scoped Topology
open Filter

/-- A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`. -/
class ContinuousEvalConst (F : Type*) (α X : outParam Type*) [FunLike F α X]
    [TopologicalSpace F] [TopologicalSpace X] : Prop where
  continuous_eval_const (x : α) : Continuous fun f : F ↦ f x

export ContinuousEvalConst (continuous_eval_const)

section ContinuousEvalConst

variable {F α X Z : Type*} [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X]
  [ContinuousEvalConst F α X] [TopologicalSpace Z] {f : Z → F} {s : Set Z} {z : Z}

/-- If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEvalConst`,
then `F'` satisfies this predicate too.

The word "forget" in the name is motivated by the term "forgetful functor". -/

Target:
theorem ContinuousEvalConst.of_continuous_forget {F' : Type*} [FunLike F' α X] [TopologicalSpace F']
    {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g :=

Failed candidate:
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp __oprover_missing_a8cfe094bc63

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:52:88: error(lean.unknownIdentifier): Unknown identifier `__oprover_missing_a8cfe094bc63`", "returncode": 1}

Corrected proof:

[Assistant]
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp hc

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.Constructions

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
-/
/-!
# Bundled morphisms with continuous evaluation at a point

In this file we define a typeclass
saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`.

## Implementation Notes

For now, we define the typeclass for non-dependent bundled functions only.
Whenever we add a type of bundled dependent functions with a topology having this property,
we may decide to generalize from `FunLike` to `DFunLike`.
-/

public section

open scoped Topology
open Filter

/-- A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`. -/
class ContinuousEvalConst (F : Type*) (α X : outParam Type*) [FunLike F α X]
    [TopologicalSpace F] [TopologicalSpace X] : Prop where
  continuous_eval_const (x : α) : Continuous fun f : F ↦ f x

export ContinuousEvalConst (continuous_eval_const)

section ContinuousEvalConst

variable {F α X Z : Type*} [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X]
  [ContinuousEvalConst F α X] [TopologicalSpace Z] {f : Z → F} {s : Set Z} {z : Z}

/-- If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEvalConst`,
then `F'` satisfies this predicate too.

The word "forget" in the name is motivated by the term "forgetful functor". -/

Target:
theorem ContinuousEvalConst.of_continuous_forget {F' : Type*} [FunLike F' α X] [TopologicalSpace F']
    {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g :=

Failed candidate:
by intro; rfl) :
    ContinuousEvalConst F' α X where
  continuous_eval_const x := by simpa only [← hf] using! (continuous_eval_const x).comp __oprover_missing_a8cfe094bc63

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:52:88: error(lean.unknownIdentifier): Unknown identifier `__oprover_missing_a8cfe094bc63`", "returncode": 1}

Corrected proof:


Response:
by
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

[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.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

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

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

-/

public section

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

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) := by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

open scoped IsMulCommutative in
/-- Any finite-dimensional irreducible representation of a commutative algebra over an algebraically
closed field is one-dimensional. -/

Target:
theorem IsSimpleModule.finrank_eq_one_of_isMulCommutative [IsMulCommutative A] :
    Module.finrank k V = 1 :=

Proof body:

[Assistant]
by
  have : Nontrivial V := IsSimpleModule.nontrivial A V
  obtain ⟨v, v_nz⟩ := exists_ne (0 : V)
  set U : Submodule A V := { Submodule.span k {v} with
    smul_mem' a w hw := by
      have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw }
  obtain hU | hU := eq_bot_or_eq_top U
  · exact (v_nz <| hU.le <| Submodule.mem_span_singleton_self v).elim
  · rw [finrank_eq_one_iff_of_nonzero v v_nz]
    rwa [← top_le_iff] at hU ⊢

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.Lie.OfAssociative
public import Mathlib.FieldTheory.IsAlgClosed.Basic

Namespace:
(root)

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

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

-/

public section

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

/-- Schur's Lemma: If `V` is a representation of an algebra `A` over an algebraically closed field
`k`, then any endomorphism of `V` is scalar. -/
theorem IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed :
    Function.Bijective (algebraMap k (Module.End A V)) := by
  have : Module.Finite k (Module.End A V) := .of_injective (LinearMap.restrictScalarsₗ k A V V k) <|
    LinearMap.restrictScalars_injective _
  classical exact IsAlgClosed.algebraMap_bijective_of_isIntegral (k := k)

variable (A V)

open scoped IsMulCommutative in
/-- Any finite-dimensional irreducible representation of a commutative algebra over an algebraically
closed field is one-dimensional. -/

Target:
theorem IsSimpleModule.finrank_eq_one_of_isMulCommutative [IsMulCommutative A] :
    Module.finrank k V = 1 :=

Proof body:


<LEAN_FILE>:106:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

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

Response:
by
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

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

Imports:
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Quotient.Basic

Namespace:
Submodule.quotientPi_aux

Local safe context:
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
/-!
# Submodule quotients and direct sums

This file contains some results on the quotient of a module by a direct sum of submodules,
and the direct sum of quotients of modules by submodules.

## Main definitions

* `Submodule.piQuotientLift`: create a map out of the direct sum of quotients
* `Submodule.quotientPiLift`: create a map out of the quotient of a direct sum
* `Submodule.quotientPi`: the quotient of a direct sum is the direct sum of quotients.

-/

@[expose] public section

namespace Submodule

open LinearMap

variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)] [∀ i, Module R (Ms i)]
variable {N : Type*} [AddCommGroup N] [Module R N]
variable {Ns : ι → Type*} [∀ i, AddCommGroup (Ns i)] [∀ i, Module R (Ns i)]

/-- Lift a family of maps to the direct sum of quotients. -/
def piQuotientLift [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i)) (q : Submodule R N)
    (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) : (∀ i, Ms i ⧸ p i) →ₗ[R] N ⧸ q :=
  lsum R (fun i => Ms i ⧸ p i) R fun i => (p i).mapQ q (f i) (hf i)

@[simp]
theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
    (piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by
  rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
  simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]

@[simp]
theorem piQuotientLift_single [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (i)
    (x : Ms i ⧸ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by
  simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
  rw [Finset.sum_eq_single i]
  · rw [Pi.single_eq_same]
  · rintro j - hj
    rw [Pi.single_eq_of_ne hj, map_zero]
  · intros
    have := Finset.mem_univ i
    contradiction

/-- Lift a family of maps to a quotient of direct sums. -/
def quotientPiLift (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) : (∀ i, Ms i) ⧸ pi Set.univ p →ₗ[R] ∀ i, Ns i :=
  (pi Set.univ p).liftQ (LinearMap.pi fun i => (f i).comp (proj i)) fun x hx =>
    mem_ker.mpr <| by
      ext i
      simpa using hf i (mem_pi.mp hx i (Set.mem_univ i))

@[simp]
theorem quotientPiLift_mk (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) (x : ∀ i, Ms i) :
    quotientPiLift p f hf (Quotient.mk x) = fun i => f i (x i) :=
  rfl

namespace quotientPi_aux

variable (p : ∀ i, Submodule R (Ms i))

@[simp]
def toFun : ((∀ i, Ms i) ⧸ pi Set.univ p) → ∀ i, Ms i ⧸ p i :=
  quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge

theorem map_add (x y : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (x + y) = toFun p x + toFun p y :=
  LinearMap.map_add (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) x y

theorem map_smul (r : R) (x : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (r • x) = (RingHom.id R r) • toFun p x :=
  LinearMap.map_smul (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) r x

variable [Fintype ι] [DecidableEq ι]

@[simp]
def invFun : (∀ i, Ms i ⧸ p i) → (∀ i, Ms i) ⧸ pi Set.univ p :=
  piQuotientLift p (pi Set.univ p) _ fun _ => le_comap_single_pi p

theorem left_inv : Function.LeftInverse (invFun p) (toFun p) := fun x =>
  Submodule.Quotient.induction_on _ x fun x' => by
    dsimp only [toFun, invFun]
    rw [quotientPiLift_mk p, funext fun i => (mkQ_apply (p i) (x' i)), piQuotientLift_mk p,
      lsum_single, id_apply]

Target:
theorem right_inv : Function.RightInverse (invFun p) (toFun p) :=

Failed candidate:
by
  let __repair_8795fe04a3 : Fact False := inferInstance
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:106:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

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

Imports:
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.LinearAlgebra.Quotient.Basic

Namespace:
Submodule.quotientPi_aux

Local safe context:
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
/-!
# Submodule quotients and direct sums

This file contains some results on the quotient of a module by a direct sum of submodules,
and the direct sum of quotients of modules by submodules.

## Main definitions

* `Submodule.piQuotientLift`: create a map out of the direct sum of quotients
* `Submodule.quotientPiLift`: create a map out of the quotient of a direct sum
* `Submodule.quotientPi`: the quotient of a direct sum is the direct sum of quotients.

-/

@[expose] public section

namespace Submodule

open LinearMap

variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)] [∀ i, Module R (Ms i)]
variable {N : Type*} [AddCommGroup N] [Module R N]
variable {Ns : ι → Type*} [∀ i, AddCommGroup (Ns i)] [∀ i, Module R (Ns i)]

/-- Lift a family of maps to the direct sum of quotients. -/
def piQuotientLift [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i)) (q : Submodule R N)
    (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) : (∀ i, Ms i ⧸ p i) →ₗ[R] N ⧸ q :=
  lsum R (fun i => Ms i ⧸ p i) R fun i => (p i).mapQ q (f i) (hf i)

@[simp]
theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
    (piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by
  rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
  simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]

@[simp]
theorem piQuotientLift_single [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
    (q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (i)
    (x : Ms i ⧸ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by
  simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
  rw [Finset.sum_eq_single i]
  · rw [Pi.single_eq_same]
  · rintro j - hj
    rw [Pi.single_eq_of_ne hj, map_zero]
  · intros
    have := Finset.mem_univ i
    contradiction

/-- Lift a family of maps to a quotient of direct sums. -/
def quotientPiLift (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) : (∀ i, Ms i) ⧸ pi Set.univ p →ₗ[R] ∀ i, Ns i :=
  (pi Set.univ p).liftQ (LinearMap.pi fun i => (f i).comp (proj i)) fun x hx =>
    mem_ker.mpr <| by
      ext i
      simpa using hf i (mem_pi.mp hx i (Set.mem_univ i))

@[simp]
theorem quotientPiLift_mk (p : ∀ i, Submodule R (Ms i)) (f : ∀ i, Ms i →ₗ[R] Ns i)
    (hf : ∀ i, p i ≤ ker (f i)) (x : ∀ i, Ms i) :
    quotientPiLift p f hf (Quotient.mk x) = fun i => f i (x i) :=
  rfl

namespace quotientPi_aux

variable (p : ∀ i, Submodule R (Ms i))

@[simp]
def toFun : ((∀ i, Ms i) ⧸ pi Set.univ p) → ∀ i, Ms i ⧸ p i :=
  quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge

theorem map_add (x y : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (x + y) = toFun p x + toFun p y :=
  LinearMap.map_add (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) x y

theorem map_smul (r : R) (x : ((i : ι) → Ms i) ⧸ pi Set.univ p) :
    toFun p (r • x) = (RingHom.id R r) • toFun p x :=
  LinearMap.map_smul (quotientPiLift p (fun i => (p i).mkQ) fun i => (ker_mkQ (p i)).ge) r x

variable [Fintype ι] [DecidableEq ι]

@[simp]
def invFun : (∀ i, Ms i ⧸ p i) → (∀ i, Ms i) ⧸ pi Set.univ p :=
  piQuotientLift p (pi Set.univ p) _ fun _ => le_comap_single_pi p

theorem left_inv : Function.LeftInverse (invFun p) (toFun p) := fun x =>
  Submodule.Quotient.induction_on _ x fun x' => by
    dsimp only [toFun, invFun]
    rw [quotientPiLift_mk p, funext fun i => (mkQ_apply (p i) (x' i)), piQuotientLift_mk p,
      lsum_single, id_apply]

Target:
theorem right_inv : Function.RightInverse (invFun p) (toFun p) :=

Failed candidate:
by
  let __repair_8795fe04a3 : Fact False := inferInstance
  dsimp only [toFun, invFun]
  rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]
  congr
  refine pi_ext fun i x ↦ ?_
  induction x using Submodule.Quotient.induction_on with | _ x'
  refine funext fun j ↦ ?_
  rw [comp_apply, piQuotientLift_single, mapQ_apply,
    quotientPiLift_mk, id_apply]
  by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]
  · subst hij
    rw [Pi.single_eq_same, Pi.single_eq_same]
  · rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:106:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:60:43: error: unsolved goals
X : Type u_1
R : SetRel X X
s : Set X
x : X
inst✝ : R.IsSymm
⊢ False

Response:
by
  have : Std.Symm fun x y ↦ ¬(x, y) ∈ R := { symm _ _ := mt R.symm }
  simpa [not_imp_not, IsSeparated] using pairwise_insert_of_symm (r := fun x y ↦ ¬(x, y) ∈ R)

[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.Pairwise.Basic
public import Mathlib.Data.Rel

Namespace:
SetRel

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

This file defines a notion of separation of a set relative to a relation.

For a relation `R`, an `R`-separated set `s` is a set such that every pair of elements of `s` is
`R`-unrelated.

The concept of uniformly separated sets is used to define two further notions of separation:
* Metric separation: `Metric.IsSeparated`, defined using the small distance relation.
* Dynamical nets: `Dynamics.IsDynNetIn`, defined using the dynamical relation.

## TODO

* Actually use `SetRel.IsSeparated` to define the above two notions.
* Link to the notion of separation given by pairwise disjoint balls.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {R S : SetRel X X} {s t : Set X} {x : X}

/-- Given a relation `R`, a set `s` is `R`-separated if its elements are pairwise `R`-unrelated from
each other. -/
def IsSeparated (R : SetRel X X) (s : Set X) : Prop := s.Pairwise fun x y ↦ ¬ x ~[R] y

protected lemma IsSeparated.empty : IsSeparated R (∅ : Set X) := pairwise_empty _
protected lemma IsSeparated.singleton : IsSeparated R {x} := pairwise_singleton ..

@[simp] lemma IsSeparated.of_subsingleton (hs : s.Subsingleton) : IsSeparated R s := hs.pairwise _

alias _root_.Set.Subsingleton.relIsSeparated := IsSeparated.of_subsingleton

nonrec lemma IsSeparated.mono_left (hUV : R ⊆ S) (hs : IsSeparated S s) : IsSeparated R s :=
  hs.mono' fun _x _y hxy h ↦ hxy <| hUV h

lemma IsSeparated.mono_right (hst : s ⊆ t) (ht : IsSeparated R t) : IsSeparated R s := ht.mono hst

lemma isSeparated_insert' :
    IsSeparated R (insert x s) ↔ IsSeparated R s ∧ (∀ y ∈ s, x ~[R] y → x = y) ∧
        ∀ y ∈ s, y ~[R] x → x = y := by
  simp [IsSeparated, pairwise_insert, not_imp_comm (a := _ = _), -not_and, forall_and]

Target:
lemma isSeparated_insert [R.IsSymm] :
    IsSeparated R (insert x s) ↔ IsSeparated R s ∧ ∀ y ∈ s, x ~[R] y → x = y :=

Failed candidate:
by
  have __repair_plan_5216cced2c : False := by
    skip
  have : Std.Symm fun x y ↦ ¬(x, y) ∈ R := { symm _ _ := mt R.symm }
  simpa [not_imp_not, IsSeparated] using pairwise_insert_of_symm (r := fun x y ↦ ¬(x, y) ∈ R)

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:60:43: error: unsolved goals\nX : Type u_1\nR : SetRel X X\ns : Set X\nx : X\ninst✝ : R.IsSymm\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  have : Std.Symm fun x y ↦ ¬(x, y) ∈ R := { symm _ _ := mt R.symm }
  simpa [not_imp_not, IsSeparated] using pairwise_insert_of_symm (r := fun x y ↦ ¬(x, y) ∈ R)

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.Pairwise.Basic
public import Mathlib.Data.Rel

Namespace:
SetRel

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

This file defines a notion of separation of a set relative to a relation.

For a relation `R`, an `R`-separated set `s` is a set such that every pair of elements of `s` is
`R`-unrelated.

The concept of uniformly separated sets is used to define two further notions of separation:
* Metric separation: `Metric.IsSeparated`, defined using the small distance relation.
* Dynamical nets: `Dynamics.IsDynNetIn`, defined using the dynamical relation.

## TODO

* Actually use `SetRel.IsSeparated` to define the above two notions.
* Link to the notion of separation given by pairwise disjoint balls.
-/

@[expose] public section

open Set

namespace SetRel
variable {X : Type*} {R S : SetRel X X} {s t : Set X} {x : X}

/-- Given a relation `R`, a set `s` is `R`-separated if its elements are pairwise `R`-unrelated from
each other. -/
def IsSeparated (R : SetRel X X) (s : Set X) : Prop := s.Pairwise fun x y ↦ ¬ x ~[R] y

protected lemma IsSeparated.empty : IsSeparated R (∅ : Set X) := pairwise_empty _
protected lemma IsSeparated.singleton : IsSeparated R {x} := pairwise_singleton ..

@[simp] lemma IsSeparated.of_subsingleton (hs : s.Subsingleton) : IsSeparated R s := hs.pairwise _

alias _root_.Set.Subsingleton.relIsSeparated := IsSeparated.of_subsingleton

nonrec lemma IsSeparated.mono_left (hUV : R ⊆ S) (hs : IsSeparated S s) : IsSeparated R s :=
  hs.mono' fun _x _y hxy h ↦ hxy <| hUV h

lemma IsSeparated.mono_right (hst : s ⊆ t) (ht : IsSeparated R t) : IsSeparated R s := ht.mono hst

lemma isSeparated_insert' :
    IsSeparated R (insert x s) ↔ IsSeparated R s ∧ (∀ y ∈ s, x ~[R] y → x = y) ∧
        ∀ y ∈ s, y ~[R] x → x = y := by
  simp [IsSeparated, pairwise_insert, not_imp_comm (a := _ = _), -not_and, forall_and]

Target:
lemma isSeparated_insert [R.IsSymm] :
    IsSeparated R (insert x s) ↔ IsSeparated R s ∧ ∀ y ∈ s, x ~[R] y → x = y :=

Failed candidate:
by
  have __repair_plan_5216cced2c : False := by
    skip
  have : Std.Symm fun x y ↦ ¬(x, y) ∈ R := { symm _ _ := mt R.symm }
  simpa [not_imp_not, IsSeparated] using pairwise_insert_of_symm (r := fun x y ↦ ¬(x, y) ∈ R)

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:60:43: error: unsolved goals\nX : Type u_1\nR : SetRel X X\ns : Set X\nx : X\ninst✝ : R.IsSymm\n⊢ False", "returncode": 1}

Corrected proof:


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

Namespace:
BoxIntegral.Prepartition

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

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

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

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

namespace TaggedPrepartition

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

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

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

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

Proof body:

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

Namespace:
BoxIntegral.Prepartition

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

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

## Tags

rectangular box, box partition
-/

@[expose] public section

noncomputable section

open Finset Function ENNReal NNReal Set

namespace BoxIntegral

variable {ι : Type*}

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

namespace TaggedPrepartition

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end TaggedPrepartition

namespace Prepartition

variable {I J : Box ι}

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

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

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

Proof body:


<LEAN_FILE>:83:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

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

Imports:
public import Mathlib.LinearAlgebra.BilinearForm.Hom
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.TensorProduct.Finite

Namespace:
LinearMap.BilinMap

Local safe context:
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
/-!
# The bilinear form on a tensor product

## Main definitions

* `LinearMap.BilinMap.tensorDistrib (B₁ ⊗ₜ B₂)`: the bilinear form on `M₁ ⊗ M₂` constructed by
  applying `B₁` on `M₁` and `B₂` on `M₂`.
* `LinearMap.BilinMap.tensorDistribEquiv`: `BilinForm.tensorDistrib` as an equivalence on finite
  free modules.

-/

@[expose] public section

universe u v w uR uA uM₁ uM₂ uN₁ uN₂

variable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}

open TensorProduct

namespace LinearMap

open LinearMap (BilinMap BilinForm)

section CommSemiring

variable [CommSemiring R] [CommSemiring A]
variable [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid N₁] [AddCommMonoid N₂]
variable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]
variable [SMulCommClass R A M₁] [IsScalarTower R A M₁]
variable [SMulCommClass R A N₁] [IsScalarTower R A N₁]
variable [Module R M₂] [Module R N₂]

namespace BilinMap

variable (R A) in
/-- The tensor product of two bilinear maps injects into bilinear maps on tensor products.

Note this is heterobasic; the bilinear map on the left can take values in a module over a
(commutative) algebra over the ring of the module in which the right bilinear map is valued. -/
def tensorDistrib :
    (BilinMap A M₁ N₁ ⊗[R] BilinMap R M₂ N₂) →ₗ[A] BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  (TensorProduct.lift.equiv (.id A) (M₁ ⊗[R] M₂) (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂)).symm.toLinearMap ∘ₗ
  ((LinearMap.llcomp A _ _ _).flip
    (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R R A A M₁ M₂ M₁ M₂).toLinearMap)
  ∘ₗ TensorProduct.AlgebraTensorModule.homTensorHomMap R _ _ _ _ _ _
  ∘ₗ (TensorProduct.AlgebraTensorModule.congr
    (TensorProduct.lift.equiv (.id A) M₁ M₁ N₁)
    (TensorProduct.lift.equiv (.id R) _ _ _)).toLinearMap

@[simp]
theorem tensorDistrib_tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) (m₁ : M₁) (m₂ : M₂)
    (m₁' : M₁) (m₂' : M₂) :
    tensorDistrib R A (B₁ ⊗ₜ B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂')
      = B₁ m₁ m₁' ⊗ₜ B₂ m₂ m₂' :=
  rfl

/-- The tensor product of two bilinear forms, a shorthand for dot notation. -/
protected abbrev tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) :
    BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  tensorDistrib R A (B₁ ⊗ₜ[R] B₂)

attribute [local ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear maps is symmetric. -/

Target:
lemma tmul_isSymm {B₁ : BilinMap A M₁ N₁} {B₂ : BilinMap R M₂ N₂}
    (hB₁ : ∀ x y, B₁ x y = B₁ y x) (hB₂ : ∀ x y, B₂ x y = B₂ y x)
    (x y : M₁ ⊗[R] M₂) :
    B₁.tmul B₂ x y = B₁.tmul B₂ y x :=

Failed candidate:
by
  have __repair_74f9707b79 : False := True.intro
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:83:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

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

Imports:
public import Mathlib.LinearAlgebra.BilinearForm.Hom
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.TensorProduct.Finite

Namespace:
LinearMap.BilinMap

Local safe context:
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
/-!
# The bilinear form on a tensor product

## Main definitions

* `LinearMap.BilinMap.tensorDistrib (B₁ ⊗ₜ B₂)`: the bilinear form on `M₁ ⊗ M₂` constructed by
  applying `B₁` on `M₁` and `B₂` on `M₂`.
* `LinearMap.BilinMap.tensorDistribEquiv`: `BilinForm.tensorDistrib` as an equivalence on finite
  free modules.

-/

@[expose] public section

universe u v w uR uA uM₁ uM₂ uN₁ uN₂

variable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}

open TensorProduct

namespace LinearMap

open LinearMap (BilinMap BilinForm)

section CommSemiring

variable [CommSemiring R] [CommSemiring A]
variable [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid N₁] [AddCommMonoid N₂]
variable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]
variable [SMulCommClass R A M₁] [IsScalarTower R A M₁]
variable [SMulCommClass R A N₁] [IsScalarTower R A N₁]
variable [Module R M₂] [Module R N₂]

namespace BilinMap

variable (R A) in
/-- The tensor product of two bilinear maps injects into bilinear maps on tensor products.

Note this is heterobasic; the bilinear map on the left can take values in a module over a
(commutative) algebra over the ring of the module in which the right bilinear map is valued. -/
def tensorDistrib :
    (BilinMap A M₁ N₁ ⊗[R] BilinMap R M₂ N₂) →ₗ[A] BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  (TensorProduct.lift.equiv (.id A) (M₁ ⊗[R] M₂) (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂)).symm.toLinearMap ∘ₗ
  ((LinearMap.llcomp A _ _ _).flip
    (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R R A A M₁ M₂ M₁ M₂).toLinearMap)
  ∘ₗ TensorProduct.AlgebraTensorModule.homTensorHomMap R _ _ _ _ _ _
  ∘ₗ (TensorProduct.AlgebraTensorModule.congr
    (TensorProduct.lift.equiv (.id A) M₁ M₁ N₁)
    (TensorProduct.lift.equiv (.id R) _ _ _)).toLinearMap

@[simp]
theorem tensorDistrib_tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) (m₁ : M₁) (m₂ : M₂)
    (m₁' : M₁) (m₂' : M₂) :
    tensorDistrib R A (B₁ ⊗ₜ B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂')
      = B₁ m₁ m₁' ⊗ₜ B₂ m₂ m₂' :=
  rfl

/-- The tensor product of two bilinear forms, a shorthand for dot notation. -/
protected abbrev tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) :
    BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  tensorDistrib R A (B₁ ⊗ₜ[R] B₂)

attribute [local ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear maps is symmetric. -/

Target:
lemma tmul_isSymm {B₁ : BilinMap A M₁ N₁} {B₂ : BilinMap R M₂ N₂}
    (hB₁ : ∀ x y, B₁ x y = B₁ y x) (hB₂ : ∀ x y, B₂ x y = B₂ y x)
    (x y : M₁ ⊗[R] M₂) :
    B₁.tmul B₂ x y = B₁.tmul B₂ y x :=

Failed candidate:
by
  have __repair_74f9707b79 : False := True.intro
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:83:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof: