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

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

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

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

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

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

## References

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

@[expose] public section

open Set

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Proof body:

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

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

Imports:
public import Mathlib.Data.Rel.Separated

Namespace:
SetRel

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

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

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

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

## References

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

@[expose] public section

open Set

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Proof body:


Response:
by
  revert h
  induction x using q.P.wCases
  case ih a₀ f'₀ f₀ =>
    induction y using q.P.wCases
    apply WEquiv.abs

[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.Data.PFunctor.Multivariate.W
public import Mathlib.Data.QPF.Multivariate.Basic

Namespace:
MvQPF

Local context:
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
/-!
# The initial algebra of a multivariate qpf is again a qpf.

For an `(n+1)`-ary QPF `F (α₀,..,αₙ)`, we take the least fixed point of `F` with
regards to its last argument `αₙ`. The result is an `n`-ary functor: `Fix F (α₀,..,αₙ₋₁)`.
Making `Fix F` into a functor allows us to take the fixed point, compose with other functors
and take a fixed point again.

## Main definitions

* `Fix.mk`     - constructor
* `Fix.dest`    - destructor
* `Fix.rec`    - recursor: basis for defining functions by structural recursion on `Fix F α`
* `Fix.drec`   - dependent recursor: generalization of `Fix.rec` where
                  the result type of the function is allowed to depend on the `Fix F α` value
* `Fix.rec_eq` - defining equation for `recursor`
* `Fix.ind`    - induction principle for `Fix F α`

## Implementation notes

For `F` a `QPF`, we define `Fix F α` in terms of the W-type of the polynomial functor `P` of `F`.
We define the relation `WEquiv` and take its quotient as the definition of `Fix F α`.

See [avigad-carneiro-hudon2019] for more details.

## Reference

* Jeremy Avigad, Mario M. Carneiro and Simon Hudon.
  [*Data Types as Quotients of Polynomial Functors*][avigad-carneiro-hudon2019]
-/

@[expose] public section

universe u v

namespace MvQPF

open TypeVec

open MvFunctor (LiftP LiftR)

open MvFunctor

variable {n : ℕ} {F : TypeVec.{u} (n + 1) → Type u} [q : MvQPF F]

/-- `recF` is used as a basis for defining the recursor on `Fix F α`. `recF`
traverses recursively the W-type generated by `q.P` using a function on `F`
as a recursive step -/
def recF {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) : q.P.W α → β :=
  q.P.wRec fun a f' _f rec => g (abs ⟨a, splitFun f' rec⟩)

theorem recF_eq {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (a : q.P.A)
    (f' : q.P.drop.B a ⟹ α) (f : q.P.last.B a → q.P.W α) :
    recF g (q.P.wMk a f' f) = g (abs ⟨a, splitFun f' (recF g ∘ f)⟩) := by
  rw [recF, MvPFunctor.wRec_eq]; rfl

set_option backward.isDefEq.respectTransparency false in
theorem recF_eq' {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (x : q.P.W α) :
    recF g x = g (abs (appendFun id (recF g) <$$> q.P.wDest' x)) := by
  induction x using q.P.wCases
  case ih a f' f =>
    rw [recF_eq, q.P.wDest'_wMk, MvPFunctor.map_eq, appendFun_comp_splitFun, TypeVec.id_comp]

/-- Equivalence relation on W-types that represent the same `Fix F`
value -/
inductive WEquiv {α : TypeVec n} : q.P.W α → q.P.W α → Prop
  | ind (a : q.P.A) (f' : q.P.drop.B a ⟹ α) (f₀ f₁ : q.P.last.B a → q.P.W α) :
    (∀ x, WEquiv (f₀ x) (f₁ x)) → WEquiv (q.P.wMk a f' f₀) (q.P.wMk a f' f₁)
  | abs (a₀ : q.P.A) (f'₀ : q.P.drop.B a₀ ⟹ α) (f₀ : q.P.last.B a₀ → q.P.W α) (a₁ : q.P.A)
    (f'₁ : q.P.drop.B a₁ ⟹ α) (f₁ : q.P.last.B a₁ → q.P.W α) :
    abs ⟨a₀, q.P.appendContents f'₀ f₀⟩ = abs ⟨a₁, q.P.appendContents f'₁ f₁⟩ →
      WEquiv (q.P.wMk a₀ f'₀ f₀) (q.P.wMk a₁ f'₁ f₁)
  | trans (u v w : q.P.W α) : WEquiv u v → WEquiv v w → WEquiv u w

theorem recF_eq_of_wEquiv (α : TypeVec n) {β : Type u} (u : F (α.append1 β) → β) (x y : q.P.W α) :
    WEquiv x y → recF u x = recF u y := by
  induction x using q.P.wCases
  case ih a₀ f'₀ f₀ =>
    induction y using q.P.wCases
    case ih a₁ f'₁ f₁ =>
      intro h
      -- Porting note: induction on h doesn't work.
      refine @WEquiv.recOn _ _ _ _ (fun a a' _ ↦ recF u a = recF u a') _ _ h ?_ ?_ ?_
      · intro a f' f₀ f₁ _h ih; simp only [recF_eq]
        congr 4; funext; apply ih
      · intro a₀ f'₀ f₀ a₁ f'₁ f₁ h; simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]
      · intro x y z _e₁ _e₂ ih₁ ih₂; exact Eq.trans ih₁ ih₂

Target:
theorem wEquiv.abs' {α : TypeVec n} (x y : q.P.W α)
    (h : MvQPF.abs (q.P.wDest' x) = MvQPF.abs (q.P.wDest' y)) :
    WEquiv x y :=

Proof body:

[Assistant]
by
  revert h
  induction x using q.P.wCases
  case ih a₀ f'₀ f₀ =>
    induction y using q.P.wCases
    apply WEquiv.abs

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.Data.PFunctor.Multivariate.W
public import Mathlib.Data.QPF.Multivariate.Basic

Namespace:
MvQPF

Local context:
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
/-!
# The initial algebra of a multivariate qpf is again a qpf.

For an `(n+1)`-ary QPF `F (α₀,..,αₙ)`, we take the least fixed point of `F` with
regards to its last argument `αₙ`. The result is an `n`-ary functor: `Fix F (α₀,..,αₙ₋₁)`.
Making `Fix F` into a functor allows us to take the fixed point, compose with other functors
and take a fixed point again.

## Main definitions

* `Fix.mk`     - constructor
* `Fix.dest`    - destructor
* `Fix.rec`    - recursor: basis for defining functions by structural recursion on `Fix F α`
* `Fix.drec`   - dependent recursor: generalization of `Fix.rec` where
                  the result type of the function is allowed to depend on the `Fix F α` value
* `Fix.rec_eq` - defining equation for `recursor`
* `Fix.ind`    - induction principle for `Fix F α`

## Implementation notes

For `F` a `QPF`, we define `Fix F α` in terms of the W-type of the polynomial functor `P` of `F`.
We define the relation `WEquiv` and take its quotient as the definition of `Fix F α`.

See [avigad-carneiro-hudon2019] for more details.

## Reference

* Jeremy Avigad, Mario M. Carneiro and Simon Hudon.
  [*Data Types as Quotients of Polynomial Functors*][avigad-carneiro-hudon2019]
-/

@[expose] public section

universe u v

namespace MvQPF

open TypeVec

open MvFunctor (LiftP LiftR)

open MvFunctor

variable {n : ℕ} {F : TypeVec.{u} (n + 1) → Type u} [q : MvQPF F]

/-- `recF` is used as a basis for defining the recursor on `Fix F α`. `recF`
traverses recursively the W-type generated by `q.P` using a function on `F`
as a recursive step -/
def recF {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) : q.P.W α → β :=
  q.P.wRec fun a f' _f rec => g (abs ⟨a, splitFun f' rec⟩)

theorem recF_eq {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (a : q.P.A)
    (f' : q.P.drop.B a ⟹ α) (f : q.P.last.B a → q.P.W α) :
    recF g (q.P.wMk a f' f) = g (abs ⟨a, splitFun f' (recF g ∘ f)⟩) := by
  rw [recF, MvPFunctor.wRec_eq]; rfl

set_option backward.isDefEq.respectTransparency false in
theorem recF_eq' {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (x : q.P.W α) :
    recF g x = g (abs (appendFun id (recF g) <$$> q.P.wDest' x)) := by
  induction x using q.P.wCases
  case ih a f' f =>
    rw [recF_eq, q.P.wDest'_wMk, MvPFunctor.map_eq, appendFun_comp_splitFun, TypeVec.id_comp]

/-- Equivalence relation on W-types that represent the same `Fix F`
value -/
inductive WEquiv {α : TypeVec n} : q.P.W α → q.P.W α → Prop
  | ind (a : q.P.A) (f' : q.P.drop.B a ⟹ α) (f₀ f₁ : q.P.last.B a → q.P.W α) :
    (∀ x, WEquiv (f₀ x) (f₁ x)) → WEquiv (q.P.wMk a f' f₀) (q.P.wMk a f' f₁)
  | abs (a₀ : q.P.A) (f'₀ : q.P.drop.B a₀ ⟹ α) (f₀ : q.P.last.B a₀ → q.P.W α) (a₁ : q.P.A)
    (f'₁ : q.P.drop.B a₁ ⟹ α) (f₁ : q.P.last.B a₁ → q.P.W α) :
    abs ⟨a₀, q.P.appendContents f'₀ f₀⟩ = abs ⟨a₁, q.P.appendContents f'₁ f₁⟩ →
      WEquiv (q.P.wMk a₀ f'₀ f₀) (q.P.wMk a₁ f'₁ f₁)
  | trans (u v w : q.P.W α) : WEquiv u v → WEquiv v w → WEquiv u w

theorem recF_eq_of_wEquiv (α : TypeVec n) {β : Type u} (u : F (α.append1 β) → β) (x y : q.P.W α) :
    WEquiv x y → recF u x = recF u y := by
  induction x using q.P.wCases
  case ih a₀ f'₀ f₀ =>
    induction y using q.P.wCases
    case ih a₁ f'₁ f₁ =>
      intro h
      -- Porting note: induction on h doesn't work.
      refine @WEquiv.recOn _ _ _ _ (fun a a' _ ↦ recF u a = recF u a') _ _ h ?_ ?_ ?_
      · intro a f' f₀ f₁ _h ih; simp only [recF_eq]
        congr 4; funext; apply ih
      · intro a₀ f'₀ f₀ a₁ f'₁ f₁ h; simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]
      · intro x y z _e₁ _e₂ ih₁ ih₂; exact Eq.trans ih₁ ih₂

Target:
theorem wEquiv.abs' {α : TypeVec n} (x y : q.P.W α)
    (h : MvQPF.abs (q.P.wDest' x) = MvQPF.abs (q.P.wDest' y)) :
    WEquiv x y :=

Proof body:


<LEAN_FILE>:193:43: 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)
⊢ False

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_plan_9960b3dd0a : False := by
    skip
  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": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:193:43: 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⊢ False", "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_plan_9960b3dd0a : False := by
    skip
  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": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:193:43: 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⊢ False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:80:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

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

Response:
by
  refine ⟨fun _ ↦ ⟨F.ringFilterBasis, rfl⟩, fun ⟨B, hB⟩ ↦ ⟨fun {I} hI a ↦ ?_⟩⟩
  obtain ⟨V, hbasis, hsub⟩ := B.mul_right a (U := I) (hB.ge (by simpa))
  obtain ⟨J, hJ, rfl⟩ := hB.le hbasis
  exact Order.PFilter.mem_of_le (fun x hx ↦ by simpa using (hsub hx)) hJ

[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.IdealFilter.Basic
public import Mathlib.Topology.Algebra.LinearTopology
public import Mathlib.Topology.Algebra.FilterBasis

Namespace:
IdealFilter

Local safe context:
/-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
/-!
# Topologies associated to ideal filters

This file constructs topological structures on a ring from an `IdealFilter` and characterizes
uniform ideal filters in terms of ring filter bases.

## Main definitions
* `WithIdealFilter`: Type synonym for a ring that depends on a choice of ideal filter. This can be
  used to assign and infer instances on a ring that depend on an ideal filter.
* `IdealFilter.addGroupFilterBasis`: the `AddGroupFilterBasis` with sets the ideals of `F`.
* `IdealFilter.ringFilterBasis`: under `[F.IsUniform]`, the `RingFilterBasis` with sets the ideals
  of `F`.

## Main statements

* `IdealFilter.isUniform_iff_exists_ringFilterBasis`: An `IdealFilter` on a ring `A` is uniform if
  and only if its ideals form a `RingFilterBasis` for `A`.

## References

* [nLab: Uniform filter](https://ncatlab.org/nlab/show/uniform+filter)

## Tags

ring theory, ideal, filter, linear topology
-/

@[expose] public section

open scoped Pointwise Topology

namespace IdealFilter

/-- The additive-group filter basis whose sets are the ideals belonging to the ideal filter `F`. -/
@[implicit_reducible]
def addGroupFilterBasis {A : Type*} [Ring A] (F : IdealFilter A) : AddGroupFilterBasis A where
  sets := {(I : Set A) | I ∈ F}
  nonempty := ⟨_, ⟨_, F.nonempty.choose_spec, rfl⟩⟩
  inter_sets := by
    rintro s t ⟨I, hI, rfl⟩ ⟨J, hJ, rfl⟩
    exact ⟨I ⊓ J, ⟨I ⊓ J, Order.PFilter.inf_mem hI hJ, rfl⟩, fun _ h ↦ h⟩
  zero' := by aesop
  add' := by aesop
  neg' := by aesop
  conj' := by aesop

/-- Under `[F.IsUniform]`, the ring filter basis obtained from `addGroupFilterBasis`. -/
@[simps! -isSimp sets, implicit_reducible]
def ringFilterBasis {A : Type*} [Ring A] {F : IdealFilter A} [F.IsUniform] :
    RingFilterBasis A where
  __ := F.addGroupFilterBasis
  mul' := by
    rintro U ⟨I, hI, rfl⟩
    exact ⟨I, ⟨I, hI, rfl⟩, Set.mul_subset_iff.mpr fun _ h₁ _ h₂ ↦ mul_mem h₁ h₂⟩
  mul_left' := by
    rintro x₀ U ⟨I, hI, rfl⟩
    exact ⟨I, ⟨I, hI, rfl⟩, fun a ha ↦ Ideal.mul_mem_left I x₀ ha⟩
  mul_right' := by
    rintro x₀ U ⟨I, hI, rfl⟩
    refine ⟨I.colon {x₀}, ⟨I.colon {x₀}, IsUniform.colon_mem hI x₀, rfl⟩,
      fun a ha ↦ Set.mem_preimage.mpr (Submodule.mem_colon_singleton.mp ha)⟩

/-- An `IdealFilter` on a ring `A` is uniform if and only if its ideals form a `RingFilterBasis`
for `A`. -/

Target:
theorem isUniform_iff_exists_ringFilterBasis {A : Type*} [Ring A] {F : IdealFilter A} :
    F.IsUniform ↔ ∃ B : RingFilterBasis A, B.sets = {(I : Set A) | I ∈ F} :=

Failed candidate:
by
  let __repair_437277eba5 : Fact False := inferInstance
  refine ⟨fun _ ↦ ⟨F.ringFilterBasis, rfl⟩, fun ⟨B, hB⟩ ↦ ⟨fun {I} hI a ↦ ?_⟩⟩
  obtain ⟨V, hbasis, hsub⟩ := B.mul_right a (U := I) (hB.ge (by simpa))
  obtain ⟨J, hJ, rfl⟩ := hB.le hbasis
  exact Order.PFilter.mem_of_le (fun x hx ↦ by simpa using (hsub hx)) hJ

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:80:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine ⟨fun _ ↦ ⟨F.ringFilterBasis, rfl⟩, fun ⟨B, hB⟩ ↦ ⟨fun {I} hI a ↦ ?_⟩⟩
  obtain ⟨V, hbasis, hsub⟩ := B.mul_right a (U := I) (hB.ge (by simpa))
  obtain ⟨J, hJ, rfl⟩ := hB.le hbasis
  exact Order.PFilter.mem_of_le (fun x hx ↦ by simpa using (hsub hx)) hJ

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.IdealFilter.Basic
public import Mathlib.Topology.Algebra.LinearTopology
public import Mathlib.Topology.Algebra.FilterBasis

Namespace:
IdealFilter

Local safe context:
/-
Copyright (c) 2025 Blake Farman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Blake Farman
-/
/-!
# Topologies associated to ideal filters

This file constructs topological structures on a ring from an `IdealFilter` and characterizes
uniform ideal filters in terms of ring filter bases.

## Main definitions
* `WithIdealFilter`: Type synonym for a ring that depends on a choice of ideal filter. This can be
  used to assign and infer instances on a ring that depend on an ideal filter.
* `IdealFilter.addGroupFilterBasis`: the `AddGroupFilterBasis` with sets the ideals of `F`.
* `IdealFilter.ringFilterBasis`: under `[F.IsUniform]`, the `RingFilterBasis` with sets the ideals
  of `F`.

## Main statements

* `IdealFilter.isUniform_iff_exists_ringFilterBasis`: An `IdealFilter` on a ring `A` is uniform if
  and only if its ideals form a `RingFilterBasis` for `A`.

## References

* [nLab: Uniform filter](https://ncatlab.org/nlab/show/uniform+filter)

## Tags

ring theory, ideal, filter, linear topology
-/

@[expose] public section

open scoped Pointwise Topology

namespace IdealFilter

/-- The additive-group filter basis whose sets are the ideals belonging to the ideal filter `F`. -/
@[implicit_reducible]
def addGroupFilterBasis {A : Type*} [Ring A] (F : IdealFilter A) : AddGroupFilterBasis A where
  sets := {(I : Set A) | I ∈ F}
  nonempty := ⟨_, ⟨_, F.nonempty.choose_spec, rfl⟩⟩
  inter_sets := by
    rintro s t ⟨I, hI, rfl⟩ ⟨J, hJ, rfl⟩
    exact ⟨I ⊓ J, ⟨I ⊓ J, Order.PFilter.inf_mem hI hJ, rfl⟩, fun _ h ↦ h⟩
  zero' := by aesop
  add' := by aesop
  neg' := by aesop
  conj' := by aesop

/-- Under `[F.IsUniform]`, the ring filter basis obtained from `addGroupFilterBasis`. -/
@[simps! -isSimp sets, implicit_reducible]
def ringFilterBasis {A : Type*} [Ring A] {F : IdealFilter A} [F.IsUniform] :
    RingFilterBasis A where
  __ := F.addGroupFilterBasis
  mul' := by
    rintro U ⟨I, hI, rfl⟩
    exact ⟨I, ⟨I, hI, rfl⟩, Set.mul_subset_iff.mpr fun _ h₁ _ h₂ ↦ mul_mem h₁ h₂⟩
  mul_left' := by
    rintro x₀ U ⟨I, hI, rfl⟩
    exact ⟨I, ⟨I, hI, rfl⟩, fun a ha ↦ Ideal.mul_mem_left I x₀ ha⟩
  mul_right' := by
    rintro x₀ U ⟨I, hI, rfl⟩
    refine ⟨I.colon {x₀}, ⟨I.colon {x₀}, IsUniform.colon_mem hI x₀, rfl⟩,
      fun a ha ↦ Set.mem_preimage.mpr (Submodule.mem_colon_singleton.mp ha)⟩

/-- An `IdealFilter` on a ring `A` is uniform if and only if its ideals form a `RingFilterBasis`
for `A`. -/

Target:
theorem isUniform_iff_exists_ringFilterBasis {A : Type*} [Ring A] {F : IdealFilter A} :
    F.IsUniform ↔ ∃ B : RingFilterBasis A, B.sets = {(I : Set A) | I ∈ F} :=

Failed candidate:
by
  let __repair_437277eba5 : Fact False := inferInstance
  refine ⟨fun _ ↦ ⟨F.ringFilterBasis, rfl⟩, fun ⟨B, hB⟩ ↦ ⟨fun {I} hI a ↦ ?_⟩⟩
  obtain ⟨V, hbasis, hsub⟩ := B.mul_right a (U := I) (hB.ge (by simpa))
  obtain ⟨J, hJ, rfl⟩ := hB.le hbasis
  exact Order.PFilter.mem_of_le (fun x hx ↦ by simpa using (hsub hx)) hJ

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:80:42: 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>:240:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_fcdc84d7d9`
<LEAN_FILE>:215:0: error: unsolved goals
R : Type u_1
inst✝ : Semiring R
motive : SkewPolynomial R → Prop
p : SkewPolynomial R
h0 : motive 0
ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)
⊢ motive p

Response:
by
  apply SkewMonoidAlgebra.induction <;> aesop

[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.Algebra.Defs
public import Mathlib.Algebra.SkewMonoidAlgebra.Single
public import Mathlib.Algebra.SkewMonoidAlgebra.Support

Namespace:
SkewPolynomial

Local safe context:
/-
Copyright (c) 2025 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos-Fernández
-/
/-!
# Univariate skew polynomials

Given a ring `R` and an endomorphism `φ` on `R` the skew polynomials over `R`
are polynomials
$$\sum_{i= 0}^n a_iX^n, n\geq 0, a_i\in R$$
where the addition is the usual addition of polynomials
$$\sum_{i= 0}^n a_iX^n + \sum_{i= 0}^n b_iX^n= \sum_{i= 0}^n (a_i + b_i)X^n.$$
The multiplication, however, is determined by
$$Xa = \varphi (a)X$$
by extending it to all polynomials in the obvious way.

Skew polynomials are represented as `SkewMonoidAlgebra R (Multiplicative ℕ)`,
where `R` is usually at least a Semiring. In this file, we define `SkewPolynomial`
and provide basic instances.

**Note**: To register the endomorphism `φ` see notation below.

## Notation

The endomorphism `φ` is implemented using some action of `Multiplicative ℕ` on `R`.
From this action, `φ` is an `abbrev` denoting $(\text{ofAdd } 1) \cdot a := \varphi(a)$.

Users that want to work with a specific map `φ` should introduce an action of
`Multiplicative ℕ` on `R`. Specifying that this action is a `MulSemiringAction` amounts
to saying that `φ` is an endomorphism.

Furthermore, with this notation `φ^[n](a) = (ofAdd n) • a`, see `φ_iterate_apply`.

## Main definitions

* `SkewPolynomial.monomial n a` is the skew polynomial `a X ^ n`. Note that
  `SkewPolynomial.monomial n` is defined as an `R`-linear map.
* `SkewPolynomial.C a` is the constant skew polynomial `a`. Note that `C` is defined as an additive
   homomorphism.
* `SkewPolynomial.CRingHom a` is the constant skew polynomial `a`, as a ring homomorphism. This
  requires to assume `[MulSemiringAction (Multiplicative ℕ) R]`.
* `SkewPolynomial.X` is the skew polynomial `X`, i.e., `SkewPolynomial.monomial 1 1`.
* `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`.
* `SkewPolynomial.coeff p n` is the coefficient of `X ^ n` in `p`.
* `SkewPolynomial.erase p n` is the skew polynomial `p` in which one removes the monomial in
  degree `n`.
* `SkewPolynomial.update p n a` is the skew polynomial obtained by replacing the coefficient of
  degree `n` by a given value `a : R`.  If `a = 0`, this is equal to `p.erase n` If
  `p.natDegree < n` and `a ≠ 0`, this increases the degree of `p` to `n`.

## Implementation notes

The implementation uses `Multiplicative ℕ` instead of `ℕ`, since Mathlib does not contain an
additive version of `SkewMonoidAlgebra`.

This decision was made because we use the type class `MulSemiringAction` to specify the properties
the action needs to respect for associativity. There is no version of this in Mathlib that
uses an acting `AddMonoid M` and so we need to use `Multiplicative ℕ` for the action.

For associativity to hold, there should be an instance of
`MulSemiringAction (Multiplicative ℕ) R` present in the context.
For example, in the context of $\mathbb{F}_q$-linear polynomials, this can be the
$q$-th Frobenius endomorphism - so $\varphi(a) = a^q$.

## Reference

The definition is inspired by Chapter 3 of [Papikian2023].

## Tags

Skew Polynomials, Twisted Polynomials.

Note that [ore33] proposes a more general definition of skew polynomial ring, where the
multiplication is determined by  $Xa = \varphi (a)X + δ (a)$, where `φ` is as above and
`δ` is a derivation.

-/

@[expose] public section

noncomputable section

open Function Multiplicative SkewMonoidAlgebra

/-- The skew polynomials over `R` is the type of univariate polynomials over `R`
endowed with a skewed convolution product. -/
abbrev SkewPolynomial (R : Type*) [AddCommMonoid R] := SkewMonoidAlgebra R (Multiplicative ℕ)

namespace SkewPolynomial

variable {R : Type*} {m n : ℕ}

section Semiring

variable [Semiring R] {p q : SkewPolynomial R}

lemma zero_def : (0 : SkewPolynomial R) = (0 : SkewMonoidAlgebra R (Multiplicative ℕ)) := rfl

variable {S S₁ S₂ : Type*}

/--
The set of all `n` such that `X^n` has a non-zero coefficient.
-/
def support (p : SkewPolynomial R) : Finset ℕ :=
  Finset.map ⟨toAdd, toAdd.injective⟩ (SkewMonoidAlgebra.support p)

/-- Though `SkewPolynomial.support` is not definiyionally equal to `SkewMonoidAlgebra.support` we
  can relate them using the following lemma. -/
lemma support_eq_skewMonoidAlgebra_support (p : SkewPolynomial R) :
    p.support = Finset.map (Multiplicative.toAdd (α := ℕ)) (SkewMonoidAlgebra.support p) := by
  simp only [support]

@[simp] lemma support_zero : (0 : SkewPolynomial R).support = ∅ := rfl

@[simp] lemma support_eq_empty : p.support = ∅ ↔ p = 0 := by simp [support]

lemma card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by simp

lemma support_add : (p + q).support ⊆ p.support ∪ q.support := by
  simpa [support, ← Finset.map_union, Finset.map_subset_map] using SkewMonoidAlgebra.support_add

/-- `coeff p n` is the coefficient of `X ^ n` in `p`. -/
def coeff (p : SkewPolynomial R) : ℕ → R := fun n ↦ (SkewMonoidAlgebra.coeff p (ofAdd n))

@[simp]
lemma mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 := by
  simp [support, coeff]

lemma notMem_support_iff : n ∉ p.support ↔ p.coeff n = 0 := by simp

/-- `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`. -/
def sum {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) : S :=
  SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r)

/-- For a skew polynomial `p`, `p.sum f` can be written in terms of `SkewMonoidAlgebra.sum p`. -/
lemma sum_def' {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r) := rfl

lemma sum_def {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = ∑ n ∈ p.support, f n (p.coeff n) := by
  simp only [sum_def', SkewMonoidAlgebra.sum_def, Finsupp.sum, toFinsupp_apply]
  apply Finset.sum_of_injOn (toAdd) (Injective.injOn fun ⦃a₁ a₂⦄ a ↦ a) (fun _ ↦ ?_) <;>
  simp +contextual [coeff]

lemma sum_sum_index {R' P : Type*} [AddCommMonoid P] [Semiring R']
    {f : SkewPolynomial R} {g : ℕ → R → SkewPolynomial R'} {h : ℕ → R' → P}
    (h_zero : ∀ (a : ℕ), h a 0 = 0)
    (h_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂) :
    sum (sum f g) h = sum f fun (a : ℕ) (b : R) ↦ sum (g a b) h := by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

@[simp]
lemma sum_zero {N : Type*} [AddCommMonoid N] {f : SkewPolynomial R} :
    (f.sum fun (_ : ℕ) _ ↦ (0 : N)) = 0 :=
  SkewMonoidAlgebra.sum_zero

section Monomial

variable (n)

/-- `monomial s a` is the monomial `a * X ^ s`. -/
def monomial : R →ₗ[R] SkewPolynomial R := lsingle R (ofAdd n)

lemma monomial_zero_right : monomial n (0 : R) = 0 := single_zero _

lemma monomial_zero_one [MulSemiringAction (Multiplicative ℕ) R] : monomial 0 (1 : R) = 1 :=
  rfl

lemma monomial_def (a : R) : monomial n a = single (ofAdd n) a := rfl

lemma monomial_add (r s : R) : monomial n (r + s) = monomial n r + monomial n s :=
  single_add ..

lemma smul_monomial {S} [Semiring S] [Module S R] (a : S) (b : R) :
    a • monomial n b = monomial n (a • b) :=
  smul_single ..

@[simp]
lemma sum_monomial (f : SkewPolynomial R) : f.sum (fun (a : ℕ) ↦ monomial a) = f :=
 SkewMonoidAlgebra.sum_single _

@[simp]
lemma sum_monomial_index {N} [AddCommMonoid N] {n : ℕ} {b : R} {h : ℕ → R → N}
    (h_zero : h n 0 = 0) : (monomial n b).sum h = h n b :=
  SkewMonoidAlgebra.sum_single_index h_zero

lemma monomial_injective : Function.Injective (monomial n : R → SkewPolynomial R) :=
  single_injective (ofAdd n)

@[simp]
lemma monomial_eq_zero_iff (t : R) : monomial n t = 0 ↔ t = 0 :=
  LinearMap.map_eq_zero_iff _ (SkewPolynomial.monomial_injective n)

lemma monomial_eq_monomial_iff {m n : ℕ} {a b : R} :
    monomial m a = monomial n b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0 := by
  rw [← Finsupp.single_eq_single_iff m n a b]
  simp only [monomial_def, ← toFinsupp_single, toFinsupp_inj]
  simp only [← ofFinsupp_single, SkewMonoidAlgebra.ofFinsupp_inj, Finsupp.single_eq_single_iff,
    EmbeddingLike.apply_eq_iff_eq]

Target:
lemma induction {motive : SkewPolynomial R → Prop} (p : SkewPolynomial R) (h0 : motive 0)
  (ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q →
    motive (SkewPolynomial.monomial n r + q)) : motive p :=

Failed candidate:
by
  have __repair_pad_0_fcdc84d7d9 : True := by trivial
  have __repair_pad_1_fcdc84d7d9 : True := by trivial
  have __repair_pad_2_fcdc84d7d9 : True := by trivial
  have __repair_pad_3_fcdc84d7d9 : True := by trivial
  have __repair_pad_4_fcdc84d7d9 : True := by trivial
  have __repair_pad_5_fcdc84d7d9 : True := by trivial
  have __repair_pad_6_fcdc84d7d9 : True := by trivial
  have __repair_pad_7_fcdc84d7d9 : True := by trivial
  have __repair_pad_8_fcdc84d7d9 : True := by trivial
  have __repair_pad_9_fcdc84d7d9 : True := by trivial
  have __repair_pad_10_fcdc84d7d9 : True := by trivial
  have __repair_pad_11_fcdc84d7d9 : True := by trivial
  have __repair_pad_12_fcdc84d7d9 : True := by trivial
  have __repair_pad_13_fcdc84d7d9 : True := by trivial
  have __repair_pad_14_fcdc84d7d9 : True := by trivial
  have __repair_pad_15_fcdc84d7d9 : True := by trivial
  have __repair_pad_16_fcdc84d7d9 : True := by trivial
  have __repair_pad_17_fcdc84d7d9 : True := by trivial
  have __repair_pad_18_fcdc84d7d9 : True := by trivial
  have __repair_pad_19_fcdc84d7d9 : True := by trivial
  have __repair_pad_20_fcdc84d7d9 : True := by trivial
  have __repair_pad_21_fcdc84d7d9 : True := by trivial
  have __repair_pad_22_fcdc84d7d9 : True := by trivial
  have __repair_pad_23_fcdc84d7d9 : True := by trivial
  have __repair_fail_fcdc84d7d9 := __oprover_overlong_fcdc84d7d9
  apply SkewMonoidAlgebra.induction <;> aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:240:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_fcdc84d7d9`\n<LEAN_FILE>:215:0: error: unsolved goals\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ motive p", "returncode": 1}

Corrected proof:

[Assistant]
by
  apply SkewMonoidAlgebra.induction <;> aesop

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.Algebra.Defs
public import Mathlib.Algebra.SkewMonoidAlgebra.Single
public import Mathlib.Algebra.SkewMonoidAlgebra.Support

Namespace:
SkewPolynomial

Local safe context:
/-
Copyright (c) 2025 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, María Inés de Frutos-Fernández
-/
/-!
# Univariate skew polynomials

Given a ring `R` and an endomorphism `φ` on `R` the skew polynomials over `R`
are polynomials
$$\sum_{i= 0}^n a_iX^n, n\geq 0, a_i\in R$$
where the addition is the usual addition of polynomials
$$\sum_{i= 0}^n a_iX^n + \sum_{i= 0}^n b_iX^n= \sum_{i= 0}^n (a_i + b_i)X^n.$$
The multiplication, however, is determined by
$$Xa = \varphi (a)X$$
by extending it to all polynomials in the obvious way.

Skew polynomials are represented as `SkewMonoidAlgebra R (Multiplicative ℕ)`,
where `R` is usually at least a Semiring. In this file, we define `SkewPolynomial`
and provide basic instances.

**Note**: To register the endomorphism `φ` see notation below.

## Notation

The endomorphism `φ` is implemented using some action of `Multiplicative ℕ` on `R`.
From this action, `φ` is an `abbrev` denoting $(\text{ofAdd } 1) \cdot a := \varphi(a)$.

Users that want to work with a specific map `φ` should introduce an action of
`Multiplicative ℕ` on `R`. Specifying that this action is a `MulSemiringAction` amounts
to saying that `φ` is an endomorphism.

Furthermore, with this notation `φ^[n](a) = (ofAdd n) • a`, see `φ_iterate_apply`.

## Main definitions

* `SkewPolynomial.monomial n a` is the skew polynomial `a X ^ n`. Note that
  `SkewPolynomial.monomial n` is defined as an `R`-linear map.
* `SkewPolynomial.C a` is the constant skew polynomial `a`. Note that `C` is defined as an additive
   homomorphism.
* `SkewPolynomial.CRingHom a` is the constant skew polynomial `a`, as a ring homomorphism. This
  requires to assume `[MulSemiringAction (Multiplicative ℕ) R]`.
* `SkewPolynomial.X` is the skew polynomial `X`, i.e., `SkewPolynomial.monomial 1 1`.
* `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`.
* `SkewPolynomial.coeff p n` is the coefficient of `X ^ n` in `p`.
* `SkewPolynomial.erase p n` is the skew polynomial `p` in which one removes the monomial in
  degree `n`.
* `SkewPolynomial.update p n a` is the skew polynomial obtained by replacing the coefficient of
  degree `n` by a given value `a : R`.  If `a = 0`, this is equal to `p.erase n` If
  `p.natDegree < n` and `a ≠ 0`, this increases the degree of `p` to `n`.

## Implementation notes

The implementation uses `Multiplicative ℕ` instead of `ℕ`, since Mathlib does not contain an
additive version of `SkewMonoidAlgebra`.

This decision was made because we use the type class `MulSemiringAction` to specify the properties
the action needs to respect for associativity. There is no version of this in Mathlib that
uses an acting `AddMonoid M` and so we need to use `Multiplicative ℕ` for the action.

For associativity to hold, there should be an instance of
`MulSemiringAction (Multiplicative ℕ) R` present in the context.
For example, in the context of $\mathbb{F}_q$-linear polynomials, this can be the
$q$-th Frobenius endomorphism - so $\varphi(a) = a^q$.

## Reference

The definition is inspired by Chapter 3 of [Papikian2023].

## Tags

Skew Polynomials, Twisted Polynomials.

Note that [ore33] proposes a more general definition of skew polynomial ring, where the
multiplication is determined by  $Xa = \varphi (a)X + δ (a)$, where `φ` is as above and
`δ` is a derivation.

-/

@[expose] public section

noncomputable section

open Function Multiplicative SkewMonoidAlgebra

/-- The skew polynomials over `R` is the type of univariate polynomials over `R`
endowed with a skewed convolution product. -/
abbrev SkewPolynomial (R : Type*) [AddCommMonoid R] := SkewMonoidAlgebra R (Multiplicative ℕ)

namespace SkewPolynomial

variable {R : Type*} {m n : ℕ}

section Semiring

variable [Semiring R] {p q : SkewPolynomial R}

lemma zero_def : (0 : SkewPolynomial R) = (0 : SkewMonoidAlgebra R (Multiplicative ℕ)) := rfl

variable {S S₁ S₂ : Type*}

/--
The set of all `n` such that `X^n` has a non-zero coefficient.
-/
def support (p : SkewPolynomial R) : Finset ℕ :=
  Finset.map ⟨toAdd, toAdd.injective⟩ (SkewMonoidAlgebra.support p)

/-- Though `SkewPolynomial.support` is not definiyionally equal to `SkewMonoidAlgebra.support` we
  can relate them using the following lemma. -/
lemma support_eq_skewMonoidAlgebra_support (p : SkewPolynomial R) :
    p.support = Finset.map (Multiplicative.toAdd (α := ℕ)) (SkewMonoidAlgebra.support p) := by
  simp only [support]

@[simp] lemma support_zero : (0 : SkewPolynomial R).support = ∅ := rfl

@[simp] lemma support_eq_empty : p.support = ∅ ↔ p = 0 := by simp [support]

lemma card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by simp

lemma support_add : (p + q).support ⊆ p.support ∪ q.support := by
  simpa [support, ← Finset.map_union, Finset.map_subset_map] using SkewMonoidAlgebra.support_add

/-- `coeff p n` is the coefficient of `X ^ n` in `p`. -/
def coeff (p : SkewPolynomial R) : ℕ → R := fun n ↦ (SkewMonoidAlgebra.coeff p (ofAdd n))

@[simp]
lemma mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 := by
  simp [support, coeff]

lemma notMem_support_iff : n ∉ p.support ↔ p.coeff n = 0 := by simp

/-- `p.sum f` is `∑ n ∈ p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`. -/
def sum {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) : S :=
  SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r)

/-- For a skew polynomial `p`, `p.sum f` can be written in terms of `SkewMonoidAlgebra.sum p`. -/
lemma sum_def' {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = SkewMonoidAlgebra.sum p (fun n r ↦ f (toAdd n : ℕ) r) := rfl

lemma sum_def {S : Type*} [AddCommMonoid S] (p : SkewPolynomial R) (f : ℕ → R → S) :
    p.sum f = ∑ n ∈ p.support, f n (p.coeff n) := by
  simp only [sum_def', SkewMonoidAlgebra.sum_def, Finsupp.sum, toFinsupp_apply]
  apply Finset.sum_of_injOn (toAdd) (Injective.injOn fun ⦃a₁ a₂⦄ a ↦ a) (fun _ ↦ ?_) <;>
  simp +contextual [coeff]

lemma sum_sum_index {R' P : Type*} [AddCommMonoid P] [Semiring R']
    {f : SkewPolynomial R} {g : ℕ → R → SkewPolynomial R'} {h : ℕ → R' → P}
    (h_zero : ∀ (a : ℕ), h a 0 = 0)
    (h_add : ∀ (a : ℕ) (b₁ b₂ : R'), h a (b₁ + b₂) = h a b₁ + h a b₂) :
    sum (sum f g) h = sum f fun (a : ℕ) (b : R) ↦ sum (g a b) h := by
  simp only [sum_def', SkewMonoidAlgebra.sum_sum_index (fun a ↦ h_zero (toAdd a))
    (fun a ↦ h_add (toAdd a))]

@[simp]
lemma sum_zero {N : Type*} [AddCommMonoid N] {f : SkewPolynomial R} :
    (f.sum fun (_ : ℕ) _ ↦ (0 : N)) = 0 :=
  SkewMonoidAlgebra.sum_zero

section Monomial

variable (n)

/-- `monomial s a` is the monomial `a * X ^ s`. -/
def monomial : R →ₗ[R] SkewPolynomial R := lsingle R (ofAdd n)

lemma monomial_zero_right : monomial n (0 : R) = 0 := single_zero _

lemma monomial_zero_one [MulSemiringAction (Multiplicative ℕ) R] : monomial 0 (1 : R) = 1 :=
  rfl

lemma monomial_def (a : R) : monomial n a = single (ofAdd n) a := rfl

lemma monomial_add (r s : R) : monomial n (r + s) = monomial n r + monomial n s :=
  single_add ..

lemma smul_monomial {S} [Semiring S] [Module S R] (a : S) (b : R) :
    a • monomial n b = monomial n (a • b) :=
  smul_single ..

@[simp]
lemma sum_monomial (f : SkewPolynomial R) : f.sum (fun (a : ℕ) ↦ monomial a) = f :=
 SkewMonoidAlgebra.sum_single _

@[simp]
lemma sum_monomial_index {N} [AddCommMonoid N] {n : ℕ} {b : R} {h : ℕ → R → N}
    (h_zero : h n 0 = 0) : (monomial n b).sum h = h n b :=
  SkewMonoidAlgebra.sum_single_index h_zero

lemma monomial_injective : Function.Injective (monomial n : R → SkewPolynomial R) :=
  single_injective (ofAdd n)

@[simp]
lemma monomial_eq_zero_iff (t : R) : monomial n t = 0 ↔ t = 0 :=
  LinearMap.map_eq_zero_iff _ (SkewPolynomial.monomial_injective n)

lemma monomial_eq_monomial_iff {m n : ℕ} {a b : R} :
    monomial m a = monomial n b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0 := by
  rw [← Finsupp.single_eq_single_iff m n a b]
  simp only [monomial_def, ← toFinsupp_single, toFinsupp_inj]
  simp only [← ofFinsupp_single, SkewMonoidAlgebra.ofFinsupp_inj, Finsupp.single_eq_single_iff,
    EmbeddingLike.apply_eq_iff_eq]

Target:
lemma induction {motive : SkewPolynomial R → Prop} (p : SkewPolynomial R) (h0 : motive 0)
  (ha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q →
    motive (SkewPolynomial.monomial n r + q)) : motive p :=

Failed candidate:
by
  have __repair_pad_0_fcdc84d7d9 : True := by trivial
  have __repair_pad_1_fcdc84d7d9 : True := by trivial
  have __repair_pad_2_fcdc84d7d9 : True := by trivial
  have __repair_pad_3_fcdc84d7d9 : True := by trivial
  have __repair_pad_4_fcdc84d7d9 : True := by trivial
  have __repair_pad_5_fcdc84d7d9 : True := by trivial
  have __repair_pad_6_fcdc84d7d9 : True := by trivial
  have __repair_pad_7_fcdc84d7d9 : True := by trivial
  have __repair_pad_8_fcdc84d7d9 : True := by trivial
  have __repair_pad_9_fcdc84d7d9 : True := by trivial
  have __repair_pad_10_fcdc84d7d9 : True := by trivial
  have __repair_pad_11_fcdc84d7d9 : True := by trivial
  have __repair_pad_12_fcdc84d7d9 : True := by trivial
  have __repair_pad_13_fcdc84d7d9 : True := by trivial
  have __repair_pad_14_fcdc84d7d9 : True := by trivial
  have __repair_pad_15_fcdc84d7d9 : True := by trivial
  have __repair_pad_16_fcdc84d7d9 : True := by trivial
  have __repair_pad_17_fcdc84d7d9 : True := by trivial
  have __repair_pad_18_fcdc84d7d9 : True := by trivial
  have __repair_pad_19_fcdc84d7d9 : True := by trivial
  have __repair_pad_20_fcdc84d7d9 : True := by trivial
  have __repair_pad_21_fcdc84d7d9 : True := by trivial
  have __repair_pad_22_fcdc84d7d9 : True := by trivial
  have __repair_pad_23_fcdc84d7d9 : True := by trivial
  have __repair_fail_fcdc84d7d9 := __oprover_overlong_fcdc84d7d9
  apply SkewMonoidAlgebra.induction <;> aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:240:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_fcdc84d7d9`\n<LEAN_FILE>:215:0: error: unsolved goals\nR : Type u_1\ninst✝ : Semiring R\nmotive : SkewPolynomial R → Prop\np : SkewPolynomial R\nh0 : motive 0\nha : ∀ (n : ℕ) (r : R) (q : SkewPolynomial R), n ∉ q.support → r ≠ 0 → motive q → motive ((monomial n) r + q)\n⊢ motive p", "returncode": 1}

Corrected proof:


<LEAN_FILE>:202:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 11096365) 0
in the target expression
  SeparableSpace ↑s

X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
⊢ SeparableSpace ↑s

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

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

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

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11096365) 0) = ((fun x : Nat => x + 11096365) 0) by rfl]
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:202:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11096365) 0\nin the target expression\n  SeparableSpace ↑s\n\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof:

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

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

Imports:
public import Mathlib.Topology.UniformSpace.Pi

Namespace:
TopologicalSpace

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

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

-- don't import the real numbers
assert_not_exists AddMonoidWithOne

public section

open Filter Set Topology Uniformity UniformSpace SetRel

namespace TopologicalSpace

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11096365) 0) = ((fun x : Nat => x + 11096365) 0) by rfl]
  rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
  lift t to Set s using hts
  refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
  rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:202:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11096365) 0\nin the target expression\n  SeparableSpace ↑s\n\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s", "returncode": 1}

Corrected proof: