awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : (i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι)
f : HasQuotient.Quotient R (B := Ideal R) (∏ i ∈ s, P i ^ e i) ≃+* ((i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι)) :=
  quotientEquivPiOfFinsetProdEq (∏ i ∈ s, P i ^ e i) P e prime coprime ⋯
⊢ ∃ (y : R),
    ∀ (i : ι) (hi : i ∈ s),
      Eq (α := R ⧸ P i ^ e i) ((Ideal.Quotient.mk (P i ^ e i) : (a : R) → R ⧸ P i ^ e i) y : R ⧸ P i ^ e i)
        (x (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) :
          R ⧸
            P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
              e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι))
have obtain : ∀ (y : HasQuotient.Quotient R (B := Ideal R) (∏ i ∈ s, P i ^ e i)),
  ∃ (y_1 : R),
    ∀ (i : ι) (hi : i ∈ s),
      Eq (α := R ⧸ P i ^ e i) ((Ideal.Quotient.mk (P i ^ e i) : (a : R) → R ⧸ P i ^ e i) y_1 : R ⧸ P i ^ e i)
        ((f : (a : R ⧸ ∏ i ∈ s, P i ^ e i) → (i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι)) y
            (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) :
          R ⧸
            P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
              e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι))
Tactic: obtain ⟨y, rfl⟩ := f.surjective x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
f : HasQuotient.Quotient R (B := Ideal R) (∏ i ∈ s, P i ^ e i) ≃+* ((i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι)) :=
  quotientEquivPiOfFinsetProdEq (∏ i ∈ s, P i ^ e i) P e prime coprime ⋯
y : HasQuotient.Quotient R (B := Ideal R) (∏ i ∈ s, P i ^ e i)
⊢ ∃ (y_1 : R),
    ∀ (i : ι) (hi : i ∈ s),
      Eq (α := R ⧸ P i ^ e i) ((Ideal.Quotient.mk (P i ^ e i) : (a : R) → R ⧸ P i ^ e i) y_1 : R ⧸ P i ^ e i)
        ((f : (a : R ⧸ ∏ i ∈ s, P i ^ e i) → (i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι)) y
            (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) :
          R ⧸
            P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
              e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι))
have obtain : ∀ (z : R),
  ∃ (y : R),
    ∀ (i : ι) (hi : i ∈ s),
      Eq (α := R ⧸ P i ^ e i) ((Ideal.Quotient.mk (P i ^ e i) : (a : R) → R ⧸ P i ^ e i) y : R ⧸ P i ^ e i)
        ((f : (a : R ⧸ ∏ i ∈ s, P i ^ e i) → (i : ↥s) → R ⧸ P (↑i : ι) ^ e (↑i : ι))
            ((Ideal.Quotient.mk (∏ i ∈ s, P i ^ e i) : (a : R) → R ⧸ ∏ i ∈ s, P i ^ e i) z :
              HasQuotient.Quotient R (B := Ideal R) (∏ i ∈ s, P i ^ e i))
            (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) :
          R ⧸
            P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
              e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι))
Tactic: obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : ↥s → R
⊢ ∃ (y : R), ∀ (i : ι) (hi : i ∈ s), y - x (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) ∈ P i ^ e i
have obtain : ∀ (y : R)
  (hy :
    ∀ (i : ι) (hi : i ∈ s),
      Eq (α := R ⧸ P i ^ e i) ((Ideal.Quotient.mk (P i ^ e i) : (a : R) → R ⧸ P i ^ e i) y : R ⧸ P i ^ e i)
        ((Ideal.Quotient.mk
                (P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
                  e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι)) :
              (a : R) → R ⧸ P (↑⟨i, hi⟩ : ι) ^ e (↑⟨i, hi⟩ : ι))
            (x (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi)) :
          R ⧸
            P (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι) ^
              e (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) : ι))),
  ∃ (y : R), ∀ (i : ι) (hi : i ∈ s), y - x (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) i hi) ∈ P i ^ e i
Tactic: obtain ⟨y, hy⟩ :=
    exists_representative_mod_finset P e prime coprime fun i =>
      Ideal.Quotient.mk _ (x i)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a : R
⊢ Squarefree (R := Ideal R) (span {a}) ↔ Squarefree a
have refine_1 : ∀ (h : Squarefree (R := Ideal R) (span {a})) (x : R) (hx : x * x ∣ a), IsUnit x
Tactic: refine ⟨fun h x hx ↦ ?_, fun h I hI ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a : R
refine_1 : ∀ (h : Squarefree (R := Ideal R) (span {a})) (x : R) (hx : x * x ∣ a), IsUnit x
⊢ Squarefree (R := Ideal R) (span {a}) ↔ Squarefree a
have refine_2 : ∀ (h : Squarefree a) (I : Ideal R) (hI : I * I ∣ span {a}), IsUnit I
Tactic: refine ⟨fun h x hx ↦ ?_, fun h I hI ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a : R
h : Squarefree (R := Ideal R) (span {a})
x : R
hx : x * x ∣ a
⊢ IsUnit x
have hx₁ : HMul.hMul (α := Ideal R) (span {x}) (span {x}) ∣ span {a}
Tactic: rw [← span_singleton_dvd_span_singleton_iff_dvd, ← span_singleton_mul_span_singleton] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a : R
h : Squarefree a
I : Ideal R
hI : I * I ∣ span {a}
⊢ IsUnit I
have hI₁ : Submodule.IsPrincipal.generator I * Submodule.IsPrincipal.generator I ∣ a
Tactic: rw [← span_singleton_generator I, span_singleton_mul_span_singleton,
      span_singleton_dvd_span_singleton_iff_dvd] at hI
Cleared hypotheses: hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have pos : ∀ (hb : b = 0), Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
Tactic: by_cases hb : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
pos : ∀ (hb : b = 0), Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have neg : ∀ (hb : ¬b = 0), Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
Tactic: by_cases hb : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : b = 0
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have pos : Membership.mem (γ := Multiset (Ideal R)) 0 (span {a})
Tactic: rw [span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : b = 0
⊢ Membership.mem (γ := Multiset (Ideal R)) 0 (span {a})
have ha₁ : Membership.mem (γ := Multiset R) 0 a
Tactic: rw [hb, normalizedFactors_zero] at ha
Cleared hypotheses: ha, hb
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (M := Ideal R) (span {a})
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have obtain : ∀ (c : Ideal R) (hc : Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) c)
  (hc' : Associated (span {a}) c), Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
Tactic: obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have neg : ∀ (this : Prime (M := Ideal R) (span {a})), Ne (α := Ideal R) (span {b}) 0
Tactic: suffices Prime (span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
neg : ∀ (this : Prime (M := Ideal R) (span {a})), Ne (α := Ideal R) (span {b}) 0
⊢ Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {b})) (span {a})
have neg₁ : Prime (M := Ideal R) (span {a})
Tactic: suffices Prime (span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ Prime (M := Ideal R) (span {a})
have neg : IsPrime (α := R) (span {a})
Tactic: rw [prime_iff_isPrime]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
neg : IsPrime (α := R) (span {a})
⊢ Prime (M := Ideal R) (span {a})
have neg₁ : Ne (α := Ideal R) (span {a}) ⊥
Tactic: rw [prime_iff_isPrime]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have pos : ∀ (h : FiniteMultiplicity a b), emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
Tactic: by_cases h : FiniteMultiplicity a b
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
pos : ∀ (h : FiniteMultiplicity a b), emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have neg : ∀ (h : ¬FiniteMultiplicity a b), emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
Tactic: by_cases h : FiniteMultiplicity a b
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have pos : emultiplicity (α := Ideal R) (span {a}) (span {b}) = (↑(multiplicity a b) : ℕ∞)
Tactic: rw [h.emultiplicity_eq_multiplicity]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = (↑(multiplicity a b) : ℕ∞)
have pos : HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b) ∣ span {b}
Tactic: apply emultiplicity_eq_of_dvd_of_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b) ∣ span {b}
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = (↑(multiplicity a b) : ℕ∞)
have pos₁ : ¬HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b + 1) ∣ span {b}
Tactic: apply emultiplicity_eq_of_dvd_of_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b) ∣ span {b}
have pos : a ^ multiplicity a b ∣ b
Tactic: rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ ¬HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b + 1) ∣ span {b}
have pos : ¬a ^ (multiplicity a b + 1) ∣ b
Tactic: rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ ¬a ^ (multiplicity a b + 1) ∣ b
have pos : multiplicity a b < multiplicity a b + 1
Tactic: apply h.not_pow_dvd_of_multiplicity_lt
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
this : ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have rw : emultiplicity (α := Ideal R) (span {a}) (span {b}) = ⊤
Tactic: rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
this : ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})
rw : emultiplicity (α := Ideal R) (span {a}) (span {b}) = ⊤
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have rw₁ : Eq (α := ℕ∞) ⊤ ⊤
Tactic: rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
have neg : ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})
Tactic: suffices ¬FiniteMultiplicity (span ({a} : Set R)) (span ({b} : Set R)) by
      rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
n : ℕ
⊢ HPow.hPow (α := Ideal R) (span {a}) n ∣ span {b}
have rw : a ^ n ∣ b
Tactic: rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
have refine_1 : (↑{d : R | d ∈ normalizedFactors r} : Type u_1) →
  (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
Tactic: refine Equiv.ofBijective ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
refine_1 :
  (↑{d : R | d ∈ normalizedFactors r} : Type u_1) →
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
⊢ (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
have refine_2 : Function.Bijective (β :=
  (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
  fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
  Subtype.mk (p := fun (x : Ideal R) ↦
    Membership.mem (γ := Set (Ideal R))
      {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
    (span {(↑d : R)}) sorry
Tactic: refine Equiv.ofBijective ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ Function.Bijective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
have refine_2 : Function.Injective (β :=
  (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
  fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
  Subtype.mk (p := fun (x : Ideal R) ↦
    Membership.mem (γ := Set (Ideal R))
      {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
    (span {(↑d : R)}) sorry
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
refine_2 :
  Function.Injective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
⊢ Function.Bijective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
have refine_2₁ : Function.Surjective (β :=
  (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
  fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
  Subtype.mk (p := fun (x : Ideal R) ↦
    Membership.mem (γ := Set (Ideal R))
      {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
    (span {(↑d : R)}) sorry
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a
b : R
hb : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} b
h :
  Eq.{u_1 + 1} (α :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    ((fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
        (Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
            (span {(↑d : R)}) ⋯ :
          { x : Ideal R //
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha))
    ((fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
        (Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
            (span {(↑d : R)}) ⋯ :
          { x : Ideal R //
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) b hb))
⊢ Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha =
    Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) b hb
have h₁ : Associated a b
Tactic: rw [Subtype.mk_eq_mk, span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
have this : i.IsPrime
Tactic: have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)