awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ (sSup S).carrier =
    sSup
      (Set.image (α := (↑(Set.Iic I) : Type u_1)) (val (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I))
        (Set.range fun (a : hI.SubDPIdeal) ↦
          ⨆ (_ : a ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) a.carrier ⋯))
have refine_1 : Eq.{u_1 + 1} (α := Ideal A) (⨆ a ∈ S, (↑a : Ideal A))
  (⨆ (i : hI.SubDPIdeal), (↑(⨆ (_ : i ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) i.carrier sorry) : Ideal A))
Tactic: rw [sSup_carrier_def, sSup_image, sSup_image, iSup_range]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
⊢ Eq.{u_1 + 1} (α := Ideal A)
    (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
    (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A))
have pos : ∀ (hJ : J ∈ S),
  Eq.{u_1 + 1} (α := Ideal A)
    (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
    (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A))
Tactic: by_cases hJ : J ∈ S
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
pos :
  ∀ (hJ : J ∈ S),
    Eq.{u_1 + 1} (α := Ideal A)
      (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
      (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A))
⊢ Eq.{u_1 + 1} (α := Ideal A)
    (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
    (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A))
have neg : ∀ (hJ : J ∉ S),
  Eq.{u_1 + 1} (α := Ideal A)
    (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
    (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A))
Tactic: by_cases hJ : J ∈ S
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ Eq.{u_1 + 1} (α := Ideal A) (⨆ a ∈ S, (↑a : Ideal A))
    (⨆ (i : hI.SubDPIdeal),
      (↑(⨆ (_ : i ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) i.carrier ⋯) : Ideal A))
have this : ∀ (J : hI.SubDPIdeal),
  Eq.{u_1 + 1} (α := Ideal A)
    (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
    (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A))
Tactic: have (J : hI.SubDPIdeal) :
      ((⨆ (_ : J ∈ S), (J : Set.Iic I) : Set.Iic I) : Ideal A) = ⨆ (_ : J ∈ S), (J : Ideal A) := by
      by_cases hJ : J ∈ S
      · simp [ciSup_pos hJ]
      · simp [hJ, not_false_eq_true, iSup_neg, Set.Iic.coe_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
this :
  ∀ (J : hI.SubDPIdeal),
    Eq.{u_1 + 1} (α := Ideal A)
      (↑(⨆ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
      (⨆ (_ : J ∈ S), (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A))
⊢ Eq.{u_1 + 1} (α := Ideal A) (⨆ a ∈ S, (↑a : Ideal A))
    (⨆ (i : hI.SubDPIdeal),
      (↑(⨆ (_ : i ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) i.carrier ⋯) : Ideal A))
have refine_1 : Eq.{u_1 + 1} (α := Ideal A) (⨆ a ∈ S, (↑a : Ideal A)) (⨆ i ∈ S, i.carrier)
Tactic: simp_rw [this]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ Eq.{u_1 + 1} (α := { x : Ideal A // x ∈ Set.Iic I })
    ((fun (J : hI.SubDPIdeal) ↦
        (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
      (sInf S))
    (sInf
      (Set.range fun (a : hI.SubDPIdeal) ↦
        ⨅ (_ : a ∈ S),
          (fun (J : hI.SubDPIdeal) ↦
              (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯ : { x : Ideal A // x ∈ Set.Iic I }))
            a))
have refine_2 : (↑((fun (J : hI.SubDPIdeal) ↦
          (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
        (sInf S)) :
    Ideal A) =
  I ⊓
    sInf
      (Set.image (α := (↑(Set.Iic I) : Type u_1)) (val (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I))
        (Set.range fun (a : hI.SubDPIdeal) ↦
          ⨅ (_ : a ∈ S),
            (fun (J : hI.SubDPIdeal) ↦
                (Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry : { x : Ideal A // x ∈ Set.Iic I }))
              a))
Tactic: rw [Subtype.ext_iff, Set.Iic.coe_sInf]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ (sInf S).carrier =
    I ⊓
      sInf
        (Set.image (α := (↑(Set.Iic I) : Type u_1)) (val (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I))
          (Set.range fun (a : hI.SubDPIdeal) ↦
            ⨅ (_ : a ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) a.carrier ⋯))
have refine_2 : Eq.{u_1 + 1} (α := Ideal A) (⨅ (x : hI.SubDPIdeal), carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : x ∈ S), x.carrier)
  (⨅ (x : hI.SubDPIdeal),
    I ⊓ (↑(⨅ (_ : x ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) x.carrier sorry) : Ideal A))
Tactic: rw [sInf_carrier_def, sInf_image, iInf_range, inf_iInf, iInf_insert, inf_iInf]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
⊢ Eq.{u_1 + 1} (α := Ideal A) (⨅ (x : hI.SubDPIdeal), carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : x ∈ S), x.carrier)
    (⨅ (x : hI.SubDPIdeal),
      I ⊓ (↑(⨅ (_ : x ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) x.carrier ⋯) : Ideal A))
have apply : ∀ (J : hI.SubDPIdeal),
  carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
Tactic: apply iInf_congr (fun J ↦ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
⊢ carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
have pos : ∀ (hJ : J ∈ S),
  carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
Tactic: by_cases hJ : J ∈ S
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
pos :
  ∀ (hJ : J ∈ S),
    carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
      I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
⊢ carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
have neg : ∀ (hJ : J ∉ S),
  carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
Tactic: by_cases hJ : J ∈ S
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
hJ : J ∈ S
⊢ carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
have pos : carrier (I := I) (hI := hI) ⊤ ⊓ J.carrier =
  I ⊓ (↑(Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier sorry) : Ideal A)
Tactic: rw [ciInf_pos hJ, ciInf_pos hJ]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set hI.SubDPIdeal
J : hI.SubDPIdeal
hJ : J ∉ S
⊢ carrier (I := I) (hI := hI) ⊤ ⊓ ⨅ (_ : J ∈ S), J.carrier =
    I ⊓ (↑(⨅ (_ : J ∈ S), Subtype.mk (p := fun (x : Ideal A) ↦ x ∈ Set.Iic I) J.carrier ⋯) : Ideal A)
have neg : carrier (I := I) (hI := hI) ⊤ = I
Tactic: simp [hJ, iInf_neg, le_top, inf_of_le_left, Set.Iic.coe_top]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
⊢ span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
have rw : {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ⊆ (↑I : Set A)
Tactic: rw [span_le]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow k z)
have hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
Tactic: have hSI := hI.dpow_span_isSubideal hS
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
⊢ ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n z)
have refine_1 : ∀ (x : A) (h : Membership.mem (γ := Set A) {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} x)
  (n : ℕ),
  n ≠ 0 →
    Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
      (hI.dpow n x)
Tactic: refine Submodule.span_induction ?_ ?_ ?_ ?_ hz
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
refine_1 :
  ∀ (x : A) (h : Membership.mem (γ := Set A) {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} x)
    (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
⊢ ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n z)
have refine_2 : ∀ (n : ℕ),
  n ≠ 0 →
    Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
      (hI.dpow n 0)
Tactic: refine Submodule.span_induction ?_ ?_ ?_ ?_ hz
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
refine_1 :
  ∀ (x : A) (h : Membership.mem (γ := Set A) {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} x)
    (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
refine_2 :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n 0)
⊢ ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n z)
have refine_3 : ∀ (x y : A)
  (hx :
    Membership.mem (γ := Submodule A A)
      (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x)
  (hy :
    Membership.mem (γ := Submodule A A)
      (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y),
  (∀ (n : ℕ),
      n ≠ 0 →
        Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
          (hI.dpow n x)) →
    (∀ (n : ℕ),
        n ≠ 0 →
          Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
            (hI.dpow n y)) →
      ∀ (n : ℕ),
        n ≠ 0 →
          Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
            (hI.dpow n (x + y))
Tactic: refine Submodule.span_induction ?_ ?_ ?_ ?_ hz
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
refine_1 :
  ∀ (x : A) (h : Membership.mem (γ := Set A) {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} x)
    (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
refine_2 :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n 0)
refine_3 :
  ∀ (x y : A)
    (hx :
      Membership.mem (γ := Submodule A A)
        (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x)
    (hy :
      Membership.mem (γ := Submodule A A)
        (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y),
    (∀ (n : ℕ),
        n ≠ 0 →
          Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
            (hI.dpow n x)) →
      (∀ (n : ℕ),
          n ≠ 0 →
            Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
              (hI.dpow n y)) →
        ∀ (n : ℕ),
          n ≠ 0 →
            Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
              (hI.dpow n (x + y))
⊢ ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n z)
have refine_4 : ∀ (a x : A)
  (hx :
    Membership.mem (γ := Submodule A A)
      (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x),
  (∀ (n : ℕ),
      n ≠ 0 →
        Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
          (hI.dpow n x)) →
    ∀ (n : ℕ),
      n ≠ 0 →
        Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
          (hI.dpow n (a • x))
Tactic: refine Submodule.span_induction ?_ ?_ ?_ ?_ hz
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
y : A
m : ℕ
hm : m ≠ 0
x : A
hxS : x ∈ S
hxy : y = hI.dpow m x
n : ℕ
hn : n ≠ 0
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow n y)
have refine_1 : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
  ((↑(n.uniformBell m) : A) * hI.dpow (n * m) x)
Tactic: rw [hxy, hI.dpow_comp hm (hS hxS)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hy :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
hy_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n y)
n : ℕ
hn : n ≠ 0
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow n (x + y))
have refine_3 : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
  (∑ k ∈ Finset.range (n + 1), hI.dpow k x * hI.dpow (n - k) y)
Tactic: rw [hI.dpow_add' (hSI hx) (hSI hy)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hy :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
hy_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n y)
n : ℕ
hn : n ≠ 0
⊢ Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (∑ k ∈ Finset.range (n + 1), hI.dpow k x * hI.dpow (n - k) y)
have refine_3 : ∀ c ∈ Finset.range (n + 1),
  Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow c x * hI.dpow (n - c) y)
Tactic: apply Submodule.sum_mem (span _)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hy :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
hy_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n y)
n : ℕ
hn : n ≠ 0
m : ℕ
h : m ∈ Finset.range (n + 1)
⊢ Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow m x * hI.dpow (n - m) y)
have pos : ∀ (hm0 : m = 0),
  Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow m x * hI.dpow (n - m) y)
Tactic: by_cases hm0 : m = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.SubDPIdeal (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
hI : DividedPowers I
S : Set A
hS : S ⊆ (↑I : Set A)
k : ℕ
hk : k ≠ 0
z : A
hz : Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) z
J : Ideal A := span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}
hSI : span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x} ≤ I
x y : A
hx :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) x
hy :
  Membership.mem (γ := Submodule A A)
    (Submodule.span A {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x}) y
hx_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n x)
hy_pow :
  ∀ (n : ℕ),
    n ≠ 0 →
      Membership.mem (γ := Ideal A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
        (hI.dpow n y)
n : ℕ
hn : n ≠ 0
m : ℕ
h : m ∈ Finset.range (n + 1)
pos :
  ∀ (hm0 : m = 0),
    Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
      (hI.dpow m x * hI.dpow (n - m) y)
⊢ Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow m x * hI.dpow (n - m) y)
have neg : ∀ (hm0 : ¬m = 0),
  Membership.mem (γ := Submodule A A) (span {y : A | ∃ (n : ℕ) (_ : n ≠ 0) (x : A) (_ : x ∈ S), y = hI.dpow n x})
    (hI.dpow m x * hI.dpow (n - m) y)
Tactic: by_cases hm0 : m = 0