awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hIJ : I ≤ J
hI : I ≠ 0
obtain : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : J * I ≠ 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
⊢ ∃ (a : R), I ⊔ Ideal.span {a} = J
have obtain₁ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : ¬J = 0 ∧ ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
Tactic: classical
  obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
  simp only [ne_eq, mul_eq_zero, not_or] at hI
  obtain ⟨hJ, hI⟩ := hI
  suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
    obtain ⟨a, K, e, e'⟩ := this
    exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
  let s := (I.finite_factors hI).toFinset
  have : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal := by
    intro p hps
    conv_rhs => rw [← mul_one (J * _)]
    rw [Finset.prod_eq_mul_prod_diff_singleton_of_mem hps, ← mul_assoc,
      mul_right_comm _ p.asIdeal]
    refine mul_lt_mul_of_pos_left ?_ ?_
    · rw [Ideal.one_eq_top, lt_top_iff_ne_top]
      exact p.2.ne_top
    · rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]
      exact ⟨hJ, fun x _ ↦ x.3⟩
  choose! a ha ha' using fun p hps ↦ SetLike.exists_of_lt (this p hps)
  obtain ⟨K, hK⟩ : J ∣ Ideal.span {∑ p ∈ s, a p} := by
    rw [Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem]
    exact sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)
  refine ⟨_, _, hK.symm, ?_⟩
  by_contra H
  obtain ⟨p, hp, h⟩ := Ideal.exists_le_maximal _ H
  let p' : HeightOneSpectrum R := ⟨p, hp.isPrime, fun e ↦ hI (by simp_all)⟩
  have hp's : p' ∈ s := by simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h
  have H₁ : J * K ≤ J * p := Ideal.mul_mono_right (le_sup_right.trans h)
  replace H₁ := hK.trans_le H₁ (Ideal.mem_span_singleton_self _)
  have H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p := by
    refine sum_mem fun q hq ↦ ?_
    rw [Finset.mem_sdiff, Finset.mem_singleton] at hq
    refine Ideal.mul_mono_right ?_ (ha q hq.1)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := p') (by simpa [hp's] using Ne.symm hq.2))
  apply ha' _ hp's
  have := IsDedekindDomain.inf_pow_eq_prod_of_prime s (fun i ↦ i.asIdeal) (fun _ ↦ 1)
    (fun i _ ↦ i.prime) (fun i _ j _ e ↦ mt HeightOneSpectrum.ext e)
  simp only [pow_one] at this
  have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩
  rw [← this, Finset.inf_eq_iInf, iInf_subtype', Ideal.mul_iInf, Ideal.mem_iInf]
  rintro ⟨q, hq⟩
  by_cases hqp : q = p'
  · subst hqp
    convert! sub_mem H₁ H₂
    rw [Finset.sum_eq_add_sum_diff_singleton_of_mem hp's, add_sub_cancel_right]
  · refine Ideal.mul_mono_right ?_ (ha p' hp's)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := q) (by simpa [hq] using hqp))
Cleared hypotheses: I, hIJ, hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hIJ : I ≤ J
hI : I ≠ 0
obtain : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : J * I ≠ 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₁ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : ¬J = 0 ∧ ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
⊢ ∃ (a : R), I ⊔ Ideal.span {a} = J
have obtain₂ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
Tactic: classical
  obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
  simp only [ne_eq, mul_eq_zero, not_or] at hI
  obtain ⟨hJ, hI⟩ := hI
  suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
    obtain ⟨a, K, e, e'⟩ := this
    exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
  let s := (I.finite_factors hI).toFinset
  have : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal := by
    intro p hps
    conv_rhs => rw [← mul_one (J * _)]
    rw [Finset.prod_eq_mul_prod_diff_singleton_of_mem hps, ← mul_assoc,
      mul_right_comm _ p.asIdeal]
    refine mul_lt_mul_of_pos_left ?_ ?_
    · rw [Ideal.one_eq_top, lt_top_iff_ne_top]
      exact p.2.ne_top
    · rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]
      exact ⟨hJ, fun x _ ↦ x.3⟩
  choose! a ha ha' using fun p hps ↦ SetLike.exists_of_lt (this p hps)
  obtain ⟨K, hK⟩ : J ∣ Ideal.span {∑ p ∈ s, a p} := by
    rw [Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem]
    exact sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)
  refine ⟨_, _, hK.symm, ?_⟩
  by_contra H
  obtain ⟨p, hp, h⟩ := Ideal.exists_le_maximal _ H
  let p' : HeightOneSpectrum R := ⟨p, hp.isPrime, fun e ↦ hI (by simp_all)⟩
  have hp's : p' ∈ s := by simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h
  have H₁ : J * K ≤ J * p := Ideal.mul_mono_right (le_sup_right.trans h)
  replace H₁ := hK.trans_le H₁ (Ideal.mem_span_singleton_self _)
  have H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p := by
    refine sum_mem fun q hq ↦ ?_
    rw [Finset.mem_sdiff, Finset.mem_singleton] at hq
    refine Ideal.mul_mono_right ?_ (ha q hq.1)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := p') (by simpa [hp's] using Ne.symm hq.2))
  apply ha' _ hp's
  have := IsDedekindDomain.inf_pow_eq_prod_of_prime s (fun i ↦ i.asIdeal) (fun _ ↦ 1)
    (fun i _ ↦ i.prime) (fun i _ j _ e ↦ mt HeightOneSpectrum.ext e)
  simp only [pow_one] at this
  have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩
  rw [← this, Finset.inf_eq_iInf, iInf_subtype', Ideal.mul_iInf, Ideal.mem_iInf]
  rintro ⟨q, hq⟩
  by_cases hqp : q = p'
  · subst hqp
    convert! sub_mem H₁ H₂
    rw [Finset.sum_eq_add_sum_diff_singleton_of_mem hp's, add_sub_cancel_right]
  · refine Ideal.mul_mono_right ?_ (ha p' hp's)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := q) (by simpa [hq] using hqp))
Cleared hypotheses: I, hIJ, hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hIJ : I ≤ J
hI : I ≠ 0
obtain : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : J * I ≠ 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₁ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : ¬J = 0 ∧ ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₂ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
⊢ ∃ (a : R), I ⊔ Ideal.span {a} = J
have obtain₃ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
  ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
Tactic: classical
  obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
  simp only [ne_eq, mul_eq_zero, not_or] at hI
  obtain ⟨hJ, hI⟩ := hI
  suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
    obtain ⟨a, K, e, e'⟩ := this
    exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
  let s := (I.finite_factors hI).toFinset
  have : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal := by
    intro p hps
    conv_rhs => rw [← mul_one (J * _)]
    rw [Finset.prod_eq_mul_prod_diff_singleton_of_mem hps, ← mul_assoc,
      mul_right_comm _ p.asIdeal]
    refine mul_lt_mul_of_pos_left ?_ ?_
    · rw [Ideal.one_eq_top, lt_top_iff_ne_top]
      exact p.2.ne_top
    · rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]
      exact ⟨hJ, fun x _ ↦ x.3⟩
  choose! a ha ha' using fun p hps ↦ SetLike.exists_of_lt (this p hps)
  obtain ⟨K, hK⟩ : J ∣ Ideal.span {∑ p ∈ s, a p} := by
    rw [Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem]
    exact sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)
  refine ⟨_, _, hK.symm, ?_⟩
  by_contra H
  obtain ⟨p, hp, h⟩ := Ideal.exists_le_maximal _ H
  let p' : HeightOneSpectrum R := ⟨p, hp.isPrime, fun e ↦ hI (by simp_all)⟩
  have hp's : p' ∈ s := by simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h
  have H₁ : J * K ≤ J * p := Ideal.mul_mono_right (le_sup_right.trans h)
  replace H₁ := hK.trans_le H₁ (Ideal.mem_span_singleton_self _)
  have H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p := by
    refine sum_mem fun q hq ↦ ?_
    rw [Finset.mem_sdiff, Finset.mem_singleton] at hq
    refine Ideal.mul_mono_right ?_ (ha q hq.1)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := p') (by simpa [hp's] using Ne.symm hq.2))
  apply ha' _ hp's
  have := IsDedekindDomain.inf_pow_eq_prod_of_prime s (fun i ↦ i.asIdeal) (fun _ ↦ 1)
    (fun i _ ↦ i.prime) (fun i _ j _ e ↦ mt HeightOneSpectrum.ext e)
  simp only [pow_one] at this
  have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩
  rw [← this, Finset.inf_eq_iInf, iInf_subtype', Ideal.mul_iInf, Ideal.mem_iInf]
  rintro ⟨q, hq⟩
  by_cases hqp : q = p'
  · subst hqp
    convert! sub_mem H₁ H₂
    rw [Finset.sum_eq_add_sum_diff_singleton_of_mem hp's, add_sub_cancel_right]
  · refine Ideal.mul_mono_right ?_ (ha p' hp's)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := q) (by simpa [hq] using hqp))
Cleared hypotheses: I, hIJ, hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
x : R
hxI : x ∈ I
hx : x ≠ 0
⊢ ∃ (y : R), I = Ideal.span {x, y}
have obtain : ∀ (y : R) (hxI : Membership.mem (γ := Ideal R) (Ideal.span {x} ⊔ Ideal.span {y}) x),
  ∃ (y_1 : R), Eq (α := Ideal R) (Ideal.span {x} ⊔ Ideal.span {y}) (Ideal.span {x, y_1})
Tactic: obtain ⟨y, rfl⟩ := exists_sup_span_eq (I.span_singleton_le_iff_mem.mpr hxI) (by simpa)
Cleared hypotheses: I, hxI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
x : R
hx : x ≠ 0
y : R
hxI : Membership.mem (γ := Ideal R) (Ideal.span {x} ⊔ Ideal.span {y}) x
⊢ ∃ (y_1 : R), Eq (α := Ideal R) (Ideal.span {x} ⊔ Ideal.span {y}) (Ideal.span {x, y_1})
have simp_rw : ∃ (y_1 : R), Eq (α := Ideal R) (Ideal.span {y, x}) (Ideal.span {y_1, x})
Tactic: simp_rw [← Ideal.span_union, Set.union_singleton, Set.pair_comm x]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
x : R
hx : x ≠ 0
y : R
hxI : Membership.mem (γ := Ideal R) (Ideal.span {x} ⊔ Ideal.span {y}) x
w : R
⊢ ∃ (y_1 : R), Eq (α := Ideal R) (Ideal.span {y, x}) (Ideal.span {y_1, x})
have h : Eq (α := Ideal R) (Ideal.span {y, x}) (Ideal.span {y, x})
Tactic: use y
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
a b c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
⊢ ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
have inr : ∀
  (this :
    ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_5 : Field K] [inst_6 : Algebra R K]
      [inst_7 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0)
      (hb : b ≠ 0) (hb' : b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c)
  (hb' : ¬b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
Tactic: wlog hb' : b = 1
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
a b c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
inr :
  ∀
    (this :
      ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_5 : Field K] [inst_6 : Algebra R K]
        [inst_7 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0)
        (hb : b ≠ 0) (hb' : b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c)
    (hb' : ¬b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
⊢ ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
have wlog : ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_5 : Field K] [inst_6 : Algebra R K]
  [inst_7 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0)
  (hb : b ≠ 0) (hb' : b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
Tactic: wlog hb' : b = 1
Cleared hypotheses: inst_4, a, b, c, hac, ha, hb
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
a b c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
this :
  ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_5 : Field K] [inst_6 : Algebra R K]
    [inst_7 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0)
    (hb : b ≠ 0) (hb' : b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
hb' : ¬b = 1
⊢ ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
have inr : ∀ (x : K)
  (e :
    b⁻¹ * a + HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (FractionalIdeal.spanSingleton R⁰ x) 1 =
      b⁻¹ * c),
  ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
Tactic: obtain ⟨x, e⟩ := this (a := b⁻¹ * a) (b := 1) (c := b⁻¹ * c) (by gcongr) (by simp [ha, hb])
      one_ne_zero rfl
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
a b c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
this :
  ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_5 : Field K] [inst_6 : Algebra R K]
    [inst_7 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K} (hac : a ≤ c) (ha : a ≠ 0)
    (hb : b ≠ 0) (hb' : b = 1), ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
hb' : ¬b = 1
x : K
e : b⁻¹ * a + HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (FractionalIdeal.spanSingleton R⁰ x) 1 = b⁻¹ * c
⊢ ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
have h : a + FractionalIdeal.spanSingleton R⁰ x * b = c
Tactic: use x
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a b c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
hb' : b = 1
⊢ ∃ (x : K), a + FractionalIdeal.spanSingleton R⁰ x * b = c
have subst : ∀ (hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0),
  ∃ (x : K), a + HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (FractionalIdeal.spanSingleton R⁰ x) 1 = c
Tactic: subst hb'
Cleared hypotheses: b, hb, hb'
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
⊢ Ideal.span {(↑c.den : R)} * a.num ≤ c.num * Ideal.span {(↑a.den : R)}
have rw : LE.le.{u_2} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
  (↑(Ideal.span {(↑c.den : R)} * a.num) : FractionalIdeal R⁰ K)
  (↑(c.num * Ideal.span {(↑a.den : R)}) : FractionalIdeal R⁰ K)
Tactic: rw [← FractionalIdeal.coeIdeal_le_coeIdeal K]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
⊢ LE.le.{u_2} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (↑(Ideal.span {(↑c.den : R)} * a.num) : FractionalIdeal R⁰ K)
    (↑(c.num * Ideal.span {(↑a.den : R)}) : FractionalIdeal R⁰ K)
have simp : HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      a) ≤
  HMul.hMul (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      c)
    (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
Tactic: simp only [FractionalIdeal.coeIdeal_mul, FractionalIdeal.coeIdeal_span_singleton, ←
      FractionalIdeal.den_mul_self_eq_num']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
⊢ HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
          FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        a) ≤
    HMul.hMul (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
          FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        c)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
have ring_nf : HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) *
    a ≤
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑c.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (↑a.den : R)) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) *
    c
Tactic: ring_nf
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
⊢ ∃ (x : K), a + HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (FractionalIdeal.spanSingleton R⁰ x) 1 = c
have H : Ideal.span {(↑c.den : R)} * a.num ≤ c.num * Ideal.span {(↑a.den : R)}
Tactic: have H : Ideal.span {c.den.1} * a.num ≤ c.num * Ideal.span {a.den.1} := by
    rw [← FractionalIdeal.coeIdeal_le_coeIdeal K]
    simp only [FractionalIdeal.coeIdeal_mul, FractionalIdeal.coeIdeal_span_singleton, ←
      FractionalIdeal.den_mul_self_eq_num']
    ring_nf
    gcongr
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
H : Ideal.span {(↑c.den : R)} * a.num ≤ c.num * Ideal.span {(↑a.den : R)}
x : R
hx : Ideal.span {(↑c.den : R)} * a.num ⊔ Ideal.span {x} = c.num * Ideal.span {(↑a.den : R)}
⊢ ∃ (x : K), a + HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (FractionalIdeal.spanSingleton R⁰ x) 1 = c
have refine : a +
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (FractionalIdeal.spanSingleton R⁰
        (HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) x : K)
          ((algebraMap R K : (a : R) → K) ((↑a.den : R) * (↑c.den : R)) : K)))
      1 =
  c
Tactic: refine ⟨algebraMap R K x / algebraMap R K (a.den.1 * c.den.1), ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R_1 : Type u_1
inst : CommRing R_1
K_1 : Type u_2
inst_1 : Field K_1
inst_2 : Algebra R_1 K_1
inst_3 : IsFractionRing R_1 K_1
R : Type u_1
inst_4 : CommRing R
K : Type u_2
inst_5 : Field K
inst_6 : Algebra R K
inst_7 : IsFractionRing R K
inst_8 : IsDedekindDomain R
a c : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 0
H : Ideal.span {(↑c.den : R)} * a.num ≤ c.num * Ideal.span {(↑a.den : R)}
x : R
hx : Ideal.span {(↑c.den : R)} * a.num ⊔ Ideal.span {x} = c.num * Ideal.span {(↑a.den : R)}
⊢ a +
      HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        (FractionalIdeal.spanSingleton R⁰
          (HDiv.hDiv (α := K) (β := K) ((algebraMap R K : (a : R) → K) x : K)
            ((algebraMap R K : (a : R) → K) ((↑a.den : R) * (↑c.den : R)) : K)))
        1 =
    c
have refine_1 : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
  (FractionalIdeal.spanSingleton R⁰ ((algebraMap R K : (a : R) → K) ((↑a.den : R) * (↑c.den : R))) :
    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
  (0 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
Tactic: refine mul_left_injective₀ (b := .spanSingleton _
    (algebraMap R K (a.den.1 * c.den.1))) ?_ ?_