awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
x : R ⧸ HPow.hPow (β := ℕ) I 2
hx : x ∈ I.cotangentIdeal • I.cotangentIdeal
refine_1 :
  ∀ r ∈ I.cotangentIdeal,
    ∀ n ∈ I.cotangentIdeal,
      Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ (r • n)
⊢ Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ x
have refine_2 : ∀ (x y : R ⧸ HPow.hPow (β := ℕ) I 2),
  Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ x →
    Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ y →
      Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ (x + y)
Tactic: refine Submodule.smul_induction_on hx ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
x_1 : R ⧸ HPow.hPow (β := ℕ) I 2
hx_1 : x_1 ∈ I.cotangentIdeal • I.cotangentIdeal
x : R
hx : x ∈ (↑I : Set R)
y : R
hy : y ∈ (↑I : Set R)
⊢ Membership.mem (γ := Submodule (R ⧸ HPow.hPow (β := ℕ) I 2) (R ⧸ HPow.hPow (β := ℕ) I 2)) ⊥
    (HSMul.hSMul (α := R ⧸ HPow.hPow (β := ℕ) I 2) (β := R ⧸ HPow.hPow (β := ℕ) I 2)
      ((RingHom.toSemilinearMap (Quotient.mk (HPow.hPow (β := ℕ) I 2)) : (a : R) → R ⧸ I ^ 2) x :
        R ⧸ HPow.hPow (β := ℕ) I 2)
      ((RingHom.toSemilinearMap (Quotient.mk (HPow.hPow (β := ℕ) I 2)) : (a : R) → R ⧸ I ^ 2) y :
        R ⧸ HPow.hPow (β := ℕ) I 2))
have apply : HSub.hSub (α := R) ((fun (x1 x2 : R) ↦ x1 * x2) x y) 0 ∈ HPow.hPow (β := ℕ) I 2
Tactic: apply (Submodule.Quotient.eq _).mpr _
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
x_1 : R ⧸ HPow.hPow (β := ℕ) I 2
hx_1 : x_1 ∈ I.cotangentIdeal • I.cotangentIdeal
x : R
hx : x ∈ (↑I : Set R)
y : R
hy : y ∈ (↑I : Set R)
⊢ HSub.hSub (α := R) ((fun (x1 x2 : R) ↦ x1 * x2) x y) 0 ∈ HPow.hPow (β := ℕ) I 2
have rw : (fun (x1 x2 : R) ↦ x1 * x2) x y ∈ I * I
Tactic: rw [sub_zero, pow_two]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
x : R
⊢ Membership.mem (α := R ⧸ HPow.hPow (β := ℕ) I 2) I.cotangentIdeal
      ((Quotient.mk (HPow.hPow (β := ℕ) I 2) : (a : R) → R ⧸ I ^ 2) x : R ⧸ HPow.hPow (β := ℕ) I 2) ↔
    x ∈ I
have refine : Membership.mem (α := R ⧸ HPow.hPow (β := ℕ) I 2) I.cotangentIdeal
    ((Quotient.mk (HPow.hPow (β := ℕ) I 2) : (a : R) → R ⧸ I ^ 2) x : R ⧸ HPow.hPow (β := ℕ) I 2) →
  ∀ (y : R) (hy : y ∈ (↑I : Set R))
    (e :
      Eq (α := R ⧸ HPow.hPow (β := ℕ) I 2)
        ((RingHom.toSemilinearMap (Quotient.mk (HPow.hPow (β := ℕ) I 2)) : (a : R) → R ⧸ I ^ 2) y :
          R ⧸ HPow.hPow (β := ℕ) I 2)
        ((Quotient.mk (HPow.hPow (β := ℕ) I 2) : (a : R) → R ⧸ I ^ 2) x : R ⧸ HPow.hPow (β := ℕ) I 2)),
    x ∈ I
Tactic: refine ⟨fun ⟨y, hy, e⟩ ↦ ?_, fun h ↦ ⟨x, h, rfl⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
⊢ I.cotangentToQuotientSquare.range = Submodule.restrictScalars R I.cotangentIdeal
have trans : I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range
Tactic: trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
trans : I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range
⊢ I.cotangentToQuotientSquare.range = Submodule.restrictScalars R I.cotangentIdeal
have trans₁ : (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range = Submodule.restrictScalars R I.cotangentIdeal
Tactic: trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
⊢ I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range
have rw : I.cotangentToQuotientSquare.range = Submodule.map I.cotangentToQuotientSquare I.toCotangent.range
Tactic: rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
rw : I.cotangentToQuotientSquare.range = Submodule.map I.cotangentToQuotientSquare I.toCotangent.range
⊢ I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range
have rw₁ : I.cotangentToQuotientSquare.range = Submodule.map I.cotangentToQuotientSquare ⊤
Tactic: rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
rw : I.cotangentToQuotientSquare.range = Submodule.map I.cotangentToQuotientSquare I.toCotangent.range
rw₁ : I.cotangentToQuotientSquare.range = Submodule.map I.cotangentToQuotientSquare ⊤
⊢ I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range
have rw₂ : I.cotangentToQuotientSquare.range = I.cotangentToQuotientSquare.range
Tactic: rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
⊢ (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range = Submodule.restrictScalars R I.cotangentIdeal
have rw : Submodule.map (Submodule.mkQ (HPow.hPow (β := ℕ) I 2)) I = Submodule.restrictScalars R I.cotangentIdeal
Tactic: rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
I : Ideal R
⊢ Submodule.map (Submodule.mkQ (HPow.hPow (β := ℕ) I 2)) I = Submodule.restrictScalars R I.cotangentIdeal
have h : ∀ (x : R ⧸ HPow.hPow (β := ℕ) I 2),
  Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
      (Submodule.map (Submodule.mkQ (HPow.hPow (β := ℕ) I 2)) I) x ↔
    Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2)) (Submodule.restrictScalars R I.cotangentIdeal) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
x : I.Cotangent
⊢ Membership.mem (α := R ⧸ HPow.hPow (β := ℕ) I 2) (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
    (Submodule.restrictScalars R I.cotangentIdeal)
    ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) x : R ⧸ HPow.hPow (β := ℕ) I 2)
have rw : Membership.mem (α := R ⧸ HPow.hPow (β := ℕ) I 2) I.cotangentToQuotientSquare.range
  ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) x : R ⧸ HPow.hPow (β := ℕ) I 2)
Tactic: rw [← range_cotangentToQuotientSquare]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
⊢ I.Cotangent ≃ₗ[R] ↥I.cotangentIdeal
have refine_1 : Function.Injective (β := ↥I.cotangentIdeal) fun (c : I.Cotangent) ↦
  Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
    Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2)) (Submodule.restrictScalars R I.cotangentIdeal) x)
    ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry
Tactic: refine
  { LinearMap.codRestrict (I.cotangentIdeal.restrictScalars R) I.cotangentToQuotientSquare
      fun x => by rw [← range_cotangentToQuotientSquare]; exact LinearMap.mem_range_self _ _,
    Equiv.ofBijective _ ⟨?_, ?_⟩ with }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
refine_1 :
  Function.Injective (β := ↥I.cotangentIdeal) fun (c : I.Cotangent) ↦
    Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
      Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2)) (Submodule.restrictScalars R I.cotangentIdeal) x)
      ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯
⊢ I.Cotangent ≃ₗ[R] ↥I.cotangentIdeal
have refine_2 : Function.Surjective (β := ↥I.cotangentIdeal) fun (c : I.Cotangent) ↦
  Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
    Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2)) (Submodule.restrictScalars R I.cotangentIdeal) x)
    ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry
Tactic: refine
  { LinearMap.codRestrict (I.cotangentIdeal.restrictScalars R) I.cotangentToQuotientSquare
      fun x => by rw [← range_cotangentToQuotientSquare]; exact LinearMap.mem_range_self _ _,
    Equiv.ofBijective _ ⟨?_, ?_⟩ with }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (hypothesis)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
x y : I.Cotangent
e :
  Eq.{u + 1} (α := ↥I.cotangentIdeal)
    ((fun (c : I.Cotangent) ↦
        (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
            Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
              (Submodule.restrictScalars R I.cotangentIdeal) x)
            ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
          ↥(Submodule.restrictScalars R I.cotangentIdeal)))
      x)
    ((fun (c : I.Cotangent) ↦
        (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
            Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
              (Submodule.restrictScalars R I.cotangentIdeal) x)
            ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
          ↥(Submodule.restrictScalars R I.cotangentIdeal)))
      y)
⊢ x = y
have e₁ : Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
  (↑((fun (c : I.Cotangent) ↦
          (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
              Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                (Submodule.restrictScalars R I.cotangentIdeal) x)
              ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
            ↥(Submodule.restrictScalars R I.cotangentIdeal)))
        x) :
    R ⧸ I ^ 2)
  (↑((fun (c : I.Cotangent) ↦
          (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
              Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                (Submodule.restrictScalars R I.cotangentIdeal) x)
              ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
            ↥(Submodule.restrictScalars R I.cotangentIdeal)))
        y) :
    R ⧸ I ^ 2)
Tactic: replace e := congr_arg Subtype.val e
Cleared hypotheses: e
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
x y : I.Cotangent
e :
  Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          x) :
      R ⧸ I ^ 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          y) :
      R ⧸ I ^ 2)
⊢ x = y
have refine_1 : ∀ (x : ↥I)
  (e :
    Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
      (↑((fun (c : I.Cotangent) ↦
              (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                  Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                    (Submodule.restrictScalars R I.cotangentIdeal) x)
                  ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
                ↥(Submodule.restrictScalars R I.cotangentIdeal)))
            ((I.toCotangent : (a : ↥I) → I.Cotangent) x)) :
        R ⧸ I ^ 2)
      (↑((fun (c : I.Cotangent) ↦
              (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                  Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                    (Submodule.restrictScalars R I.cotangentIdeal) x)
                  ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
                ↥(Submodule.restrictScalars R I.cotangentIdeal)))
            y) :
        R ⧸ I ^ 2)),
  (I.toCotangent : (a : ↥I) → I.Cotangent) x = y
Tactic: obtain ⟨x, rfl⟩ := I.toCotangent_surjective x
Cleared hypotheses: x, e
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
y : I.Cotangent
x : ↥I
e :
  Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          ((I.toCotangent : (a : ↥I) → I.Cotangent) x)) :
      R ⧸ I ^ 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          y) :
      R ⧸ I ^ 2)
⊢ (I.toCotangent : (a : ↥I) → I.Cotangent) x = y
have refine_1 : ∀ (y : ↥I)
  (e :
    Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
      (↑((fun (c : I.Cotangent) ↦
              (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                  Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                    (Submodule.restrictScalars R I.cotangentIdeal) x)
                  ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
                ↥(Submodule.restrictScalars R I.cotangentIdeal)))
            ((I.toCotangent : (a : ↥I) → I.Cotangent) x)) :
        R ⧸ I ^ 2)
      (↑((fun (c : I.Cotangent) ↦
              (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                  Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                    (Submodule.restrictScalars R I.cotangentIdeal) x)
                  ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) sorry :
                ↥(Submodule.restrictScalars R I.cotangentIdeal)))
            ((I.toCotangent : (a : ↥I) → I.Cotangent) y)) :
        R ⧸ I ^ 2)),
  Eq (α := I.Cotangent) ((I.toCotangent : (a : ↥I) → I.Cotangent) x : I.Cotangent)
    ((I.toCotangent : (a : ↥I) → I.Cotangent) y : I.Cotangent)
Tactic: obtain ⟨y, rfl⟩ := I.toCotangent_surjective y
Cleared hypotheses: y, e
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
x y : ↥I
e :
  Eq.{u + 1} (α := R ⧸ HPow.hPow (β := ℕ) I 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          ((I.toCotangent : (a : ↥I) → I.Cotangent) x)) :
      R ⧸ I ^ 2)
    (↑((fun (c : I.Cotangent) ↦
            (Subtype.mk (p := fun (x : R ⧸ HPow.hPow (β := ℕ) I 2) ↦
                Membership.mem (γ := Submodule R (R ⧸ HPow.hPow (β := ℕ) I 2))
                  (Submodule.restrictScalars R I.cotangentIdeal) x)
                ((I.cotangentToQuotientSquare : (a : I.Cotangent) → R ⧸ I ^ 2) c) ⋯ :
              ↥(Submodule.restrictScalars R I.cotangentIdeal)))
          ((I.toCotangent : (a : ↥I) → I.Cotangent) y)) :
      R ⧸ I ^ 2)
⊢ Eq (α := I.Cotangent) ((I.toCotangent : (a : ↥I) → I.Cotangent) x : I.Cotangent)
    ((I.toCotangent : (a : ↥I) → I.Cotangent) y : I.Cotangent)
have refine_1 : (↑x : R) - (↑y : R) ∈ HPow.hPow (β := ℕ) I 2
Tactic: rw [I.toCotangent_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
x y : ↥I
e : Eq (α := R ⧸ HPow.hPow (β := ℕ) I 2) (Submodule.Quotient.mk (↑x : R)) (Submodule.Quotient.mk (↑y : R))
⊢ (↑x : R) - (↑y : R) ∈ HPow.hPow (β := ℕ) I 2
have refine_1 : ∀ (e : (↑x : R) - (↑y : R) ∈ HPow.hPow (β := ℕ) I 2), (↑x : R) - (↑y : R) ∈ HPow.hPow (β := ℕ) I 2
Tactic: rwa [Submodule.Quotient.eq] at e
Cleared hypotheses: e
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
A : Type u_1
B : Type u_2
inst_7 : CommRing A
inst_8 : CommRing B
inst_9 : Algebra R A
inst_10 : Algebra R B
f : A →ₐ[R] B
⊢ AlgHom.{u, u_1, u_2} R (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) B
have refine_1 : ∀ a ∈ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2,
  Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) a : B) (0 : B)
Tactic: refine { Ideal.Quotient.lift (RingHom.ker f.toRingHom ^ 2) f.toRingHom ?_ with commutes' := ?_ }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
A : Type u_1
B : Type u_2
inst_7 : CommRing A
inst_8 : CommRing B
inst_9 : Algebra R A
inst_10 : Algebra R B
f : A →ₐ[R] B
refine_1 :
  ∀ a ∈ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2,
    Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) a : B) (0 : B)
⊢ AlgHom.{u, u_1, u_2} R (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) B
have refine_2 : ∀ (r : R),
  OneHom.toFun
      (↑(↑(Quotient.lift (HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) f.toRingHom sorry) :
            A ⧸ RingHom.ker f.toRingHom ^ 2 →* B) :
        OneHom (A ⧸ RingHom.ker f.toRingHom ^ 2) B)
      ((algebraMap R (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) :
          (a : R) → A ⧸ RingHom.ker f.toRingHom ^ 2)
        r) =
    (algebraMap R B : (a : R) → B) r
Tactic: refine { Ideal.Quotient.lift (RingHom.ker f.toRingHom ^ 2) f.toRingHom ?_ with commutes' := ?_ }
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
S : Type v
S' : Type w
inst : CommRing R
inst_1 : CommSemiring S
inst_2 : Algebra S R
inst_3 : CommSemiring S'
inst_4 : Algebra S' R
inst_5 : Algebra S S'
inst_6 : IsScalarTower S S' R
I : Ideal R
A : Type u_1
B : Type u_2
inst_7 : CommRing A
inst_8 : CommRing B
inst_9 : Algebra R A
inst_10 : Algebra R B
f : A →ₐ[R] B
r : R
⊢ OneHom.toFun
      (↑(↑(Quotient.lift (HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) f.toRingHom ⋯) :
            A ⧸ RingHom.ker f.toRingHom ^ 2 →* B) :
        OneHom (A ⧸ RingHom.ker f.toRingHom ^ 2) B)
      ((algebraMap R (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) :
          (a : R) → A ⧸ RingHom.ker f.toRingHom ^ 2)
        r) =
    (algebraMap R B : (a : R) → B) r
have refine_2 : Eq.{u_2 + 1} (α := B) ((f.toRingHom : (a : A) → B) ((algebraMap R A : (a : R) → A) r : A) : B)
  ((algebraMap R B : (a : R) → B) r : B)
Tactic: rw [IsScalarTower.algebraMap_apply R A, RingHom.toFun_eq_coe, Ideal.Quotient.algebraMap_eq,
      Ideal.Quotient.lift_mk]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
A : Type u_1
B : Type u_2
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra R A
inst_4 : Algebra R B
f : A →ₐ[R] B
⊢ RingHom.ker f.kerSquareLift.toRingHom = cotangentIdeal (RingHom.ker f.toRingHom)
have a : RingHom.ker f.kerSquareLift.toRingHom ≤ cotangentIdeal (RingHom.ker f.toRingHom)
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
A : Type u_1
B : Type u_2
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra R A
inst_4 : Algebra R B
f : A →ₐ[R] B
a : RingHom.ker f.kerSquareLift.toRingHom ≤ cotangentIdeal (RingHom.ker f.toRingHom)
⊢ RingHom.ker f.kerSquareLift.toRingHom = cotangentIdeal (RingHom.ker f.toRingHom)
have a₁ : cotangentIdeal (RingHom.ker f.toRingHom) ≤ RingHom.ker f.kerSquareLift.toRingHom
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Cotangent (goal)
Goal:
R : Type u
inst : CommRing R
A : Type u_1
B : Type u_2
inst_1 : CommRing A
inst_2 : CommRing B
inst_3 : Algebra R A
inst_4 : Algebra R B
f : A →ₐ[R] B
x : A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2
hx :
  Membership.mem (γ :=
    Submodule (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)
      (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2))
    (RingHom.ker f.kerSquareLift.toRingHom) x
⊢ Membership.mem (γ :=
    Submodule (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)
      (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2))
    (cotangentIdeal (RingHom.ker f.toRingHom)) x
have a : ∀ (x : A)
  (hx :
    Membership.mem (α := A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) (γ :=
      Submodule (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)
        (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2))
      (RingHom.ker f.kerSquareLift.toRingHom)
      ((Quotient.mk (HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) :
            (a : A) → A ⧸ RingHom.ker f.toRingHom ^ 2)
          x :
        A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)),
  Membership.mem (α := A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) (γ :=
    Submodule (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)
      (A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2))
    (cotangentIdeal (RingHom.ker f.toRingHom))
    ((Quotient.mk (HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2) :
          (a : A) → A ⧸ RingHom.ker f.toRingHom ^ 2)
        x :
      A ⧸ HPow.hPow (α := Ideal A) (β := ℕ) (RingHom.ker f.toRingHom) 2)
Tactic: obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
Cleared hypotheses: x, hx