l3lab/ntp-mathlib huggingface dataset:
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 infer_instance
    

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) = ⊤
this : (x : N) → Decidable (x = 0)
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin (Nat.succ d)), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s 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]; exact Submodule.mem_top⟩
    haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
    exact ⟨0⟩
  · have : ∀ x : N, Decidable (x = 0) := fun _ => by classical infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    

Proof state:
case h.succ.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
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin (Nat.succ d)), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    

Proof state:
case h.succ.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
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin (Nat.succ d)), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
have := IH ?_ s' ?_
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · 

Proof state:
case h.succ.intro.refine_3.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
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
⊢ ∃ k, Nonempty (N ≃ₗ[R] ⨁ (i : Fin (Nat.succ d)), R ⧸ Submodule.span R {p ^ k i})

Next tactic:
have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hN hj fi; refine' ⟨x, h0, _⟩; rw [h1]
        simp only [LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply,
          Function.comp_apply]
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := 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
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
⊢ ∀ (i : Fin d),
    ∃ x,
      p ^ k i • x = 0 ∧
        f (Submodule.Quotient.mk x) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
intro i
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        

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
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
i : Fin d
⊢ ∃ x,
    p ^ k i • x = 0 ∧
      f (Submodule.Quotient.mk x) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        

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
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
i : Fin d
fi : R ⧸ Submodule.span R {p ^ k i} →ₗ[R] N ⧸ Submodule.span R {s j} :=
  ↑(LinearEquiv.symm f) ∘ₗ DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i
⊢ ∃ x,
    p ^ k i • x = 0 ∧
      f (Submodule.Quotient.mk x) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hN hj fi
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hN hj fi; 

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
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
i : Fin d
fi : R ⧸ Submodule.span R {p ^ k i} →ₗ[R] N ⧸ Submodule.span R {s j} :=
  ↑(LinearEquiv.symm f) ∘ₗ DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i
x : N
h0 : p ^ k i • x = 0
h1 : Submodule.Quotient.mk x = fi 1
⊢ ∃ x,
    p ^ k i • x = 0 ∧
      f (Submodule.Quotient.mk x) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
refine' ⟨x, h0, _⟩
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hN hj fi; refine' ⟨x, h0, _⟩; 

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
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
i : Fin d
fi : R ⧸ Submodule.span R {p ^ k i} →ₗ[R] N ⧸ Submodule.span R {s j} :=
  ↑(LinearEquiv.symm f) ∘ₗ DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i
x : N
h0 : p ^ k i • x = 0
h1 : Submodule.Quotient.mk x = fi 1
⊢ f (Submodule.Quotient.mk x) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
rw [h1]
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 infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs
    let s' : Fin d → N ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
    -- `obtain` doesn't work with placeholders.
    have := IH ?_ s' ?_
    obtain ⟨k, ⟨f⟩⟩ := this
    clear IH
    · have : ∀ i : Fin d,
          ∃ x : N, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 := by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hN hj fi; refine' ⟨x, h0, _⟩; rw [h1]
        

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
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
d : ℕ
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) = ⊤
this : (x : N) → Decidable (x = 0)
j : Fin (Nat.succ d)
hj : IsTorsionBy R N (p ^ pOrder hN (s j))
s' : Fin d → N ⧸ Submodule.span R {s j} := Submodule.Quotient.mk ∘ s ∘ Fin.succAbove j
k : Fin d → ℕ
f : (N ⧸ Submodule.span R {s j}) ≃ₗ[R] ⨁ (i : Fin d), R ⧸ Submodule.span R {p ^ k i}
i : Fin d
fi : R ⧸ Submodule.span R {p ^ k i} →ₗ[R] N ⧸ Submodule.span R {s j} :=
  ↑(LinearEquiv.symm f) ∘ₗ DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i
x : N
h0 : p ^ k i • x = 0
h1 : Submodule.Quotient.mk x = fi 1
⊢ f (fi 1) = (DirectSum.lof R (Fin d) (fun i => R ⧸ Submodule.span R {p ^ k i}) i) 1

Next tactic:
simp only [LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply,
          Function.comp_apply]