awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ Eq (α := Ideal B) (differentIdeal R₁ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
have a : Dvd.dvd (α := Ideal B) (differentIdeal R₁ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
Tactic: apply dvd_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
a : Dvd.dvd (α := Ideal B) (differentIdeal R₁ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ Eq (α := Ideal B) (differentIdeal R₁ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
have a₁ : Dvd.dvd (α := Ideal B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)) (differentIdeal R₁ B)
Tactic: apply dvd_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (hypothesis)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
have this : differentIdeal A B =
  HMul.hMul (α := Ideal B) (differentIdeal R₂ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
Tactic: have := differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
this :
  differentIdeal A B =
    HMul.hMul (α := Ideal B) (differentIdeal R₂ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
have rwa : ∀ (this : differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₂ B) (differentIdeal R₁ B)),
  differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
Tactic: rwa [← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃,
    mul_comm] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
this :
  differentIdeal A B =
    HMul.hMul (α := Ideal B) (differentIdeal R₂ B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
rwa :
  ∀ (this : differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₂ B) (differentIdeal R₁ B)),
    differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
⊢ differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
have rwa₁ : ∀ (this : differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)),
  differentIdeal A B = HMul.hMul (α := Ideal B) (differentIdeal R₁ B) (differentIdeal R₂ B)
Tactic: rwa [← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃,
    mul_comm] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
inst_56 : IsLocalization.{u_6, u_4} (algebraMapSubmonoid.{u_1, u_6} (R := A) R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ Algebra.IsSeparable.{u_1, u_6} (FractionRing A) (FractionRing R₂)
have refine : Eq (α := RingHom.{u_3, u_6} K (FractionRing R₂))
  (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap (FractionRing A) (FractionRing R₂))
    (↑(AlgEquiv.toRingEquiv (AlgEquiv.symm (FractionRing.algEquiv A K))) : K →+* FractionRing A))
  (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
    (↑(AlgEquiv.toRingEquiv (AlgEquiv.symm (FractionRing.algEquiv R₂ ↥F₂))) : ↥F₂ →+* FractionRing R₂)
    (algebraMap K ↥F₂))
Tactic: refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv
          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (hypothesis)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
inst_56 : IsLocalization.{u_6, u_4} (algebraMapSubmonoid.{u_1, u_6} (R := A) R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
⊢ Eq (α := Submodule R₁ L) (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
have this : Algebra.IsSeparable.{u_1, u_6} (FractionRing A) (FractionRing R₂)
Tactic: have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by
    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv
          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_
    ext x
    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm
      (FractionRing.algEquiv R₂ ↥F₂).symm _
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
inst_56 : IsLocalization.{u_6, u_4} (algebraMapSubmonoid.{u_1, u_6} (R := A) R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
this_1 : Algebra.IsSeparable.{u_1, u_6} (FractionRing A) (FractionRing R₂)
this :
  LE.le (α := Submodule B L) (span B ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (traceDual R₁ (↥F₁) 1)
⊢ Eq (α := Submodule R₁ L) (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
have a : LE.le (α := Submodule R₁ L) (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
  (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.LinearDisjoint (goal)
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
inst_56 : IsLocalization.{u_6, u_4} (algebraMapSubmonoid.{u_1, u_6} (R := A) R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
this_1 : Algebra.IsSeparable.{u_1, u_6} (FractionRing A) (FractionRing R₂)
this :
  LE.le (α := Submodule B L) (span B ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (traceDual R₁ (↥F₁) 1)
a :
  LE.le (α := Submodule R₁ L) (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
⊢ Eq (α := Submodule R₁ L) (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
have a₁ : LE.le (α := Submodule R₁ L) (restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (traceDual R₁ (↥F₁) 1))
  (span R₁ ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
Tactic: apply le_antisymm