awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ 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
have 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
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : 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
have H₁ : top = J.adicTopology
Tactic: change _ = _ at H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
R : Type u_1
inst : CommRing R
top : TopologicalSpace R
inst_1 : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
⊢ (∀ (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
have 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
Tactic: rw [H]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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)) ∧
    ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
have mp : ∀ (n : ℕ), IsOpen[J.adicTopology] (↑(J ^ n) : Set R)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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)
⊢ (∀ (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
have mp₁ : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n : ℕ), (↑(J ^ n) : Set R) ⊆ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ IsAdic J
have mpr : IsTopologicalAddGroup R
Tactic: apply IsTopologicalAddGroup.ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ IsAdic J
have mpr₁ : IsTopologicalAddGroup R
Tactic: apply IsTopologicalAddGroup.ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ IsAdic J
have mpr₂ : Eq.{u_1 + 1} (α := Filter R) (𝓝 0) (𝓝 0)
Tactic: apply IsTopologicalAddGroup.ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ Eq.{u_1 + 1} (α := Filter R) (𝓝 0) (𝓝 0)
have mpr : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s ↔ Membership.mem (γ := Filter R) (𝓝 0) s
Tactic: ext s
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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) ⊤ := ⋯
⊢ Membership.mem (γ := Filter R) (𝓝 0) s ↔ Membership.mem (γ := Filter R) (𝓝 0) s
have mpr : Membership.mem (γ := Filter R) (𝓝 0) s ↔ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
Tactic: rw [J.hasBasis_nhds_zero_adic.mem_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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) ⊤ := ⋯
⊢ Membership.mem (γ := Filter R) (𝓝 0) s ↔ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
have mpr : Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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 ↔ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
have mpr₁ : (∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s) → Membership.mem (γ := Filter R) (𝓝 0) s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
have mpr : ∀ (n : ℕ) (h : (↑(J ^ n) : Set R) ⊆ s), ∃ (i : ℕ), True ∧ (↑(J ^ i) : Set R) ⊆ s
Tactic: rcases H₂ s H with ⟨n, h⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ Membership.mem (γ := Filter R) (𝓝 0) s
have mpr : ∀ (n : ℕ) (hn : (↑(J ^ n) : Set R) ⊆ s), Membership.mem (γ := Filter R) (𝓝 0) s
Tactic: rcases H with ⟨n, -, hn⟩
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ Membership.mem (γ := Filter R) (𝓝 0) s
have mpr : ∃ t ⊆ s, IsOpen[top] t ∧ 0 ∈ t
Tactic: rw [mem_nhds_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ IsAdic (J ^ n)
have 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
Tactic: rw [isAdic_iff] at h ⊢
Cleared hypotheses: h, n, hn
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ (∀ (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
have left : ∀ (n_1 : ℕ), IsOpen[inst_1] (↑((J ^ n) ^ n_1) : Set R)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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)
⊢ (∀ (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
have right : ∀ (s : Set R), Membership.mem (γ := Filter R) (𝓝 0) s → ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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)
have left : IsOpen[inst_1] (↑(J ^ (n * m)) : Set R)
Tactic: rw [← pow_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V
have right : ∀ (m : ℕ) (hm : (↑(J ^ m) : Set R) ⊆ V), ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V
Tactic: obtain ⟨m, hm⟩ := h.right V hV
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ ∃ (n_1 : ℕ), (↑((J ^ n) ^ n_1) : Set R) ⊆ V
have h₁ : (↑((J ^ n) ^ m) : Set R) ⊆ V
Tactic: use m
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
have h₁ : (↑((J ^ n) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
Tactic: refine Set.Subset.trans ?_ hm
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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) ⊆ (↑(J ^ m) : Set R)
have h₁ : ∀ (hn : LT.lt (α := ℕ) 0 0), (↑(HPow.hPow (β := ℕ) J 0 ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
Tactic: cases n
Cleared hypotheses: n, hn
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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)
⊢ (↑((J ^ n) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
have h₂ : ∀ (n : ℕ) (hn : 0 < n + 1), (↑((J ^ (n + 1)) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
Tactic: cases n
Cleared hypotheses: n, hn
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ (↑(HPow.hPow (β := ℕ) J 0 ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
have h₁ : False
Tactic: exfalso
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ (↑((J ^ (n + 1)) ^ m) : Set R) ⊆ (↑(J ^ m) : Set R)
have h₁ : (↑(J ^ (n * m + m)) : Set R) ⊆ (↑(J ^ m) : Set R)
Tactic: rw [← pow_mul, Nat.succ_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
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
⊢ (↑(J ^ (n * m + m)) : Set R) ⊆ (↑(J ^ m) : Set R)
have h₁ : m ≤ n * m + m
Tactic: apply Ideal.pow_le_pow_right
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
⊢ IsAdic.{u_2} (R := A) ⊥ ↔ DiscreteTopology A
have 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
Tactic: rw [isAdic_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
⊢ ((∀ (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
have mp : ((∀ (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
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
mp :
  ((∀ (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
⊢ ((∀ (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
have mpr : DiscreteTopology A →
  (∀ (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
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
h : ∀ (n : ℕ), IsOpen[inst_1] (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A)
_h' : ∀ (s : Set A), Membership.mem (γ := Filter A) (𝓝 0) s → ∃ (n : ℕ), (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A) ⊆ s
⊢ DiscreteTopology A
have mp : IsOpen[inst_1] {0}
Tactic: rw [discreteTopology_iff_isOpen_singleton_zero]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
h : DiscreteTopology A
⊢ (∀ (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
have mpr : ∀ (n : ℕ), IsOpen[inst_1] (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
h : DiscreteTopology A
mpr : ∀ (n : ℕ), IsOpen[inst_1] (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A)
⊢ (∀ (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
have mpr₁ : ∀ (s : Set A), Membership.mem (γ := Filter A) (𝓝 0) s → ∃ (n : ℕ), (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A) ⊆ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : TopologicalSpace A
inst_2 : IsTopologicalRing A
h : DiscreteTopology A
U : Set A
U_nhds : Membership.mem (γ := Filter A) (𝓝 0) U
⊢ ∃ (n : ℕ), (↑(HPow.hPow (α := Ideal A) ⊥ n) : Set A) ⊆ U
have h : (↑(HPow.hPow (α := Ideal A) (β := ℕ) ⊥ 1) : Set A) ⊆ U
Tactic: use 1
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : WithIdeal R
S : Type u_2
inst_2 : CommRing S
inst_3 : WithIdeal S
f : R →+* S
hf : LE.le.{u_2} (α := Ideal S) (Ideal.map (R := R) f i) i
⊢ ContinuousAt.{u_1, u_2} (X := R) (Y := S) (⇑f : (a : R) → S) 0
have rw : ∀ (ib : ℕ),
  True →
    ∃ (ia : ℕ),
      True ∧
        ∀ x ∈ (↑(HPow.hPow (α := Ideal R) i ia) : Set R),
          Membership.mem (α := S) (↑(HPow.hPow (α := Ideal S) i ib) : Set S) ((f : (a : R) → S) x : S)
Tactic: rw [ContinuousAt, map_zero, i.hasBasis_nhds_zero_adic.tendsto_iff i.hasBasis_nhds_zero_adic]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : WithIdeal R
S : Type u_2
inst_2 : CommRing S
inst_3 : WithIdeal S
f : R →+* S
hf : LE.le.{u_2} (α := Ideal S) (Ideal.map (R := R) f i) i
⊢ ∀ (ib : ℕ),
    True →
      ∃ (ia : ℕ),
        True ∧
          ∀ x ∈ (↑(HPow.hPow (α := Ideal R) i ia) : Set R),
            Membership.mem (α := S) (↑(HPow.hPow (α := Ideal S) i ib) : Set S) ((f : (a : R) → S) x : S)
have refine : ∀ (n : ℕ), True → Ideal.map f (HPow.hPow (α := Ideal R) i n) ≤ HPow.hPow (α := Ideal S) i n
Tactic: refine fun n _ ↦ ⟨n, trivial, Ideal.map_le_iff_le_comap.mp ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : WithIdeal R
S : Type u_2
inst_2 : CommRing S
inst_3 : WithIdeal S
e : R ≃+* S
h : Eq.{u_2 + 1} (α := Ideal S) (Ideal.map (R := R) e.toRingHom i) i
⊢ LE.le.{u_2} (α := Ideal S) (Ideal.map (R := R) e.toRingHom i) i
have rw : LE.le.{u_2} (α := Ideal S) i i
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : WithIdeal R
a : R
ha : Membership.mem (γ := Ideal R) i a
⊢ IsTopologicallyNilpotent a
have suffices : ∀ (m : ℕ), ∃ (n₀ : ℕ), ∀ (n : ℕ), n₀ ≤ n → a ^ n ∈ HPow.hPow (α := Ideal R) i m
Tactic: suffices ∀ m : ℕ, ∃ n₀, ∀ n, n₀ ≤ n → a ^ n ∈ i ^ m by
    simpa [IsTopologicallyNilpotent, i.hasBasis_nhds_zero_adic.tendsto_right_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.Bases (goal)
Goal:
A : Type u_1
ι : Type u_2
inst : Ring A
inst_1 : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ Set.Nonempty.{u_1} (α := Set A) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)}
have inhabit : ∀ (inhabited_h : Inhabited ι), Set.Nonempty.{u_1} (α := Set A) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)}
Tactic: inhabit ι
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.Bases (goal)
Goal:
A : Type u_1
ι : Type u_2
inst : Ring A
inst_1 : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i j : ι
⊢ ∃ (z : Set A),
    Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} z ∧
      z ⊆ (↑(B i) : Set A) ∩ (↑(B j) : Set A)
have obtain : ∀ (k : ι) (hk : B k ≤ B i ⊓ B j),
  ∃ (z : Set A),
    Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} z ∧
      z ⊆ (↑(B i) : Set A) ∩ (↑(B j) : Set A)
Tactic: obtain ⟨k, hk⟩ := hB.inter i j
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.Bases (goal)
Goal:
A : Type u_1
ι : Type u_2
inst : Ring A
inst_1 : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ ∃ (z : Set A),
    Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} z ∧
      z ⊆ (↑(B i) : Set A) ∩ (↑(B j) : Set A)
have h : Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} (↑(B k) : Set A) ∧
  (↑(B k) : Set A) ⊆ (↑(B i) : Set A) ∩ (↑(B j) : Set A)
Tactic: use B k
-- Lean 4 proof step from Mathlib.Topology.Algebra.Nonarchimedean.Bases (goal)
Goal:
A : Type u_1
ι : Type u_2
inst : Ring A
inst_1 : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} (↑(B k) : Set A) ∧
    (↑(B k) : Set A) ⊆ (↑(B i) : Set A) ∩ (↑(B j) : Set A)
have h : Membership.mem (γ := Set (Set A)) {U : Set A | ∃ (i : ι), U = (↑(B i) : Set A)} (↑(B k) : Set A)
Tactic: constructor