awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x y : Associates R
⊢ (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (HMul.hMul (α := Associates R) (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y)))) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
        Ideal R)
have simp_rw : (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
        (Associates.mk
          (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x *
            Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
    Ideal R) =
  HMul.hMul (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
      Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
      Ideal R)
Tactic: simp_rw [mk_mul_mk, associatesEquivIsPrincipal_apply, span_singleton_mul_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x y : Associates R
simp_rw :
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (Associates.mk
            (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x *
              Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
        Ideal R)
⊢ (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (HMul.hMul (α := Associates R) (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y)))) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y))) :
        Ideal R)
have simp_rw₁ : span
    {Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x *
        Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y} =
  HMul.hMul (α := Ideal R) (span {Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x})
    (span {Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) y})
Tactic: simp_rw [mk_mul_mk, associatesEquivIsPrincipal_apply, span_singleton_mul_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 0) : Ideal R) 0
have rw : Eq.{u_1 + 1} (α := Ideal R)
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 0)) :
    Ideal R)
  0
Tactic: rw [← mk_zero, associatesEquivIsPrincipal_apply, Submodule.zero_eq_bot, span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 0)) :
      Ideal R)
    0
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 0) : Ideal R) 0
have rw₁ : Eq (α := Ideal R) (span {0}) 0
Tactic: rw [← mk_zero, associatesEquivIsPrincipal_apply, Submodule.zero_eq_bot, span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 0)) :
      Ideal R)
    0
rw₁ : Eq (α := Ideal R) (span {0}) 0
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 0) : Ideal R) 0
have rw₂ : Eq (α := Ideal R) (span {0}) ⊥
Tactic: rw [← mk_zero, associatesEquivIsPrincipal_apply, Submodule.zero_eq_bot, span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 0)) :
      Ideal R)
    0
rw₁ : Eq (α := Ideal R) (span {0}) 0
rw₂ : Eq (α := Ideal R) (span {0}) ⊥
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 0) : Ideal R) 0
have rw₃ : Eq.{u_1 + 1} (α := R) 0 0
Tactic: rw [← mk_zero, associatesEquivIsPrincipal_apply, Submodule.zero_eq_bot, span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
have rw : Eq.{u_1 + 1} (α := Ideal R)
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 1)) :
    Ideal R)
  1
Tactic: rw [one_eq_mk_one, associatesEquivIsPrincipal_apply, span_singleton_one, one_eq_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 1)) :
      Ideal R)
    1
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
have rw₁ : Eq (α := Ideal R) (span {1}) 1
Tactic: rw [one_eq_mk_one, associatesEquivIsPrincipal_apply, span_singleton_one, one_eq_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 1)) :
      Ideal R)
    1
rw₁ : Eq (α := Ideal R) (span {1}) 1
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
have rw₂ : Eq.{u_1 + 1} (α := Ideal R) ⊤ 1
Tactic: rw [one_eq_mk_one, associatesEquivIsPrincipal_apply, span_singleton_one, one_eq_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq.{u_1 + 1} (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (Associates.mk 1)) :
      Ideal R)
    1
rw₁ : Eq (α := Ideal R) (span {1}) 1
rw₂ : Eq.{u_1 + 1} (α := Ideal R) ⊤ 1
⊢ Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
have rw₃ : Eq.{u_1 + 1} (α := Ideal R) ⊤ ⊤
Tactic: rw [one_eq_mk_one, associatesEquivIsPrincipal_apply, span_singleton_one, one_eq_top]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x x_1 : Associates R
⊢ Equiv.toFun (associatesEquivIsPrincipal R) (x * x_1) =
    Equiv.toFun (associatesEquivIsPrincipal R) x * Equiv.toFun (associatesEquivIsPrincipal R) x_1
have rw : (↑(Equiv.toFun (associatesEquivIsPrincipal R) (x * x_1)) : Ideal R) =
  (↑(Equiv.toFun (associatesEquivIsPrincipal R) x * Equiv.toFun (associatesEquivIsPrincipal R) x_1) : Ideal R)
Tactic: rw [Subtype.ext_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x x_1 : Associates R
⊢ (↑(Equiv.toFun (associatesEquivIsPrincipal R) (x * x_1)) : Ideal R) =
    (↑(Equiv.toFun (associatesEquivIsPrincipal R) x * Equiv.toFun (associatesEquivIsPrincipal R) x_1) : Ideal R)
have erw : HMul.hMul (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x) : Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x_1) : Ideal R) =
  (↑(Equiv.toFun (associatesEquivIsPrincipal R) x * Equiv.toFun (associatesEquivIsPrincipal R) x_1) : Ideal R)
Tactic: erw [associatesEquivIsPrincipal_mul]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x : Associates R
⊢ Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰ x ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x) : Ideal R)
have rw : Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰
    (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x)) ↔
  Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
      Ideal R)
Tactic: rw [← quot_out x, mk_mem_nonZeroDivisors_associates,
          associatesEquivIsPrincipal_apply, span_singleton_nonZeroDivisors]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x : Associates R
rw :
  Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰
      (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x)) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
⊢ Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰ x ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x) : Ideal R)
have rw₁ : Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
  Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
      Ideal R)
Tactic: rw [← quot_out x, mk_mem_nonZeroDivisors_associates,
          associatesEquivIsPrincipal_apply, span_singleton_nonZeroDivisors]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x : Associates R
rw :
  Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰
      (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x)) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
rw₁ :
  Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
⊢ Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰ x ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x) : Ideal R)
have rw₂ : Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
  Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
    (span {Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x})
Tactic: rw [← quot_out x, mk_mem_nonZeroDivisors_associates,
          associatesEquivIsPrincipal_apply, span_singleton_nonZeroDivisors]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x : Associates R
rw :
  Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰
      (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x)) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
rw₁ :
  Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (Associates.mk (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x))) :
        Ideal R)
rw₂ :
  Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (span {Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x})
⊢ Membership.mem (γ := Submonoid (Associates R)) (Associates R)⁰ x ↔
    Membership.mem (γ := Submonoid (Ideal R)) (Ideal R)⁰
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) x) : Ideal R)
have rw₃ : Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x) ↔
  Membership.mem (γ := Submonoid R) R⁰ (Quot.out (r := (⇑(Associated.setoid R) : R → R → Prop)) x)
Tactic: rw [← quot_out x, mk_mem_nonZeroDivisors_associates,
          associatesEquivIsPrincipal_apply, span_singleton_nonZeroDivisors]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x y : Associates ↥R⁰
⊢ (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              (x * y)) :
          ↥(Ideal R)⁰) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                  (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
                x) :
            ↥(Ideal R)⁰) :
        Ideal R)
      (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                  (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
                y) :
            ↥(Ideal R)⁰) :
        Ideal R)
have simp_rw : (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
        (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) (x * y)) :
          Associates R)) :
    Ideal R) =
  HMul.hMul (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) x) :
            Associates R)) :
      Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) y) :
            Associates R)) :
      Ideal R)
Tactic: simp_rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_mul, Submonoid.coe_mul,
    associatesEquivIsPrincipal_mul]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
x y : Associates ↥R⁰
simp_rw :
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) (x * y)) :
            Associates R)) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) x) :
              Associates R)) :
        Ideal R)
      (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
            (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) y) :
              Associates R)) :
        Ideal R)
⊢ (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              (x * y)) :
          ↥(Ideal R)⁰) :
      Ideal R) =
    HMul.hMul (α := Ideal R)
      (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                  (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
                x) :
            ↥(Ideal R)⁰) :
        Ideal R)
      (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                  (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
                y) :
            ↥(Ideal R)⁰) :
        Ideal R)
have simp_rw₁ : (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
        (↑(HMul.hMul (α := ↥(Associates R)⁰) (β := ↥(Associates R)⁰)
              ((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) x :
                ↥(Associates R)⁰)
              ((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) y :
                ↥(Associates R)⁰)) :
          Associates R)) :
    Ideal R) =
  HMul.hMul (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) x) :
            Associates R)) :
      Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) y) :
            Associates R)) :
      Ideal R)
Tactic: simp_rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_mul, Submonoid.coe_mul,
    associatesEquivIsPrincipal_mul]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
⊢ Eq (α := Ideal R)
    (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              1) :
          ↥(Ideal R)⁰) :
      Ideal R)
    1
have rw : Eq (α := Ideal R)
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
        (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) 1) :
          Associates R)) :
    Ideal R)
  1
Tactic: rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_one, OneMemClass.coe_one,
    associatesEquivIsPrincipal_map_one]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) 1) :
            Associates R)) :
      Ideal R)
    1
⊢ Eq (α := Ideal R)
    (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              1) :
          ↥(Ideal R)⁰) :
      Ideal R)
    1
have rw₁ : Eq (α := Ideal R)
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (↑1 : Associates R)) :
    Ideal R)
  1
Tactic: rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_one, OneMemClass.coe_one,
    associatesEquivIsPrincipal_map_one]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) 1) :
            Associates R)) :
      Ideal R)
    1
rw₁ :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (↑1 : Associates R)) :
      Ideal R)
    1
⊢ Eq (α := Ideal R)
    (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              1) :
          ↥(Ideal R)⁰) :
      Ideal R)
    1
have rw₂ : Eq (α := Ideal R)
  (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
Tactic: rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_one, OneMemClass.coe_one,
    associatesEquivIsPrincipal_map_one]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.IsPrincipal (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
rw :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I })
          (↑((MulEquiv.symm associatesNonZeroDivisorsEquiv : (a : Associates ↥R⁰) → ↥(Associates R)⁰) 1) :
            Associates R)) :
      Ideal R)
    1
rw₁ :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) (↑1 : Associates R)) :
      Ideal R)
    1
rw₂ :
  Eq (α := Ideal R)
    (↑((associatesEquivIsPrincipal R : (a : Associates R) → { I : Ideal R // IsPrincipal I }) 1) : Ideal R) 1
⊢ Eq (α := Ideal R)
    (↑(↑((associatesNonZeroDivisorsEquivIsPrincipal R :
                (a : Associates ↥R⁰) → { I : ↥(Ideal R)⁰ // IsPrincipal (↑I : Ideal R) })
              1) :
          ↥(Ideal R)⁰) :
      Ideal R)
    1
have rw₃ : Eq.{u_1 + 1} (α := Ideal R) 1 1
Tactic: rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_one, OneMemClass.coe_one,
    associatesEquivIsPrincipal_map_one]