l3lab/ntp-mathlib huggingface dataset:
theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
    {k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
    ∃ x : M, p ^ k • x = 0 ∧ Submodule.Quotient.mk (p := span R {z}) x = f 1 := by
  have f1 := mk_surjective (R ∙ z) (f 1)
  have : p ^ k • f1.choose ∈ R ∙ z := by
    rw [← Quotient.mk_eq_zero, mk_smul, f1.choose_spec, ← f.map_smul]
    convert f.map_zero; change _ • Submodule.Quotient.mk _ = _
    rw [← mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
    exact Submodule.mem_span_singleton_self _
  obtain ⟨a, ha⟩ := p_pow_smul_lift hp hM hz this
  refine' ⟨f1.choose - a • z, by 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
z : M
hz : IsTorsionBy R M (p ^ pOrder hM z)
k : ℕ
f : R ⧸ Submodule.span R {p ^ k} →ₗ[R] M ⧸ Submodule.span R {z}
f1 : ∃ a, Submodule.Quotient.mk a = f 1
this : p ^ k • Exists.choose f1 ∈ Submodule.span R {z}
a : R
ha : p ^ k • Exists.choose f1 = p ^ k • a • z
⊢ p ^ k • (Exists.choose f1 - a • z) = 0

Next tactic:
rw [smul_sub, sub_eq_zero, ha]
theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
    {k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
    ∃ x : M, p ^ k • x = 0 ∧ Submodule.Quotient.mk (p := span R {z}) x = f 1 := by
  have f1 := mk_surjective (R ∙ z) (f 1)
  have : p ^ k • f1.choose ∈ R ∙ z := by
    rw [← Quotient.mk_eq_zero, mk_smul, f1.choose_spec, ← f.map_smul]
    convert f.map_zero; change _ • Submodule.Quotient.mk _ = _
    rw [← mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
    exact Submodule.mem_span_singleton_self _
  obtain ⟨a, ha⟩ := p_pow_smul_lift hp hM hz this
  refine' ⟨f1.choose - a • z, by rw [smul_sub, sub_eq_zero, ha], _⟩
  

Proof state:
case intro
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
z : M
hz : IsTorsionBy R M (p ^ pOrder hM z)
k : ℕ
f : R ⧸ Submodule.span R {p ^ k} →ₗ[R] M ⧸ Submodule.span R {z}
f1 : ∃ a, Submodule.Quotient.mk a = f 1
this : p ^ k • Exists.choose f1 ∈ Submodule.span R {z}
a : R
ha : p ^ k • Exists.choose f1 = p ^ k • a • z
⊢ Submodule.Quotient.mk (Exists.choose f1 - a • z) = f 1

Next tactic:
rw [mk_sub, mk_smul, (Quotient.mk_eq_zero _).mpr <| Submodule.mem_span_singleton_self _,
    smul_zero, sub_zero, f1.choose_spec]
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
hN : IsTorsion' N ↥(Submonoid.powers p)
h' : Finite R N
⊢ ∃ d k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; 

Proof state:
case intro.intro
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
hN : IsTorsion' N ↥(Submonoid.powers p)
h' : Finite R N
d : ℕ
s : Fin d → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ d k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
use d
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; 

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
hN : IsTorsion' N ↥(Submonoid.powers p)
h' : Finite R N
d : ℕ
s : Fin d → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
clear h'
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
hN : IsTorsion' N ↥(Submonoid.powers p)
d : ℕ
s : Fin d → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
induction' d with d IH generalizing N
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · 

Proof state:
case h.zero
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
simp only [Nat.zero_eq] at *
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    

Proof state:
case h.zero
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
use finZeroElim
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ finZeroElim i})

Next tactic:
rw [Set.range_eq_empty, Submodule.span_empty] at hs
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
⊢ Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ finZeroElim i})

Next tactic:
haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
x : N
⊢ x = default

Next tactic:
dsimp
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
x : N
⊢ x = 0

Next tactic:
rw [← Submodule.mem_bot R, hs]
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
x : N
⊢ x ∈ ⊤

Next tactic:
exact Submodule.mem_top
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
this : Unique N
⊢ Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ finZeroElim i})

Next tactic:
haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
    

Proof state:
case h
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin Nat.zero → N
hs : ⊥ = ⊤
this✝ : Unique N
this : IsEmpty (Fin Nat.zero)
⊢ Nonempty (N ≃ₗ[R] ⨁ (i : Fin Nat.zero), R ⧸ Submodule.span R {p ^ finZeroElim i})

Next tactic:
exact ⟨0⟩
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
    exact ⟨0⟩
  · 

Proof state:
case h.succ
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
IH :
  ∀ {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],
    IsTorsion' N ↥(Submonoid.powers p) →
      ∀ (s : Fin d → N),
        Submodule.span R (Set.range s) = ⊤ → ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin (Nat.succ d) → N
hs : Submodule.span R (Set.range s) = ⊤
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin (Nat.succ d)), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
have : ∀ x : N, Decidable (x = 0) := fun _ => by classical infer_instance
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
    exact ⟨0⟩
  · have : ∀ x : N, Decidable (x = 0) := fun _ => by 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
IH :
  ∀ {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],
    IsTorsion' N ↥(Submonoid.powers p) →
      ∀ (s : Fin d → N),
        Submodule.span R (Set.range s) = ⊤ → ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin (Nat.succ d) → N
hs : Submodule.span R (Set.range s) = ⊤
x✝ : N
⊢ Decidable (x✝ = 0)

Next tactic:
classical infer_instance
set_option maxHeartbeats 400000 in
/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`.-/
theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoid.powers p))
    [h' : Module.Finite R N] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| N ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) := by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction' d with d IH generalizing N
  · simp only [Nat.zero_eq] at *
    -- Porting note: was `use fun i => finZeroElim i`
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique N :=
      ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
    exact ⟨0⟩
  · have : ∀ x : N, Decidable (x = 0) := fun _ => by classical 

Proof state:
R : Type u
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type v
inst✝³ : AddCommGroup M
inst✝² : Module R M
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
IH :
  ∀ {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],
    IsTorsion' N ↥(Submonoid.powers p) →
      ∀ (s : Fin d → N),
        Submodule.span R (Set.range s) = ⊤ → ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i})
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hN : IsTorsion' N ↥(Submonoid.powers p)
s : Fin (Nat.succ d) → N
hs : Submodule.span R (Set.range s) = ⊤
x✝ : N
⊢ Decidable (x✝ = 0)

Next tactic:
infer_instance