awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R₁ : Type u_1
A : Type u_3
inst : CommSemiring R₁
inst_1 : Ring A
inst_2 : Algebra R₁ A
I J : Ideal A
inst_3 : I.IsTwoSided
inst_4 : J.IsTwoSided
h : I = J
⊢ Eq (α := (A ⧸ J) ≃ₐ[R₁] A ⧸ I) (AlgEquiv.symm (quotientEquivAlgOfEq R₁ h)) (quotientEquivAlgOfEq R₁ ⋯)
have h₁ : ∀ (a : A ⧸ J),
  Eq (α := A ⧸ I) ((AlgEquiv.symm (quotientEquivAlgOfEq R₁ h) : (a : A ⧸ J) → A ⧸ I) a : A ⧸ I)
    ((quotientEquivAlgOfEq R₁ sorry : (a : A ⧸ J) → A ⧸ I) a : A ⧸ I)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : Ring R
I J : Ideal R
inst_1 : I.IsTwoSided
⊢ comap (F := R →+* R ⧸ I) (Quotient.mk I) (map (F := R →+* R ⧸ I) (Quotient.mk I) J) = I ⊔ J
have h : ∀ (x : R),
  Membership.mem (γ := Ideal R) (comap (F := R →+* R ⧸ I) (Quotient.mk I) (map (F := R →+* R ⧸ I) (Quotient.mk I) J))
      x ↔
    x ∈ I ⊔ J
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : Ring R
I : Ideal R
inst_1 : I.IsTwoSided
p : Ideal R
inst_2 : p.IsPrime
hIP : I ≤ p
⊢ IsPrime (α := R ⧸ I) (map (F := R →+* R ⧸ I) (Quotient.mk I) p)
have hf : Surjective (α := R) (β := R ⧸ I) (⇑(Quotient.mk I) : (a : R) → R ⧸ I)
Tactic: apply Ideal.map_isPrime_of_surjective
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : Ring R
I : Ideal R
inst_1 : I.IsTwoSided
p : Ideal R
inst_2 : p.IsPrime
hIP : I ≤ p
hf : Surjective (α := R) (β := R ⧸ I) (⇑(Quotient.mk I) : (a : R) → R ⧸ I)
⊢ IsPrime (α := R ⧸ I) (map (F := R →+* R ⧸ I) (Quotient.mk I) p)
have hk : ker (F := R →+* R ⧸ I) (Quotient.mk I) ≤ p
Tactic: apply Ideal.map_isPrime_of_surjective
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₁ : comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
    (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
  I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₂ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
    (comap (F :=
      R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
  I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₃ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
    (RingHom.ker (F :=
      R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
  I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
rw₃ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (RingHom.ker (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
    I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₄ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J) = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
rw₃ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (RingHom.ker (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
    I ⊔ J
rw₄ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J) = I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₅ : J ⊔ comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) ⊥ = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
rw₃ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (RingHom.ker (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
    I ⊔ J
rw₄ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J) = I ⊔ J
rw₅ : J ⊔ comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) ⊥ = I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₆ : J ⊔ RingHom.ker (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
rw₃ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (RingHom.ker (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
    I ⊔ J
rw₄ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J) = I ⊔ J
rw₅ : J ⊔ comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) ⊥ = I ⊔ J
rw₆ : J ⊔ RingHom.ker (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) = I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₇ : J ⊔ I = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
rw : comap (quotQuotMk I J) ⊥ = I ⊔ J
rw₁ :
  comap (F := R →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
      (RingHom.comp (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) (Ideal.Quotient.mk I)) ⊥ =
    I ⊔ J
rw₂ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (comap (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) ⊥) =
    I ⊔ J
rw₃ :
  comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I)
      (RingHom.ker (F :=
        R ⧸ I →+* HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))) =
    I ⊔ J
rw₄ : comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J) = I ⊔ J
rw₅ : J ⊔ comap (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) ⊥ = I ⊔ J
rw₆ : J ⊔ RingHom.ker (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) = I ⊔ J
rw₇ : J ⊔ I = I ⊔ J
⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J
have rw₈ : I ⊔ J = I ⊔ J
Tactic: rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
    comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
    mk_ker, sup_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ (quotQuotToQuotSup I J).comp (liftSupQuotQuotMk I J) = RingHom.id (R ⧸ I ⊔ J)
have h : ((quotQuotToQuotSup I J).comp (liftSupQuotQuotMk I J)).comp (Ideal.Quotient.mk (I ⊔ J)) =
  RingHom.comp (RingHom.id (R ⧸ I ⊔ J)) (Ideal.Quotient.mk (I ⊔ J))
Tactic: apply Ideal.Quotient.ringHom_ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ (quotQuotToQuotSup I J).comp (liftSupQuotQuotMk I J) = RingHom.id (R ⧸ I ⊔ J)
have h : ((quotQuotToQuotSup I J).comp (liftSupQuotQuotMk I J)).comp (Ideal.Quotient.mk (I ⊔ J)) =
  RingHom.comp (RingHom.id (R ⧸ I ⊔ J)) (Ideal.Quotient.mk (I ⊔ J))
Tactic: repeat apply Ideal.Quotient.ringHom_ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ (liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J) =
    RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
have h : ((liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J)).comp
    (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) =
  RingHom.comp
    (RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
    (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
Tactic: apply Ideal.Quotient.ringHom_ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ ((liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J)).comp
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)) =
    RingHom.comp
      (RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
have h : (((liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J)).comp
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))).comp
    (Ideal.Quotient.mk I) =
  RingHom.comp
    (RingHom.comp
      (RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
    (Ideal.Quotient.mk I)
Tactic: apply Ideal.Quotient.ringHom_ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
⊢ (liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J) =
    RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
have h : (((liftSupQuotQuotMk I J).comp (quotQuotToQuotSup I J)).comp
        (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))).comp
    (Ideal.Quotient.mk I) =
  RingHom.comp
    (RingHom.comp
      (RingHom.id (HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
      (Ideal.Quotient.mk (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J)))
    (Ideal.Quotient.mk I)
Tactic: repeat apply Ideal.Quotient.ringHom_ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
h : I ≤ J
⊢ Eq (α := R →+* R ⧸ J)
    (RingHom.comp (↑(quotQuotEquivQuotOfLE h) : (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J →+* R ⧸ J) (quotQuotMk I J))
    (Ideal.Quotient.mk J)
have a : ∀ (x : R),
  Eq (α := R ⧸ J)
    ((RingHom.comp (↑(quotQuotEquivQuotOfLE h) : (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J →+* R ⧸ J) (quotQuotMk I J) :
          (a : R) → R ⧸ J)
        x :
      R ⧸ J)
    ((Ideal.Quotient.mk J : (a : R) → R ⧸ J) x : R ⧸ J)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
inst : CommRing R
I J : Ideal R
h : I ≤ J
⊢ @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
      (↑(quotQuotEquivQuotOfLE h).symm : R ⧸ J →+* (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J) (Ideal.Quotient.mk J) =
    quotQuotMk I J
have a : ∀ (x : R),
  Eq (α := HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
    ((@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
            (↑(quotQuotEquivQuotOfLE h).symm : R ⧸ J →+* (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J) (Ideal.Quotient.mk J) :
          (a : R) → (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J)
        x :
      HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
    ((quotQuotMk I J : (a : R) → (R ⧸ I) ⧸ map (Ideal.Quotient.mk I) J) x :
      HasQuotient.Quotient (R ⧸ I) (B := Ideal (R ⧸ I)) (map (F := R →+* R ⧸ I) (Ideal.Quotient.mk I) J))
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
A : Type u_1
inst : CommSemiring R
inst_1 : CommRing A
inst_2 : Algebra R A
I J : Ideal A
h : I ≤ J
⊢ Eq (α := A ⧸ I →ₐ[R] A ⧸ J)
    (AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h) : (A ⧸ I) ⧸ map (Quotient.mkₐ R I) J →ₐ[R] A ⧸ J)
      (Quotient.mkₐ R (map (F := A →ₐ[R] A ⧸ I) (Quotient.mkₐ R I) J)))
    (Quotient.factorₐ R h)
have H : ∀ (x : A ⧸ I),
  Eq (α := A ⧸ J)
    ((AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h) : (A ⧸ I) ⧸ map (Quotient.mkₐ R I) J →ₐ[R] A ⧸ J)
            (Quotient.mkₐ R (map (F := A →ₐ[R] A ⧸ I) (Quotient.mkₐ R I) J)) :
          (a : A ⧸ I) → A ⧸ J)
        x :
      A ⧸ J)
    ((Quotient.factorₐ R h : (a : A ⧸ I) → A ⧸ J) x : A ⧸ J)
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u
A : Type u_1
inst : CommSemiring R
inst_1 : CommRing A
inst_2 : Algebra R A
I J : Ideal A
h : I ≤ J
x : A ⧸ I
⊢ Eq (α := A ⧸ J)
    ((AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h) : (A ⧸ I) ⧸ map (Quotient.mkₐ R I) J →ₐ[R] A ⧸ J)
            (Quotient.mkₐ R (map (F := A →ₐ[R] A ⧸ I) (Quotient.mkₐ R I) J)) :
          (a : A ⧸ I) → A ⧸ J)
        x :
      A ⧸ J)
    ((Quotient.factorₐ R h : (a : A ⧸ I) → A ⧸ J) x : A ⧸ J)
have H : ∀ (x : A),
  Eq (α := A ⧸ J)
    ((AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h) : (A ⧸ I) ⧸ map (Quotient.mkₐ R I) J →ₐ[R] A ⧸ J)
            (Quotient.mkₐ R (map (F := A →ₐ[R] A ⧸ I) (Quotient.mkₐ R I) J)) :
          (a : A ⧸ I) → A ⧸ J)
        ((Ideal.Quotient.mk I : (a : A) → A ⧸ I) x) :
      A ⧸ J)
    ((Quotient.factorₐ R h : (a : A ⧸ I) → A ⧸ J) ((Ideal.Quotient.mk I : (a : A) → A ⧸ I) x) : A ⧸ J)
Tactic: obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
n : ℕ
⊢ (↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤) ≃ₗ[R]
    ↥(map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n))
have refine_1 : HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤ ≤ Submodule.comap (Submodule.subtype (I ^ n)) (I ^ (n + 1))
Tactic: refine { LinearMap.codRestrict
    (Submodule.restrictScalars _ (Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n)))
    (Submodule.mapQ (I • ⊤) (I ^ (n + 1)) (Submodule.subtype (I ^ n)) ?_) ?_,
    Equiv.ofBijective _ ⟨?_, ?_⟩ with }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
n : ℕ
refine_1 : HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤ ≤ Submodule.comap (Submodule.subtype (I ^ n)) (I ^ (n + 1))
⊢ (↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤) ≃ₗ[R]
    ↥(map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n))
have refine_2 : ∀ (c : ↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤),
  Membership.mem (α := R ⧸ I ^ (n + 1)) (γ := Submodule R (R ⧸ I ^ (n + 1)))
    (Submodule.restrictScalars R (map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n)))
    (((HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤).mapQ (I ^ (n + 1)) (Submodule.subtype (I ^ n)) sorry :
          (a : ↥(I ^ n) ⧸ I • ⊤) → R ⧸ I ^ (n + 1))
        c :
      R ⧸ I ^ (n + 1))
Tactic: refine { LinearMap.codRestrict
    (Submodule.restrictScalars _ (Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n)))
    (Submodule.mapQ (I • ⊤) (I ^ (n + 1)) (Submodule.subtype (I ^ n)) ?_) ?_,
    Equiv.ofBijective _ ⟨?_, ?_⟩ with }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Operations (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
n : ℕ
refine_1 : HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤ ≤ Submodule.comap (Submodule.subtype (I ^ n)) (I ^ (n + 1))
refine_2 :
  ∀ (c : ↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤),
    Membership.mem (α := R ⧸ I ^ (n + 1)) (γ := Submodule R (R ⧸ I ^ (n + 1)))
      (Submodule.restrictScalars R (map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n)))
      (((HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤).mapQ (I ^ (n + 1)) (Submodule.subtype (I ^ n)) ⋯ :
            (a : ↥(I ^ n) ⧸ I • ⊤) → R ⧸ I ^ (n + 1))
          c :
        R ⧸ I ^ (n + 1))
⊢ (↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤) ≃ₗ[R]
    ↥(map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n))
have refine_3 : Function.Injective (β := ↥(map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n)))
  fun (c : ↥(I ^ n) ⧸ HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤) ↦
  Subtype.mk (p := fun (x : R ⧸ I ^ (n + 1)) ↦
    Membership.mem (γ := Submodule R (R ⧸ I ^ (n + 1)))
      (Submodule.restrictScalars R (map (F := R →+* R ⧸ I ^ (n + 1)) (Quotient.mk (I ^ (n + 1))) (I ^ n))) x)
    (((HSMul.hSMul (β := Submodule R ↥(I ^ n)) I ⊤).mapQ (I ^ (n + 1)) (Submodule.subtype (I ^ n)) sorry :
        (a : ↥(I ^ n) ⧸ I • ⊤) → R ⧸ I ^ (n + 1))
      c)
    sorry
Tactic: refine { LinearMap.codRestrict
    (Submodule.restrictScalars _ (Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n)))
    (Submodule.mapQ (I • ⊤) (I ^ (n + 1)) (Submodule.subtype (I ^ n)) ?_) ?_,
    Equiv.ofBijective _ ⟨?_, ?_⟩ with }