<LEAN_FILE>:113:2: warning: declaration uses `sorry`
<LEAN_FILE>:118:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_notMem]
  on_goal 4 => simpa using mul_notMem_of_exponent_two (by simp) hx hy hxy
  all_goals simp_all

[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.GroupTheory.SpecificGroups.Cyclic

Namespace:
IsKleinFour

Local safe context:
/-
Copyright (c) 2023 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen
-/
/-!
# Klein Four Group

The Klein (Vierergruppe) four-group is a non-cyclic abelian group with four elements, in which
each element is self-inverse and in which composing any two of the three non-identity elements
produces the third one.

## Main definitions

* `IsKleinFour` : A mixin class which states that the group has order four and exponent two.
* `mulEquiv'` : An equivalence between a Klein four-group and a group of exponent two which
  preserves the identity is in fact an isomorphism.
* `mulEquiv`: Any two Klein four-groups are isomorphic via any identity-preserving equivalence.

## References

* https://en.wikipedia.org/wiki/Klein_four-group
* https://en.wikipedia.org/wiki/Alternating_group

## TODO

* Prove an `IsKleinFour` group is isomorphic to the normal subgroup of `alternatingGroup (Fin 4)`
  with the permutation cycles `V = {(), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}`.  This is the kernel
  of the surjection of `alternatingGroup (Fin 4)` onto `alternatingGroup (Fin 3) ≃ (ZMod 3)`.
  In other words, we have the exact sequence `V → A₄ → A₃`.

* The outer automorphism group of `A₆` is the Klein four-group `V = (ZMod 2) × (ZMod 2)`,
  and is related to the outer automorphism of `S₆`. The extra outer automorphism in `A₆`
  swaps the 3-cycles (like `(1 2 3)`) with elements of shape `3²` (like `(1 2 3)(4 5 6)`).

## Tags
non-cyclic abelian group
-/

@[expose] public section

/-! ### Klein four-groups as a mixin class -/

/-- An (additive) Klein four-group is an (additive) group of cardinality four and exponent two. -/
class IsAddKleinFour (G : Type*) [AddGroup G] : Prop where
  card_four : Nat.card G = 4
  exponent_two : AddMonoid.exponent G = 2

/-- A Klein four-group is a group of cardinality four and exponent two. -/
@[to_additive existing IsAddKleinFour]
class IsKleinFour (G : Type*) [Group G] : Prop where
  card_four : Nat.card G = 4
  exponent_two : Monoid.exponent G = 2

attribute [simp] IsKleinFour.card_four IsKleinFour.exponent_two
  IsAddKleinFour.card_four IsAddKleinFour.exponent_two

instance : IsAddKleinFour (ZMod 2 × ZMod 2) where
  card_four := by simp
  exponent_two := by simp [AddMonoid.exponent_prod]

instance {G : Type*} [Group G] [IsKleinFour G] : IsAddKleinFour (Additive G) where
  card_four := by rw [← IsKleinFour.card_four (G := G)]; congr!
  exponent_two := by simp

instance {G : Type*} [AddGroup G] [IsAddKleinFour G] : IsKleinFour (Multiplicative G) where
  card_four := by rw [← IsAddKleinFour.card_four (G := G)]; congr!
  exponent_two := by simp

namespace IsKleinFour

@[to_additive]
theorem isMulCommutative {G : Type*} [Group G] [IsKleinFour G] :
    IsMulCommutative G where
  is_comm.comm := mul_comm_of_exponent_two exponent_two

/-- This instance is scoped, because it always applies (which makes linting and typeclass inference
potentially *a lot* slower). -/
@[to_additive]
scoped instance instFinite {G : Type*} [Group G] [IsKleinFour G] : Finite G :=
  Nat.finite_of_card_ne_zero <| by simp [IsKleinFour.card_four]

@[to_additive (attr := simp)]
lemma card_four' {G : Type*} [Group G] [Fintype G] [IsKleinFour G] :
    Fintype.card G = 4 :=
  Nat.card_eq_fintype_card (α := G).symm ▸ IsKleinFour.card_four

open Finset

variable {G : Type*} [Group G] [IsKleinFour G]

@[to_additive]
lemma not_isCyclic : ¬IsCyclic G :=
  fun h ↦ by simpa using h.exponent_eq_card

@[to_additive]
lemma inv_eq_self (x : G) : x⁻¹ = x := inv_eq_self_of_exponent_two (by simp) x

/- this is not an appropriate global `simp` lemma for a `Prop`-mixin class. Indeed, if it were
then every time Lean sees `·⁻¹` it would try to apply `inv_eq_self` which would trigger
type class inference to try and synthesize an `IsKleinFour` instance. -/
scoped[IsKleinFour] attribute [simp] inv_eq_self
scoped[IsAddKleinFour] attribute [simp] neg_eq_self

@[to_additive]
lemma mul_self (x : G) : x * x = 1 := by
  rw [mul_eq_one_iff_eq_inv, inv_eq_self]

@[to_additive]

Target:
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ :=

Failed candidate:
by
  have __repair_25f5553809 : False := True.intro
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_notMem]
  on_goal 4 => simpa using mul_notMem_of_exponent_two (by simp) hx hy hxy
  all_goals simp_all

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:113:2: warning: declaration uses `sorry`\n<LEAN_FILE>:118:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_notMem]
  on_goal 4 => simpa using mul_notMem_of_exponent_two (by simp) hx hy hxy
  all_goals simp_all

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.GroupTheory.SpecificGroups.Cyclic

Namespace:
IsKleinFour

Local safe context:
/-
Copyright (c) 2023 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen
-/
/-!
# Klein Four Group

The Klein (Vierergruppe) four-group is a non-cyclic abelian group with four elements, in which
each element is self-inverse and in which composing any two of the three non-identity elements
produces the third one.

## Main definitions

* `IsKleinFour` : A mixin class which states that the group has order four and exponent two.
* `mulEquiv'` : An equivalence between a Klein four-group and a group of exponent two which
  preserves the identity is in fact an isomorphism.
* `mulEquiv`: Any two Klein four-groups are isomorphic via any identity-preserving equivalence.

## References

* https://en.wikipedia.org/wiki/Klein_four-group
* https://en.wikipedia.org/wiki/Alternating_group

## TODO

* Prove an `IsKleinFour` group is isomorphic to the normal subgroup of `alternatingGroup (Fin 4)`
  with the permutation cycles `V = {(), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}`.  This is the kernel
  of the surjection of `alternatingGroup (Fin 4)` onto `alternatingGroup (Fin 3) ≃ (ZMod 3)`.
  In other words, we have the exact sequence `V → A₄ → A₃`.

* The outer automorphism group of `A₆` is the Klein four-group `V = (ZMod 2) × (ZMod 2)`,
  and is related to the outer automorphism of `S₆`. The extra outer automorphism in `A₆`
  swaps the 3-cycles (like `(1 2 3)`) with elements of shape `3²` (like `(1 2 3)(4 5 6)`).

## Tags
non-cyclic abelian group
-/

@[expose] public section

/-! ### Klein four-groups as a mixin class -/

/-- An (additive) Klein four-group is an (additive) group of cardinality four and exponent two. -/
class IsAddKleinFour (G : Type*) [AddGroup G] : Prop where
  card_four : Nat.card G = 4
  exponent_two : AddMonoid.exponent G = 2

/-- A Klein four-group is a group of cardinality four and exponent two. -/
@[to_additive existing IsAddKleinFour]
class IsKleinFour (G : Type*) [Group G] : Prop where
  card_four : Nat.card G = 4
  exponent_two : Monoid.exponent G = 2

attribute [simp] IsKleinFour.card_four IsKleinFour.exponent_two
  IsAddKleinFour.card_four IsAddKleinFour.exponent_two

instance : IsAddKleinFour (ZMod 2 × ZMod 2) where
  card_four := by simp
  exponent_two := by simp [AddMonoid.exponent_prod]

instance {G : Type*} [Group G] [IsKleinFour G] : IsAddKleinFour (Additive G) where
  card_four := by rw [← IsKleinFour.card_four (G := G)]; congr!
  exponent_two := by simp

instance {G : Type*} [AddGroup G] [IsAddKleinFour G] : IsKleinFour (Multiplicative G) where
  card_four := by rw [← IsAddKleinFour.card_four (G := G)]; congr!
  exponent_two := by simp

namespace IsKleinFour

@[to_additive]
theorem isMulCommutative {G : Type*} [Group G] [IsKleinFour G] :
    IsMulCommutative G where
  is_comm.comm := mul_comm_of_exponent_two exponent_two

/-- This instance is scoped, because it always applies (which makes linting and typeclass inference
potentially *a lot* slower). -/
@[to_additive]
scoped instance instFinite {G : Type*} [Group G] [IsKleinFour G] : Finite G :=
  Nat.finite_of_card_ne_zero <| by simp [IsKleinFour.card_four]

@[to_additive (attr := simp)]
lemma card_four' {G : Type*} [Group G] [Fintype G] [IsKleinFour G] :
    Fintype.card G = 4 :=
  Nat.card_eq_fintype_card (α := G).symm ▸ IsKleinFour.card_four

open Finset

variable {G : Type*} [Group G] [IsKleinFour G]

@[to_additive]
lemma not_isCyclic : ¬IsCyclic G :=
  fun h ↦ by simpa using h.exponent_eq_card

@[to_additive]
lemma inv_eq_self (x : G) : x⁻¹ = x := inv_eq_self_of_exponent_two (by simp) x

/- this is not an appropriate global `simp` lemma for a `Prop`-mixin class. Indeed, if it were
then every time Lean sees `·⁻¹` it would try to apply `inv_eq_self` which would trigger
type class inference to try and synthesize an `IsKleinFour` instance. -/
scoped[IsKleinFour] attribute [simp] inv_eq_self
scoped[IsAddKleinFour] attribute [simp] neg_eq_self

@[to_additive]
lemma mul_self (x : G) : x * x = 1 := by
  rw [mul_eq_one_iff_eq_inv, inv_eq_self]

@[to_additive]

Target:
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ :=

Failed candidate:
by
  have __repair_25f5553809 : False := True.intro
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_notMem]
  on_goal 4 => simpa using mul_notMem_of_exponent_two (by simp) hx hy hxy
  all_goals simp_all

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:113:2: warning: declaration uses `sorry`\n<LEAN_FILE>:118:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


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

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

Namespace:
DividedPowers

Local 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)) :=

Proof body:

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

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

Namespace:
DividedPowers

Local 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)) :=

Proof body:


Response:
by
  rw [range_eq_top, ker_eq_bot, injective_iff_surjective_of_finrank_eq_finrank H]

/-- Given a linear map `f` between two vector spaces with the same dimension, if
`ker f = ⊥` then `linearEquivOfInjective` is the induced isomorphism
between the two vector spaces. -/
noncomputable def linearEquivOfInjective [FiniteDimensional K V] [FiniteDimensional K V₂]
    (f : V →ₗ[K] V₂) (hf : Injective f) (hdim : finrank K V = finrank K V₂) : V ≃ₗ[K] V₂ :=
  LinearEquiv.ofBijective f
    ⟨hf, (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hdim).mp hf⟩

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

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

Namespace:
LinearMap

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

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

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

@[expose] public section

assert_not_exists Monoid.exponent Module.IsTorsion

universe u v v'

open Cardinal Submodule Module Function

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

namespace Submodule

open IsNoetherian Module

section DivisionRing

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

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

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

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

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

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

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

theorem isCompl_iff_disjoint [FiniteDimensional K V] (s t : Submodule K V)
    (hdim : finrank K V ≤ finrank K s + finrank K t) :
    IsCompl s t ↔ Disjoint s t :=
  ⟨fun h ↦ h.1, fun h ↦ ⟨h, codisjoint_iff.mpr <| eq_top_of_disjoint s t hdim h⟩⟩

theorem sup_span_singleton_eq_top_iff [Module.Finite K V] {W : Submodule K V} {v : V} (hv : v ∉ W) :
    W ⊔ span K {v} = ⊤ ↔ finrank K (V ⧸ W) = 1 := by
  refine ⟨fun hW ↦ ?_, fun hW ↦ ?_⟩
  · suffices W ⊓ span K {v} = ⊥ by
      have hv₀ : v ≠ 0 := by aesop
      have aux := finrank_sup_add_finrank_inf_eq W (span K {v})
      rw [hW, finrank_span_singleton hv₀, this, finrank_bot, finrank_top,
        ← finrank_quotient_add_finrank W] at aux
      lia
    refine (Submodule.eq_bot_iff _).mpr fun w hw ↦ ?_
    obtain ⟨ht, t, rfl⟩ : w ∈ W ∧ ∃ t : K, t • v = w := by simpa [mem_span_singleton] using hw
    rcases eq_or_ne t 0 with rfl | ht₀; · simp
    rw [Submodule.smul_mem_iff _ ht₀] at ht
    contradiction
  · apply Submodule.eq_top_of_disjoint
    · rw [← W.finrank_quotient_add_finrank, add_comm, add_le_add_iff_left, hW]
      aesop
    · exact Submodule.disjoint_span_singleton_of_notMem hv

theorem finrank_sup_span_singleton [Module.Finite K V] {p : Submodule K V} {v : V} (hv : v ∉ p) :
    finrank K (p ⊔ Submodule.span K {v} : Submodule K V) = finrank K p + 1 := by
  rw [← Nat.add_left_inj, finrank_sup_add_finrank_inf_eq, add_assoc,
    Nat.add_left_cancel_iff, finrank_span_singleton (by aesop),
    Nat.left_eq_add, Submodule.finrank_eq_zero, eq_bot_iff]
  intro x
  simp only [mem_inf, mem_span_singleton]
  rintro ⟨hx, ⟨a, hx'⟩⟩
  rw [← hx'] at hx
  suffices a = 0 by simp [← hx', this]
  contrapose hv
  simpa [smul_mem_iff p hv] using hx

theorem eq_top_iff_finrank_eq [Module.Finite K V] {W : Submodule K V} :
    W = ⊤ ↔ finrank K W = finrank K V := by
  refine ⟨fun h ↦ by rw [h, finrank_top], fun h ↦ ?_⟩
  apply eq_of_le_of_finrank_eq le_top
  rw [finrank_top, h]

end DivisionRing

end Submodule

namespace FiniteDimensional

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

variable [FiniteDimensional K V] [FiniteDimensional K V₂]

/-- Given isomorphic subspaces `p q` of vector spaces `V` and `V₁` respectively,
  `p.quotient` is isomorphic to `q.quotient`. -/
noncomputable def LinearEquiv.quotEquivOfEquiv {p : Subspace K V} {q : Subspace K V₂}
    (f₁ : p ≃ₗ[K] q) (f₂ : V ≃ₗ[K] V₂) : (V ⧸ p) ≃ₗ[K] V₂ ⧸ q :=
  LinearEquiv.ofFinrankEq _ _
    (by
      rw [← @add_right_cancel_iff _ _ _ (finrank K p), Submodule.finrank_quotient_add_finrank,
        LinearEquiv.finrank_eq f₁, Submodule.finrank_quotient_add_finrank,
        LinearEquiv.finrank_eq f₂])

-- TODO: generalize to the case where one of `p` and `q` is finite-dimensional.
/-- Given the subspaces `p q`, if `p.quotient ≃ₗ[K] q`, then `q.quotient ≃ₗ[K] p` -/
noncomputable def LinearEquiv.quotEquivOfQuotEquiv {p q : Subspace K V} (f : (V ⧸ p) ≃ₗ[K] q) :
    (V ⧸ q) ≃ₗ[K] p :=
  LinearEquiv.ofFinrankEq _ _ <| by
    rw [← add_right_cancel_iff, Submodule.finrank_quotient_add_finrank, ← LinearEquiv.finrank_eq f,
      add_comm, Submodule.finrank_quotient_add_finrank]

end DivisionRing

end FiniteDimensional

namespace LinearMap

open Module

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

/-- rank-nullity theorem : the dimensions of the kernel and the range of a linear map add up to
the dimension of the source space. -/
theorem finrank_range_add_finrank_ker [FiniteDimensional K V] (f : V →ₗ[K] V₂) :
    finrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K V := by
  rw [← f.quotKerEquivRange.finrank_eq]
  exact Submodule.finrank_quotient_add_finrank _

lemma ker_ne_bot_of_finrank_lt [FiniteDimensional K V] [FiniteDimensional K V₂] {f : V →ₗ[K] V₂}
    (h : finrank K V₂ < finrank K V) :
    LinearMap.ker f ≠ ⊥ := by
  have h₁ := f.finrank_range_add_finrank_ker
  have h₂ : finrank K (LinearMap.range f) ≤ finrank K V₂ := (LinearMap.range f).finrank_le
  suffices 0 < finrank K (LinearMap.ker f) from Submodule.one_le_finrank_iff.mp this
  lia

end DivisionRing

end LinearMap

open Module

namespace LinearMap

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

theorem injective_iff_surjective_of_finrank_eq_finrank [FiniteDimensional K V]
    [FiniteDimensional K V₂] (H : finrank K V = finrank K V₂) {f : V →ₗ[K] V₂} :
    Function.Injective f ↔ Function.Surjective f := by
  have := finrank_range_add_finrank_ker f
  rw [← ker_eq_bot, ← range_eq_top]; refine ⟨fun h => ?_, fun h => ?_⟩
  · rw [h, finrank_bot, add_zero, H] at this
    exact eq_top_of_finrank_eq this
  · rw [h, finrank_top, H] at this
    exact Submodule.finrank_eq_zero.1 (add_right_injective _ this)

Target:
theorem ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank [FiniteDimensional K V]
    [FiniteDimensional K V₂] (H : finrank K V = finrank K V₂) {f : V →ₗ[K] V₂} :
    LinearMap.ker f = ⊥ ↔ LinearMap.range f = ⊤ :=

Proof body:

[Assistant]
by
  rw [range_eq_top, ker_eq_bot, injective_iff_surjective_of_finrank_eq_finrank H]

/-- Given a linear map `f` between two vector spaces with the same dimension, if
`ker f = ⊥` then `linearEquivOfInjective` is the induced isomorphism
between the two vector spaces. -/
noncomputable def linearEquivOfInjective [FiniteDimensional K V] [FiniteDimensional K V₂]
    (f : V →ₗ[K] V₂) (hf : Injective f) (hdim : finrank K V = finrank K V₂) : V ≃ₗ[K] V₂ :=
  LinearEquiv.ofBijective f
    ⟨hf, (LinearMap.injective_iff_surjective_of_finrank_eq_finrank hdim).mp hf⟩

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

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

Namespace:
LinearMap

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

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

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

@[expose] public section

assert_not_exists Monoid.exponent Module.IsTorsion

universe u v v'

open Cardinal Submodule Module Function

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

namespace Submodule

open IsNoetherian Module

section DivisionRing

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

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

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

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

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

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

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

theorem isCompl_iff_disjoint [FiniteDimensional K V] (s t : Submodule K V)
    (hdim : finrank K V ≤ finrank K s + finrank K t) :
    IsCompl s t ↔ Disjoint s t :=
  ⟨fun h ↦ h.1, fun h ↦ ⟨h, codisjoint_iff.mpr <| eq_top_of_disjoint s t hdim h⟩⟩

theorem sup_span_singleton_eq_top_iff [Module.Finite K V] {W : Submodule K V} {v : V} (hv : v ∉ W) :
    W ⊔ span K {v} = ⊤ ↔ finrank K (V ⧸ W) = 1 := by
  refine ⟨fun hW ↦ ?_, fun hW ↦ ?_⟩
  · suffices W ⊓ span K {v} = ⊥ by
      have hv₀ : v ≠ 0 := by aesop
      have aux := finrank_sup_add_finrank_inf_eq W (span K {v})
      rw [hW, finrank_span_singleton hv₀, this, finrank_bot, finrank_top,
        ← finrank_quotient_add_finrank W] at aux
      lia
    refine (Submodule.eq_bot_iff _).mpr fun w hw ↦ ?_
    obtain ⟨ht, t, rfl⟩ : w ∈ W ∧ ∃ t : K, t • v = w := by simpa [mem_span_singleton] using hw
    rcases eq_or_ne t 0 with rfl | ht₀; · simp
    rw [Submodule.smul_mem_iff _ ht₀] at ht
    contradiction
  · apply Submodule.eq_top_of_disjoint
    · rw [← W.finrank_quotient_add_finrank, add_comm, add_le_add_iff_left, hW]
      aesop
    · exact Submodule.disjoint_span_singleton_of_notMem hv

theorem finrank_sup_span_singleton [Module.Finite K V] {p : Submodule K V} {v : V} (hv : v ∉ p) :
    finrank K (p ⊔ Submodule.span K {v} : Submodule K V) = finrank K p + 1 := by
  rw [← Nat.add_left_inj, finrank_sup_add_finrank_inf_eq, add_assoc,
    Nat.add_left_cancel_iff, finrank_span_singleton (by aesop),
    Nat.left_eq_add, Submodule.finrank_eq_zero, eq_bot_iff]
  intro x
  simp only [mem_inf, mem_span_singleton]
  rintro ⟨hx, ⟨a, hx'⟩⟩
  rw [← hx'] at hx
  suffices a = 0 by simp [← hx', this]
  contrapose hv
  simpa [smul_mem_iff p hv] using hx

theorem eq_top_iff_finrank_eq [Module.Finite K V] {W : Submodule K V} :
    W = ⊤ ↔ finrank K W = finrank K V := by
  refine ⟨fun h ↦ by rw [h, finrank_top], fun h ↦ ?_⟩
  apply eq_of_le_of_finrank_eq le_top
  rw [finrank_top, h]

end DivisionRing

end Submodule

namespace FiniteDimensional

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

variable [FiniteDimensional K V] [FiniteDimensional K V₂]

/-- Given isomorphic subspaces `p q` of vector spaces `V` and `V₁` respectively,
  `p.quotient` is isomorphic to `q.quotient`. -/
noncomputable def LinearEquiv.quotEquivOfEquiv {p : Subspace K V} {q : Subspace K V₂}
    (f₁ : p ≃ₗ[K] q) (f₂ : V ≃ₗ[K] V₂) : (V ⧸ p) ≃ₗ[K] V₂ ⧸ q :=
  LinearEquiv.ofFinrankEq _ _
    (by
      rw [← @add_right_cancel_iff _ _ _ (finrank K p), Submodule.finrank_quotient_add_finrank,
        LinearEquiv.finrank_eq f₁, Submodule.finrank_quotient_add_finrank,
        LinearEquiv.finrank_eq f₂])

-- TODO: generalize to the case where one of `p` and `q` is finite-dimensional.
/-- Given the subspaces `p q`, if `p.quotient ≃ₗ[K] q`, then `q.quotient ≃ₗ[K] p` -/
noncomputable def LinearEquiv.quotEquivOfQuotEquiv {p q : Subspace K V} (f : (V ⧸ p) ≃ₗ[K] q) :
    (V ⧸ q) ≃ₗ[K] p :=
  LinearEquiv.ofFinrankEq _ _ <| by
    rw [← add_right_cancel_iff, Submodule.finrank_quotient_add_finrank, ← LinearEquiv.finrank_eq f,
      add_comm, Submodule.finrank_quotient_add_finrank]

end DivisionRing

end FiniteDimensional

namespace LinearMap

open Module

section DivisionRing

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

/-- rank-nullity theorem : the dimensions of the kernel and the range of a linear map add up to
the dimension of the source space. -/
theorem finrank_range_add_finrank_ker [FiniteDimensional K V] (f : V →ₗ[K] V₂) :
    finrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K V := by
  rw [← f.quotKerEquivRange.finrank_eq]
  exact Submodule.finrank_quotient_add_finrank _

lemma ker_ne_bot_of_finrank_lt [FiniteDimensional K V] [FiniteDimensional K V₂] {f : V →ₗ[K] V₂}
    (h : finrank K V₂ < finrank K V) :
    LinearMap.ker f ≠ ⊥ := by
  have h₁ := f.finrank_range_add_finrank_ker
  have h₂ : finrank K (LinearMap.range f) ≤ finrank K V₂ := (LinearMap.range f).finrank_le
  suffices 0 < finrank K (LinearMap.ker f) from Submodule.one_le_finrank_iff.mp this
  lia

end DivisionRing

end LinearMap

open Module

namespace LinearMap

variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
  [Module K V₂]

theorem injective_iff_surjective_of_finrank_eq_finrank [FiniteDimensional K V]
    [FiniteDimensional K V₂] (H : finrank K V = finrank K V₂) {f : V →ₗ[K] V₂} :
    Function.Injective f ↔ Function.Surjective f := by
  have := finrank_range_add_finrank_ker f
  rw [← ker_eq_bot, ← range_eq_top]; refine ⟨fun h => ?_, fun h => ?_⟩
  · rw [h, finrank_bot, add_zero, H] at this
    exact eq_top_of_finrank_eq this
  · rw [h, finrank_top, H] at this
    exact Submodule.finrank_eq_zero.1 (add_right_injective _ this)

Target:
theorem ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank [FiniteDimensional K V]
    [FiniteDimensional K V₂] (H : finrank K V = finrank K V₂) {f : V →ₗ[K] V₂} :
    LinearMap.ker f = ⊥ ↔ LinearMap.range f = ⊤ :=

Proof body:


<LEAN_FILE>:56:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge5cf1492d8.value`
<LEAN_FILE>:55:0: error: unsolved goals
C : Type u
inst✝² : Category.{v, u} C
J : Type w
inst✝¹ : Category.{v_1, w} J
inst✝ : HasLimitsOfShape J C
X : TopCat
F : J ⥤ Presheaf C X
H : ∀ (j : J), (F.obj j).IsSheaf
c : Cone F
hc : IsLimit c
⊢ c.pt.IsSheaf

Response:
by
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

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

Imports:
public import Mathlib.Topology.Sheaves.Sheaf
public import Mathlib.CategoryTheory.Sites.Limits
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Namespace:
TopCat

Local safe context:
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
/-!
# Presheaves in `C` have limits and colimits when `C` does.
-/

public section

noncomputable section

universe v u w t

open CategoryTheory

open CategoryTheory.Limits

variable {C : Type u} [Category.{v} C] {J : Type w} [Category* J]

namespace TopCat

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Presheaf C X) :=
  functorCategoryHasLimitsOfShape

instance [HasLimits C] (X : TopCat.{v}) : HasLimits.{v} (Presheaf C X) where

instance [HasColimitsOfShape J C] (X : TopCat) : HasColimitsOfShape J (Presheaf C X) :=
  functorCategoryHasColimitsOfShape

instance [HasColimits.{v, u} C] (X : TopCat.{t}) : HasColimitsOfSize.{v, v} (Presheaf C X) where

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : CreatesLimitsOfShape J (Sheaf.forget C X) :=
  inferInstanceAs <| CreatesLimitsOfShape J (sheafToPresheaf _ _)

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Sheaf C X) :=
  hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape (Sheaf.forget C X)

instance [HasLimits C] (X : TopCat) : CreatesLimits.{v, v} (Sheaf.forget C X) where

instance [HasLimits C] (X : TopCat.{v}) : HasLimitsOfSize.{v, v} (Sheaf.{v} C X) where

set_option backward.isDefEq.respectTransparency false in

Target:
theorem isSheaf_of_isLimit [HasLimitsOfShape J C] {X : TopCat.{v}} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf :=

Failed candidate:
by
  have __repair_e5cf1492d8 := OProverMissinge5cf1492d8.value
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:56:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge5cf1492d8.value`\n<LEAN_FILE>:55:0: error: unsolved goals\nC : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : Category.{v_1, w} J\ninst✝ : HasLimitsOfShape J C\nX : TopCat\nF : J ⥤ Presheaf C X\nH : ∀ (j : J), (F.obj j).IsSheaf\nc : Cone F\nhc : IsLimit c\n⊢ c.pt.IsSheaf", "returncode": 1}

Corrected proof:

[Assistant]
by
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

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

Imports:
public import Mathlib.Topology.Sheaves.Sheaf
public import Mathlib.CategoryTheory.Sites.Limits
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Namespace:
TopCat

Local safe context:
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
/-!
# Presheaves in `C` have limits and colimits when `C` does.
-/

public section

noncomputable section

universe v u w t

open CategoryTheory

open CategoryTheory.Limits

variable {C : Type u} [Category.{v} C] {J : Type w} [Category* J]

namespace TopCat

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Presheaf C X) :=
  functorCategoryHasLimitsOfShape

instance [HasLimits C] (X : TopCat.{v}) : HasLimits.{v} (Presheaf C X) where

instance [HasColimitsOfShape J C] (X : TopCat) : HasColimitsOfShape J (Presheaf C X) :=
  functorCategoryHasColimitsOfShape

instance [HasColimits.{v, u} C] (X : TopCat.{t}) : HasColimitsOfSize.{v, v} (Presheaf C X) where

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : CreatesLimitsOfShape J (Sheaf.forget C X) :=
  inferInstanceAs <| CreatesLimitsOfShape J (sheafToPresheaf _ _)

instance [HasLimitsOfShape J C] (X : TopCat.{t}) : HasLimitsOfShape J (Sheaf C X) :=
  hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape (Sheaf.forget C X)

instance [HasLimits C] (X : TopCat) : CreatesLimits.{v, v} (Sheaf.forget C X) where

instance [HasLimits C] (X : TopCat.{v}) : HasLimitsOfSize.{v, v} (Sheaf.{v} C X) where

set_option backward.isDefEq.respectTransparency false in

Target:
theorem isSheaf_of_isLimit [HasLimitsOfShape J C] {X : TopCat.{v}} (F : J ⥤ Presheaf.{v} C X)
    (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf :=

Failed candidate:
by
  have __repair_e5cf1492d8 := OProverMissinge5cf1492d8.value
  let F' : J ⥤ Sheaf C X :=
    { obj := fun j => ⟨F.obj j, H j⟩
      map := fun f => ⟨F.map f⟩ }
  let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _
  exact Presheaf.isSheaf_of_iso
    ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)
    (limit F').2

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:56:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissinge5cf1492d8.value`\n<LEAN_FILE>:55:0: error: unsolved goals\nC : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : Category.{v_1, w} J\ninst✝ : HasLimitsOfShape J C\nX : TopCat\nF : J ⥤ Presheaf C X\nH : ∀ (j : J), (F.obj j).IsSheaf\nc : Cone F\nhc : IsLimit c\n⊢ c.pt.IsSheaf", "returncode": 1}

Corrected proof:


<LEAN_FILE>:247:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingfb89c2c2cb.value`
<LEAN_FILE>:246:0: error: unsolved goals
C : Type u
inst✝ : Category.{v, u} C
F G : C ⥤ Type w
⊢ Limits.coprod.inr ≫ (binaryCoproductIso F G).hom = coprod.inr

Response:
by
  simp [binaryCoproductIso]
  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.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Types.Limits
public import Mathlib.CategoryTheory.Limits.Types.Colimits

Namespace:
CategoryTheory.FunctorToTypes

Local safe context:
/-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
/-!
# Binary (co)products of type-valued functors

Defines an explicit construction of binary products and coproducts of type-valued functors.

Also defines isomorphisms to the categorical product and coproduct, respectively.
-/

@[expose] public section

open CategoryTheory Limits ConcreteCategory

universe w v u

namespace CategoryTheory.FunctorToTypes

variable {C : Type u} [Category.{v} C]
variable (F G : C ⥤ Type w)

section prod

/-- `prod F G` is the explicit binary product of type-valued functors `F` and `G`. -/
@[simps obj map]
def prod : C ⥤ Type w where
  obj a := F.obj a × G.obj a
  map f := ↾fun a ↦ (F.map f a.1, G.map f a.2)

variable {F G}

/-- The first projection of `prod F G`, onto `F`. -/
@[simps app]
def prod.fst : prod F G ⟶ F where
  app _ := ↾fun a ↦ a.1

/-- The second projection of `prod F G`, onto `G`. -/
@[simps app]
def prod.snd : prod F G ⟶ G where
  app _ := ↾fun a ↦ a.2

/-- Given natural transformations `F ⟶ F₁` and `F ⟶ F₂`, construct
a natural transformation `F ⟶ prod F₁ F₂`. -/
@[simps]
def prod.lift {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    F ⟶ prod F₁ F₂ where
  app x := ↾fun y ↦ ⟨τ₁.app x y, τ₂.app x y⟩

@[simp]
lemma prod.lift_fst {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    prod.lift τ₁ τ₂ ≫ prod.fst = τ₁ := rfl

@[simp]
lemma prod.lift_snd {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    prod.lift τ₁ τ₂ ≫ prod.snd = τ₂ := rfl

variable (F G)

/-- The binary fan whose point is `prod F G`. -/
@[simps!]
def binaryProductCone : BinaryFan F G :=
  BinaryFan.mk prod.fst prod.snd

/-- `prod F G` is a limit cone. -/
@[simps]
def binaryProductLimit : IsLimit (binaryProductCone F G) where
  lift (s : BinaryFan F G) := prod.lift s.fst s.snd
  fac _ := fun ⟨j⟩ ↦ WalkingPair.casesOn j rfl rfl
  uniq _ _ h := by
    simp only [← h ⟨WalkingPair.right⟩, ← h ⟨WalkingPair.left⟩]
    congr

/-- `prod F G` is a binary product for `F` and `G`. -/
def binaryProductLimitCone : Limits.LimitCone (pair F G) :=
  ⟨_, binaryProductLimit F G⟩

/-- The categorical binary product of type-valued functors is `prod F G`. -/
noncomputable def binaryProductIso : F ⨯ G ≅ prod F G :=
  limit.isoLimitCone (binaryProductLimitCone F G)

@[simp]
lemma binaryProductIso_hom_comp_fst :
    (binaryProductIso F G).hom ≫ prod.fst = Limits.prod.fst := rfl

@[simp]
lemma binaryProductIso_hom_comp_snd :
    (binaryProductIso F G).hom ≫ prod.snd = Limits.prod.snd := rfl

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma binaryProductIso_inv_comp_fst :
    (binaryProductIso F G).inv ≫ Limits.prod.fst = prod.fst := by
  simp [binaryProductIso, binaryProductLimitCone]

@[simp]
lemma binaryProductIso_inv_comp_fst_apply (a : C) (z : (prod F G).obj a) :
    dsimp% (Limits.prod.fst (X := F)).app a ((binaryProductIso F G).inv.app a z) = z.1 :=
  congr_hom (congr_app (binaryProductIso_inv_comp_fst F G) a) z

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma binaryProductIso_inv_comp_snd :
    (binaryProductIso F G).inv ≫ Limits.prod.snd = prod.snd := by
  simp [binaryProductIso, binaryProductLimitCone]

@[simp]
lemma binaryProductIso_inv_comp_snd_apply (a : C) (z : (prod F G).obj a) :
    dsimp% (Limits.prod.snd (X := F)).app a ((binaryProductIso F G).inv.app a z) = z.2 :=
  congr_hom (congr_app (binaryProductIso_inv_comp_snd F G) a) z

variable {F G}

/-- Construct an element of `(F ⨯ G).obj a` from an element of `F.obj a` and
an element of `G.obj a`. -/
noncomputable
def prodMk {a : C} (x : F.obj a) (y : G.obj a) : (F ⨯ G).obj a :=
  ((binaryProductIso F G).inv).app a ⟨x, y⟩

@[simp]
lemma prodMk_fst {a : C} (x : F.obj a) (y : G.obj a) :
    (Limits.prod.fst (X := F)).app a (prodMk x y) = x := by
  simp [prodMk]

@[simp]
lemma prodMk_snd {a : C} (x : F.obj a) (y : G.obj a) :
    (Limits.prod.snd (X := F)).app a (prodMk x y) = y := by
  simp [prodMk]

@[ext]
lemma prod_ext {a : C} (z w : (prod F G).obj a) (h1 : z.1 = w.1) (h2 : z.2 = w.2) :
    z = w := Prod.ext h1 h2

variable (F G)

/-- `(F ⨯ G).obj a` is in bijection with the product of `F.obj a` and `G.obj a`. -/
@[simps]
noncomputable
def binaryProductEquiv (a : C) : (F ⨯ G).obj a ≃ (F.obj a) × (G.obj a) where
  toFun z := ⟨((binaryProductIso F G).hom.app a z).1, ((binaryProductIso F G).hom.app a z).2⟩
  invFun z := prodMk z.1 z.2
  left_inv _ := by simp [-prod_obj, prodMk]
  right_inv _ := by simp [-prod_obj, prodMk]

@[ext]
lemma prod_ext' (a : C) (z w : (F ⨯ G).obj a)
    (h1 : (Limits.prod.fst (X := F)).app a z = (Limits.prod.fst (X := F)).app a w)
    (h2 : (Limits.prod.snd (X := F)).app a z = (Limits.prod.snd (X := F)).app a w) :
    z = w := by
  apply Equiv.injective (binaryProductEquiv F G a)
  aesop

end prod

section coprod

/-- `coprod F G` is the explicit binary coproduct of type-valued functors `F` and `G`. -/
@[simps obj map]
def coprod : C ⥤ Type w where
  obj a := F.obj a ⊕ G.obj a
  map f := ↾(Sum.map (F.map f) (G.map f))
  map_id _ := by ext ⟨⟩ <;> simp
  map_comp _ _ := by ext ⟨⟩ <;> simp

variable {F G}

/-- The left inclusion of `F` into `coprod F G`. -/
@[simps]
def coprod.inl : F ⟶ coprod F G where
  app _ := ↾fun x ↦ .inl x

/-- The right inclusion of `G` into `coprod F G`. -/
@[simps]
def coprod.inr : G ⟶ coprod F G where
  app _ := ↾fun x ↦ .inr x

set_option backward.defeqAttrib.useBackward true in
/-- Given natural transformations `F₁ ⟶ F` and `F₂ ⟶ F`, construct
a natural transformation `coprod F₁ F₂ ⟶ F`. -/
@[simps]
def coprod.desc {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod F₁ F₂ ⟶ F where
  app a := ↾(Sum.elim (τ₁.app a) (τ₂.app a))
  naturality _ _ _ := by ext ⟨⟩ <;> simp

@[simp]
lemma coprod.desc_inl {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod.inl ≫ coprod.desc τ₁ τ₂ = τ₁ := rfl

@[simp]
lemma coprod.desc_inr {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod.inr ≫ coprod.desc τ₁ τ₂ = τ₂ := rfl

variable (F G)

/-- The binary cofan whose point is `coprod F G`. -/
@[simps!]
def binaryCoproductCocone : BinaryCofan F G :=
  BinaryCofan.mk coprod.inl coprod.inr

set_option backward.defeqAttrib.useBackward true in
/-- `coprod F G` is a colimit cocone. -/
@[simps]
def binaryCoproductColimit : IsColimit (binaryCoproductCocone F G) where
  desc (s : BinaryCofan F G) := coprod.desc s.inl s.inr
  fac _ := fun ⟨j⟩ ↦ WalkingPair.casesOn j rfl rfl
  uniq _ _ h := by
    ext _ x
    cases x with | _ => simp [← h ⟨WalkingPair.right⟩, ← h ⟨WalkingPair.left⟩]

/-- `coprod F G` is a binary coproduct for `F` and `G`. -/
def binaryCoproductColimitCocone : Limits.ColimitCocone (pair F G) :=
  ⟨_, binaryCoproductColimit F G⟩

/-- The categorical binary coproduct of type-valued functors is `coprod F G`. -/
noncomputable def binaryCoproductIso : F ⨿ G ≅ coprod F G :=
  colimit.isoColimitCocone (binaryCoproductColimitCocone F G)

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma inl_comp_binaryCoproductIso_hom :
    Limits.coprod.inl ≫ (binaryCoproductIso F G).hom = coprod.inl := by
  simp only [binaryCoproductIso]
  aesop

@[simp]
lemma inl_comp_binaryCoproductIso_hom_apply (a : C) (x : F.obj a) :
    dsimp% (binaryCoproductIso F G).hom.app a ((Limits.coprod.inl (X := F)).app a x) = .inl x :=
  congr_hom (congr_app (inl_comp_binaryCoproductIso_hom F G) a) x

set_option backward.isDefEq.respectTransparency false in
@[simp]

Target:
lemma inr_comp_binaryCoproductIso_hom :
    Limits.coprod.inr ≫ (binaryCoproductIso F G).hom = coprod.inr :=

Failed candidate:
by
  have __repair_fb89c2c2cb := OProverMissingfb89c2c2cb.value
  simp [binaryCoproductIso]
  aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:247:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingfb89c2c2cb.value`\n<LEAN_FILE>:246:0: error: unsolved goals\nC : Type u\ninst✝ : Category.{v, u} C\nF G : C ⥤ Type w\n⊢ Limits.coprod.inr ≫ (binaryCoproductIso F G).hom = coprod.inr", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp [binaryCoproductIso]
  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.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Types.Limits
public import Mathlib.CategoryTheory.Limits.Types.Colimits

Namespace:
CategoryTheory.FunctorToTypes

Local safe context:
/-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
/-!
# Binary (co)products of type-valued functors

Defines an explicit construction of binary products and coproducts of type-valued functors.

Also defines isomorphisms to the categorical product and coproduct, respectively.
-/

@[expose] public section

open CategoryTheory Limits ConcreteCategory

universe w v u

namespace CategoryTheory.FunctorToTypes

variable {C : Type u} [Category.{v} C]
variable (F G : C ⥤ Type w)

section prod

/-- `prod F G` is the explicit binary product of type-valued functors `F` and `G`. -/
@[simps obj map]
def prod : C ⥤ Type w where
  obj a := F.obj a × G.obj a
  map f := ↾fun a ↦ (F.map f a.1, G.map f a.2)

variable {F G}

/-- The first projection of `prod F G`, onto `F`. -/
@[simps app]
def prod.fst : prod F G ⟶ F where
  app _ := ↾fun a ↦ a.1

/-- The second projection of `prod F G`, onto `G`. -/
@[simps app]
def prod.snd : prod F G ⟶ G where
  app _ := ↾fun a ↦ a.2

/-- Given natural transformations `F ⟶ F₁` and `F ⟶ F₂`, construct
a natural transformation `F ⟶ prod F₁ F₂`. -/
@[simps]
def prod.lift {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    F ⟶ prod F₁ F₂ where
  app x := ↾fun y ↦ ⟨τ₁.app x y, τ₂.app x y⟩

@[simp]
lemma prod.lift_fst {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    prod.lift τ₁ τ₂ ≫ prod.fst = τ₁ := rfl

@[simp]
lemma prod.lift_snd {F₁ F₂ : C ⥤ Type w} (τ₁ : F ⟶ F₁) (τ₂ : F ⟶ F₂) :
    prod.lift τ₁ τ₂ ≫ prod.snd = τ₂ := rfl

variable (F G)

/-- The binary fan whose point is `prod F G`. -/
@[simps!]
def binaryProductCone : BinaryFan F G :=
  BinaryFan.mk prod.fst prod.snd

/-- `prod F G` is a limit cone. -/
@[simps]
def binaryProductLimit : IsLimit (binaryProductCone F G) where
  lift (s : BinaryFan F G) := prod.lift s.fst s.snd
  fac _ := fun ⟨j⟩ ↦ WalkingPair.casesOn j rfl rfl
  uniq _ _ h := by
    simp only [← h ⟨WalkingPair.right⟩, ← h ⟨WalkingPair.left⟩]
    congr

/-- `prod F G` is a binary product for `F` and `G`. -/
def binaryProductLimitCone : Limits.LimitCone (pair F G) :=
  ⟨_, binaryProductLimit F G⟩

/-- The categorical binary product of type-valued functors is `prod F G`. -/
noncomputable def binaryProductIso : F ⨯ G ≅ prod F G :=
  limit.isoLimitCone (binaryProductLimitCone F G)

@[simp]
lemma binaryProductIso_hom_comp_fst :
    (binaryProductIso F G).hom ≫ prod.fst = Limits.prod.fst := rfl

@[simp]
lemma binaryProductIso_hom_comp_snd :
    (binaryProductIso F G).hom ≫ prod.snd = Limits.prod.snd := rfl

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma binaryProductIso_inv_comp_fst :
    (binaryProductIso F G).inv ≫ Limits.prod.fst = prod.fst := by
  simp [binaryProductIso, binaryProductLimitCone]

@[simp]
lemma binaryProductIso_inv_comp_fst_apply (a : C) (z : (prod F G).obj a) :
    dsimp% (Limits.prod.fst (X := F)).app a ((binaryProductIso F G).inv.app a z) = z.1 :=
  congr_hom (congr_app (binaryProductIso_inv_comp_fst F G) a) z

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma binaryProductIso_inv_comp_snd :
    (binaryProductIso F G).inv ≫ Limits.prod.snd = prod.snd := by
  simp [binaryProductIso, binaryProductLimitCone]

@[simp]
lemma binaryProductIso_inv_comp_snd_apply (a : C) (z : (prod F G).obj a) :
    dsimp% (Limits.prod.snd (X := F)).app a ((binaryProductIso F G).inv.app a z) = z.2 :=
  congr_hom (congr_app (binaryProductIso_inv_comp_snd F G) a) z

variable {F G}

/-- Construct an element of `(F ⨯ G).obj a` from an element of `F.obj a` and
an element of `G.obj a`. -/
noncomputable
def prodMk {a : C} (x : F.obj a) (y : G.obj a) : (F ⨯ G).obj a :=
  ((binaryProductIso F G).inv).app a ⟨x, y⟩

@[simp]
lemma prodMk_fst {a : C} (x : F.obj a) (y : G.obj a) :
    (Limits.prod.fst (X := F)).app a (prodMk x y) = x := by
  simp [prodMk]

@[simp]
lemma prodMk_snd {a : C} (x : F.obj a) (y : G.obj a) :
    (Limits.prod.snd (X := F)).app a (prodMk x y) = y := by
  simp [prodMk]

@[ext]
lemma prod_ext {a : C} (z w : (prod F G).obj a) (h1 : z.1 = w.1) (h2 : z.2 = w.2) :
    z = w := Prod.ext h1 h2

variable (F G)

/-- `(F ⨯ G).obj a` is in bijection with the product of `F.obj a` and `G.obj a`. -/
@[simps]
noncomputable
def binaryProductEquiv (a : C) : (F ⨯ G).obj a ≃ (F.obj a) × (G.obj a) where
  toFun z := ⟨((binaryProductIso F G).hom.app a z).1, ((binaryProductIso F G).hom.app a z).2⟩
  invFun z := prodMk z.1 z.2
  left_inv _ := by simp [-prod_obj, prodMk]
  right_inv _ := by simp [-prod_obj, prodMk]

@[ext]
lemma prod_ext' (a : C) (z w : (F ⨯ G).obj a)
    (h1 : (Limits.prod.fst (X := F)).app a z = (Limits.prod.fst (X := F)).app a w)
    (h2 : (Limits.prod.snd (X := F)).app a z = (Limits.prod.snd (X := F)).app a w) :
    z = w := by
  apply Equiv.injective (binaryProductEquiv F G a)
  aesop

end prod

section coprod

/-- `coprod F G` is the explicit binary coproduct of type-valued functors `F` and `G`. -/
@[simps obj map]
def coprod : C ⥤ Type w where
  obj a := F.obj a ⊕ G.obj a
  map f := ↾(Sum.map (F.map f) (G.map f))
  map_id _ := by ext ⟨⟩ <;> simp
  map_comp _ _ := by ext ⟨⟩ <;> simp

variable {F G}

/-- The left inclusion of `F` into `coprod F G`. -/
@[simps]
def coprod.inl : F ⟶ coprod F G where
  app _ := ↾fun x ↦ .inl x

/-- The right inclusion of `G` into `coprod F G`. -/
@[simps]
def coprod.inr : G ⟶ coprod F G where
  app _ := ↾fun x ↦ .inr x

set_option backward.defeqAttrib.useBackward true in
/-- Given natural transformations `F₁ ⟶ F` and `F₂ ⟶ F`, construct
a natural transformation `coprod F₁ F₂ ⟶ F`. -/
@[simps]
def coprod.desc {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod F₁ F₂ ⟶ F where
  app a := ↾(Sum.elim (τ₁.app a) (τ₂.app a))
  naturality _ _ _ := by ext ⟨⟩ <;> simp

@[simp]
lemma coprod.desc_inl {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod.inl ≫ coprod.desc τ₁ τ₂ = τ₁ := rfl

@[simp]
lemma coprod.desc_inr {F₁ F₂ : C ⥤ Type w} (τ₁ : F₁ ⟶ F) (τ₂ : F₂ ⟶ F) :
    coprod.inr ≫ coprod.desc τ₁ τ₂ = τ₂ := rfl

variable (F G)

/-- The binary cofan whose point is `coprod F G`. -/
@[simps!]
def binaryCoproductCocone : BinaryCofan F G :=
  BinaryCofan.mk coprod.inl coprod.inr

set_option backward.defeqAttrib.useBackward true in
/-- `coprod F G` is a colimit cocone. -/
@[simps]
def binaryCoproductColimit : IsColimit (binaryCoproductCocone F G) where
  desc (s : BinaryCofan F G) := coprod.desc s.inl s.inr
  fac _ := fun ⟨j⟩ ↦ WalkingPair.casesOn j rfl rfl
  uniq _ _ h := by
    ext _ x
    cases x with | _ => simp [← h ⟨WalkingPair.right⟩, ← h ⟨WalkingPair.left⟩]

/-- `coprod F G` is a binary coproduct for `F` and `G`. -/
def binaryCoproductColimitCocone : Limits.ColimitCocone (pair F G) :=
  ⟨_, binaryCoproductColimit F G⟩

/-- The categorical binary coproduct of type-valued functors is `coprod F G`. -/
noncomputable def binaryCoproductIso : F ⨿ G ≅ coprod F G :=
  colimit.isoColimitCocone (binaryCoproductColimitCocone F G)

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma inl_comp_binaryCoproductIso_hom :
    Limits.coprod.inl ≫ (binaryCoproductIso F G).hom = coprod.inl := by
  simp only [binaryCoproductIso]
  aesop

@[simp]
lemma inl_comp_binaryCoproductIso_hom_apply (a : C) (x : F.obj a) :
    dsimp% (binaryCoproductIso F G).hom.app a ((Limits.coprod.inl (X := F)).app a x) = .inl x :=
  congr_hom (congr_app (inl_comp_binaryCoproductIso_hom F G) a) x

set_option backward.isDefEq.respectTransparency false in
@[simp]

Target:
lemma inr_comp_binaryCoproductIso_hom :
    Limits.coprod.inr ≫ (binaryCoproductIso F G).hom = coprod.inr :=

Failed candidate:
by
  have __repair_fb89c2c2cb := OProverMissingfb89c2c2cb.value
  simp [binaryCoproductIso]
  aesop

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:247:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingfb89c2c2cb.value`\n<LEAN_FILE>:246:0: error: unsolved goals\nC : Type u\ninst✝ : Category.{v, u} C\nF G : C ⥤ Type w\n⊢ Limits.coprod.inr ≫ (binaryCoproductIso F G).hom = coprod.inr", "returncode": 1}

Corrected proof:


<LEAN_FILE>:76:38: error: Type mismatch
  True.intro
has type
  True
but is expected to have type
  False

Response:
by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

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

Imports:
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Ring.Defs

Namespace:
IsDedekindFiniteMonoid

Local safe context:
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
/-!
# Lemmas about regular elements in rings.
-/

public section

variable {α : Type*}

/-- Left `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isLeftRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, k * x = 0 → x = 0) : IsLeftRegular k := by
  refine fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ ?_)
  rw [mul_sub, sub_eq_zero, h']

/-- Right `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isRightRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, x * k = 0 → x = 0) : IsRightRegular k := by
  refine fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ ?_)
  rw [sub_mul, sub_eq_zero, h']

theorem IsRegular.of_ne_zero' [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} (hk : k ≠ 0) :
    IsRegular k :=
  ⟨isLeftRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left hk,
    isRightRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk⟩

@[deprecated (since := "2026-01-21")] alias isRegular_of_ne_zero' := IsRegular.of_ne_zero'

theorem isRegular_iff_ne_zero' [Nontrivial α] [NonUnitalNonAssocRing α] [NoZeroDivisors α]
    {k : α} : IsRegular k ↔ k ≠ 0 :=
  ⟨fun h => by
    rintro rfl
    exact not_not.mpr h.left not_isLeftRegular_zero, .of_ne_zero'⟩

/-- A ring with no zero divisors is a cancellative `MonoidWithZero`.

Note this is not an instance as it forms a typeclass loop. -/
lemma NoZeroDivisors.toIsCancelMulZero [NonUnitalNonAssocRing α] [NoZeroDivisors α] :
    IsCancelMulZero α where
  mul_left_cancel_of_ne_zero ha := (IsRegular.of_ne_zero' ha).1
  mul_right_cancel_of_ne_zero hb := (IsRegular.of_ne_zero' hb).2

namespace IsDedekindFiniteMonoid

variable [Ring α]

/-- A ring is Dedekind-finite if and only if every element has at most one right inverse. -/
theorem iff_eq_of_mul_left_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * y = 1 → x * z = 1 → y = z := by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxy hxz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [← mul_assoc, h hxz] using congr_arg (z * ·) hxy
  have := h _ _ (1 - y * x + y) eq <| by
    rw [mul_add, mul_sub, ← mul_assoc, eq, mul_one, one_mul, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

/-- A ring is Dedekind-finite if and only if every element has at most one left inverse. -/

Target:
theorem iff_eq_of_mul_right_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * z = 1 → y * z = 1 → x = y :=

Failed candidate:
by
  have __repair_52bacc44f4 : False := True.intro
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:76:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

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

Imports:
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Ring.Defs

Namespace:
IsDedekindFiniteMonoid

Local safe context:
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
/-!
# Lemmas about regular elements in rings.
-/

public section

variable {α : Type*}

/-- Left `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isLeftRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, k * x = 0 → x = 0) : IsLeftRegular k := by
  refine fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ ?_)
  rw [mul_sub, sub_eq_zero, h']

/-- Right `Mul` by a `k : α` over `[Ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `NoZeroDivisors`. -/
theorem isRightRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
    (h : ∀ x : α, x * k = 0 → x = 0) : IsRightRegular k := by
  refine fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ ?_)
  rw [sub_mul, sub_eq_zero, h']

theorem IsRegular.of_ne_zero' [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} (hk : k ≠ 0) :
    IsRegular k :=
  ⟨isLeftRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left hk,
    isRightRegular_of_non_zero_divisor k fun _ h =>
      (NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk⟩

@[deprecated (since := "2026-01-21")] alias isRegular_of_ne_zero' := IsRegular.of_ne_zero'

theorem isRegular_iff_ne_zero' [Nontrivial α] [NonUnitalNonAssocRing α] [NoZeroDivisors α]
    {k : α} : IsRegular k ↔ k ≠ 0 :=
  ⟨fun h => by
    rintro rfl
    exact not_not.mpr h.left not_isLeftRegular_zero, .of_ne_zero'⟩

/-- A ring with no zero divisors is a cancellative `MonoidWithZero`.

Note this is not an instance as it forms a typeclass loop. -/
lemma NoZeroDivisors.toIsCancelMulZero [NonUnitalNonAssocRing α] [NoZeroDivisors α] :
    IsCancelMulZero α where
  mul_left_cancel_of_ne_zero ha := (IsRegular.of_ne_zero' ha).1
  mul_right_cancel_of_ne_zero hb := (IsRegular.of_ne_zero' hb).2

namespace IsDedekindFiniteMonoid

variable [Ring α]

/-- A ring is Dedekind-finite if and only if every element has at most one right inverse. -/
theorem iff_eq_of_mul_left_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * y = 1 → x * z = 1 → y = z := by
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxy hxz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [← mul_assoc, h hxz] using congr_arg (z * ·) hxy
  have := h _ _ (1 - y * x + y) eq <| by
    rw [mul_add, mul_sub, ← mul_assoc, eq, mul_one, one_mul, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

/-- A ring is Dedekind-finite if and only if every element has at most one left inverse. -/

Target:
theorem iff_eq_of_mul_right_eq_one :
    IsDedekindFiniteMonoid α ↔ ∀ x y z : α, x * z = 1 → y * z = 1 → x = y :=

Failed candidate:
by
  have __repair_52bacc44f4 : False := True.intro
  refine (isDedekindFiniteMonoid_iff _).trans ⟨fun h x y z hxz hyz ↦ ?_, fun h x y eq ↦ ?_⟩
  · simpa [mul_assoc, h hyz] using congr_arg (· * y) hxz
  have := h _ (1 - y * x + x) _ eq <| by
    rw [add_mul, sub_mul, mul_assoc, eq, one_mul, mul_one, sub_self, zero_add]
  rwa [right_eq_add, sub_eq_zero, eq_comm] at this

Structured Lean error:
{"kind": "type_mismatch_wrong_shape", "message": "<LEAN_FILE>:76:38: error: Type mismatch\n  True.intro\nhas type\n  True\nbut is expected to have type\n  False", "returncode": 1}

Corrected proof:


<LEAN_FILE>:60:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing5202302049.value`
<LEAN_FILE>:59:0: error: unsolved goals
S : Type u_1
inst✝³ : CommRing S
q : Ideal S
R : Type u_2
inst✝² : CommRing R
inst✝¹ : Algebra R S
hq : q.IsPrime
inst✝ : Module.Finite R S
⊢ 0 < q.inertiaDeg' R

Response:
by
  let := Localization.AtPrime.algebraOfLiesOver (q.under R) q
  rw [inertiaDeg'_def]
  apply Module.finrank_pos

[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.NumberTheory.RamificationInertia.Inertia
public import Mathlib.RingTheory.QuasiFinite.Basic

Namespace:
Ideal

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

Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

## Main definitions

* `Ideal.inertiaDeg' q R`: The inertia degree of `q` over `R`.

## Main statements

* `inertiaDeg_eq_inertiaDeg'`: The inertia degree agrees with the usual definition in the case of
  maximal ideals.
* `inertiaDeg'_tower`: Inertia degree is multiplicative in towers.
-/

@[expose] public section

namespace Ideal

section

variable {S : Type*} [CommRing S] (q : Ideal S) (R : Type*) [CommRing R] [Algebra R S]

open Classical in
/-- Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

When `q` is not prime, we use a junk value of `0`.

This will eventually replace the existing definition of `Ideal.inertiaDeg`. -/
noncomputable def inertiaDeg' : ℕ :=
  if _ : q.IsPrime then
    letI := Localization.AtPrime.algebraOfLiesOver (q.under R) q
    Module.finrank (q.under R).ResidueField q.ResidueField else 0

theorem inertiaDeg'_def [hq : q.IsPrime]
    [Algebra (Localization.AtPrime (q.under R)) (Localization.AtPrime q)]
    [Localization.AtPrime.IsLiesOverAlgebra (q.under R) q] :
    q.inertiaDeg' R = Module.finrank (q.under R).ResidueField q.ResidueField := by
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

theorem inertiaDeg'_of_not_isPrime (hq : ¬ q.IsPrime) : q.inertiaDeg' R = 0 :=
  dif_neg hq

Target:
theorem inertiaDeg'_pos [hq : q.IsPrime] [Module.Finite R S] : 0 < q.inertiaDeg' R :=

Failed candidate:
by
  have __repair_5202302049 := OProverMissing5202302049.value
  let := Localization.AtPrime.algebraOfLiesOver (q.under R) q
  rw [inertiaDeg'_def]
  apply Module.finrank_pos

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:60:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing5202302049.value`\n<LEAN_FILE>:59:0: error: unsolved goals\nS : Type u_1\ninst✝³ : CommRing S\nq : Ideal S\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\nhq : q.IsPrime\ninst✝ : Module.Finite R S\n⊢ 0 < q.inertiaDeg' R", "returncode": 1}

Corrected proof:

[Assistant]
by
  let := Localization.AtPrime.algebraOfLiesOver (q.under R) q
  rw [inertiaDeg'_def]
  apply Module.finrank_pos

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.NumberTheory.RamificationInertia.Inertia
public import Mathlib.RingTheory.QuasiFinite.Basic

Namespace:
Ideal

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

Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

## Main definitions

* `Ideal.inertiaDeg' q R`: The inertia degree of `q` over `R`.

## Main statements

* `inertiaDeg_eq_inertiaDeg'`: The inertia degree agrees with the usual definition in the case of
  maximal ideals.
* `inertiaDeg'_tower`: Inertia degree is multiplicative in towers.
-/

@[expose] public section

namespace Ideal

section

variable {S : Type*} [CommRing S] (q : Ideal S) (R : Type*) [CommRing R] [Algebra R S]

open Classical in
/-- Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined
to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.

When `q` is not prime, we use a junk value of `0`.

This will eventually replace the existing definition of `Ideal.inertiaDeg`. -/
noncomputable def inertiaDeg' : ℕ :=
  if _ : q.IsPrime then
    letI := Localization.AtPrime.algebraOfLiesOver (q.under R) q
    Module.finrank (q.under R).ResidueField q.ResidueField else 0

theorem inertiaDeg'_def [hq : q.IsPrime]
    [Algebra (Localization.AtPrime (q.under R)) (Localization.AtPrime q)]
    [Localization.AtPrime.IsLiesOverAlgebra (q.under R) q] :
    q.inertiaDeg' R = Module.finrank (q.under R).ResidueField q.ResidueField := by
  convert! dif_pos hq
  simp [Algebra.algebra_ext_iff, Localization.AtPrime.IsLiesOverAlgebra.algebraMap_eq]

theorem inertiaDeg'_of_not_isPrime (hq : ¬ q.IsPrime) : q.inertiaDeg' R = 0 :=
  dif_neg hq

Target:
theorem inertiaDeg'_pos [hq : q.IsPrime] [Module.Finite R S] : 0 < q.inertiaDeg' R :=

Failed candidate:
by
  have __repair_5202302049 := OProverMissing5202302049.value
  let := Localization.AtPrime.algebraOfLiesOver (q.under R) q
  rw [inertiaDeg'_def]
  apply Module.finrank_pos

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:60:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing5202302049.value`\n<LEAN_FILE>:59:0: error: unsolved goals\nS : Type u_1\ninst✝³ : CommRing S\nq : Ideal S\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\nhq : q.IsPrime\ninst✝ : Module.Finite R S\n⊢ 0 < q.inertiaDeg' R", "returncode": 1}

Corrected proof:


<LEAN_FILE>:45:43: error: unsolved goals
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
f : R[X] →ₐ[R] S
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : Algebra.WeaklyQuasiFiniteAt R P
⊢ False

Response:
by
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

[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.LocalRing.ResidueField.Polynomial
public import Mathlib.RingTheory.QuasiFinite.Weakly

Namespace:
Polynomial

Local safe context:
/-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-! # Quasi-finite primes in polynomial algebras -/

@[expose] public section

variable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
  [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T]

namespace Polynomial

attribute [local instance] Algebra.WeaklyQuasiFiniteAt.finite_locoalization in
lemma not_weaklyQuasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.WeaklyQuasiFiniteAt R P := by
  intro H
  wlog hR : IsField R
  · let p := P.under R
    obtain ⟨Q, hQ⟩ := (PrimeSpectrum.preimageEquivFiber R R[X]
        ⟨p, inferInstance⟩).symm.surjective ⟨⟨P, ‹_›⟩, rfl⟩
    have inst : Algebra.WeaklyQuasiFiniteAt p.ResidueField Q.asIdeal :=
      .baseChange P Q.asIdeal congr($(hQ.symm).1.1)
    exact this (Q.asIdeal.comap (polyEquivTensor' R p.ResidueField).toRingHom)
      inferInstance (Field.toIsField _)
  let := hR.toField
  have := Module.Finite.of_injective
    (IsScalarTower.toAlgHom R R[X] (Localization.AtPrime P)).toLinearMap
    (IsLocalization.injective _ P.primeCompl_le_nonZeroDivisors)
  exact transcendental_X R (Algebra.IsIntegral.isIntegral X).isAlgebraic

/-- `R[X]` is not quasi-finite over `R` at any prime. -/
lemma not_quasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.QuasiFiniteAt R P :=
  fun _ ↦ not_weaklyQuasiFiniteAt P inferInstance

Target:
lemma map_under_lt_comap_of_weaklyQuasiFiniteAt
    (f : R[X] →ₐ[R] S) (P : Ideal S) [P.IsPrime] [Algebra.WeaklyQuasiFiniteAt R P] :
    (P.under R).map C < P.comap (f : R[X] →+* S) :=

Failed candidate:
by
  have __repair_plan_763c2fc0e6 : False := by
    skip
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:45:43: error: unsolved goals\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\n⊢ False", "returncode": 1}

Corrected proof:

[Assistant]
by
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

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.LocalRing.ResidueField.Polynomial
public import Mathlib.RingTheory.QuasiFinite.Weakly

Namespace:
Polynomial

Local safe context:
/-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
/-! # Quasi-finite primes in polynomial algebras -/

@[expose] public section

variable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
  [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T]

namespace Polynomial

attribute [local instance] Algebra.WeaklyQuasiFiniteAt.finite_locoalization in
lemma not_weaklyQuasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.WeaklyQuasiFiniteAt R P := by
  intro H
  wlog hR : IsField R
  · let p := P.under R
    obtain ⟨Q, hQ⟩ := (PrimeSpectrum.preimageEquivFiber R R[X]
        ⟨p, inferInstance⟩).symm.surjective ⟨⟨P, ‹_›⟩, rfl⟩
    have inst : Algebra.WeaklyQuasiFiniteAt p.ResidueField Q.asIdeal :=
      .baseChange P Q.asIdeal congr($(hQ.symm).1.1)
    exact this (Q.asIdeal.comap (polyEquivTensor' R p.ResidueField).toRingHom)
      inferInstance (Field.toIsField _)
  let := hR.toField
  have := Module.Finite.of_injective
    (IsScalarTower.toAlgHom R R[X] (Localization.AtPrime P)).toLinearMap
    (IsLocalization.injective _ P.primeCompl_le_nonZeroDivisors)
  exact transcendental_X R (Algebra.IsIntegral.isIntegral X).isAlgebraic

/-- `R[X]` is not quasi-finite over `R` at any prime. -/
lemma not_quasiFiniteAt (P : Ideal R[X]) [P.IsPrime] : ¬ Algebra.QuasiFiniteAt R P :=
  fun _ ↦ not_weaklyQuasiFiniteAt P inferInstance

Target:
lemma map_under_lt_comap_of_weaklyQuasiFiniteAt
    (f : R[X] →ₐ[R] S) (P : Ideal S) [P.IsPrime] [Algebra.WeaklyQuasiFiniteAt R P] :
    (P.under R).map C < P.comap (f : R[X] →+* S) :=

Failed candidate:
by
  have __repair_plan_763c2fc0e6 : False := by
    skip
  algebraize [f.toRingHom]
  refine lt_of_le_of_ne (Ideal.map_le_iff_le_comap.mpr ?_) fun e ↦ ?_
  · rw [Ideal.comap_comap, ← algebraMap_eq, f.comp_algebraMap]
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) (P.under R[X])
  let := Localization.AtPrime.algebraOfLiesOver (P.under R[X]) P
  let := Localization.AtPrime.algebraOfLiesOver (P.under R) P
  have : Module.Finite (Ideal.under R P).ResidueField P.ResidueField :=
    Algebra.WeaklyQuasiFiniteAt.finite_residueField ..
  have : Module.Finite (P.under R).ResidueField (P.under R[X]).ResidueField :=
    .of_injective (IsScalarTower.toAlgHom _ _ P.ResidueField).toLinearMap
      (algebraMap (P.under R[X]).ResidueField P.ResidueField).injective
  have : Module.Finite (P.under R).ResidueField (RatFunc (P.under R).ResidueField) :=
    .of_surjective (residueFieldMapCAlgEquiv _ (P.under _) e.symm).toLinearMap
      (residueFieldMapCAlgEquiv _ (P.under _) e.symm).surjective
  exact RatFunc.transcendental_X (K := (P.under R).ResidueField)
    (Algebra.IsIntegral.isIntegral _).isAlgebraic

Structured Lean error:
{"kind": "unsolved_goals_multistep_planning", "message": "<LEAN_FILE>:45:43: error: unsolved goals\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\n⊢ False", "returncode": 1}

Corrected proof: