l3lab/ntp-mathlib huggingface dataset:
theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
    [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
    (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
    generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ := by
  let a : R := generator (ϕ.submoduleImage N)
  let d : R := IsPrincipal.generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})
  have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
  have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
  refine' dvd_trans _ d_dvd_right
  rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
    exact ⟨r₁, r₂, hr₁⟩
  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
  have : span R {d} ≤ ψ'.submoduleImage N := 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
N O : Submodule R M
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
inst✝ : IsPrincipal (LinearMap.submoduleImage ϕ N)
y : M
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ O) } = generator (LinearMap.submoduleImage ϕ N)
ψ : ↥O →ₗ[R] R
a : R := generator (LinearMap.submoduleImage ϕ N)
d : R := generator (span R {a, ψ { val := y, property := (_ : y ∈ O) }})
d_dvd_left : d ∣ a
d_dvd_right : d ∣ ψ { val := y, property := (_ : y ∈ O) }
r₁ r₂ : R
d_eq : d = r₁ * a + r₂ * ψ { val := y, property := (_ : y ∈ O) }
ψ' : ↥O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
⊢ span R {d} ≤ LinearMap.submoduleImage ψ' N

Next tactic:
rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
    [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
    (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
    generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ := by
  let a : R := generator (ϕ.submoduleImage N)
  let d : R := IsPrincipal.generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})
  have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
  have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
  refine' dvd_trans _ d_dvd_right
  rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
    exact ⟨r₁, r₂, hr₁⟩
  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
  have : span R {d} ≤ ψ'.submoduleImage N := by
    rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
    

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
N O : Submodule R M
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
inst✝ : IsPrincipal (LinearMap.submoduleImage ϕ N)
y : M
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ O) } = generator (LinearMap.submoduleImage ϕ N)
ψ : ↥O →ₗ[R] R
a : R := generator (LinearMap.submoduleImage ϕ N)
d : R := generator (span R {a, ψ { val := y, property := (_ : y ∈ O) }})
d_dvd_left : d ∣ a
d_dvd_right : d ∣ ψ { val := y, property := (_ : y ∈ O) }
r₁ r₂ : R
d_eq : d = r₁ * a + r₂ * ψ { val := y, property := (_ : y ∈ O) }
ψ' : ↥O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
⊢ ∃ y, ∃ (yN : y ∈ N), ψ' { val := y, property := (_ : y ∈ O) } = d

Next tactic:
refine' ⟨y, yN, _⟩
theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
    [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
    (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
    generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ := by
  let a : R := generator (ϕ.submoduleImage N)
  let d : R := IsPrincipal.generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})
  have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
  have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
  refine' dvd_trans _ d_dvd_right
  rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
    exact ⟨r₁, r₂, hr₁⟩
  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
  have : span R {d} ≤ ψ'.submoduleImage N := by
    rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
    refine' ⟨y, yN, _⟩
    

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
N O : Submodule R M
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
inst✝ : IsPrincipal (LinearMap.submoduleImage ϕ N)
y : M
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ O) } = generator (LinearMap.submoduleImage ϕ N)
ψ : ↥O →ₗ[R] R
a : R := generator (LinearMap.submoduleImage ϕ N)
d : R := generator (span R {a, ψ { val := y, property := (_ : y ∈ O) }})
d_dvd_left : d ∣ a
d_dvd_right : d ∣ ψ { val := y, property := (_ : y ∈ O) }
r₁ r₂ : R
d_eq : d = r₁ * a + r₂ * ψ { val := y, property := (_ : y ∈ O) }
ψ' : ↥O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
⊢ ψ' { val := y, property := (_ : y ∈ O) } = d

Next tactic:
change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
    [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
    (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
    generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ := by
  let a : R := generator (ϕ.submoduleImage N)
  let d : R := IsPrincipal.generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})
  have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
  have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
  refine' dvd_trans _ d_dvd_right
  rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
    exact ⟨r₁, r₂, hr₁⟩
  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
  have : span R {d} ≤ ψ'.submoduleImage N := by
    rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
    refine' ⟨y, yN, _⟩
    change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
    

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
N O : Submodule R M
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
inst✝ : IsPrincipal (LinearMap.submoduleImage ϕ N)
y : M
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ O) } = generator (LinearMap.submoduleImage ϕ N)
ψ : ↥O →ₗ[R] R
a : R := generator (LinearMap.submoduleImage ϕ N)
d : R := generator (span R {a, ψ { val := y, property := (_ : y ∈ O) }})
d_dvd_left : d ∣ a
d_dvd_right : d ∣ ψ { val := y, property := (_ : y ∈ O) }
r₁ r₂ : R
d_eq : d = r₁ * a + r₂ * ψ { val := y, property := (_ : y ∈ O) }
ψ' : ↥O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
⊢ r₁ * ϕ { val := y, property := (_ : y ∈ O) } + r₂ * ψ { val := y, property := (_ : y ∈ O) } = d

Next tactic:
rw [d_eq, ϕy_eq]
theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
    [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
    (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
    generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ := by
  let a : R := generator (ϕ.submoduleImage N)
  let d : R := IsPrincipal.generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})
  have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
  have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
  refine' dvd_trans _ d_dvd_right
  rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
    exact ⟨r₁, r₂, hr₁⟩
  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
  have : span R {d} ≤ ψ'.submoduleImage N := by
    rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
    refine' ⟨y, yN, _⟩
    change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
    rw [d_eq, ϕy_eq]
  refine'
    le_antisymm (this.trans (le_of_eq _)) (Ideal.span_singleton_le_span_singleton.mpr d_dvd_left)
  rw [span_singleton_generator]
  apply (le_trans _ this).eq_of_not_gt (hϕ ψ')
  rw [← span_singleton_generator (ϕ.submoduleImage N)]
  exact Ideal.span_singleton_le_span_singleton.mpr d_dvd_left
  · 

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
N O : Submodule R M
hNO : N ≤ O
ϕ : ↥O →ₗ[R] R
hϕ : ∀ (ψ : ↥O →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N
inst✝ : IsPrincipal (LinearMap.submoduleImage ϕ N)
y : M
yN : y ∈ N
ϕy_eq : ϕ { val := y, property := (_ : y ∈ O) } = generator (LinearMap.submoduleImage ϕ N)
ψ : ↥O →ₗ[R] R
a : R := generator (LinearMap.submoduleImage ϕ N)
d : R := generator (span R {a, ψ { val := y, property := (_ : y ∈ O) }})
d_dvd_left : d ∣ a
d_dvd_right : d ∣ ψ { val := y, property := (_ : y ∈ O) }
⊢ generator (LinearMap.submoduleImage ϕ N) ∈ span R {a, ψ { val := y, property := (_ : y ∈ O) }}

Next tactic:
exact subset_span (mem_insert _ _)
/-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.

Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
submodules `N' ≤ M'` of strictly smaller rank, whose basis we can extend to get a basis
of `N` and `M`. Moreover, if the basis for `M'` is up to scalars a basis for `N'`,
then the basis we find for `M` is up to scalars a basis for `N`.

For `basis_of_pid` we only need the first half and can fix `M = ⊤`,
for `smith_normal_form` we need the full statement,
but must also feed in a basis for `M` using `basis_of_pid` to keep the induction going.
-/
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
    (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
    ∃ y ∈ M,
      ∃ (a : R) (_ : a • y ∈ N),
        ∃ M' ≤ M,
          ∃ N' ≤ N,
            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
              ∀ (n') (bN' : Basis (Fin n') R N'),
                ∃ bN : Basis (Fin (n' + 1)) R N,
                  ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                      ∀ (as : Fin n' → R)
                        (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                        ∃ as' : Fin (n' + 1) → R,
                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
  -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
  -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
  have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N := by
    

Proof state:
ι : Type u_1
R : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M✝ : Type u_3
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R M✝
b : ι → M✝
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
⊢ ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N

Next tactic:
obtain ⟨P, P_eq, P_max⟩ :=
      set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _
        (show (Set.range fun ψ : M →ₗ[R] R ↦ ψ.submoduleImage N).Nonempty from
          ⟨_, Set.mem_range.mpr ⟨0, rfl⟩⟩)
/-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.

Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
submodules `N' ≤ M'` of strictly smaller rank, whose basis we can extend to get a basis
of `N` and `M`. Moreover, if the basis for `M'` is up to scalars a basis for `N'`,
then the basis we find for `M` is up to scalars a basis for `N`.

For `basis_of_pid` we only need the first half and can fix `M = ⊤`,
for `smith_normal_form` we need the full statement,
but must also feed in a basis for `M` using `basis_of_pid` to keep the induction going.
-/
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
    (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
    ∃ y ∈ M,
      ∃ (a : R) (_ : a • y ∈ N),
        ∃ M' ≤ M,
          ∃ N' ≤ N,
            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
              ∀ (n') (bN' : Basis (Fin n') R N'),
                ∃ bN : Basis (Fin (n' + 1)) R N,
                  ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                      ∀ (as : Fin n' → R)
                        (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                        ∃ as' : Fin (n' + 1) → R,
                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
  -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
  -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
  have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N := by
    obtain ⟨P, P_eq, P_max⟩ :=
      set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _
        (show (Set.range fun ψ : M →ₗ[R] R ↦ ψ.submoduleImage N).Nonempty from
          ⟨_, Set.mem_range.mpr ⟨0, rfl⟩⟩)
    

Proof state:
case intro.intro
ι : Type u_1
R : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M✝ : Type u_3
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R M✝
b : ι → M✝
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
P : Submodule R R
P_eq : P ∈ range fun ψ => LinearMap.submoduleImage ψ N
P_max : ∀ I ∈ range fun ψ => LinearMap.submoduleImage ψ N, ¬P < I
⊢ ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N

Next tactic:
obtain ⟨ϕ, rfl⟩ := Set.mem_range.mp P_eq
/-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.

Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
submodules `N' ≤ M'` of strictly smaller rank, whose basis we can extend to get a basis
of `N` and `M`. Moreover, if the basis for `M'` is up to scalars a basis for `N'`,
then the basis we find for `M` is up to scalars a basis for `N`.

For `basis_of_pid` we only need the first half and can fix `M = ⊤`,
for `smith_normal_form` we need the full statement,
but must also feed in a basis for `M` using `basis_of_pid` to keep the induction going.
-/
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
    (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
    ∃ y ∈ M,
      ∃ (a : R) (_ : a • y ∈ N),
        ∃ M' ≤ M,
          ∃ N' ≤ N,
            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
              ∀ (n') (bN' : Basis (Fin n') R N'),
                ∃ bN : Basis (Fin (n' + 1)) R N,
                  ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                      ∀ (as : Fin n' → R)
                        (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                        ∃ as' : Fin (n' + 1) → R,
                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
  -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
  -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
  have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N := by
    obtain ⟨P, P_eq, P_max⟩ :=
      set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _
        (show (Set.range fun ψ : M →ₗ[R] R ↦ ψ.submoduleImage N).Nonempty from
          ⟨_, Set.mem_range.mpr ⟨0, rfl⟩⟩)
    obtain ⟨ϕ, rfl⟩ := Set.mem_range.mp P_eq
    

Proof state:
case intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M✝ : Type u_3
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R M✝
b : ι → M✝
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
ϕ : ↥M →ₗ[R] R
P_eq : LinearMap.submoduleImage ϕ N ∈ range fun ψ => LinearMap.submoduleImage ψ N
P_max : ∀ I ∈ range fun ψ => LinearMap.submoduleImage ψ N, ¬LinearMap.submoduleImage ϕ N < I
⊢ ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬LinearMap.submoduleImage ϕ N < LinearMap.submoduleImage ψ N

Next tactic:
exact ⟨ϕ, fun ψ hψ ↦ P_max _ ⟨_, rfl⟩ hψ⟩
/-- 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
  

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 : Submodule R M
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Next tactic:
haveI := Classical.decEq M
/-- 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
  

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 : Submodule R M
this : DecidableEq M
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Next tactic:
cases nonempty_fintype ι
/-- 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
  

Proof state:
case 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
N : Submodule R M
this : DecidableEq M
val✝ : Fintype ι
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Next tactic:
induction' N using inductionOnRank with N ih
/-- 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
  · 

Proof state:
case intro.b
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
⊢ Basis ?m.115820 R M

Next tactic:
exact b
/-- 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
  

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')
⊢ ∃ 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:
let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm