-- Mathlib/RingTheory/FiniteLength.lean
theorem IsSemisimpleModule.finite_tfae [IsSemisimpleModule R M] :
    List.TFAE [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,
      ∃ s : Set (Submodule R M), s.Finite ∧ sSupIndep s ∧
        sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R m]

Proof state:
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s✝ : Set (Submodule R M)
hs : sSupIndep s✝ ∧ sSup s✝ = ⊤ ∧ ∀ m ∈ s✝, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
x✝ : ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
s : Set (Submodule R M)
fin : s.Finite
left✝ : sSupIndep s
sSup_eq_top : sSup s = ⊤
simple : ∀ (x : ↑s), IsSimpleModule R ↥↑x
⊢ IsNoetherian R ↥(⨆ i, ↑i) ∧ IsArtinian R ↥(⨆ i, ↑i)

Tactic:
have := fin.to_subtype

Result:
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s✝ : Set (Submodule R M)
hs : sSupIndep s✝ ∧ sSup s✝ = ⊤ ∧ ∀ m ∈ s✝, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
x✝ : ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
s : Set (Submodule R M)
fin : s.Finite
left✝ : sSupIndep s
sSup_eq_top : sSup s = ⊤
simple : ∀ (x : ↑s), IsSimpleModule R ↥↑x
this : Finite ↑s
⊢ IsNoetherian R ↥(⨆ i, ↑i) ∧ IsArtinian R ↥(⨆ i, ↑i)

Proof state:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_5_to_4 :
  (∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m) → IsNoetherian R M ∧ IsArtinian R M
tfae_4_to_2 : IsNoetherian R M ∧ IsArtinian R M → IsNoetherian R M
tfae_4_to_3 : IsNoetherian R M ∧ IsArtinian R M → IsArtinian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Tactic:
tfae_finish

Result:
no goals

Proof state:
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Tactic:
rw [isFiniteLength_iff_isNoetherian_isArtinian]

Result:
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Proof state:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Tactic:
tfae_have 3 → 5 := fun _ ↦ ⟨s, WellFoundedLT.finite_of_sSupIndep hs.1, hs⟩

Result:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Proof state:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Tactic:
tfae_have 2 → 5 := fun _ ↦ ⟨s, WellFoundedGT.finite_of_sSupIndep hs.1, hs⟩

Result:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Proof state:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_5_to_4 :
  (∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m) → IsNoetherian R M ∧ IsArtinian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE

Tactic:
tfae_have 4 → 2 := And.left

Result:
case intro
R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s : Set (Submodule R M)
hs : sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_3_to_5 : IsArtinian R M → ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m
tfae_5_to_4 :
  (∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m) → IsNoetherian R M ∧ IsArtinian R M
tfae_4_to_2 : IsNoetherian R M ∧ IsArtinian R M → IsNoetherian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M,
      ∃ s, s.Finite ∧ sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m].TFAE