awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
⊢ ∃ (ι : Type u) (x : Fintype ι) (x : DecidableEq ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    DirectSum.IsInternal (S := Submodule R M) fun (i : ι) ↦ torsionBy R M (p i ^ e i)
have refine_1 : Fintype ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))
Tactic: refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
refine_1 : Fintype ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))
⊢ ∃ (ι : Type u) (x : Fintype ι) (x : DecidableEq ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    DirectSum.IsInternal (S := Submodule R M) fun (i : ι) ↦ torsionBy R M (p i ^ e i)
have refine_2 : ∀ (i : ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))),
  Irreducible (IsPrincipal.generator (↑i : Ideal R))
Tactic: refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
p : Ideal R
hp : Membership.mem (γ := Finset (Ideal R)) (Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤))) p
⊢ Irreducible
    (IsPrincipal.generator
      (↑(Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Finset (Ideal R)) (Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤))) x) p
            hp) :
        Ideal R))
have hP : Prime p
Tactic: have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
p : Ideal R
hp : Membership.mem (γ := Finset (Ideal R)) (Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤))) p
hP : Prime p
⊢ Irreducible
    (IsPrincipal.generator
      (↑(Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Finset (Ideal R)) (Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤))) x) p
            hp) :
        Ideal R))
have this : p.IsPrime
Tactic: haveI := Ideal.isPrime_of_prime hP
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
⊢ ∃ (ι : Type u) (x : Fintype ι) (x : DecidableEq ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    DirectSum.IsInternal (S := Submodule R M) fun (i : ι) ↦ torsionBy R M (p i ^ e i)
have refine_1 : Fintype ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))
Tactic: classical
  refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
  · exact Finset.fintypeCoeSort _
  · rintro ⟨p, hp⟩
    have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
    haveI := Ideal.isPrime_of_prime hP
    exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
inst_5 : Module.Finite R M
hM : Module.IsTorsion R M
refine_1 : Fintype ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))
⊢ ∃ (ι : Type u) (x : Fintype ι) (x : DecidableEq ι) (p : ι → R) (_ : ∀ (i : ι), Irreducible (p i)) (e : ι → ℕ),
    DirectSum.IsInternal (S := Submodule R M) fun (i : ι) ↦ torsionBy R M (p i ^ e i)
have refine_2 : ∀ (i : ↥(Multiset.toFinset (factors (annihilator.{u, v} (M := M) ⊤)))),
  Irreducible (IsPrincipal.generator (↑i : Ideal R))
Tactic: classical
  refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
  · exact Finset.fintypeCoeSort _
  · rintro ⟨p, hp⟩
    have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
    haveI := Ideal.isPrime_of_prime hP
    exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry})
Tactic: dsimp only [pOrder]
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯})
have rw : Associates.mk (generator (torsionOf R M x)) =
  HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry)
Tactic: rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
⊢ Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
    Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
have h : ∀ (n : ℕ),
  p ^ n • x = 0 ↔ Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
Tactic: ext n
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
n : ℕ
⊢ p ^ n • x = 0 ↔ Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
have h : p ^ n • x = 0 ↔ Membership.mem (γ := Ideal R) (torsionOf R M x) (p ^ n)
Tactic: rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
⊢ Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
have prop : Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
  Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
Tactic: have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
prop :
  Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
    Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
⊢ Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
have this : ∃ (n : ℕ), p ^ n • x = 0
Tactic: have := (isTorsion'_powers_iff p).mp hM x
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
prop :
  Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
    Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
this : ∃ (n : ℕ), p ^ n • x = 0
⊢ Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
have this₁ : ∃ (n : ℕ), Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n
Tactic: rw [prop] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry})
Tactic: classical
  dsimp only [pOrder]
  rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
  have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
  have := (isTorsion'_powers_iff p).mp hM x; rw [prop] at this
  convert!
    Associates.eq_pow_find_of_dvd_irreducible_pow (Associates.irreducible_mk.mpr hp)
      this.choose_spec
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯})
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp₁ : Associates.mk (generator (torsionOf R M x)) =
  HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry)
Tactic: classical
  dsimp only [pOrder]
  rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
  have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
  have := (isTorsion'_powers_iff p).mp hM x; rw [prop] at this
  convert!
    Associates.eq_pow_find_of_dvd_irreducible_pow (Associates.irreducible_mk.mpr hp)
      this.choose_spec
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯})
dsimp₁ :
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp₂ : ∀
  (prop :
    Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
      Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n),
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry)
Tactic: classical
  dsimp only [pOrder]
  rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
  have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
  have := (isTorsion'_powers_iff p).mp hM x; rw [prop] at this
  convert!
    Associates.eq_pow_find_of_dvd_irreducible_pow (Associates.irreducible_mk.mpr hp)
      this.choose_spec
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯})
dsimp₁ :
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
dsimp₂ :
  ∀
    (prop :
      Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
        Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n),
    Associates.mk (generator (torsionOf R M x)) =
      HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp₃ : ∀
  (prop :
    Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
      Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n)
  (this : ∃ (n : ℕ), p ^ n • x = 0),
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry)
Tactic: classical
  dsimp only [pOrder]
  rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
  have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
  have := (isTorsion'_powers_iff p).mp hM x; rw [prop] at this
  convert!
    Associates.eq_pow_find_of_dvd_irreducible_pow (Associates.irreducible_mk.mpr hp)
      this.choose_spec
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x : M
dsimp : Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯})
dsimp₁ :
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
dsimp₂ :
  ∀
    (prop :
      Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
        Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n),
    Associates.mk (generator (torsionOf R M x)) =
      HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
dsimp₃ dsimp₄ :
  ∀
    (prop :
      Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
        Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n)
    (this : ∃ (n : ℕ), p ^ n • x = 0),
    Associates.mk (generator (torsionOf R M x)) =
      HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) ⋯)
⊢ Eq (α := Ideal R) (torsionOf R M x) (Ideal.span {p ^ pOrder hM x})
have dsimp₅ : ∀
  (prop :
    Eq (α := (n : ℕ) → Prop) (fun (n : ℕ) ↦ p ^ n • x = 0) fun (n : ℕ) ↦
      Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n)
  (this : ∃ (n : ℕ), Associates.mk (generator (torsionOf R M x)) ∣ HPow.hPow (α := Associates R) (Associates.mk p) n),
  Associates.mk (generator (torsionOf R M x)) =
    HPow.hPow (α := Associates R) (Associates.mk p) (Nat.find (p := fun (n : ℕ) ↦ p ^ n • x = 0) sorry)
Tactic: classical
  dsimp only [pOrder]
  rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
    Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
  have prop :
    (fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
      (Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
    ext n; rw [← Associates.mk_pow, Associates.mk_dvd_mk, ← mem_iff_generator_dvd]; rfl
  have := (isTorsion'_powers_iff p).mp hM x; rw [prop] at this
  convert!
    Associates.eq_pow_find_of_dvd_irreducible_pow (Associates.irreducible_mk.mpr hp)
      this.choose_spec
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x y : M
k : ℕ
hM' : IsTorsionBy R M (p ^ pOrder hM y)
h : Membership.mem (γ := Submodule R M) (R ∙ y) (p ^ k • x)
⊢ ∃ (a : R), p ^ k • x = p ^ k • a • y
have pos : ∀ (hk : k ≤ pOrder hM y), ∃ (a : R), p ^ k • x = p ^ k • a • y
Tactic: by_cases! hk : k ≤ pOrder hM y
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x y : M
k : ℕ
hM' : IsTorsionBy R M (p ^ pOrder hM y)
h : Membership.mem (γ := Submodule R M) (R ∙ y) (p ^ k • x)
pos : ∀ (hk : k ≤ pOrder hM y), ∃ (a : R), p ^ k • x = p ^ k • a • y
⊢ ∃ (a : R), p ^ k • x = p ^ k • a • y
have neg : ∀ (hk : pOrder hM y < k), ∃ (a : R), p ^ k • x = p ^ k • a • y
Tactic: by_cases! hk : k ≤ pOrder hM y
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x y : M
k : ℕ
hM' : IsTorsionBy R M (p ^ pOrder hM y)
h : Membership.mem (γ := Submodule R M) (R ∙ y) (p ^ k • x)
hk : k ≤ pOrder hM y
pos : ∃ (a : R), p ^ k • x = p ^ k • a • y
⊢ ∃ (a : R), p ^ k • x = p ^ k • a • y
have pos₁ : Eq (α := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k) (torsionOf R M y)
Tactic: let f :=
      ((R ∙ p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans
        (quotTorsionOfEquivSpanSingleton R M y)
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x y : M
k : ℕ
hM' : IsTorsionBy R M (p ^ pOrder hM y)
h : Membership.mem (γ := Submodule R M) (R ∙ y) (p ^ k • x)
hk : k ≤ pOrder hM y
f : LinearEquiv.{u, u, u, v} (RingHom.id R)
  (HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)) ↥(R ∙ y) :=
  Submodule.quotEquivOfEq (R ∙ p ^ (pOrder hM y - k) * p ^ k) (torsionOf R M y) ⋯ ≪≫ₗ
    quotTorsionOfEquivSpanSingleton R M y
⊢ Membership.mem (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)) (γ :=
    Submodule R (HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)))
    (Submodule.span R
      (singleton (α := HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
        ((Ideal.Quotient.mk (R ∙ p ^ (pOrder hM y - k) * p ^ k) : (a : R) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
            (p ^ k) :
          HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))))
    ((f.symm : (a : ↥(R ∙ y)) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
        (Subtype.mk (p := fun (x : M) ↦ Membership.mem (γ := Submodule R M) (R ∙ y) x) (p ^ k • x) h) :
      HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
have rw : Eq.{u + 1} (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
  ((f.symm : (a : ↥(R ∙ y)) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
      (p ^ (pOrder hM y - k) •
        Subtype.mk (p := fun (x : M) ↦ Membership.mem (γ := Submodule R M) (R ∙ y) x) (p ^ k • x) h) :
    HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
  (0 : HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
Tactic: rw [← Quotient.torsionBy_eq_span_singleton, mem_torsionBy_iff, ← f.symm.map_smul]
-- Lean 4 proof step from Mathlib.Algebra.Module.PID (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsPrincipalIdealRing R
M : Type v
inst_2 : AddCommGroup M
inst_3 : Module R M
inst_4 : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion'.{v, u} M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
x y : M
k : ℕ
hM' : IsTorsionBy R M (p ^ pOrder hM y)
h : Membership.mem (γ := Submodule R M) (R ∙ y) (p ^ k • x)
hk : k ≤ pOrder hM y
f : LinearEquiv.{u, u, u, v} (RingHom.id R)
  (HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)) ↥(R ∙ y) :=
  Submodule.quotEquivOfEq (R ∙ p ^ (pOrder hM y - k) * p ^ k) (torsionOf R M y) ⋯ ≪≫ₗ
    quotTorsionOfEquivSpanSingleton R M y
rw :
  Eq.{u + 1} (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
    ((f.symm : (a : ↥(R ∙ y)) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
        (p ^ (pOrder hM y - k) •
          Subtype.mk (p := fun (x : M) ↦ Membership.mem (γ := Submodule R M) (R ∙ y) x) (p ^ k • x) h) :
      HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
    (0 : HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
⊢ Membership.mem (α := HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)) (γ :=
    Submodule R (HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k)))
    (Submodule.span R
      (singleton (α := HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
        ((Ideal.Quotient.mk (R ∙ p ^ (pOrder hM y - k) * p ^ k) : (a : R) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
            (p ^ k) :
          HasQuotient.Quotient R (B := Ideal R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))))
    ((f.symm : (a : ↥(R ∙ y)) → R ⧸ R ∙ p ^ (pOrder hM y - k) * p ^ k)
        (Subtype.mk (p := fun (x : M) ↦ Membership.mem (γ := Submodule R M) (R ∙ y) x) (p ^ k • x) h) :
      HasQuotient.Quotient R (B := Submodule R R) (R ∙ p ^ (pOrder hM y - k) * p ^ k))
have ha : Membership.mem (γ := Submonoid R) (nonZeroDivisors R) (p ^ (pOrder hM y - k))
Tactic: rw [← Quotient.torsionBy_eq_span_singleton, mem_torsionBy_iff, ← f.symm.map_smul]