awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
this :
  NeZero
    (s.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b}))
h1 :
  ⊥ <
    (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal
p : ℕ
hp : Nat.Prime p
hsp :
  (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal =
    span {p}
refine_1 :
  ∀
    (hmax :
      (s.toFun (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b})
            2).asIdeal =
        maximalIdeal ℕ),
    False
⊢ False
have refine_1₁ : ∀ (q : ℕ) (hq : Nat.Prime q)
  (hsq :
    (s.toFun (r :=
          {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
            match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
            | (a, b) => a < b})
          2).asIdeal =
      span {q}),
  False
Tactic: obtain hmax | ⟨q, hq, hsq⟩ :=
      (Ideal.isPrime_nat_iff.mp (s 2).2).resolve_left (h1.trans (s.step 1)).ne'
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
⊢ 2 ≤ ringKrullDim ℕ
have refine_2 : ∀ (i : Fin 2),
  Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}
    (![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
        i.castSucc,
      ![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
        i.succ)
Tactic: refine le_iSup_of_le ⟨2, ![⊥, ⟨_, (span_singleton_prime two_ne_zero).mpr <| Nat.prime_iff.mp
      Nat.prime_two⟩, ⟨_, (maximalIdeal.isMaximal ℕ).isPrime⟩], fun i ↦ ?_⟩ le_rfl
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
i : Fin 2
⊢ Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}
    (![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        i.castSucc,
      ![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        i.succ)
have refine_2 : Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b}
  (![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.castSucc ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))),
    ![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.succ ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))))
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
i : Fin 2
refine_2 :
  Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}
    (![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        (Fin.castSucc ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))),
      ![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        (Fin.succ ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))))
⊢ Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}
    (![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        i.castSucc,
      ![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        i.succ)
have refine_2₁ : Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b}
  (![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.castSucc ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))),
    ![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := maximalIdeal ℕ, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.succ ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))))
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
⊢ Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ))
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}
    (![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        (Fin.castSucc ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2))),
      ![⊥, { asIdeal := span {2}, isPrime := ⋯ : PrimeSpectrum ℕ },
          { asIdeal := maximalIdeal ℕ, isPrime := ⋯ : PrimeSpectrum ℕ }]
        (Fin.succ ((fun (i : Fin 2) ↦ i) (⟨1, ⋯⟩ : Fin 2))))
have refine_2 : Membership.mem (γ := SetRel (PrimeSpectrum ℕ) (PrimeSpectrum ℕ)) {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ | x.1 < x.2}
  (![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := span {2, 3}, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.castSucc (⟨1, sorry⟩ : Fin 2)),
    ![⊥, { asIdeal := span {2}, isPrime := sorry : PrimeSpectrum ℕ },
        { asIdeal := span {2, 3}, isPrime := sorry : PrimeSpectrum ℕ }]
      (Fin.succ (⟨1, sorry⟩ : Fin 2)))
Tactic: simp_rw [Nat.maximalIdeal_eq_span_two_three]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (hypothesis)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
H : Eq (α := Ideal α) (Ideal.span {a}) ⊤
⊢ False
have H₁ : IsUnit a
Tactic: rw [Ideal.span_singleton_eq_top] at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (hypothesis)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
⊢ ∃ (I : Ideal α), I.IsMaximal ∧ a ∈ I
have this : Ne (α := Ideal α) (Ideal.span {a}) ⊤
Tactic: have : Ideal.span ({a} : Set α) ≠ ⊤ := by
    intro H
    rw [Ideal.span_singleton_eq_top] at H
    contradiction
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
this : Ne (α := Ideal α) (Ideal.span {a}) ⊤
⊢ ∃ (I : Ideal α), I.IsMaximal ∧ a ∈ I
have rcases : ∀ (I : Ideal α) (Imax : I.IsMaximal) (H : Ideal.span {a} ≤ I), ∃ (I : Ideal α), I.IsMaximal ∧ a ∈ I
Tactic: rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
this : Ne (α := Ideal α) (Ideal.span {a}) ⊤
I : Ideal α
Imax : I.IsMaximal
H : Ideal.span {a} ≤ I
⊢ ∃ (I : Ideal α), I.IsMaximal ∧ a ∈ I
have right : a ∈ I
Tactic: use I, Imax
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
this : Ne (α := Ideal α) (Ideal.span {a}) ⊤
I : Ideal α
Imax : I.IsMaximal
H : Ideal.span {a} ≤ I
⊢ a ∈ I
have right : Membership.mem (γ := Submodule α α) (Ideal.span {a}) a
Tactic: apply H
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
a : α
inst : CommSemiring α
h : a ∈ nonunits α
this : Ne (α := Ideal α) (Ideal.span {a}) ⊤
I : Ideal α
Imax : I.IsMaximal
H : Ideal.span {a} ≤ I
⊢ Membership.mem (γ := Submodule α α) (Ideal.span {a}) a
have right : Membership.mem (γ := Set α) {a} a
Tactic: apply Ideal.subset_span
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : DivisionMonoid α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
ha : IsUnit a
⊢ (↑a : α)⁻¹ ∈ S
have obtain : ∀ (u : (↥S)ˣ), (↑(↑u : ↥S) : α)⁻¹ ∈ S
Tactic: obtain ⟨u, rfl⟩ := ha
Cleared hypotheses: a, ha
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : DivisionMonoid α
inst_2 : SubmonoidClass C α
S : C
u : (↥S)ˣ
⊢ (↑(↑u : ↥S) : α)⁻¹ ∈ S
have h : (↑(↑u : ↥S) : α)⁻¹ = (↑(↑u⁻¹ : ↥S) : α)
Tactic: convert! u⁻¹.1.2
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : Group α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α)⁻¹ ∉ S
have rw : ¬IsUnit a ↔ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : Group α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
rw : ¬IsUnit a ↔ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α)⁻¹ ∉ S
have rw₁ : (↑a : α)⁻¹ ∉ S ↔ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
ha : (↑a : α) ≠ 0
⊢ IsUnit a ↔ (↑a : α)⁻¹ ∈ S
have rw : (↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S ↔ (↑a : α)⁻¹ ∈ S
Tactic: rw [isUnit_iff_and, and_iff_right ha]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
ha : (↑a : α) ≠ 0
rw : (↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S ↔ (↑a : α)⁻¹ ∈ S
⊢ IsUnit a ↔ (↑a : α)⁻¹ ∈ S
have rw₁ : (↑a : α)⁻¹ ∈ S ↔ (↑a : α)⁻¹ ∈ S
Tactic: rw [isUnit_iff_and, and_iff_right ha]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
have rw : ¬IsUnit a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff_and, not_and_or, Ne, not_not]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
rw : ¬IsUnit a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
have rw₁ : ¬((↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S) ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff_and, not_and_or, Ne, not_not]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
rw : ¬IsUnit a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₁ : ¬((↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S) ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
have rw₂ : ¬(↑a : α) ≠ 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff_and, not_and_or, Ne, not_not]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
rw : ¬IsUnit a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₁ : ¬((↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S) ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₂ : ¬(↑a : α) ≠ 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
have rw₃ : ¬¬(↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff_and, not_and_or, Ne, not_not]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
rw : ¬IsUnit a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₁ : ¬((↑a : α) ≠ 0 ∧ (↑a : α)⁻¹ ∈ S) ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₂ : ¬(↑a : α) ≠ 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
rw₃ : ¬¬(↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
have rw₄ : (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff, isUnit_iff_and, not_and_or, Ne, not_not]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
ha : (↑a : α) ≠ 0
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α)⁻¹ ∉ S
have rw : (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff_or, or_iff_right ha]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Nonunits (goal)
Goal:
α : Type u_2
C : Type u_4
inst : SetLike C α
inst_1 : GroupWithZero α
inst_2 : SubmonoidClass C α
S : C
a : ↥S
ha : (↑a : α) ≠ 0
rw : (↑a : α) = 0 ∨ (↑a : α)⁻¹ ∉ S ↔ (↑a : α)⁻¹ ∉ S
⊢ Membership.mem (γ := Set ↥S) (nonunits ↥S) a ↔ (↑a : α)⁻¹ ∉ S
have rw₁ : (↑a : α)⁻¹ ∉ S ↔ (↑a : α)⁻¹ ∉ S
Tactic: rw [mem_nonunits_iff_or, or_iff_right ha]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
I J : Ideal S
coprime : IsCoprime I J
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (I * J) =
    cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
have rw : Nat.card.{u_1} (S ⧸ I * J) = cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
Tactic: rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,
    Nat.card_prod]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
I J : Ideal S
coprime : IsCoprime I J
rw : Nat.card.{u_1} (S ⧸ I * J) = cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (I * J) =
    cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
have rw₁ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * cardQuot.{u_1, u_1} (R := S) (M := S) J
Tactic: rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,
    Nat.card_prod]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
I J : Ideal S
coprime : IsCoprime I J
rw : Nat.card.{u_1} (S ⧸ I * J) = cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
rw₁ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * cardQuot.{u_1, u_1} (R := S) (M := S) J
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (I * J) =
    cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
have rw₂ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
Tactic: rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,
    Nat.card_prod]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
I J : Ideal S
coprime : IsCoprime I J
rw : Nat.card.{u_1} (S ⧸ I * J) = cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
rw₁ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * cardQuot.{u_1, u_1} (R := S) (M := S) J
rw₂ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (I * J) =
    cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
have rw₃ : Nat.card.{u_1} ((S ⧸ I) × S ⧸ J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
Tactic: rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,
    Nat.card_prod]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
I J : Ideal S
coprime : IsCoprime I J
rw : Nat.card.{u_1} (S ⧸ I * J) = cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
rw₁ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * cardQuot.{u_1, u_1} (R := S) (M := S) J
rw₂ : Nat.card.{u_1} (S ⧸ I * J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
rw₃ : Nat.card.{u_1} ((S ⧸ I) × S ⧸ J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (I * J) =
    cardQuot.{u_1, u_1} (R := S) (M := S) I * cardQuot.{u_1, u_1} (R := S) (M := S) J
have rw₄ : Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J) = Nat.card.{u_1} (S ⧸ I) * Nat.card.{u_1} (S ⧸ J)
Tactic: rw [cardQuot_apply, cardQuot_apply, cardQuot_apply,
    Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,
    Nat.card_prod]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ a * d - a * d' ∈ P ^ (i + 1)
have simp : a * (d - d') ∈ P ^ (i + 1)
Tactic: simp only [← mul_sub]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
⊢ a * d + e - (a * d' + e') ∈ P ^ (i + 1)
have this : a * d - a * d' ∈ P ^ (i + 1)
Tactic: have : a * d - a * d' ∈ P ^ (i + 1) := by
    simp only [← mul_sub]
    exact Ideal.mul_mem_mul a_mem h
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
i : ℕ
a d d' e e' : S
a_mem : a ∈ P ^ i
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : d - d' ∈ P
this : a * d - a * d' ∈ P ^ (i + 1)
⊢ a * d + e - (a * d' + e') ∈ P ^ (i + 1)
have h₁ : a * d + e - (a * d' + e') = a * d - a * d' + (e - e')
Tactic: convert! Ideal.add_mem _ this (Ideal.sub_mem _ e_mem e'_mem) using 1
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
eq_b : P ^ i = span {a} ⊔ P ^ (i + 1)
⊢ ∃ (d : S), ∃ e ∈ P ^ (i + 1), a * d + e = c
have c_mem₁ : c ∈ span {a} ⊔ P ^ (i + 1)
Tactic: rw [eq_b] at c_mem
Cleared hypotheses: c_mem, eq_b
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ span {a} ⊔ P ^ (i + 1)
eq_b : P ^ i = span {a} ⊔ P ^ (i + 1)
⊢ ∃ (d : S), ∃ e ∈ P ^ (i + 1), a * d + e = c
have simp : ∃ (d : S), ∃ e ∈ P ^ (i + 1), d * a + e = c
Tactic: simp only [mul_comm a]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ ∃ (d : S), ∃ e ∈ P ^ (i + 1), a * d + e = c
have suffices : P ^ i = span {a} ⊔ P ^ (i + 1)
Tactic: suffices eq_b : P ^ i = Ideal.span {a} ⊔ P ^ (i + 1) by
    rw [eq_b] at c_mem
    simp only [mul_comm a]
    exact Ideal.mem_span_singleton_sup.mp c_mem
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ P ^ i = span {a} ⊔ P ^ (i + 1)
have refine : P ^ (i + 1) ≠ span {a} ⊔ P ^ (i + 1)
Tactic: refine (Ideal.eq_prime_pow_of_succ_lt_of_le hP (lt_of_le_of_ne le_sup_right ?_)
    (sup_le (Ideal.span_le.mpr (Set.singleton_subset_iff.mpr a_mem))
      (Ideal.pow_succ_lt_pow hP i).le)).symm
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ P ^ (i + 1) ≠ span {a} ⊔ P ^ (i + 1)
have contrapose : ∀ (this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)), a ∈ P ^ (i + 1)
Tactic: contrapose a_notMem with this
Cleared hypotheses: a_notMem
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
c_mem : c ∈ P ^ i
this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)
⊢ a ∈ P ^ (i + 1)
have rw : a ∈ span {a} ⊔ P ^ (i + 1)
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : IsDedekindDomain S
P : Ideal S
P_prime : P.IsPrime
hP : P ≠ ⊥
i : ℕ
a b : S
a_notMem : a ∉ P ^ (i + 1)
ab_mem : a * b ∈ P ^ (i + 1)
⊢ b ∈ P
have simp : ∀ (a_notMem : ¬P ^ i * P ∣ span {a}) (ab_mem : P ^ i * P ∣ HMul.hMul (α := Ideal S) (span {a}) (span {b})), P ∣ span {b}
Tactic: simp only [← Ideal.span_singleton_le_iff_mem, ← Ideal.dvd_iff_le, pow_succ, ←
    Ideal.span_singleton_mul_span_singleton] at a_notMem ab_mem ⊢
Cleared hypotheses: a_notMem, ab_mem
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
⊢ a * (d - d') ∈ P ^ (i + 1)
have h₁ : a * (d - d') = a * d + e - (a * d' + e') + (e' - e)
Tactic: convert! Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem) using 1
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
⊢ d - d' ∈ P
have h' : a * (d - d') ∈ P ^ (i + 1)
Tactic: have h' : a * (d - d') ∈ P ^ (i + 1) := by
    convert! Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem) using 1
    ring