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import topology.algebra.ordered
import for_mathlib.filter
import for_mathlib.topology
import valuation.linear_ordered_comm_group_with_zero
/-!
# The topology on linearly ordered commutative groups with zero
Let Γ₀ be a linearly ordered commutative group to which we have adjoined a zero element.
Then Γ₀ may naturally be endowed with a topology that turns Γ₀ into a topological monoid.
The topology is the following:
A subset U ⊆ Γ₀ is open if 0 ∉ U or if there is an invertible γ₀ ∈ Γ₀ such that {γ | γ < γ₀} ⊆ U.
-/
local attribute [instance, priority 0] classical.DLO
open_locale topological_space
namespace linear_ordered_comm_group_with_zero
open topological_space filter set linear_ordered_structure
variables (Γ₀ : Type*) [linear_ordered_comm_group_with_zero Γ₀]
/--The neighbourhoods around γ ∈ Γ₀, used in the definition of the topology on Γ₀.
These neighbourhoods are defined as follows:
A set s is a neighbourhood of 0 if there is an invertible γ₀ ∈ Γ₀ such that {γ | γ < γ₀} ⊆ s.
If γ ≠ 0, then every set that contains γ is a neighbourhood of γ. -/
def nhds_fun : Γ₀ → filter Γ₀ :=
(λ x : Γ₀, if x = 0 then ⨅ (γ₀ : units Γ₀), principal {γ | γ < γ₀} else pure x)
/--The topology on a linearly ordered commutative group with a zero element adjoined.
A subset U is open if 0 ∉ U or if there is an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U. -/
protected def topological_space : topological_space Γ₀ :=
topological_space.mk_of_nhds (nhds_fun Γ₀)
local attribute [instance] linear_ordered_comm_group_with_zero.topological_space
/--The neighbourhoods {γ | γ < γ₀} of 0 form a directed set indexed by the invertible elements γ₀.-/
@[nolint]
lemma directed_lt : directed (≥) (λ (γ₀ : units Γ₀), principal {γ : Γ₀ | γ < ↑γ₀}) :=
begin
intros γ₁ γ₂,
use min γ₁ γ₂,
split,
{ change principal {γ : Γ₀ | γ < ↑(min γ₁ γ₂)} ≤ principal {γ : Γ₀ | γ < ↑γ₁},
rw [principal_mono, coe_min],
intros x x_in,
calc x < min ↑γ₁ ↑γ₂ : x_in
... ≤ γ₁ : min_le_left _ _ },
{ change principal {γ : Γ₀ | γ < ↑(min γ₁ γ₂)} ≤ principal {γ : Γ₀ | γ < ↑γ₂},
rw [principal_mono, coe_min],
intros x x_in,
calc x < min ↑γ₁ ↑γ₂ : x_in
... ≤ γ₂ : min_le_right _ _ }
end
-- We need two auxilliary lemmas to show that nhds_fun accurately describes the neighbourhoods
-- coming from the topology (that is defined in terms of nhds_fun).
/--At all points of a linearly ordered commutative group with a zero element adjoined,
the pure filter is smaller than the filter given by nhds_fun.-/
private lemma pure_le_nhds_fun : pure ≤ nhds_fun Γ₀ :=
λ x, by { by_cases hx : x = 0; simp [hx, nhds_fun] }
/--For every point Γ₀, and every “neighbourhood” s of it (described by nhds_fun), there is a
smaller “neighbourhood” t ⊆ s, such that s is a “neighbourhood“ of all the points in t.-/
private lemma nhds_fun_ok : ∀ (x : Γ₀) (s ∈ nhds_fun Γ₀ x),
(∃ t ∈ nhds_fun Γ₀ x, t ⊆ s ∧ ∀ y ∈ t, s ∈ nhds_fun Γ₀ y) :=
begin
intros x U U_in,
by_cases hx : x = 0,
{ simp [hx, nhds_fun] at U_in ⊢,
change U ∈ ⨅ (γ₀ : units Γ₀), principal {γ : Γ₀ | γ < ↑γ₀} at U_in,
rw mem_infi (directed_lt Γ₀) ⟨1⟩ at U_in,
cases U_in with γ₀ h,
use {γ : Γ₀ | γ < ↑γ₀},
rw mem_principal_sets at h,
split,
{ apply mem_infi_sets γ₀,
rw mem_principal_sets},
{ refine ⟨h, _⟩,
intros y y_in,
by_cases hy : y = 0 ; simp [hy, h y_in],
{ apply mem_infi_sets γ₀,
rwa mem_principal_sets } } },
{ simp [hx, nhds_fun] at U_in ⊢,
use {x},
refine ⟨mem_singleton _, singleton_subset_iff.2 U_in, _⟩,
intros y y_in,
rw mem_singleton_iff at y_in,
rw y_in,
simpa [hx] }
end
variables {Γ₀}
/--The neighbourhood filter of an invertible element consists of all sets containing that element.-/
@[simp] lemma nhds_coe (γ : units Γ₀) : 𝓝 (γ : Γ₀) = pure (γ : Γ₀) :=
calc 𝓝 (γ : Γ₀) = nhds_fun Γ₀ γ : nhds_mk_of_nhds (nhds_fun Γ₀) γ (pure_le_nhds_fun Γ₀) (nhds_fun_ok Γ₀)
... = pure (γ : Γ₀) : if_neg (group_with_zero.unit_ne_zero γ)
/--The neighbourhood filter of a nonzero element consists of all sets containing that element.-/
@[simp] lemma nhds_of_ne_zero (γ : Γ₀) (h : γ ≠ 0) :
𝓝 γ = pure γ :=
nhds_coe (group_with_zero.mk₀ _ h)
/--If γ is an invertible element of a linearly ordered group with zero element adjoined,
then {γ} is a neighbourhood of γ.-/
lemma singleton_nhds_of_units (γ : units Γ₀) : ({γ} : set Γ₀) ∈ 𝓝 (γ : Γ₀) :=
by simp
/--If γ is a nonzero element of a linearly ordered group with zero element adjoined,
then {γ} is a neighbourhood of γ.-/
lemma singleton_nhds_of_ne_zero (γ : Γ₀) (h : γ ≠ 0) : ({γ} : set Γ₀) ∈ 𝓝 (γ : Γ₀) :=
by simp [h]
/--If U is a neighbourhood of 0 in a linearly ordered group with zero element adjoined,
then there exists an invertible element γ₀ such that {γ | γ < γ₀} ⊆ U.
-/
lemma nhds_zero_mem (U : set Γ₀) : U ∈ 𝓝 (0 : Γ₀) ↔ ∃ γ₀ : units Γ₀, {γ : Γ₀ | γ < γ₀} ⊆ U :=
begin
rw nhds_mk_of_nhds (nhds_fun Γ₀) 0 (pure_le_nhds_fun Γ₀) (nhds_fun_ok Γ₀),
simp [nhds_fun],
rw mem_infi (directed_lt Γ₀) ⟨1⟩,
{ split,
{ rintro ⟨γ₀, H⟩,
rw mem_principal_sets at H,
use [γ₀, H] },
{ rintro ⟨γ₀, H⟩,
use γ₀,
rwa mem_principal_sets } }
end
/--If γ is an invertible element of a linearly ordered group with zero element adjoined,
then {x | x < γ} is a neighbourhood of 0.-/
lemma nhds_zero_of_units (γ : units Γ₀) : {x : Γ₀ | x < γ} ∈ 𝓝 (0 : Γ₀) :=
by { rw nhds_zero_mem, use γ }
/--If γ is a nonzero element of a linearly ordered group with zero element adjoined,
then {x | x < γ} is a neighbourhood of 0.-/
lemma nhds_zero_of_ne_zero (γ : Γ₀) (h : γ ≠ 0) : {x : Γ₀ | x < γ} ∈ 𝓝 (0 : Γ₀) :=
nhds_zero_of_units (group_with_zero.mk₀ _ h)
variable (Γ₀)
/--The topology on a linearly ordered group with zero element adjoined
is compatible with the order structure.-/
lemma ordered_topology : order_closed_topology Γ₀ :=
{ is_closed_le' :=
begin
show is_open {p : Γ₀ × Γ₀ | ¬p.fst ≤ p.snd},
simp only [not_le],
rw is_open_iff_mem_nhds,
rintros ⟨a,b⟩ hab,
change b < a at hab,
have ha : a ≠ 0 := ne_zero_of_lt hab,
rw [nhds_prod_eq, mem_prod_iff],
by_cases hb : b = 0,
{ subst b,
use [{a}, singleton_nhds_of_ne_zero _ ha, {x : Γ₀ | x < a}, nhds_zero_of_ne_zero _ ha],
intros p p_in,
cases mem_prod.1 p_in with h1 h2,
rw mem_singleton_iff at h1,
change p.2 < p.1,
rwa h1 },
{ use [{a}, singleton_nhds_of_ne_zero _ ha, {b}, singleton_nhds_of_ne_zero _ hb],
intros p p_in,
cases mem_prod.1 p_in with h1 h2,
rw mem_singleton_iff at h1 h2,
change p.2 < p.1,
rwa [h1, h2] }
end }
local attribute [instance] ordered_topology
/--The topology on a linearly ordered group with zero element adjoined is T₂ (aka Hausdorff).-/
lemma t2_space : t2_space Γ₀ := order_closed_topology.to_t2_space
local attribute [instance] t2_space
/--The topology on a linearly ordered group with zero element adjoined is T₃ (aka regular).-/
lemma regular_space : regular_space Γ₀ :=
begin
haveI : t1_space Γ₀ := t2_space.t1_space,
split,
intros s x s_closed x_not_in_s,
by_cases hx : x = 0,
{ refine ⟨s, _, subset.refl _, _⟩,
{ subst x,
rw is_open_iff_mem_nhds,
intros y hy,
by_cases hy' : y = 0, { subst y, contradiction },
simpa [hy'] },
{ rw inf_eq_bot_iff,
use -s,
simp only [exists_prop, mem_principal_sets],
split,
exact mem_nhds_sets (by rwa is_open_compl_iff) (by rwa mem_compl_iff),
exact ⟨s, subset.refl s, by simp⟩ } },
{ simp only [inf_eq_bot_iff, exists_prop, mem_principal_sets],
exact ⟨-{x}, is_open_compl_iff.mpr is_closed_singleton, by rwa subset_compl_singleton_iff,
{x}, singleton_nhds_of_ne_zero x hx, -{x}, by simp [subset.refl]⟩ }
end
/--The filter basis around the 0 element of a linearly ordered group with zero element adjoined.-/
def zero_filter_basis : filter_basis Γ₀ :=
{ sets := range (λ γ : units Γ₀, {x : Γ₀ | x < γ}),
ne_empty := range_nonempty _,
directed := begin
intros s t hs ht,
rw mem_range at hs ht,
rcases hs with ⟨γs, rfl⟩,
rcases ht with ⟨γt, rfl⟩,
simp only [exists_prop, mem_range],
rcases directed_lt Γ₀ γs γt with ⟨γ, hs, ht⟩,
change principal {g : Γ₀ | g < ↑γ} ≤ principal {g : Γ₀ | g < ↑γt} at ht,
change principal {g : Γ₀ | g < ↑γ} ≤ principal {g : Γ₀ | g < ↑γs} at hs,
rw [le_principal_iff, mem_principal_sets] at hs ht,
use [{x : Γ₀ | x < γ}, γ, rfl, subset_inter_iff.mpr ⟨hs, ht⟩]
end}
variable {Γ₀}
-- TODO: Generalise the following definition into something like filter_basis.pure.
/--The filter basis around nonzero elements of
a linearly ordered group with zero element adjoined.-/
@[nolint]
def ne_zero_filter_basis (x : Γ₀) : filter_basis Γ₀ :=
{ sets := ({({x} : set Γ₀)} : set (set Γ₀)),
ne_empty := by simp,
directed := by finish }
variable (Γ₀)
/--The neighbourhood basis of a linearly ordered group with zero element adjoined.-/
def nhds_basis : nhds_basis Γ₀ :=
{ B := λ x, if h : x = 0 then zero_filter_basis Γ₀ else ne_zero_filter_basis x,
is_nhds := begin
intro x,
ext s,
split_ifs with hx,
{ subst x,
rw nhds_zero_mem,
simp [zero_filter_basis, filter_basis.mem_filter, filter_basis.mem_iff],
split,
{ rintros ⟨γ₀, h⟩,
use [{x : Γ₀ | x < ↑γ₀}, γ₀, h] },
{ rintros ⟨_, ⟨γ₀, rfl⟩, h⟩,
exact ⟨γ₀, h⟩ } },
{ simp [hx, filter_basis.mem_filter, filter_basis.mem_iff, ne_zero_filter_basis], }
end }
local attribute [instance] nhds_basis
lemma mem_nhds_basis_zero {U : set Γ₀} :
U ∈ nhds_basis.B (0 : Γ₀) ↔ ∃ γ : units Γ₀, U = {x : Γ₀ | x < γ } :=
begin
dsimp [nhds_basis, zero_filter_basis],
simp only [dif_pos],
convert iff.rfl,
simp [eq_comm]
end
lemma mem_nhds_basis_ne_zero {U : set Γ₀} {γ₀ : Γ₀} (h : γ₀ ≠ 0) :
U ∈ nhds_basis.B γ₀ ↔ U = {γ₀} :=
begin
dsimp [nhds_basis],
simp only [dif_neg h],
dsimp [filter_basis.has_mem, ne_zero_filter_basis γ₀],
exact set.mem_singleton_iff
end
variable {Γ₀}
-- until the end of this section, all linearly ordered commutative groups will be endowed with
-- the discrete topology
variables (α : Type*) [linear_ordered_comm_group α]
/--The discrete topology on a linearly ordered commutative group.-/
@[nolint] def discrete_ordered_comm_group : topological_space α := ⊥
local attribute [instance] discrete_ordered_comm_group
lemma ordered_comm_group_is_discrete : discrete_topology α := ⟨rfl⟩
local attribute [instance] ordered_comm_group_is_discrete
lemma comap_coe_nhds (γ : units Γ₀) : 𝓝 γ = comap coe (𝓝 (γ : Γ₀)) :=
begin
rw [nhds_discrete, filter.comap_pure (λ _ _ h, units.ext h) γ],
change comap coe (pure (γ : Γ₀)) = comap coe (𝓝 ↑γ),
rw ← nhds_coe γ,
end
lemma tendsto_zero {α : Type*} {F : filter α} {f : α → Γ₀} :
tendsto f F (𝓝 (0 : Γ₀)) ↔ ∀ γ₀ : units Γ₀, { x : α | f x < γ₀ } ∈ F :=
begin
rw nhds_basis.tendsto_into,
simp only [mem_nhds_basis_zero, exists_imp_distrib],
split ; intro h,
{ intro γ₀,
exact h {γ | γ < ↑γ₀} γ₀ rfl },
{ rintros _ γ₀ rfl,
exact h γ₀ }
end
lemma mem_nhds_zero {s} :
s ∈ 𝓝 (0 : Γ₀) ↔ ∃ γ : units Γ₀, { x : Γ₀ | x < γ } ⊆ s :=
begin
rw nhds_basis.mem_nhds_iff,
simp only [exists_prop, mem_nhds_basis_zero],
split,
{ rintros ⟨_, ⟨⟨γ, rfl⟩, h⟩⟩,
exact ⟨γ, h⟩ },
{ rintros ⟨γ, h⟩,
exact ⟨{x : Γ₀ | x < γ}, ⟨γ, rfl⟩, h⟩ }
end
lemma mem_nhds_coe {s} {γ : Γ₀} (h : γ ≠ 0) :
s ∈ 𝓝 γ ↔ γ ∈ s :=
begin
rw nhds_basis.mem_nhds_iff,
simp only [exists_prop, mem_nhds_basis_ne_zero _ h, h],
split,
{ rintros ⟨_, rfl, h₂⟩,
rwa singleton_subset_iff at h₂ },
{ intro h,
use [{γ}, rfl],
rwa singleton_subset_iff },
end
lemma tendsto_nonzero {α : Type*} {F : filter α} {f : α → Γ₀} {γ₀ : Γ₀} (h : γ₀ ≠ 0) :
tendsto f F (𝓝 (γ₀ : Γ₀)) ↔ { x : α | f x = γ₀ } ∈ F :=
begin
rw nhds_basis.tendsto_into,
simp only [mem_nhds_basis_ne_zero _ h, forall_eq],
convert iff.rfl,
ext s,
exact mem_singleton_iff.symm
end
/--A linearly ordered commutative group with zero Γ₀ is a topological monoid
if it is endowed with the following topology:
A subset U ⊆ Γ₀ is open if 0 ∉ U or if there is an invertible γ₀ ∈ Γ₀ such that {γ | γ < γ₀} ⊆ U.
-/
instance : topological_monoid Γ₀ :=
⟨begin
rw continuous_iff_continuous_at,
rintros ⟨x, y⟩,
by_cases hx : x = 0; by_cases hy : y = 0,
all_goals {
try {subst x}, try {subst y},
intros U U_in,
rw nhds_prod_eq,
try { simp only [_root_.mul_zero, _root_.zero_mul] at U_in},
rw mem_nhds_zero at U_in <|> rw [mem_nhds_coe] at U_in,
rw mem_map,
rw mem_prod_same_iff <|> rw mem_prod_iff,
try { cases U_in with γ hγ } },
{ cases linear_ordered_structure.exists_square_le γ with γ₀ hγ₀,
simp only [mem_nhds_zero, exists_prop],
refine ⟨{x : Γ₀ | x < ↑γ₀}, ⟨γ₀, subset.refl _⟩, _⟩,
rw set.prod_subset_iff,
intros x x_in y y_in,
apply hγ,
change x*y < γ,
have := mul_lt_mul' x_in y_in,
refine lt_of_lt_of_le this _,
exact_mod_cast hγ₀ },
{ simp only [set.prod_subset_iff, mem_nhds_zero, mem_nhds_coe hy, exists_prop],
use [{x : Γ₀ | x < γ*y⁻¹}, γ * (group_with_zero.mk₀ y hy)⁻¹, subset.refl _,
{(group_with_zero.mk₀ y hy)}, mem_singleton y],
intros x x_in y' y'_in,
rw mem_singleton_iff at y'_in,
rw y'_in,
apply hγ,
change x * y < γ,
simpa [hy] using mul_lt_right' y x_in hy, },
{ simp only [set.prod_subset_iff, mem_nhds_zero, mem_nhds_coe hx, exists_prop],
use [{(group_with_zero.mk₀ x hx)}, mem_singleton _, {y : Γ₀ | y < γ*x⁻¹},
γ * (group_with_zero.mk₀ x hx)⁻¹, subset.refl _],
intros x' x'_in y y_in,
rw mem_singleton_iff at x'_in,
rw x'_in,
apply hγ,
change x * y < γ,
rw mul_comm,
simpa [hx] using mul_lt_right' x y_in hx, },
{ simp [mem_nhds_coe, hx, hy],
use [{x}, mem_singleton _, {y}, mem_singleton _],
rw set.prod_subset_iff,
intros x' x'_in y' y'_in,
rw mem_singleton_iff at *,
rw [x'_in, y'_in],
simpa using U_in },
{ assume H, simp at H, tauto }
end⟩
end linear_ordered_comm_group_with_zero
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