l3lab/ntp-mathlib huggingface dataset:
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  

Proof state:
case intro.ih
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
by_cases N_bot : N = ⊥
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · 

Proof state:
case pos
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
N_bot : N = ⊥
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Next tactic:
subst N_bot
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    

Proof state:
case pos
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
ih : ∀ N' ≤ ⊥, ∀ x ∈ ⊥, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥⊥)

Next tactic:
exact ⟨0, ⟨Basis.empty _⟩⟩
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    exact ⟨0, ⟨Basis.empty _⟩⟩
  

Proof state:
case neg
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
N_bot : ¬N = ⊥
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    exact ⟨0, ⟨Basis.empty _⟩⟩
  obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    exact ⟨0, ⟨Basis.empty _⟩⟩
  obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
  obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
n' : ℕ
bN' : Basis (Fin n') R ↥N'
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
obtain ⟨bN, _hbN⟩ := h' n' bN'
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    exact ⟨0, ⟨Basis.empty _⟩⟩
  obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
  obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
  obtain ⟨bN, _hbN⟩ := h' n' bN'
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ :=
  Basis.map (Basis.reindex b (Fintype.equivFin ι)) (LinearEquiv.symm (LinearEquiv.ofTop ⊤ (_ : ⊤ = ⊤)))
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
n' : ℕ
bN' : Basis (Fin n') R ↥N'
bN : Basis (Fin (n' + 1)) R ↥N
_hbN :
  ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
    ∃ (hnm : n' + 1 ≤ m' + 1),
      ∃ bM,
        ∀ (as : Fin n' → R),
          (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
            ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
exact ⟨n' + 1, ⟨bN⟩⟩
/-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
if `R` is a principal ideal domain.

This is a `lemma` to make the induction a bit easier. To actually access the basis,
see `Submodule.basisOfPid`.

See also the stronger version `Submodule.smithNormalForm`.
-/
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
    (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
  haveI := Classical.decEq M
  cases nonempty_fintype ι
  induction' N using inductionOnRank with N ih
  · exact b
  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
  by_cases N_bot : N = ⊥
  · subst N_bot
    exact ⟨0, ⟨Basis.empty _⟩⟩
  obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
  obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
  obtain ⟨bN, _hbN⟩ := h' n' bN'
  exact ⟨n' + 1, ⟨bN⟩⟩
  

Proof state:
case intro.inst
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ Finite ι

Next tactic:
infer_instance
theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
  

Proof state:
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
⊢ basisOfPid b ⊥ = { fst := 0, snd := Basis.empty ↥⊥ }

Next tactic:
obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
  obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
  

Proof state:
case mk
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
n : ℕ
b' : Basis (Fin n) R ↥⊥
⊢ { fst := n, snd := b' } = { fst := 0, snd := Basis.empty ↥⊥ }

Next tactic:
let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
  obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
  let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
  

Proof state:
case mk
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
n : ℕ
b' : Basis (Fin n) R ↥⊥
e : Fin n ≃ Fin 0 := Basis.indexEquiv b' (Basis.empty ↥⊥)
⊢ { fst := n, snd := b' } = { fst := 0, snd := Basis.empty ↥⊥ }

Next tactic:
obtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr ⟨e⟩
theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
  obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
  let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
  obtain rfl : n = 0 := by 

Proof state:
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
n : ℕ
b' : Basis (Fin n) R ↥⊥
e : Fin n ≃ Fin 0 := Basis.indexEquiv b' (Basis.empty ↥⊥)
⊢ n = 0

Next tactic:
simpa using Fintype.card_eq.mpr ⟨e⟩
theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
  obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
  let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
  obtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr ⟨e⟩
  

Proof state:
case mk
ι✝ : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module R M
b✝ : ι✝ → M
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
b' : Basis (Fin 0) R ↥⊥
e : Fin 0 ≃ Fin 0 := Basis.indexEquiv b' (Basis.empty ↥⊥)
⊢ { fst := 0, snd := b' } = { fst := 0, snd := Basis.empty ↥⊥ }

Next tactic:
exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
/-- A finite type torsion free module over a PID admits a basis. -/
noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
  

Proof state:
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
b : ι → M
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : NoZeroSMulDivisors R M
⊢ (n : ℕ) × Basis (Fin n) R M

Next tactic:
classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    have := exists_maximal_independent R s
    let I : Set ι := this.choose
    obtain
      ⟨indepI : LinearIndependent R (s ∘ (fun x => x) : I → M), hI :
        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
      this.choose_spec
    let N := span R (range <| (s ∘ (fun x => x) : I → M))
    -- same as `span R (s '' I)` but more convenient
    let _sI : I → N := fun i ↦ ⟨s i.1, subset_span (mem_range_self i)⟩
    -- `s` restricted to `I` is a basis of `N`
    let sI_basis : Basis I R N := Basis.span indepI
    -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
    have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N := by
      intro i
      by_cases hi : i ∈ I
      · use 1, zero_ne_one.symm
        rw [one_smul]
        exact subset_span (mem_range_self (⟨i, hi⟩ : I))
      · simpa [image_eq_range s I] using hI i hi
    choose a ha ha' using exists_a
    let A := ∏ i, a i
    have hA : A ≠ 0 := by
      rw [Finset.prod_ne_zero_iff]
      simpa using ha
    -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
    let φ : M →ₗ[R] M := LinearMap.lsmul R M A
    have : LinearMap.ker φ = ⊥ := @LinearMap.ker_lsmul R M _ _ _ _ _ hA
    let ψ := LinearEquiv.ofInjective φ (LinearMap.ker_eq_bot.mp this)
    have : LinearMap.range φ ≤ N := by
      -- as announced, `A • M ⊆ N`
      suffices ∀ i, φ (s i) ∈ N by
        rw [LinearMap.range_eq_map, ← hs, map_span_le]
        rintro _ ⟨i, rfl⟩
        apply this
      intro i
      calc
        (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
          rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
        _ ∈ N := N.smul_mem _ (ha' i)

    -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
    obtain ⟨n, b : Basis (Fin n) R (LinearMap.range φ)⟩ := Submodule.basisOfPidOfLE this sI_basis
    -- hence `M` is free.
    exact ⟨n, b.map ψ.symm⟩
/-- A finite type torsion free module over a PID admits a basis. -/
noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
  classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    

Proof state:
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
b : ι → M
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : NoZeroSMulDivisors R M
⊢ (n : ℕ) × Basis (Fin n) R M

Next tactic:
have := exists_maximal_independent R s
/-- A finite type torsion free module over a PID admits a basis. -/
noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
  classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    have := exists_maximal_independent R s
    

Proof state:
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
b : ι → M
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : NoZeroSMulDivisors R M
this : ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
⊢ (n : ℕ) × Basis (Fin n) R M

Next tactic:
let I : Set ι := this.choose
/-- A finite type torsion free module over a PID admits a basis. -/
noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
  classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    have := exists_maximal_independent R s
    let I : Set ι := this.choose
    

Proof state:
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
b : ι → M
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : NoZeroSMulDivisors R M
this : ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
I : Set ι := Exists.choose this
⊢ (n : ℕ) × Basis (Fin n) R M

Next tactic:
obtain
      ⟨indepI : LinearIndependent R (s ∘ (fun x => x) : I → M), hI :
        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
      this.choose_spec
/-- A finite type torsion free module over a PID admits a basis. -/
noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
  classical
    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
    have := exists_maximal_independent R s
    let I : Set ι := this.choose
    obtain
      ⟨indepI : LinearIndependent R (s ∘ (fun x => x) : I → M), hI :
        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
      this.choose_spec
    

Proof state:
case intro
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
b : ι → M
inst✝¹ : Fintype ι
s : ι → M
hs : span R (range s) = ⊤
inst✝ : NoZeroSMulDivisors R M
this : ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
I : Set ι := Exists.choose this
indepI : LinearIndependent (ι := ↑I) R (s ∘ fun x => ↑x)
hI : ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)
⊢ (n : ℕ) × Basis (Fin n) R M

Next tactic:
let N := span R (range <| (s ∘ (fun x => x) : I → M))