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

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

Next tactic:
refine ⟨?_, ⟨?_⟩⟩
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]
        simp only [LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply,
          Function.comp_apply]
      refine ⟨?_, ⟨?_⟩⟩
      · 

Proof state:
case h.succ.intro.refine_3.intro.intro.refine_1
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}
this :
  ∀ (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
⊢ Fin (Nat.succ d) → ℕ

Next tactic:
exact fun a => (fun i => (Option.rec (pOrder hN (s j)) k i : ℕ)) (finSuccEquiv d a)
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := 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
h' : Finite R N
hN : IsTorsion R N
⊢ ∃ ι x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  

Proof state:
case intro.intro.intro.intro.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
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
⊢ ∃ ι x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
haveI := fI
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  haveI := fI
  

Proof state:
case intro.intro.intro.intro.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
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
this : Fintype I
⊢ ∃ ι x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
    haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)
    exact fun i =>
      torsion_by_prime_power_decomposition.{u, v} (hp i)
        ((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩)
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  haveI := fI
  have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := 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
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
this : Fintype I
⊢ ∀ (i : I), ∃ d k, Nonempty (↥(torsionBy R N (p i ^ e i)) ≃ₗ[R] ⨁ (j : Fin d), R ⧸ span R {p i ^ k j})

Next tactic:
haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  haveI := fI
  have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
    haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e 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
N : Type (max u v)
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
this✝ : Fintype I
this : ∀ (i : I), IsNoetherian R ↥(torsionBy R N (p i ^ e i))
⊢ ∀ (i : I), ∃ d k, Nonempty (↥(torsionBy R N (p i ^ e i)) ≃ₗ[R] ⨁ (j : Fin d), R ⧸ span R {p i ^ k j})

Next tactic:
exact fun i =>
      torsion_by_prime_power_decomposition.{u, v} (hp i)
        ((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩)
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  haveI := fI
  have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
    haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)
    exact fun i =>
      torsion_by_prime_power_decomposition.{u, v} (hp i)
        ((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩)
  

Proof state:
case intro.intro.intro.intro.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
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
this✝ : Fintype I
this : ∀ (i : I), ∃ d k, Nonempty (↥(torsionBy R N (p i ^ e i)) ≃ₗ[R] ⨁ (j : Fin d), R ⧸ span R {p i ^ k j})
⊢ ∃ ι x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
classical
  refine'
    ⟨Σ i, Fin (this i).choose, inferInstance, fun ⟨i, _⟩ => p i, fun ⟨i, _⟩ => hp i, fun ⟨i, j⟩ =>
      (this i).choose_spec.choose j,
      ⟨(LinearEquiv.ofBijective (DirectSum.coeLinearMap _) h).symm.trans <|
          (DFinsupp.mapRange.linearEquiv fun i => (this i).choose_spec.choose_spec.some).trans <|
            (DirectSum.sigmaLcurryEquiv R).symm.trans
              (DFinsupp.mapRange.linearEquiv fun i => quotEquivOfEq _ _ _)⟩⟩
  cases' i with i j
  simp only
/-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_directSum_of_isTorsion [h' : Module.Finite R N] (hN : Module.IsTorsion R N) :
    ∃ (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  obtain ⟨I, fI, _, p, hp, e, h⟩ := Submodule.exists_isInternal_prime_power_torsion_of_pid hN
  haveI := fI
  have :
    ∀ i,
      ∃ (d : ℕ) (k : Fin d → ℕ),
        Nonempty <| torsionBy R N (p i ^ e i) ≃ₗ[R] ⨁ j, R ⧸ R ∙ p i ^ k j := by
    haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)
    exact fun i =>
      torsion_by_prime_power_decomposition.{u, v} (hp i)
        ((isTorsion'_powers_iff <| p i).mpr fun x => ⟨e i, smul_torsionBy _ _⟩)
  classical
  

Proof state:
case intro.intro.intro.intro.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
h' : Finite R N
hN : IsTorsion R N
I : Type u
fI : Fintype I
w✝ : DecidableEq I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)
this✝ : Fintype I
this : ∀ (i : I), ∃ d k, Nonempty (↥(torsionBy R N (p i ^ e i)) ≃ₗ[R] ⨁ (j : Fin d), R ⧸ span R {p i ^ k j})
⊢ ∃ ι x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
refine'
    ⟨Σ i, Fin (this i).choose, inferInstance, fun ⟨i, _⟩ => p i, fun ⟨i, _⟩ => hp i, fun ⟨i, j⟩ =>
      (this i).choose_spec.choose j,
      ⟨(LinearEquiv.ofBijective (DirectSum.coeLinearMap _) h).symm.trans <|
          (DFinsupp.mapRange.linearEquiv fun i => (this i).choose_spec.choose_spec.some).trans <|
            (DirectSum.sigmaLcurryEquiv R).symm.trans
              (DFinsupp.mapRange.linearEquiv fun i => quotEquivOfEq _ _ _)⟩⟩
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := 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
h' : Finite R N
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
haveI := isNoetherian_submodule' (torsion R N)
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  haveI := isNoetherian_submodule' (torsion R N)
  

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
h' : Finite R N
this : IsNoetherian R ↥(torsion R N)
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  haveI := isNoetherian_submodule' (torsion R N)
  haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
  

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
h' : Finite R N
this✝ : IsNoetherian R ↥(torsion R N)
this : Finite R (N ⧸ torsion R N)
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
    equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  haveI := isNoetherian_submodule' (torsion R N)
  haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
  obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
    equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)
  

Proof state:
case intro.intro.intro.intro.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
h' : Finite R N
this✝ : IsNoetherian R ↥(torsion R N)
this : Finite R (N ⧸ torsion R N)
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : ↥(torsion R N) ≃ₗ[R] ⨁ (i : I), R ⧸ span R {p i ^ e i}
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
obtain ⟨n, ⟨g⟩⟩ := @Module.basisOfFiniteTypeTorsionFree' R _ _ _ (N ⧸ torsion R N) _ _ _ _
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  haveI := isNoetherian_submodule' (torsion R N)
  haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
  obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
    equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)
  obtain ⟨n, ⟨g⟩⟩ := @Module.basisOfFiniteTypeTorsionFree' R _ _ _ (N ⧸ torsion R N) _ _ _ _
  

Proof state:
case intro.intro.intro.intro.intro.intro.mk.ofRepr
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
h' : Finite R N
this✝ : IsNoetherian R ↥(torsion R N)
this : Finite R (N ⧸ torsion R N)
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : ↥(torsion R N) ≃ₗ[R] ⨁ (i : I), R ⧸ span R {p i ^ e i}
n : ℕ
g : (N ⧸ torsion R N) ≃ₗ[R] Fin n →₀ R
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
haveI : Module.Projective R (N ⧸ torsion R N) := Module.Projective.of_basis ⟨g⟩
/-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
  module over a PID is isomorphic to the product of a free module and a direct sum of some
  `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers.-/
theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
    ∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
      Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by
  haveI := isNoetherian_submodule' (torsion R N)
  haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
  obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
    equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)
  obtain ⟨n, ⟨g⟩⟩ := @Module.basisOfFiniteTypeTorsionFree' R _ _ _ (N ⧸ torsion R N) _ _ _ _
  haveI : Module.Projective R (N ⧸ torsion R N) := Module.Projective.of_basis ⟨g⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.mk.ofRepr
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
h' : Finite R N
this✝¹ : IsNoetherian R ↥(torsion R N)
this✝ : Finite R (N ⧸ torsion R N)
I : Type u
fI : Fintype I
p : I → R
hp : ∀ (i : I), Irreducible (p i)
e : I → ℕ
h : ↥(torsion R N) ≃ₗ[R] ⨁ (i : I), R ⧸ span R {p i ^ e i}
n : ℕ
g : (N ⧸ torsion R N) ≃ₗ[R] Fin n →₀ R
this : Projective R (N ⧸ torsion R N)
⊢ ∃ n ι x p,
    ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, Nonempty (N ≃ₗ[R] (Fin n →₀ R) × ⨁ (i : ι), R ⧸ span R {p i ^ e i})

Next tactic:
obtain ⟨f, hf⟩ := Module.projective_lifting_property _ LinearMap.id (torsion R N).mkQ_surjective