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import topology.basic
import topology.algebra.ring
import algebra.group_power
import ring_theory.subring
import tactic.ring
import for_mathlib.topological_rings
import for_mathlib.nonarchimedean.adic_topology
/-!
# Power bounded elements
The theory of topologically nilpotent, bounded, and power-bounded
elements and subsets of topological rings.
-/
local attribute [instance] set.pointwise_mul_semiring
open set
variables {R : Type*} [comm_ring R] [topological_space R]
/- Note: the following definitions are made without assuming
any compatibility between the algebraic and topological structure on R.
Of course, in applications one will assume that R is a topological ring. -/
/--An element r of a topological ring is topologically nilpotent if for all neighbourhouds U of 0,
there exists a natural number N such that r^n ∈ U for all n > N.
See [Wedhorn, Def 5.25, p. 36]. -/
def is_topologically_nilpotent (r : R) : Prop :=
∀ U ∈ (nhds (0 : R)), ∃ N : ℕ, ∀ n : ℕ, n > N → r^n ∈ U
/--A subset T of a topological ring is topologically nilpotent if for all neighbourhouds U of 0,
there exists a natural number N such that r^n ∈ U for all n > N.
(Here T^n is the set {t₁ * t₂ * ⋯ * tₙ | t₁, t₂, …, tₙ ∈ T}.) -/
def is_topologically_nilpotent_subset (T : set R) : Prop :=
∀ U ∈ (nhds (0 : R)), ∃ n : ℕ, T ^ n ⊆ U
/--A subset B of a topological ring is bounded if for all neighbourhoods U of 0 ∈ R,
there exists a neighbourhood V or 0 such that for all v ∈ V and b ∈ B we have v*b ∈ U.
See [Wedhorn, Def 5.27, p. 36]. -/
def is_bounded (B : set R) : Prop :=
∀ U ∈ nhds (0 : R), ∃ V ∈ nhds (0 : R), ∀ v ∈ V, ∀ b ∈ B, v*b ∈ U
/--A subset B of a topological ring is bounded if and only if for all neighbourhoods U of 0 ∈ R,
there exists a neighbourhood V or 0 such that for all v ∈ V and b ∈ B we have V*B ⊆ U.
(Here V*B denotes the set {v * b | v ∈ V, b ∈ B}.) -/
lemma is_bounded_iff (B : set R) :
is_bounded B ↔ ∀ U ∈ nhds (0 : R), ∃ V ∈ nhds (0 : R), V * B ⊆ U :=
forall_congr $ λ U, imp_congr iff.rfl $ exists_congr $ λ V, exists_congr $ λ hV,
begin
split,
{ rintros H _ ⟨v, hv, b, hb, rfl⟩, exact H v hv b hb },
{ intros H v hv b hb, exact H ⟨v, hv, b, hb, rfl⟩ }
end
section topological_ring
variables [topological_ring R]
/--A topological ring with adic topology is bounded.-/
lemma is_adic.is_bounded (h : is_adic R) : is_bounded (univ : set R) :=
begin
intros U hU,
rw mem_nhds_sets_iff at hU,
rcases hU with ⟨V, hV₁, ⟨hV₂, h0⟩⟩,
tactic.unfreeze_local_instances,
rcases h with ⟨J, hJ⟩,
rw is_ideal_adic_iff at hJ,
have H : (∃ (n : ℕ), (J^n).carrier ⊆ V) :=
begin
apply hJ.2,
exact mem_nhds_sets hV₂ h0,
end,
rcases H with ⟨n, hn⟩,
use (J^n).carrier, -- the key step
split,
{ exact mem_nhds_sets (hJ.1 n) (J^n).zero_mem },
{ rintros a ha b hb,
apply hV₁,
exact hn ((J^n).mul_mem_right ha), }
end
section
open submodule topological_add_group
set_option class.instance_max_depth 58
/--A subset B of a nonarchimedean ring is bounded if and only if
for all neighbourhoods U of 0 ∈ R, there exists an open additive subgroup V such that
V * B generates a subgroup contained in U. -/
lemma is_bounded_add_subgroup_iff (hR : nonarchimedean R) (B : set R) :
is_bounded B ↔ ∀ U ∈ nhds (0:R), ∃ V : open_add_subgroup R,
(↑((V : set R) • span ℤ B) : set R) ⊆ U :=
begin
split,
{ rintros H U hU,
cases hR U hU with W hW,
rw is_bounded_iff at H,
rcases H _ W.mem_nhds_zero with ⟨V', hV', H'⟩,
cases hR V' hV' with V hV,
use V,
refine set.subset.trans _ hW,
change ↑(span _ _ * span _ _) ⊆ _,
rw [span_mul_span, span_int_eq_add_group_closure, add_group.closure_subset_iff],
exact set.subset.trans (set.pointwise_mul_subset_mul hV (set.subset.refl B)) H' },
{ intros H,
rw is_bounded_iff,
intros U hU,
cases H U hU with V hV,
use [V, V.mem_nhds_zero],
refine set.subset.trans _ hV,
rintros _ ⟨v, hv, b, hb, rfl⟩,
exact mul_mem_mul (subset_span hv) (subset_span hb) }
end
/--If J is an ideal in a topological ring whose topology is J-adic,
then J is topologically nilpotent.-/
lemma is_ideal_adic.topologically_nilpotent {J : ideal R} (h : is-J-adic) :
is_topologically_nilpotent_subset (↑J : set R) :=
begin
rw is_ideal_adic_iff at h,
intros U hU,
cases h.2 U hU with n hn,
use n,
exact set.subset.trans (J.pow_subset_pow) hn
end
end
end topological_ring
namespace is_bounded
open topological_add_group
/--A subset of a bounded subset is bounded. See [Wedhorn, Rmk 5.28(2)].-/
lemma subset {S₁ S₂ : set R} (h : S₁ ⊆ S₂) (H : is_bounded S₂) : is_bounded S₁ :=
begin
intros U hU,
rcases H U hU with ⟨V, hV₁, hV₂⟩,
use [V, hV₁],
intros v hv b hb,
exact hV₂ _ hv _ (h hb),
end
/--The subgroup generated by a bounded subset of a nonarchimedean ring is bounded.
See [Wedhorn, Prop 5.30(1)].-/
lemma add_group.closure [topological_ring R] (hR : nonarchimedean R) (T : set R)
(hT : is_bounded T) : is_bounded (add_group.closure T) :=
begin
intros U hU,
-- find subgroup U' in U
rcases hR U hU with ⟨U', hU'⟩,
-- U' still a nhd
-- Use boundedness hypo for T with U' to get V
rcases hT (U' : set R) U'.mem_nhds_zero with ⟨V, hV, hB⟩,
-- find subgroup V' in V
rcases hR V hV with ⟨V', hV'⟩,
-- V' works for our proof
use [V', V'.mem_nhds_zero],
intros v hv b hb,
-- Suffices to prove we're in U'
apply hU',
-- Prove the result by induction
apply add_group.in_closure.rec_on hb,
{ intros t ht,
exact hB v (hV' hv) t ht },
{ rw mul_zero, exact U'.zero_mem },
{ intros a Ha Hv,
rwa [←neg_mul_comm, neg_mul_eq_neg_mul_symm, is_add_subgroup.neg_mem_iff] },
{ intros a b ha hb Ha Hb,
rw [mul_add],
exact U'.add_mem Ha Hb }
end
end is_bounded
/--An element r of a topological ring is power bounded if the set of all positive powers of r
is a bounded subset. See [Wedhorn, Def 5.27].-/
definition is_power_bounded (r : R) : Prop := is_bounded (powers r)
/--A subset T of a topological ring is power bounded if the submonoid generated by T is bounded.
See [Wedhorn, Def 5.27].-/
definition is_power_bounded_subset (T : set R) : Prop := is_bounded (monoid.closure T)
namespace power_bounded
open topological_add_group
/-- 0 is power bounded.-/
lemma zero : is_power_bounded (0 : R) :=
λ U hU, ⟨U,
begin
split, {exact hU},
intros v hv b H,
induction H with n H,
induction n ; { simp [H.symm, pow_succ, mem_of_nhds hU], try {assumption} }
end⟩
/-- 1 is power bounded.-/
lemma one : is_power_bounded (1 : R) :=
λ U hU, ⟨U,
begin
split, {exact hU},
intros v hv b H,
cases H with n H,
simpa [H.symm]
end⟩
/-- An element r is power bounded if and only if the singleton {r} is power bounded.-/
lemma singleton (r : R) : is_power_bounded r ↔ is_power_bounded_subset ({r} : set R) :=
begin
unfold is_power_bounded,
unfold is_power_bounded_subset,
rw monoid.closure_singleton,
end
/-- A subset of a power bounded set is power bounded. See [Wedhorn, Rmk 5.28(2)].-/
lemma subset {B C : set R} (h : B ⊆ C) (hC : is_power_bounded_subset C) :
is_power_bounded_subset B :=
λ U hU, exists.elim (hC U hU) $
λ V ⟨hV, hC⟩, ⟨V, hV, λ v hv b hb, hC v hv b $ monoid.closure_mono h hb⟩
/--The union of two power bounded sets is power bounded. See [Wedhorn, Rmk 5.28(3)].-/
lemma union {S T : set R} (hS : is_power_bounded_subset S) (hT : is_power_bounded_subset T) :
is_power_bounded_subset (S ∪ T) :=
begin
intros U hU,
rcases hT U hU with ⟨V, hV, hVU⟩,
rcases hS V hV with ⟨W, hW, hWV⟩,
use [W, hW],
intros v hv b hb,
rw monoid.mem_closure_union_iff at hb,
rcases hb with ⟨y, hy, z, hz, rfl⟩,
rw [←mul_assoc],
apply hVU _ _ _ hz,
exact hWV _ hv _ hy,
end
/--The monoid generated by a power bounded subset is power bounded.-/
lemma monoid.closure {T : set R}
(hT : is_power_bounded_subset T) : is_power_bounded_subset (monoid.closure T) :=
begin
refine is_bounded.subset _ hT,
apply monoid.closure_subset,
refl
end
/--The product of two power bounded elements is power bounded.-/
lemma mul (a b : R)
(ha : is_power_bounded a) (hb : is_power_bounded b) :
is_power_bounded (a * b) :=
begin
rw singleton at ha hb ⊢,
refine subset _ (monoid.closure (union ha hb)),
rw [set.singleton_subset_iff, monoid.mem_closure_union_iff],
refine ⟨a, _, b, _, rfl⟩; exact monoid.subset_closure (set.mem_singleton _)
end
section topological_ring
variables [topological_ring R]
/--The subgroup generated by a power bounded subset of a nonarchimedean ring is power bounded.
See [Wedhorn, Prop 5.30(1)].-/
lemma add_group.closure (hR : nonarchimedean R) {T : set R}
(hT : is_power_bounded_subset T) : is_power_bounded_subset (add_group.closure T) :=
begin
refine is_bounded.subset _ (is_bounded.add_group.closure hR _ hT),
intros a ha,
apply monoid.in_closure.rec_on ha,
{ apply add_group.closure_mono,
exact monoid.subset_closure },
{ apply add_group.mem_closure,
exact monoid.in_closure.one _ },
{ intros a b ha hb Ha Hb,
show a * b ∈ ring.closure T,
exact is_submonoid.mul_mem Ha Hb }
end
/--The subring generated by a power bounded subset of a nonarchimedean ring is power bounded.
See [Wedhorn, Prop 5.30(2)].-/
lemma ring.closure (hR : nonarchimedean R) (T : set R)
(hT : is_power_bounded_subset T) : is_power_bounded_subset (ring.closure T) :=
add_group.closure hR $ monoid.closure hT
/--The subring generated by a power bounded subset of a nonarchimedean ring is bounded.-/
lemma ring.closure' (hR : nonarchimedean R) (T : set R)
(hT : is_power_bounded_subset T) : is_bounded (_root_.ring.closure T) :=
is_bounded.subset monoid.subset_closure (ring.closure hR _ hT)
/--The sum of two power bounded elements of a nonarchimedean ring is power bounded.-/
lemma add (hR : nonarchimedean R) (a b : R)
(ha : is_power_bounded a) (hb : is_power_bounded b) :
is_power_bounded (a + b) :=
begin
rw singleton at ha hb ⊢,
refine subset _ (add_group.closure hR (union ha hb)),
rw set.singleton_subset_iff,
apply is_add_submonoid.add_mem;
apply add_group.subset_closure; simp
end
end topological_ring
end power_bounded
variable (R)
/-- The subring of power bounded elements.
(Note that this definition makes sense for all R,
but this subset is only a subring for certain rings, such as non-archimedean rings.) -/
definition power_bounded_subring := {r : R | is_power_bounded r}
variable {R}
namespace power_bounded_subring
open topological_add_group
/--The coercion from the power bounded subring to the ambient ring.-/
instance : has_coe (power_bounded_subring R) R := ⟨subtype.val⟩
lemma zero_mem : (0 : R) ∈ power_bounded_subring R := power_bounded.zero
lemma one_mem : (1 : R) ∈ power_bounded_subring R := power_bounded.one
lemma mul_mem :
∀ ⦃a b : R⦄, a ∈ power_bounded_subring R → b ∈ power_bounded_subring R →
a * b ∈ power_bounded_subring R :=
power_bounded.mul
/--The subset of power bounded elements of a topological ring is a submonoid.-/
instance : is_submonoid (power_bounded_subring R) :=
{ one_mem := one_mem,
mul_mem := mul_mem }
section topological_ring
variables [topological_ring R]
lemma add_mem (h : nonarchimedean R) ⦃a b : R⦄
(ha : a ∈ power_bounded_subring R) (hb : b ∈ power_bounded_subring R) :
a + b ∈ power_bounded_subring R :=
power_bounded.add h a b ha hb
lemma neg_mem : ∀ ⦃a : R⦄, a ∈ power_bounded_subring R → -a ∈ power_bounded_subring R :=
λ a ha U hU,
begin
let Usymm := U ∩ {u | -u ∈ U},
have hUsymm : Usymm ∈ (nhds (0 : R)) :=
begin
apply filter.inter_mem_sets hU,
apply continuous.tendsto (topological_add_group.continuous_neg R) 0,
simpa
end,
rcases ha Usymm hUsymm with ⟨V, ⟨V_nhd, hV⟩⟩,
use [V, V_nhd],
rintros v hv _ ⟨n, rfl⟩,
rw show v * (-a)^n = ((-1)^n * v) * a^n,
by { rw [neg_eq_neg_one_mul, mul_pow], ring },
suffices H : (-1)^n * v * a^n ∈ Usymm,
{ exact H.left },
have H := hV v hv (a^n) ⟨n, rfl⟩,
cases (@neg_one_pow_eq_or R _ n) with h h;
simpa [h, H.1, H.2] using H
end
/--The subset of power bounded elements of a nonarchimedean ring is a subgroup.-/
instance (hR : nonarchimedean R) : is_add_subgroup (power_bounded_subring R) :=
{ zero_mem := zero_mem,
add_mem := add_mem hR,
neg_mem := neg_mem }
/--The subset of power bounded elements of a nonarchimedean ring is a subring.-/
instance (hR : nonarchimedean R) : is_subring (power_bounded_subring R) :=
{ ..power_bounded_subring.is_submonoid,
..power_bounded_subring.is_add_subgroup hR }
end topological_ring
variable (R)
/--A topological ring is uniform if the subset of power bounded elements is bounded.-/
definition is_uniform : Prop := is_bounded (power_bounded_subring R)
end power_bounded_subring
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