l3lab/ntp-mathlib huggingface dataset:
theorem isNoetherian_iff_fg_wellFounded :
    IsNoetherian R M ↔
      WellFounded
        ((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by
  let α := { N : Submodule R M // N.FG }
  constructor
  · intro H
    let f : α ↪o Submodule R M := OrderEmbedding.subtype _
    exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
  · intro H
    constructor
    intro N
    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
      WellFounded.has_min H { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
    convert h₁
    refine' (e.antisymm _).symm
    by_contra h₃
    

Proof state:
case h.e'_6
R : Type u_1
M : Type u_2
P : Type u_3
N✝ : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N✝
inst✝² : Module R N✝
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
α : Type u_2 := { N // FG N }
H : WellFounded fun x x_1 => x > x_1
N N₀ : Submodule R M
h₁ : FG N₀
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > { val := N₀, property := h₁ }
h₃ : ¬N ≤ N₀
⊢ False

Next tactic:
obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
theorem isNoetherian_iff_fg_wellFounded :
    IsNoetherian R M ↔
      WellFounded
        ((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by
  let α := { N : Submodule R M // N.FG }
  constructor
  · intro H
    let f : α ↪o Submodule R M := OrderEmbedding.subtype _
    exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
  · intro H
    constructor
    intro N
    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
      WellFounded.has_min H { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
    convert h₁
    refine' (e.antisymm _).symm
    by_contra h₃
    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
    

Proof state:
case h.e'_6.intro.intro
R : Type u_1
M : Type u_2
P : Type u_3
N✝ : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N✝
inst✝² : Module R N✝
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
α : Type u_2 := { N // FG N }
H : WellFounded fun x x_1 => x > x_1
N N₀ : Submodule R M
h₁ : FG N₀
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > { val := N₀, property := h₁ }
h₃ : ¬N ≤ N₀
x : M
hx₁ : x ∈ N
hx₂ : x ∉ N₀
⊢ False

Next tactic:
apply hx₂
theorem isNoetherian_iff_fg_wellFounded :
    IsNoetherian R M ↔
      WellFounded
        ((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by
  let α := { N : Submodule R M // N.FG }
  constructor
  · intro H
    let f : α ↪o Submodule R M := OrderEmbedding.subtype _
    exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
  · intro H
    constructor
    intro N
    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
      WellFounded.has_min H { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
    convert h₁
    refine' (e.antisymm _).symm
    by_contra h₃
    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
    apply hx₂
    

Proof state:
case h.e'_6.intro.intro
R : Type u_1
M : Type u_2
P : Type u_3
N✝ : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N✝
inst✝² : Module R N✝
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
α : Type u_2 := { N // FG N }
H : WellFounded fun x x_1 => x > x_1
N N₀ : Submodule R M
h₁ : FG N₀
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > { val := N₀, property := h₁ }
h₃ : ¬N ≤ N₀
x : M
hx₁ : x ∈ N
hx₂ : x ∉ N₀
⊢ x ∈ N₀

Next tactic:
rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂
      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|
      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]
theorem isNoetherian_iff_fg_wellFounded :
    IsNoetherian R M ↔
      WellFounded
        ((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by
  let α := { N : Submodule R M // N.FG }
  constructor
  · intro H
    let f : α ↪o Submodule R M := OrderEmbedding.subtype _
    exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
  · intro H
    constructor
    intro N
    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
      WellFounded.has_min H { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
    convert h₁
    refine' (e.antisymm _).symm
    by_contra h₃
    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
    apply hx₂
    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂
      ⟨_, Submodule.FG.sup ⟨{x}, by 

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N✝ : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N✝
inst✝² : Module R N✝
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
α : Type u_2 := { N // FG N }
H : WellFounded fun x x_1 => x > x_1
N N₀ : Submodule R M
h₁ : FG N₀
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > { val := N₀, property := h₁ }
h₃ : ¬N ≤ N₀
x : M
hx₁ : x ∈ N
hx₂ : x ∉ N₀
⊢ span R ↑{x} = span R {x}

Next tactic:
rw [Finset.coe_singleton]
theorem isNoetherian_iff_fg_wellFounded :
    IsNoetherian R M ↔
      WellFounded
        ((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by
  let α := { N : Submodule R M // N.FG }
  constructor
  · intro H
    let f : α ↪o Submodule R M := OrderEmbedding.subtype _
    exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
  · intro H
    constructor
    intro N
    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
      WellFounded.has_min H { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩
    convert h₁
    refine' (e.antisymm _).symm
    by_contra h₃
    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃
    apply hx₂
    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂
      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|
      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]
    

Proof state:
case h.e'_6.intro.intro
R : Type u_1
M : Type u_2
P : Type u_3
N✝ : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N✝
inst✝² : Module R N✝
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
α : Type u_2 := { N // FG N }
H : WellFounded fun x x_1 => x > x_1
N N₀ : Submodule R M
h₁ : FG N₀
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > { val := N₀, property := h₁ }
h₃ : ¬N ≤ N₀
x : M
hx₁ : x ∈ N
hx₂ : x ∉ N₀
⊢ x ∈ span R {x} ⊔ N₀

Next tactic:
exact (le_sup_left : (R ∙ x) ≤ _) (Submodule.mem_span_singleton_self _)
/-- A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them.
-/
theorem set_has_maximal_iff_noetherian :
    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
⊢ (∀ (a : Set (Submodule R M)), Set.Nonempty a → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M

Next tactic:
rw [isNoetherian_iff_wellFounded, WellFounded.wellFounded_iff_has_min]
/-- A module is Noetherian iff every increasing chain of submodules stabilizes. -/
theorem monotone_stabilizes_iff_noetherian :
    (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
⊢ (∀ (f : ℕ →o Submodule R M), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m) ↔ IsNoetherian R M

Next tactic:
rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
theorem eventuallyConst_of_isNoetherian [IsNoetherian R M] (f : ℕ →o Submodule R M) :
    atTop.EventuallyConst f := by
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
inst✝² : AddCommMonoid P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ →o Submodule R M
⊢ EventuallyConst (⇑f) atTop

Next tactic:
simp_rw [eventuallyConst_atTop, eq_comm]
theorem eventuallyConst_of_isNoetherian [IsNoetherian R M] (f : ℕ →o Submodule R M) :
    atTop.EventuallyConst f := by
  simp_rw [eventuallyConst_atTop, eq_comm]
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R N
inst✝² : AddCommMonoid P
inst✝¹ : Module R P
inst✝ : IsNoetherian R M
f : ℕ →o Submodule R M
⊢ ∃ i, ∀ (j : ℕ), i ≤ j → f i = f j

Next tactic:
exact (monotone_stabilizes_iff_noetherian.mpr inferInstance) f
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
⊢ Set.Finite s

Next tactic:
refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
⊢ (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1

Next tactic:
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
⊢ (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1

Next tactic:
have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
⊢ ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s

Next tactic:
rintro n x ⟨y, _, rfl⟩
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    

Proof state:
case intro.intro
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
n y : ℕ
left✝ : y ∈ {m | m ≤ n}
⊢ (Subtype.val ∘ ⇑f) y ∈ s

Next tactic:
exact (f y).2
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
⊢ (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1

Next tactic:
let coe' : s → M := (↑)
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
⊢ (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1

Next tactic:
have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
    exact ⟨fun hab x (hxa : x ≤ a) => le_trans hxa hab, fun hx => hx a (le_refl a)⟩
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
⊢ ∀ (a b : ℕ), a ≤ b ↔ span R (coe' ∘ ⇑f '' {m | m ≤ a}) ≤ span R (Subtype.val ∘ ⇑f '' {m | m ≤ b})

Next tactic:
intro a b
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
a b : ℕ
⊢ a ≤ b ↔ span R (coe' ∘ ⇑f '' {m | m ≤ a}) ≤ span R (Subtype.val ∘ ⇑f '' {m | m ≤ b})

Next tactic:
rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
    

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
a b : ℕ
⊢ a ≤ b ↔ ∀ x ∈ {m | m ≤ a}, x ∈ {m | m ≤ b}

Next tactic:
exact ⟨fun hab x (hxa : x ≤ a) => le_trans hxa hab, fun hx => hx a (le_refl a)⟩
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
    exact ⟨fun hab x (hxa : x ≤ a) => le_trans hxa hab, fun hx => hx a (le_refl a)⟩
  

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this✝ : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
this : ∀ (a b : ℕ), a ≤ b ↔ span R (coe' ∘ ⇑f '' {m | m ≤ a}) ≤ span R (Subtype.val ∘ ⇑f '' {m | m ≤ b})
⊢ (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1

Next tactic:
exact
    ⟨⟨fun n => span R (coe' ∘ f '' { m | m ≤ n }), fun x y => by
        rw [le_antisymm_iff, (this x y).symm, (this y x).symm, ← le_antisymm_iff, imp_self]
        trivial⟩,
      by dsimp [GT.gt]; simp only [lt_iff_le_not_le, (this _ _).symm]; tauto⟩
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
    exact ⟨fun hab x (hxa : x ≤ a) => le_trans hxa hab, fun hx => hx a (le_refl a)⟩
  exact
    ⟨⟨fun n => span R (coe' ∘ f '' { m | m ≤ n }), fun x y => by
        

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this✝ : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
this : ∀ (a b : ℕ), a ≤ b ↔ span R (coe' ∘ ⇑f '' {m | m ≤ a}) ≤ span R (Subtype.val ∘ ⇑f '' {m | m ≤ b})
x y : ℕ
⊢ (fun n => span R (coe' ∘ ⇑f '' {m | m ≤ n})) x = (fun n => span R (coe' ∘ ⇑f '' {m | m ≤ n})) y → x = y

Next tactic:
rw [le_antisymm_iff, (this x y).symm, (this y x).symm, ← le_antisymm_iff, imp_self]
theorem finite_of_linearIndependent [Nontrivial R] {s : Set M}
    (hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
  refine'
    by_contradiction fun hf =>
      (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_gt R M)).elim' _
  have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
  have : ∀ n, (↑) ∘ f '' { m | m ≤ n } ⊆ s := by
    rintro n x ⟨y, _, rfl⟩
    exact (f y).2
  let coe' : s → M := (↑)
  have : ∀ a b : ℕ, a ≤ b ↔
    span R (coe' ∘ f '' { m | m ≤ a }) ≤ span R ((↑) ∘ f '' { m | m ≤ b }) := by
    intro a b
    rw [span_le_span_iff hs (this a) (this b),
      Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
    exact ⟨fun hab x (hxa : x ≤ a) => le_trans hxa hab, fun hx => hx a (le_refl a)⟩
  exact
    ⟨⟨fun n => span R (coe' ∘ f '' { m | m ≤ n }), fun x y => by
        rw [le_antisymm_iff, (this x y).symm, (this y x).symm, ← le_antisymm_iff, imp_self]
        

Proof state:
R : Type u_1
M : Type u_2
P : Type u_3
N : Type w
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherian R M
inst✝ : Nontrivial R
s : Set M
hs : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val
hf : ¬Set.Finite s
f : ℕ ↪ ↑s
this✝ : ∀ (n : ℕ), Subtype.val ∘ ⇑f '' {m | m ≤ n} ⊆ s
coe' : ↑s → M := Subtype.val
this : ∀ (a b : ℕ), a ≤ b ↔ span R (coe' ∘ ⇑f '' {m | m ≤ a}) ≤ span R (Subtype.val ∘ ⇑f '' {m | m ≤ b})
x y : ℕ
⊢ True

Next tactic:
trivial