awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
m : M
i : ℕ
a : R
a_in : a ∈ (↑(HSMul.hSMul (β := Ideal R) (I ^ i) ⊤) : Set R)
⊢ a ∈ I ^ i

Tactic:
simpa [(I ^ i).mul_top] using a_in

Premises used: HPow.hPow, Ideal.mul_top
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
m : M
i : ℕ
a : R
a_in : a ∈ (↑(HSMul.hSMul (β := Ideal R) (I ^ i) ⊤) : Set R)
⊢ a ∈ I ^ i

Tactic:
replace a_in : a ∈ I ^ i := by simpa [(I ^ i).mul_top] using a_in
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
m : M
i : ℕ
a : R
a_in : a ∈ I ^ i
⊢ Membership.mem (γ := Set R) ((fun (x : R) ↦ x • m) ⁻¹' (↑(HSMul.hSMul (β := Submodule R M) (I ^ i) ⊤) : Set M)) a

Tactic:
smul_mem_smul a_in mem_top

Premises used: Set.Mem, setOf, Set.preimage, Submodule.smul_mem_smul, Submodule.mem_top
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
m : M
i : ℕ
a : R
a_in : a ∈ I ^ i
⊢ Membership.mem (γ := Set R) ((fun (x : R) ↦ x • m) ⁻¹' (↑(HSMul.hSMul (β := Submodule R M) (I ^ i) ⊤) : Set M)) a

Tactic:
exact smul_mem_smul a_in mem_top

Premises used: Submodule.smul_mem_smul, Submodule.mem_top
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
refine :
  IsOpen[I.adicTopology]
    (AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup (AddSubgroup.toAddSubmonoid (toAddSubgroup (I ^ n))) : AddSubsemigroup R))
⊢ OpenAddSubgroup R

Tactic:
refine ⟨(I ^ n).toAddSubgroup, ?_⟩
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
⊢ IsOpen[⋯.topology] (↑(⋯.openAddSubgroup n) : Set R)

Tactic:
(I.adic_basis.toRing_subgroups_basis.openAddSubgroup n).isOpen

Premises used: RingSubgroupsBasis.openAddSubgroup, SubmodulesRingBasis.toRing_subgroups_basis, OpenAddSubgroup.isOpen, Ideal.adic_basis
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
h :
  AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup (AddSubgroup.toAddSubmonoid (toAddSubgroup (I ^ n))) : AddSubsemigroup R) =
    (↑(⋯.openAddSubgroup n) : Set R)
⊢ IsOpen[I.adicTopology]
    (AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup (AddSubgroup.toAddSubmonoid (toAddSubgroup (I ^ n))) : AddSubsemigroup R))

Tactic:
convert! (I.adic_basis.toRing_subgroups_basis.openAddSubgroup n).isOpen

Premises used: RingSubgroupsBasis.openAddSubgroup, SubmodulesRingBasis.toRing_subgroups_basis, OpenAddSubgroup.isOpen, Ideal.adic_basis
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
h : (↑(I ^ n) : Set R) = (↑(HSMul.hSMul (β := Ideal R) (I ^ n) ⊤) : Set R)
⊢ AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup (AddSubgroup.toAddSubmonoid (toAddSubgroup (I ^ n))) : AddSubsemigroup R) =
    (↑(⋯.openAddSubgroup n) : Set R)

Tactic:
change (↑(I ^ n) : Set R) = ↑(I ^ n • (⊤ : Ideal R))

Premises used: SetLike.coe, HPow.hPow, HSMul.hSMul, Top.top
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
n : ℕ
this : TopologicalSpace R := I.adicTopology
⊢ (↑(I ^ n) : Set R) = (↑(HSMul.hSMul (β := Ideal R) (I ^ n) ⊤) : Set R)

Tactic:
simp
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
mp :
  IsAdic J →
    (∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)) ∧
      ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
mpr :
  ((∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)) ∧
      ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s) →
    IsAdic J
⊢ IsAdic J ↔
    (∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)) ∧
      ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s

Tactic:
constructor
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : IsAdic J
⊢ top = J.adicTopology

Tactic:
change _ = _ at H
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
mp :
  (∀ (n : ℕ), IsOpen[J.adicTopology] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
⊢ (∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s

Tactic:
rw [H]
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
this : TopologicalSpace R := J.adicTopology
mp : ∀ (n : ℕ), IsOpen[J.adicTopology] (↑(J ^ n) : Set R)
mp₁ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
⊢ (∀ (n : ℕ), IsOpen[J.adicTopology] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s

Tactic:
constructor
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
this : TopologicalSpace R := J.adicTopology
n : ℕ
⊢ IsOpen[J.adicTopology] (↑(J ^ n) : Set R)

Tactic:
(J.openAddSubgroup n).isOpen'

Premises used: OpenAddSubgroup.isOpen', Ideal.openAddSubgroup
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
this : TopologicalSpace R := J.adicTopology
n : ℕ
⊢ IsOpen[J.adicTopology] (↑(J ^ n) : Set R)

Tactic:
exact (J.openAddSubgroup n).isOpen'

Premises used: OpenAddSubgroup.isOpen', Ideal.openAddSubgroup
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
this : TopologicalSpace R := J.adicTopology
s : Set R
hs : Membership.mem (γ := Filter R) (𝓝 0) s
⊢ ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s

Tactic:
simpa using J.hasBasis_nhds_zero_adic.mem_iff.mp hs

Premises used: Filter.HasBasis.mem_iff, Ideal.hasBasis_nhds_zero_adic, Iff.mp
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
mpr : IsTopologicalAddGroup R
mpr₁ : IsTopologicalAddGroup R
mpr₂ : Eq.{u_1 + 1} (α := Filter R) (𝓝 0) (𝓝 0)
⊢ IsAdic J

Tactic:
apply IsTopologicalAddGroup.ext

Premises used: IsTopologicalAddGroup.ext
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
⊢ IsTopologicalAddGroup R

Tactic:
apply @IsTopologicalRing.to_topologicalAddGroup

Premises used: IsTopologicalRing.to_topologicalAddGroup
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
⊢ IsTopologicalAddGroup R

Tactic:
apply (RingSubgroupsBasis.toRingFilterBasis _).toAddGroupFilterBasis.isTopologicalAddGroup

Premises used: AddGroupFilterBasis.isTopologicalAddGroup, RingFilterBasis.toAddGroupFilterBasis
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
mpr : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s ↔ Membership.mem (γ := Filter R) (𝓝 0) s
⊢ Eq.{u_1 + 1} (α := Filter R) (𝓝 0) (𝓝 0)

Tactic:
ext s
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
mpr : Membership.mem (γ := Filter R) (𝓝 0) s ↔ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
⊢ Membership.mem (γ := Filter R) (𝓝 0) s ↔ Membership.mem (γ := Filter R) (𝓝 0) s

Tactic:
rw [J.hasBasis_nhds_zero_adic.mem_iff]

Premises used: Filter.HasBasis.mem_iff, Ideal.hasBasis_nhds_zero_adic
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
mpr : Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
mpr₁ : (∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s) → Membership.mem (γ := Filter R) (𝓝 0) s
⊢ Membership.mem (γ := Filter R) (𝓝 0) s ↔ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s

Tactic:
constructor
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
H : Membership.mem (γ := Filter R) (𝓝 0) s
mpr : ∀ (n : ℕ) (h : (↑(J ^ n) : Set R) ⊆ s), ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
⊢ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s

Tactic:
rcases H₂ s H with ⟨n, h⟩
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
H : Membership.mem (γ := Filter R) (𝓝 0) s
n : ℕ
h : (↑(J ^ n) : Set R) ⊆ s
⊢ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s

Tactic:
⟨n, trivial, h⟩

Premises used: trivial, And.intro, Exists.intro
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
H : Membership.mem (γ := Filter R) (𝓝 0) s
n : ℕ
h : (↑(J ^ n) : Set R) ⊆ s
⊢ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s

Tactic:
exact ⟨n, trivial, h⟩
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
H : ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
mpr : ∀ (n : ℕ) (hn : (↑(J ^ n) : Set R) ⊆ s), Membership.mem (γ := Filter R) (𝓝 0) s
⊢ Membership.mem (γ := Filter R) (𝓝 0) s

Tactic:
rcases H with ⟨n, -, hn⟩
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
n : ℕ
hn : (↑(J ^ n) : Set R) ⊆ s
mpr : ∃ t ⊆ s, IsOpen[top] t ∧ 0 ∈ t
⊢ Membership.mem (γ := Filter R) (𝓝 0) s

Tactic:
rw [mem_nhds_iff]

Premises used: mem_nhds_iff
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
n : ℕ
hn : (↑(J ^ n) : Set R) ⊆ s
⊢ ∃ t ⊆ s, IsOpen[top] t ∧ 0 ∈ t

Tactic:
⟨_, hn, H₁ n, (J ^ n).zero_mem⟩

Premises used: Ideal.zero_mem, HPow.hPow, And.intro, Exists.intro
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] (↑(J ^ n) : Set R)
H₂ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
s : Set R
this : SubmodulesRingBasis.{0, u_1, u_1} (R := R) (A := R) fun (n : ℕ) ↦ HSMul.hSMul (β := Ideal R) (J ^ n) ⊤ := ⋯
n : ℕ
hn : (↑(J ^ n) : Set R) ⊆ s
⊢ ∃ t ⊆ s, IsOpen[top] t ∧ 0 ∈ t

Tactic:
exact ⟨_, hn, H₁ n, (J ^ n).zero_mem⟩
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h : IsAdic J
n : ℕ
hn : 0 < n
rw :
  ∀
    (h :
      (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
        ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s)
    {n : ℕ} (hn : 0 < n),
    (∀ (n_1 : ℕ), IsOpen[inst_1] (↑((J ^ n) ^ n_1) : Set R)) ∧
      ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ s
⊢ IsAdic (J ^ n)

Tactic:
rw [isAdic_iff] at h ⊢

Premises used: isAdic_iff
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
left : ∀ (n_1 : ℕ), IsOpen[inst_1] (↑((J ^ n) ^ n_1) : Set R)
right : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ s
⊢ (∀ (n_1 : ℕ), IsOpen[inst_1] (↑((J ^ n) ^ n_1) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ s

Tactic:
constructor
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
m : ℕ
left : IsOpen[inst_1] (↑(J ^ (n * m)) : Set R)
⊢ IsOpen[inst_1] (↑((J ^ n) ^ m) : Set R)

Tactic:
rw [← pow_mul]

Premises used: pow_mul
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
m : ℕ
⊢ IsOpen[inst_1] (↑(J ^ (n * m)) : Set R)

Tactic:
apply h.left

Premises used: And.left
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
right : ∀ (m : ℕ) (hm : (↑(J ^ m) : Set R) ⊆ V), ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V
⊢ ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V

Tactic:
obtain ⟨m, hm⟩ := h.right V hV

Premises used: And.right
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
h₁ : (↑((J ^ n) ^ m) : Set R) ⊆ V
⊢ ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V

Tactic:
use m
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
⊢ (↑((J ^ n) ^ m) : Set R) ⊆ V

Tactic:
Set.Subset.trans ?_ hm

Premises used: Set.Subset.trans
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
h₁ : (↑((J ^ n) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
⊢ (↑((J ^ n) ^ m) : Set R) ⊆ V

Tactic:
refine Set.Subset.trans ?_ hm

Premises used: Set.Subset.trans
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
h₁ : ∀ (hn : LT.lt (α := ℕ) 0 0), (↑(HPow.hPow (β := ℕ) J 0 ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
h₂ : ∀ (n : ℕ) (hn : 0 < n + 1), (↑((J ^ (n + 1)) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
⊢ (↑((J ^ n) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)

Tactic:
cases n
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
hn : LT.lt (α := ℕ) 0 0
h₁ : False
⊢ (↑(HPow.hPow (β := ℕ) J 0 ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)

Tactic:
exfalso
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
hn : LT.lt (α := ℕ) 0 0
⊢ False

Tactic:
Nat.not_succ_le_zero 0 hn

Premises used: Nat.not_succ_le_zero, OfNat.ofNat
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
hn : LT.lt (α := ℕ) 0 0
⊢ False

Tactic:
exact Nat.not_succ_le_zero 0 hn

Premises used: Nat.not_succ_le_zero, OfNat.ofNat
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
n : ℕ
hn : 0 < n + 1
h₁ : (↑(J ^ (n * m + m)) : Set R) ⊆ (↑(J ^ m) : Set R)
⊢ (↑((J ^ (n + 1)) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)

Tactic:
rw [← pow_mul, Nat.succ_mul]

Premises used: pow_mul, Nat.succ_mul
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
n : ℕ
hn : 0 < n + 1
h₁ : m ≤ n * m + m
⊢ (↑(J ^ (n * m + m)) : Set R) ⊆ (↑(J ^ m) : Set R)

Tactic:
apply Ideal.pow_le_pow_right

Premises used: Ideal.pow_le_pow_right
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
inst_2 : IsTopologicalRing R
J : Ideal R
h :
  (∀ (n : ℕ), IsOpen[inst_1] (↑(J ^ n) : Set R)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
V : Set R
hV : Membership.mem (γ := Filter R) (𝓝 0) V
m : ℕ
hm : (↑(J ^ m) : Set R) ⊆ V
n : ℕ
hn : 0 < n + 1
⊢ m ≤ n * m + m

Tactic:
apply Nat.le_add_left

Premises used: Nat.le_add_left
-- Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
rw :
  ((∀ (n : ℕ), IsOpen[inst_1] (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A)) ∧
      ∀ (s : Set A),
        Membership.mem (γ := Filter A) (𝓝 0) s → ∃ (n : ℕ), (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A) ⊆ s) ↔
    DiscreteTopology A
⊢ IsAdic.{u_2} (R := A) ⊥ ↔ DiscreteTopology A

Tactic:
rw [isAdic_iff]

Premises used: isAdic_iff