awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      ((FractionalIdeal.dual A K 1)⁻¹ : FractionalIdeal (R := B) B⁰ L) (↑Q : FractionalIdeal B⁰ L) ≤
    (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
⊢ False
have H₁ : HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
    (HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      ((FractionalIdeal.dual A K 1)⁻¹ : FractionalIdeal (R := B) B⁰ L) (↑Q : FractionalIdeal B⁰ L)) ≤
  HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
    (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
    (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
Tactic: replace H := mul_le_mul_right H (FractionalIdeal.dual A K 1)
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
        ((FractionalIdeal.dual A K 1)⁻¹ : FractionalIdeal (R := B) B⁰ L) (↑Q : FractionalIdeal B⁰ L)) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
⊢ False
have hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
Tactic: have hne : (1 : FractionalIdeal B⁰ L) ≠ 0 := one_ne_zero
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
        ((FractionalIdeal.dual A K 1)⁻¹ : FractionalIdeal (R := B) B⁰ L) (↑Q : FractionalIdeal B⁰ L)) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
⊢ False
have H₁ : (↑Q : FractionalIdeal B⁰ L) ≤
  HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
    (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
    (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
Tactic: rw [mul_inv_cancel_left₀ (FractionalIdeal.dual_ne_zero A K hne)] at H
Cleared hypotheses: H, hne
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
⊢ False
have neg : Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
Tactic: apply hx
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this_3 : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
this :
  Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)
⊢ Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
have obtain : ∀ (y : A) (hy : y ∈ (↑p : Set A))
  (e :
    Eq (α := K) ((Algebra.linearMap A K : (a : A) → K) y : K)
      (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)),
  Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
Tactic: obtain ⟨y, hy, e⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
y : A
hy : y ∈ (↑p : Set A)
e :
  Eq (α := K) ((Algebra.linearMap A K : (a : A) → K) y : K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)
⊢ Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
have e₁ : y = (intTrace A B : (a : B) → A) x
Tactic: rw [← Algebra.algebraMap_intTrace (A := A), Algebra.linearMap_apply,
      (IsLocalization.injective _ le_rfl).eq_iff] at e
Cleared hypotheses: e
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
⊢ Membership.mem (α := A) p ((intTrace A B : (a : B) → A) x : A)
have neg : Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
  (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)
Tactic: suffices Algebra.trace K L (algebraMap B L x) ∈ (p : FractionalIdeal A⁰ K) by
    obtain ⟨y, hy, e⟩ := this
    rw [← Algebra.algebraMap_intTrace (A := A), Algebra.linearMap_apply,
      (IsLocalization.injective _ le_rfl).eq_iff] at e
    exact e ▸ hy
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)
have neg : ∀ (i : L),
  Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) i →
    ∀ (j : L),
      Membership.mem (γ := FractionalIdeal (R := B) B⁰ L)
          (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L) j →
        Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
          (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) (i * j) : K)
Tactic: refine FractionalIdeal.mul_induction_on (H ⟨_, hxQ, rfl⟩) ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x : B
hxQ : x ∈ Q
hx : ((intTrace A B : (a : B) → A) x : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
neg :
  ∀ (i : L),
    Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) i →
      ∀ (j : L),
        Membership.mem (γ := FractionalIdeal (R := B) B⁰ L)
            (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L) j →
          Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
            (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) (i * j) : K)
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) ((algebraMap B L : (a : B) → L) x) : K)
have neg₁ : ∀ (x y : L),
  Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
      (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) x : K) →
    Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
        (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) y : K) →
      Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
        (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) (x + y) : K)
Tactic: refine FractionalIdeal.mul_induction_on (H ⟨_, hxQ, rfl⟩) ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x_1 : B
hxQ : x_1 ∈ Q
hx_1 : ((intTrace A B : (a : B) → A) x_1 : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
y_1 y : B
h : Membership.mem (γ := Set B) ((⇑(algebraMap A B) : (a : A) → B) '' (↑p : Set A)) y
x : L
hx : Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) x
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) y : L)) :
      K)
have neg : ∀ (y : A) (hy : y ∈ (↑p : Set A)),
  Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
      K)
Tactic: obtain ⟨y, hy, rfl⟩ := h
Cleared hypotheses: y, h
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x_1 : B
hxQ : x_1 ∈ Q
hx_1 : ((intTrace A B : (a : B) → A) x_1 : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
y_1 : B
x : L
hx : Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) x
y : A
hy : y ∈ (↑p : Set A)
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
      K)
have neg : ∀ (z : A)
  (hz :
    Eq (α := K) (((algebraMap A K : A →+* K) : (a : A) → K) z : K)
      ((((traceForm K L : (a : L) → L →ₗ[K] K) x : L →ₗ[K] K) : (a : L) → K) 1 : K)),
  Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
      K)
Tactic: obtain ⟨z, hz⟩ :=
        (FractionalIdeal.mem_dual (by simp)).mp hx 1 ⟨1, trivial, (algebraMap B L).map_one⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (hypothesis)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x_1 : B
hxQ : x_1 ∈ Q
hx_1 : ((intTrace A B : (a : B) → A) x_1 : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
y_1 : B
x : L
hx : Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) x
y : A
hy : y ∈ (↑p : Set A)
z : A
hz :
  Eq (α := K) (((algebraMap A K : A →+* K) : (a : A) → K) z : K)
    ((((traceForm K L : (a : L) → L →ₗ[K] K) x : L →ₗ[K] K) : (a : L) → K) 1 : K)
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
      K)
have hz₁ : Eq (α := K) (((algebraMap A K : A →+* K) : (a : A) → K) z : K) (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) x : K)
Tactic: simp only [Algebra.traceForm_apply, mul_one] at hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Different (goal)
Goal:
A : Type u_1
B : Type u_3
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
inst_3 : IsDomain A
inst_4 : IsDedekindDomain A
inst_5 : IsDedekindDomain B
inst_6 : IsTorsionFree A B
inst_7 : Module.Finite A B
inst_8 : Algebra.IsSeparable.{u_1, u_3} (FractionRing A) (FractionRing B)
p : Ideal A
P Q : Ideal B
hP : P * Q = Ideal.map (F := A →+* B) (algebraMap A B) p
x_1 : B
hxQ : x_1 ∈ Q
hx_1 : ((intTrace A B : (a : B) → A) x_1 : A) ∉ p
hp : ¬p = ⊥
this_1 : Algebra.{u_1, u_3} (A ⧸ p) (B ⧸ Q) := Ideal.Quotient.algebraQuotientOfLEComap ⋯
K : Type u_1 := FractionRing A
L : Type u_3 := FractionRing B
this_2 : IsLocalization (algebraMapSubmonoid.{u_1, u_3} (R := A) B A⁰) L
this : FiniteDimensional K L
H :
  (↑Q : FractionalIdeal B⁰ L) ≤
    HMul.hMul (α := FractionalIdeal (R := B) B⁰ L) (β := FractionalIdeal (R := B) B⁰ L)
      (FractionalIdeal.dual A K 1 : FractionalIdeal (R := B) B⁰ L)
      (↑(Ideal.map (F := A →+* B) (algebraMap A B) p) : FractionalIdeal B⁰ L)
hne : Ne.{u_3 + 1} (α := FractionalIdeal (R := B) B⁰ L) 1 0
y_1 : B
x : L
hx : Membership.mem (γ := FractionalIdeal (R := B) B⁰ L) (FractionalIdeal.dual A K 1) x
y : A
hy : y ∈ (↑p : Set A)
z : A
hz :
  Eq (α := K) (((algebraMap A K : A →+* K) : (a : A) → K) z : K) (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K) x : K)
⊢ Membership.mem (α := K) (γ := FractionalIdeal (R := A) A⁰ K) (↑p : FractionalIdeal A⁰ K)
    (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
        (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
      K)
have neg : Eq (α := K) ((Algebra.linearMap A K : (a : A) → K) (z * y) : K)
  (((Algebra.trace K L : L →ₗ[K] K) : (a : L) → K)
      (HMul.hMul (β := L) x ((Algebra.linearMap B L : (a : B) → L) ((algebraMap A B : (a : A) → B) y) : L)) :
    K)
Tactic: refine ⟨z * y, Ideal.mul_mem_left _ _ hy, ?_⟩