awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hrn :
  N ≤
    comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
      (HSMul.hSMul (β := Submodule R M) I (R ∙ i) ⊔ HSMul.hSMul (β := Submodule R M) I (span R s))
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
⊢ ∃ (r : R),
    r - 1 ∈ I ∧
      N ≤
          comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
            (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
        s ⊆ (↑N : Set M)
have insert : c * r - 1 ∈ I
Tactic: refine ⟨c * r, ?_, ?_, hs.2⟩
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hrn :
  N ≤
    comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
      (HSMul.hSMul (β := Submodule R M) I (R ∙ i) ⊔ HSMul.hSMul (β := Submodule R M) I (span R s))
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
insert : c * r - 1 ∈ I
⊢ ∃ (r : R),
    r - 1 ∈ I ∧
      N ≤
          comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
            (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
        s ⊆ (↑N : Set M)
have insert₁ : N ≤
  comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
    (HSMul.hSMul (β := Submodule R M) I (span R s))
Tactic: refine ⟨c * r, ?_, ?_, hs.2⟩
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hrn :
  N ≤
    comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
      (HSMul.hSMul (β := Submodule R M) I (R ∙ i) ⊔ HSMul.hSMul (β := Submodule R M) I (span R s))
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have hrn₁ : Membership.mem (γ := Submodule R M)
  (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
    (HSMul.hSMul (β := Submodule R M) I (R ∙ i) ⊔ HSMul.hSMul (β := Submodule R M) I (span R s)))
  n
Tactic: specialize hrn hn
Cleared hypotheses: hrn
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
hrn :
  Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
      (HSMul.hSMul (β := Submodule R M) I (R ∙ i) ⊔ HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have hrn₁ : ∃ y ∈ HSMul.hSMul (β := Submodule R M) I (R ∙ i),
  ∃ z ∈ HSMul.hSMul (β := Submodule R M) I (span R s),
    y + z = (((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r : M →ₗ[R] M) : (a : M) → M) n
Tactic: rw [mem_comap, mem_sup] at hrn
Cleared hypotheses: hrn
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
hrn :
  ∃ y ∈ HSMul.hSMul (β := Submodule R M) I (R ∙ i),
    ∃ z ∈ HSMul.hSMul (β := Submodule R M) I (span R s),
      y + z = (((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r : M →ₗ[R] M) : (a : M) → M) n
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have insert : ∀ (y : M) (hy : y ∈ HSMul.hSMul (β := Submodule R M) I (R ∙ i)) (z : M)
  (hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s))
  (hyz : y + z = (((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r : M →ₗ[R] M) : (a : M) → M) n),
  Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
Tactic: rcases hrn with ⟨y, hy, z, hz, hyz⟩
Cleared hypotheses: hrn
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
y : M
hy : y ∈ HSMul.hSMul (β := Submodule R M) I (R ∙ i)
z : M
hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
hyz : y + z = r • n
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have insert : ∀ (hy : ∃ y_1 ∈ I, y_1 • i = y) (z : M) (hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)) (hyz : y + z = r • n),
  Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
Tactic: rw [mem_smul_span_singleton] at hy
Cleared hypotheses: hy, z, hz, hyz
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
y : M
hy : ∃ y_1 ∈ I, y_1 • i = y
z : M
hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
hyz : y + z = r • n
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have insert : ∀ d ∈ I,
  ∀ (hyz : d • i + z = r • n),
    Membership.mem (γ := Submodule R M)
      (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
        (HSMul.hSMul (β := Submodule R M) I (span R s)))
      n
Tactic: rcases hy with ⟨d, _, rfl⟩
Cleared hypotheses: y, hy, hyz
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
z : M
hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
d : R
left : d ∈ I
hyz : d • i + z = r • n
⊢ Membership.mem (γ := Submodule R M)
    (comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) (c * r))
      (HSMul.hSMul (β := Submodule R M) I (span R s)))
    n
have insert : (c * r) • n ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
Tactic: simp only [mem_comap, LinearMap.lsmul_apply]
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s_1 : Set M
i : M
s : Set M
h : i ∉ s
hs_1 : s.Finite
ih :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
            s ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ (↑N : Set M) ∧ s ⊆ (↑N : Set M)
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
n : M
hn : n ∈ N
z : M
hz : z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
d : R
left : d ∈ I
hyz : d • i + z = r • n
⊢ (c * r) • n ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
have insert : d • c • i + c • z ∈ HSMul.hSMul (β := Submodule R M) I (span R s)
Tactic: rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s : Set M
hfs : s.Finite
H :
  ∃ (r : R),
    r - 1 ∈ I ∧
      N ≤
          comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
            (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
        s ⊆ (↑N : Set M)
⊢ ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
have empty : ∀
  (H :
    ∃ (r : R),
      r - 1 ∈ I ∧
        N ≤
            comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
              (HSMul.hSMul (β := Submodule R M) I (span R ∅)) ∧
          ∅ ⊆ (↑N : Set M)),
  ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
Tactic: induction s, hfs using Set.Finite.induction_on with
  | empty =>
    rcases H with ⟨r, hr1, hrn, _⟩
    refine ⟨r, hr1, fun n hn => ?_⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  | @insert i s _ _ ih =>
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel_left]
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine ⟨c * r, ?_, ?_, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    simp only [mem_comap, LinearMap.lsmul_apply]
    rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
    exact add_mem (smul_mem _ _ hci) (smul_mem _ _ hz)
Cleared hypotheses: hfs, H
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
s : Set M
hfs : s.Finite
H :
  ∃ (r : R),
    r - 1 ∈ I ∧
      N ≤
          comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
            (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
        s ⊆ (↑N : Set M)
empty :
  ∀
    (H :
      ∃ (r : R),
        r - 1 ∈ I ∧
          N ≤
              comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                (HSMul.hSMul (β := Submodule R M) I (span R ∅)) ∧
            ∅ ⊆ (↑N : Set M)),
    ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
⊢ ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
have insert : ∀ {i : M} {s : Set M},
  i ∉ s →
    s.Finite →
      ∀
        (ih :
          ∀
            (H :
              ∃ (r : R),
                r - 1 ∈ I ∧
                  N ≤
                      comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                        (HSMul.hSMul (β := Submodule R M) I (span R s)) ∧
                    s ⊆ (↑N : Set M)),
            ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0)
        (H :
          ∃ (r : R),
            r - 1 ∈ I ∧
              N ≤
                  comap (σ₁₂ := RingHom.id R) ((LinearMap.lsmul R M : (a : R) → M →ₗ[R] M) r)
                    (HSMul.hSMul (β := Submodule R M) I (span R (insert i s))) ∧
                insert i s ⊆ (↑N : Set M)),
        ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
Tactic: induction s, hfs using Set.Finite.induction_on with
  | empty =>
    rcases H with ⟨r, hr1, hrn, _⟩
    refine ⟨r, hr1, fun n hn => ?_⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  | @insert i s _ _ ih =>
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel_left]
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine ⟨c * r, ?_, ?_, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    simp only [mem_comap, LinearMap.lsmul_apply]
    rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
    exact add_mem (smul_mem _ _ hci) (smul_mem _ _ hz)
Cleared hypotheses: hfs, H
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nakayama (goal)
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
hn : N.FG
hin : N ≤ I • N
⊢ ∃ r ∈ I, ∀ n ∈ N, r • n = n
have obtain : ∀ (r : R) (hr : r - 1 ∈ I) (hr' : ∀ n ∈ N, r • n = 0), ∃ r ∈ I, ∀ n ∈ N, r • n = n
Tactic: obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
⊢ IsNilpotent f ↔ ∀ (m : M), ∃ (n : ℕ), Eq (α := M) ((f ^ n : (a : M) → M) m : M) (0 : M)
have refine : ∀ (h : ∀ (m : M), ∃ (n : ℕ), Eq (α := M) ((f ^ n : (a : M) → M) m : M) (0 : M)), IsNilpotent f
Tactic: refine ⟨fun ⟨n, hn⟩ m ↦ ⟨n, by simp [hn]⟩, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
h : ∀ (m : M), ∃ (n : ℕ), Eq (α := M) ((f ^ n : (a : M) → M) m : M) (0 : M)
⊢ IsNilpotent f
have rcases : ∀ (S : Finset M) (hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤), IsNilpotent f
Tactic: rcases Module.Finite.fg_top (R := R) (M := M) with ⟨S, hS⟩
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
h : ∀ (m : M), ∃ (n : ℕ), Eq (α := M) ((f ^ n : (a : M) → M) m : M) (0 : M)
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
⊢ IsNilpotent f
have choose : ∀ (g : (m : M) → ℕ) (hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)), IsNilpotent f
Tactic: choose g hg using h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
⊢ IsNilpotent f
have h : f ^ S.sup g = 0
Tactic: use Finset.sup S g
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
⊢ f ^ S.sup g = 0
have h : ∀ (m : M), Eq (α := M) ((f ^ S.sup g : (a : M) → M) m : M) ((0 : (a : M) → M) m : M)
Tactic: ext m
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
m : M
⊢ Eq (α := M) ((f ^ S.sup g : (a : M) → M) m : M) ((0 : (a : M) → M) m : M)
have hm : Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) m
Tactic: have hm : m ∈ Submodule.span R S := by simp [hS]
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
m : M
hm : Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) m
⊢ Eq (α := M) ((f ^ S.sup g : (a : M) → M) m : M) ((0 : (a : M) → M) m : M)
have h : ∀ (x : M) (hx : x ∈ (↑S : Set M)), Eq (α := M) ((f ^ S.sup g : (a : M) → M) x : M) ((0 : (a : M) → M) x : M)
Tactic: induction hm using Submodule.span_induction with
  | mem x hx => exact pow_map_zero_of_le (Finset.le_sup hx) (hg x)
  | zero => simp
  | add => simp_all
  | smul => simp_all
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
m : M
hm : Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) m
h : ∀ (x : M) (hx : x ∈ (↑S : Set M)), Eq (α := M) ((f ^ S.sup g : (a : M) → M) x : M) ((0 : (a : M) → M) x : M)
⊢ Eq (α := M) ((f ^ S.sup g : (a : M) → M) m : M) ((0 : (a : M) → M) m : M)
have h₁ : Eq (α := M) ((f ^ S.sup g : (a : M) → M) 0 : M) ((0 : (a : M) → M) 0 : M)
Tactic: induction hm using Submodule.span_induction with
  | mem x hx => exact pow_map_zero_of_le (Finset.le_sup hx) (hg x)
  | zero => simp
  | add => simp_all
  | smul => simp_all
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.RingTheory.Finiteness.Nilpotent (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
inst_3 : Module.Finite R M
f : End R M
S : Finset M
hS : Eq (α := Submodule R M) (Submodule.span R (↑S : Set M)) ⊤
g : (m : M) → ℕ
hg : ∀ (m : M), Eq (α := M) ((f ^ g m : (a : M) → M) m : M) (0 : M)
m : M
hm : Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) m
h : ∀ (x : M) (hx : x ∈ (↑S : Set M)), Eq (α := M) ((f ^ S.sup g : (a : M) → M) x : M) ((0 : (a : M) → M) x : M)
h₁ : Eq (α := M) ((f ^ S.sup g : (a : M) → M) 0 : M) ((0 : (a : M) → M) 0 : M)
⊢ Eq (α := M) ((f ^ S.sup g : (a : M) → M) m : M) ((0 : (a : M) → M) m : M)
have h₂ : ∀ (x y : M),
  Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) x →
    Membership.mem (γ := Submodule R M) (Submodule.span R (↑S : Set M)) y →
      Eq (α := M) ((f ^ S.sup g : (a : M) → M) x : M) ((0 : (a : M) → M) x : M) →
        Eq (α := M) ((f ^ S.sup g : (a : M) → M) y : M) ((0 : (a : M) → M) y : M) →
          Eq (α := M) ((f ^ S.sup g : (a : M) → M) (x + y) : M) ((0 : (a : M) → M) (x + y) : M)
Tactic: induction hm using Submodule.span_induction with
  | mem x hx => exact pow_map_zero_of_le (Finset.le_sup hx) (hg x)
  | zero => simp
  | add => simp_all
  | smul => simp_all
Cleared hypotheses: hm