l3lab/ntp-mathlib huggingface dataset:
/-- This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below.
-/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
    I / J = I * J⁻¹ := by
  by_cases hJ : J = 0
  · rw [hJ, div_zero, inv_zero', mul_zero]
  refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
  · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
    

Proof state:
case neg.refine'_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
I J : FractionalIdeal A⁰ K
hJ : ¬J = 0
⊢ ∀ i ∈ I / J, ∀ j ∈ J, i * j ∈ I

Next tactic:
intro x hx y hy
/-- This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below.
-/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
    I / J = I * J⁻¹ := by
  by_cases hJ : J = 0
  · rw [hJ, div_zero, inv_zero', mul_zero]
  refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
  · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
    intro x hx y hy
    

Proof state:
case neg.refine'_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
I J : FractionalIdeal A⁰ K
hJ : ¬J = 0
x : K
hx : x ∈ I / J
y : K
hy : y ∈ J
⊢ x * y ∈ I

Next tactic:
rw [mem_div_iff_of_nonzero hJ] at hx
/-- This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below.
-/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
    I / J = I * J⁻¹ := by
  by_cases hJ : J = 0
  · rw [hJ, div_zero, inv_zero', mul_zero]
  refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
  · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
    intro x hx y hy
    rw [mem_div_iff_of_nonzero hJ] at hx
    

Proof state:
case neg.refine'_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
I J : FractionalIdeal A⁰ K
hJ : ¬J = 0
x : K
hx : ∀ y ∈ J, x * y ∈ I
y : K
hy : y ∈ J
⊢ x * y ∈ I

Next tactic:
exact hx y hy
/-- This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below.
-/
protected theorem div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
    I / J = I * J⁻¹ := by
  by_cases hJ : J = 0
  · rw [hJ, div_zero, inv_zero', mul_zero]
  refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)
  · rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
    intro x hx y hy
    rw [mem_div_iff_of_nonzero hJ] at hx
    exact hx y hy
  

Proof state:
case neg.refine'_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
I J : FractionalIdeal A⁰ K
hJ : ¬J = 0
⊢ I * J⁻¹ * J ≤ I

Next tactic:
rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]
/-- Fractional ideals have cancellative multiplication in a Dedekind domain.

Although this instance is a direct consequence of the instance
`FractionalIdeal.semifield`, we define this instance to provide
a computable alternative.
-/
-- Porting note: added noncomputable because otherwise it fails, so it seems that the goal
-- is not achieved...
noncomputable instance FractionalIdeal.cancelCommMonoidWithZero :
    CancelCommMonoidWithZero (FractionalIdeal A⁰ K) :=
  { @FractionalIdeal.commSemiring A _ A⁰ K _ _,  -- Project out the computable fields first.
    (by 

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
src✝ : CommSemiring (FractionalIdeal A⁰ K) := commSemiring
⊢ CancelCommMonoidWithZero (FractionalIdeal A⁰ K)

Next tactic:
infer_instance
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
⊢ I ∣ J

Next tactic:
by_cases hI : I = ⊥
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : I = ⊥
⊢ I ∣ J

Next tactic:
have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by 

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : I = ⊥
⊢ J = ⊥

Next tactic:
rwa [hI, ← eq_bot_iff] at h
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : I = ⊥
hJ : J = ⊥
⊢ I ∣ J

Next tactic:
rw [hI, hJ]
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
⊢ I ∣ J

Next tactic:
have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
    

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
⊢ I ∣ J

Next tactic:
have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
        (le_of_eq (inv_mul_cancel hI'))
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
    have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
        (le_of_eq (inv_mul_cancel hI'))
    

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
this : (↑I)⁻¹ * ↑J ≤ 1
⊢ I ∣ J

Next tactic:
obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
    have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
        (le_of_eq (inv_mul_cancel hI'))
    obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this
    

Proof state:
case neg.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
this : (↑I)⁻¹ * ↑J ≤ 1
H : Ideal A
hH : ↑H = (↑I)⁻¹ * ↑J
⊢ I ∣ J

Next tactic:
use H
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
    have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
        (le_of_eq (inv_mul_cancel hI'))
    obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this
    use H
    

Proof state:
case h
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
this : (↑I)⁻¹ * ↑J ≤ 1
H : Ideal A
hH : ↑H = (↑I)⁻¹ * ↑J
⊢ J = I * H

Next tactic:
refine coeIdeal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from ?_)
/-- For ideals in a Dedekind domain, to divide is to contain. -/
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
  ⟨Ideal.le_of_dvd, fun h => by
    by_cases hI : I = ⊥
    · have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
      rw [hI, hJ]
    have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
    have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
      le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
        (le_of_eq (inv_mul_cancel hI'))
    obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this
    use H
    refine coeIdeal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from ?_)
    

Proof state:
case h
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
this : (↑I)⁻¹ * ↑J ≤ 1
H : Ideal A
hH : ↑H = (↑I)⁻¹ * ↑J
⊢ ↑J = ↑(I * H)

Next tactic:
rw [coeIdeal_mul, hH, ← mul_assoc, mul_inv_cancel hI', one_mul]
theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
  ⟨fun ⟨hI, H, hunit, hmul⟩ =>
    lt_of_le_of_ne (Ideal.dvd_iff_le.mp ⟨H, hmul⟩)
      (mt
        (fun h =>
          have : H = 1 := mul_left_cancel₀ hI (by 

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : Ideal A
x✝ : DvdNotUnit I J
hI : I ≠ 0
H : Ideal A
hunit : ¬IsUnit H
hmul : J = I * H
h : J = I
⊢ I * H = I * 1

Next tactic:
rw [← hmul, h, mul_one]
instance : WfDvdMonoid (Ideal A) where
  wellFounded_dvdNotUnit := by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ WellFounded DvdNotUnit

Next tactic:
have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindRing.toIsNoetherian)
instance : WfDvdMonoid (Ideal A) where
  wellFounded_dvdNotUnit := by
    have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindRing.toIsNoetherian)
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
this : WellFounded fun x x_1 => x > x_1
⊢ WellFounded DvdNotUnit

Next tactic:
convert this
instance : WfDvdMonoid (Ideal A) where
  wellFounded_dvdNotUnit := by
    have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindRing.toIsNoetherian)
    convert this
    

Proof state:
case h.e'_2.h.h.h.e
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
this : WellFounded fun x x_1 => x > x_1
x✝¹ x✝ : Ideal A
⊢ DvdNotUnit = GT.gt

Next tactic:
ext
instance : WfDvdMonoid (Ideal A) where
  wellFounded_dvdNotUnit := by
    have : WellFounded ((· > ·) : Ideal A → Ideal A → Prop) :=
      isNoetherian_iff_wellFounded.mp (isNoetherianRing_iff.mp IsDedekindRing.toIsNoetherian)
    convert this
    ext
    

Proof state:
case h.e'_2.h.h.h.e.h.h.a
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
this : WellFounded fun x x_1 => x > x_1
x✝³ x✝² x✝¹ x✝ : Ideal A
⊢ DvdNotUnit x✝¹ x✝ ↔ x✝¹ > x✝

Next tactic:
rw [Ideal.dvdNotUnit_iff_lt]
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ ∀ {a : Ideal A}, Irreducible a ↔ Prime a

Next tactic:
intro P
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
⊢ Irreducible P ↔ Prime P

Next tactic:
exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
        rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
          SetLike.le_def]
        contrapose!
        rintro ⟨⟨x, x_mem, x_not_mem⟩, ⟨y, y_mem, y_not_mem⟩⟩
        exact
          ⟨x * y, Ideal.mul_mem_mul x_mem y_mem,
            mt this.isPrime.mem_or_mem (not_or_of_not x_not_mem y_not_mem)⟩⟩, Prime.irreducible⟩
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
⊢ P ∣ I * J → P ∣ I ∨ P ∣ J

Next tactic:
have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
⊢ IsMaximal P

Next tactic:
refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
⊢ ∀ (b : Ideal A), P < b → b = ⊤

Next tactic:
intro J hJ
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J✝ J : Ideal A
hJ : P < J
⊢ J = ⊤

Next tactic:
obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          

Proof state:
case intro.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J✝ J : Ideal A
hJ : P < J
_J_ne : J ≠ 0
H : Ideal A
hunit : ¬IsUnit H
P_eq : P = J * H
⊢ J = ⊤

Next tactic:
exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
        

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
this : IsMaximal P
⊢ P ∣ I * J → P ∣ I ∨ P ∣ J

Next tactic:
rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
          SetLike.le_def]
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
        rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
          SetLike.le_def]
        

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
this : IsMaximal P
⊢ (∀ ⦃x : A⦄, x ∈ I * J → x ∈ P) → (∀ ⦃x : A⦄, x ∈ I → x ∈ P) ∨ ∀ ⦃x : A⦄, x ∈ J → x ∈ P

Next tactic:
contrapose!
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
        rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
          SetLike.le_def]
        contrapose!
        

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
this : IsMaximal P
⊢ ((∃ x ∈ I, x ∉ P) ∧ ∃ x ∈ J, x ∉ P) → ∃ x ∈ I * J, x ∉ P

Next tactic:
rintro ⟨⟨x, x_mem, x_not_mem⟩, ⟨y, y_mem, y_not_mem⟩⟩
instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
  { irreducible_iff_prime := by
      intro P
      exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_unit, fun I J => by
        have : P.IsMaximal := by
          refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_unit, ?_⟩⟩
          intro J hJ
          obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
          exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
        rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
          SetLike.le_def]
        contrapose!
        rintro ⟨⟨x, x_mem, x_not_mem⟩, ⟨y, y_mem, y_not_mem⟩⟩
        

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
this : IsMaximal P
x : A
x_mem : x ∈ I
x_not_mem : x ∉ P
y : A
y_mem : y ∈ J
y_not_mem : y ∉ P
⊢ ∃ x ∈ I * J, x ∉ P

Next tactic:
exact
          ⟨x * y, Ideal.mul_mem_mul x_mem y_mem,
            mt this.isPrime.mem_or_mem (not_or_of_not x_not_mem y_not_mem)⟩
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
h : Prime P
⊢ IsPrime P

Next tactic:
refine ⟨?_, fun hxy => ?_⟩
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  refine ⟨?_, fun hxy => ?_⟩
  · 

Proof state:
case refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
h : Prime P
⊢ P ≠ ⊤

Next tactic:
rintro rfl