awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_1 :
  Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
have refine_2 : ∀ ⦃J : Submodule R M⦄,
  Membership.mem (γ := Finset (Submodule R M))
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      J →
    J.IsPrimary
Tactic: refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_1 :
  Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N
refine_2 :
  ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
have refine_3 : (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M)).Pairwise
  ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
Tactic: refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
⊢ Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N
have refine_1 : ∀ (x : M),
  Membership.mem (γ := Submodule R M)
      (Finset.inf
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        id)
      x ↔
    x ∈ N
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
⊢ ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary
have refine_2 : ∀ a ∈ s, ({I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id).IsPrimary
Tactic: simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ ({I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}.inf id).IsPrimary
have refine_2 : J ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}
Tactic: refine isPrimary_finsetInf (i := J) ?_ ?_ (by simp)
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
refine_2 : J ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}
⊢ ({I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}.inf id).IsPrimary
have refine_2₁ : ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary
Tactic: refine isPrimary_finsetInf (i := J) ?_ ?_ (by simp)
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary
have refine_2 : ∀ ⦃y : Submodule R M⦄, y ∈ s → (y.colon Set.univ).radical = (J.colon Set.univ).radical → y.IsPrimary
Tactic: simp only [Finset.mem_filter, id_eq, and_imp]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I : Submodule R M
hI :
  I ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
J : Submodule R M
hJ :
  J ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
hIJ : I ≠ J
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
have hI₁ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I
Tactic: simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ
Cleared hypotheses: hI, hJ
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I : Submodule R M
hI :
  I ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
J : Submodule R M
hJ :
  J ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
hIJ : I ≠ J
hI₁ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
have hJ₁ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J
Tactic: simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ
Cleared hypotheses: hI, hJ
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
hI : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I
hJ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
have refine_3 : ∀ (I' : Submodule R M) (hI' : I' ∈ s)
  (hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I),
  ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
Tactic: obtain ⟨I', hI', hI⟩ := hI
Cleared hypotheses: hI
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
hJ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J
I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
have refine_3 : ∀ (J' : Submodule R M) (hJ' : J' ∈ s)
  (hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J),
  ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
Tactic: obtain ⟨J', hJ', hJ⟩ := hJ
Cleared hypotheses: hJ
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
⊢ ¬(I.colon Set.univ).radical = (J.colon Set.univ).radical
have refine_3 : ∀ (hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical), I = J
Tactic: contrapose hIJ
Cleared hypotheses: hIJ
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
⊢ I = J
have rw : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = J
Tactic: rw [← hI, ← hJ, this]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
rw : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = J
⊢ I = J
have rw₁ : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id =
  {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
Tactic: rw [← hI, ← hJ, this]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
rw : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = J
rw₁ :
  {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id =
    {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
⊢ I = J
have rw₂ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id =
  {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
Tactic: rw [← hI, ← hJ, this]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
⊢ I = J
have refine_3 : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
Tactic: suffices (I'.colon Set.univ).radical = (J'.colon Set.univ).radical by
      rw [← hI, ← hJ, this]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have hIJ₁ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
Tactic: rw [← hI, colon_finsetInf,
        radical_finset_inf (i := I') (by simp [hI']) (by simp), id_eq] at hIJ
Cleared hypotheses: hIJ
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have refine_3 : (J.colon Set.univ).radical = (J'.colon Set.univ).radical
Tactic: rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
refine_3 : (J.colon Set.univ).radical = (J'.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have refine_3₁ : (({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id).colon Set.univ).radical =
  (J'.colon Set.univ).radical
Tactic: rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
refine_3 : (J.colon Set.univ).radical = (J'.colon Set.univ).radical
refine_3₁ :
  (({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id).colon Set.univ).radical =
    (J'.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have refine_3₂ : radical
    ({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf fun (i : Submodule R M) ↦
      (id i).colon Set.univ) =
  (J'.colon Set.univ).radical
Tactic: rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
refine_3 : (J.colon Set.univ).radical = (J'.colon Set.univ).radical
refine_3₁ :
  (({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id).colon Set.univ).radical =
    (J'.colon Set.univ).radical
refine_3₂ :
  radical
      ({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf fun (i : Submodule R M) ↦
        (id i).colon Set.univ) =
    (J'.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have refine_3₃ : ((id J').colon Set.univ).radical = (J'.colon Set.univ).radical
Tactic: rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I'.colon Set.univ).radical = (J.colon Set.univ).radical
refine_3 : (J.colon Set.univ).radical = (J'.colon Set.univ).radical
refine_3₁ :
  (({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id).colon Set.univ).radical =
    (J'.colon Set.univ).radical
refine_3₂ :
  radical
      ({I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf fun (i : Submodule R M) ↦
        (id i).colon Set.univ) =
    (J'.colon Set.univ).radical
refine_3₃ : ((id J').colon Set.univ).radical = (J'.colon Set.univ).radical
⊢ (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
have refine_3₄ : (J'.colon Set.univ).radical = (J'.colon Set.univ).radical
Tactic: rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Lasker (goal)
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
have refine_1 : Finset.inf
    (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
      (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
    id =
  N
Tactic: classical
  refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩
  · ext
    grind [Finset.inf_image, Submodule.mem_finsetInf]
  · simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂]
    intro J hJ
    refine isPrimary_finsetInf (i := J) ?_ ?_ (by simp)
    · simp [hJ]
    · simp only [Finset.mem_filter, id_eq, and_imp]
      intro y hy
      simp [hs' hy]
  · intro I hI J hJ hIJ
    simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ
    obtain ⟨I', hI', hI⟩ := hI
    obtain ⟨J', hJ', hJ⟩ := hJ
    simp only [Function.onFun, ne_eq]
    contrapose hIJ
    suffices (I'.colon Set.univ).radical = (J'.colon Set.univ).radical by
      rw [← hI, ← hJ, this]
    · rw [← hI, colon_finsetInf,
        radical_finset_inf (i := I') (by simp [hI']) (by simp), id_eq] at hIJ
      rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]