awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (y : L),
    Membership.mem (γ := FractionalIdeal.{u_3, u} (R := B) B⁰ L)
        (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L) y →
      Membership.mem (γ := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L) (x * y)
⊢ Membership.mem (γ := Submodule A L)
    (comap
      ((↑A :
          {S : Type u_2} →
            {M : Type u} →
              {M₂ : Type u_2} →
                [inst : Semiring A] →
                  [inst_22 : Semiring S] →
                    [inst_23 : AddCommMonoid M] →
                      [inst_24 : AddCommMonoid M₂] →
                        [inst_25 : Module A M] →
                          [inst_26 : Module A M₂] →
                            [inst_27 : Module S M] →
                              [inst_28 : Module S M₂] →
                                [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
        (Algebra.trace K L))
      1)
    x
have rw : ∀ (y : K),
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K) y →
    Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
      (HMul.hMul (α := K) ((Algebra.trace K L : (a : L) → K) x : K) y)
Tactic: rw [Submodule.mem_comap, LinearMap.coe_restrictScalars, ← FractionalIdeal.coe_one,
      ← div_self (G₀ := FractionalIdeal A⁰ K) (a := p) (by simpa using hp),
      FractionalIdeal.mem_coe, FractionalIdeal.mem_div_iff_of_ne_zero (by simpa using hp)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (y : L),
    Membership.mem (γ := FractionalIdeal.{u_3, u} (R := B) B⁰ L)
        (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L) y →
      Membership.mem (γ := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L) (x * y)
⊢ ∀ (y : K),
    Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K) y →
      Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
        (HMul.hMul (α := K) ((Algebra.trace K L : (a : L) → K) x : K) y)
have hx₁ : ∀ (a_1 : B),
  Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
    ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
Tactic: simp only [FractionalIdeal.mem_coeIdeal, forall_exists_index, and_imp,
      forall_apply_eq_imp_iff₂] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : K
hy' : Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K) y
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) ((Algebra.trace K L : (a : L) → K) x : K) y)
have obtain : ∀ (y : A) (hy : y ∈ p)
  (hy' :
    Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
      ((algebraMap A K : (a : A) → K) y : K)),
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
Tactic: obtain ⟨y, hy, rfl : algebraMap A K _ = _⟩ := (FractionalIdeal.mem_coeIdeal _).mp hy'
Cleared hypotheses: y, hy'
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
have obtain : ∀ (z : B) (hz : z ∈ a)
  (hz' :
    (algebraMap B L : (a : B) → L) z =
      HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)),
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
Tactic: obtain ⟨z, hz, hz'⟩ := hx _ (Ideal.mem_map_of_mem _ hy)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
z : B
hz : z ∈ a
hz' :
  (algebraMap B L : (a : B) → L) z =
    HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)
⊢ Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((Algebra.trace K L : (a : L) → K) ((algebraMap B L : (a : B) → L) z : L) : K)
have rw : Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  ((algebraMap A K : (a : A) → K) ((intTrace A B : (a : B) → A) z : A) : K)
Tactic: rw [← algebraMap_intTrace (A := A)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
z : B
hz : z ∈ a
hz' :
  (algebraMap B L : (a : B) → L) z =
    HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
have this₁ : Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  ((Algebra.trace K L : (a : L) → K) ((algebraMap B L : (a : B) → L) z : L) : K)
Tactic: have : trace K L (algebraMap B L z) ∈ (p : FractionalIdeal A⁰ K) := by
      rw [← algebraMap_intTrace (A := A)]
      exact ⟨intTrace A B z, this z hz, rfl⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this_2 : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
z : B
hz : z ∈ a
hz' :
  (algebraMap B L : (a : B) → L) z =
    HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)
this :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((Algebra.trace K L : (a : L) → K) ((algebraMap B L : (a : B) → L) z : L) : K)
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
have rwa : Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  (HMul.hMul (α := K) (β := K) ((algebraMap A K : (a : A) → K) y : K) ((Algebra.trace K L : (a : L) → K) x : K))
Tactic: rwa [mul_comm, ← smul_eq_mul, ← map_smul, Algebra.smul_def, mul_comm,
      ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L, ← hz']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this_2 : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
z : B
hz : z ∈ a
hz' :
  (algebraMap B L : (a : B) → L) z =
    HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)
this :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((Algebra.trace K L : (a : L) → K) ((algebraMap B L : (a : B) → L) z : L) : K)
rwa :
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((algebraMap A K : (a : A) → K) y : K) ((Algebra.trace K L : (a : L) → K) x : K))
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
have rwa₁ : Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  (HSMul.hSMul (α := K) (β := K) ((algebraMap A K : (a : A) → K) y : K) ((Algebra.trace K L : (a : L) → K) x : K))
Tactic: rwa [mul_comm, ← smul_eq_mul, ← map_smul, Algebra.smul_def, mul_comm,
      ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L, ← hz']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : CommRing B
inst_3 : Field L
inst_4 : Algebra A K
inst_5 : Algebra B L
inst_6 : Algebra A B
inst_7 : Algebra K L
inst_8 : Algebra A L
inst_9 : IsScalarTower A K L
inst_10 : IsScalarTower A B L
inst_11 : IsDomain A
inst_12 : IsFractionRing A K
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
inst_15 : IsIntegralClosure B A L
inst_16 : IsFractionRing B L
inst_17 : IsDedekindDomain A
inst_18 : IsDedekindDomain B
inst_19 : IsTorsionFree A B
inst_20 : Module.Finite A B
p : Ideal A
inst_21 : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP_1 : P ^ e ∣ Ideal.map (F := A →+* B) (algebraMap A B) p
a : Ideal B
ha : Ideal.map (F := A →+* B) (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Eq.{u_3 + 1} (α := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this_1 : Ideal.map (F := A →+* B) (algebraMap A B) p ∣ a ^ e
this_2 : ∀ x ∈ a, Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
hP :
  (↑(P ^ (e - 1)) : FractionalIdeal B⁰ L)⁻¹ =
    HDiv.hDiv (α := FractionalIdeal.{u_3, u} (R := B) B⁰ L) (↑a : FractionalIdeal B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
x : L
hx :
  ∀ (a_1 : B),
    Membership.mem (γ := Ideal B) (Ideal.map (F := A →+* B) (algebraMap A B) p) a_1 →
      ∃ x' ∈ a, (algebraMap B L : (a : B) → L) x' = HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) a_1 : L)
y : A
hy : y ∈ p
hy' :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((algebraMap A K : (a : A) → K) y : K)
z : B
hz : z ∈ a
hz' :
  (algebraMap B L : (a : B) → L) z =
    HMul.hMul (β := L) x ((algebraMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y : B) : L)
this :
  Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    ((Algebra.trace K L : (a : L) → K) ((algebraMap B L : (a : B) → L) z : L) : K)
rwa :
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((algebraMap A K : (a : A) → K) y : K) ((Algebra.trace K L : (a : L) → K) x : K))
rwa₁ :
  Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HSMul.hSMul (α := K) (β := K) ((algebraMap A K : (a : A) → K) y : K) ((Algebra.trace K L : (a : L) → K) x : K))
⊢ Membership.mem (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (HMul.hMul (α := K) (β := K) ((Algebra.trace K L : (a : L) → K) x : K) ((algebraMap A K : (a : A) → K) y : K))
have rwa₂ : Membership.mem.{u_2, u_2} (α := K) (γ := FractionalIdeal.{u_1, u_2} (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  ((Algebra.trace K L : (a : L) → K) (HSMul.hSMul.{u_2, u, u} (α := K) ((algebraMap A K : (a : A) → K) y : K) x) : K)
Tactic: rwa [mul_comm, ← smul_eq_mul, ← map_smul, Algebra.smul_def, mul_comm,
      ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L, ← hz']