(Co)product of a family of filters

In this file we define two filters on Π i, α i and prove some basic properties of these filters.

• Filter.pi (f : Π i, Filter (α i)) to be the maximal filter on Π i, α i such that ∀ i, Filter.Tendsto (Function.eval i) (Filter.pi f) (f i). It is defined as Π i, Filter.comap (Function.eval i) (f i). This is a generalization of Filter.prod to indexed products.

• Filter.coprodᵢ (f : Π i, Filter (α i)): a generalization of Filter.coprod; it is the supremum of comap (eval i) (f i).

theorem Filter.tendsto_eval_pi {ι : Type u_1} {α : ι → Type u_2} (f : (i : ι) → Filter (α i)) (i : ι) : Tendsto (Function.eval i) (pi f) (f i)

theorem Filter.tendsto_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {β : Type u_3} {m : β → (i : ι) → α i} {l : Filter β} : Tendsto m l (pi f) ↔ ∀ (i : ι), Tendsto (fun (x : β) => m x i) l (f i)

theorem Filter.Tendsto.apply {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {β : Type u_3} {m : β → (i : ι) → α i} {l : Filter β} : Tendsto m l (pi f) → ∀ (i : ι), Tendsto (fun (x : β) => m x i) l (f i)

If a function tends to a product Filter.pi f of filters, then its i-th component tends to f i. See also Filter.Tendsto.apply_nhds for the special case of converging to a point in a product of topological spaces.

theorem Filter.le_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {g : Filter ((i : ι) → α i)} : g ≤ pi f ↔ ∀ (i : ι), Tendsto (Function.eval i) g (f i)

theorem Filter.pi_mono {ι : Type u_1} {α : ι → Type u_2} {f₁ f₂ : (i : ι) → Filter (α i)} (h : ∀ (i : ι), f₁ i ≤ f₂ i) : pi f₁ ≤ pi f₂

theorem Filter.mem_pi_of_mem {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} (i : ι) {s : Set (α i)} (hs : s ∈ f i) : Function.eval i ⁻¹' s ∈ pi f

theorem Filter.pi_mem_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} {I : Set ι} (hI : I.Finite) (h : ∀ i ∈ I, s i ∈ f i) : I.pi s ∈ pi f

theorem Filter.mem_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : Set ((i : ι) → α i)} : s ∈ pi f ↔ ∃ (I : Set ι), I.Finite ∧ ∃ (t : (i : ι) → Set (α i)), (∀ (i : ι), t i ∈ f i) ∧ I.pi t ⊆ s

theorem Filter.mem_pi' {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : Set ((i : ι) → α i)} : s ∈ pi f ↔ ∃ (I : Finset ι) (t : (i : ι) → Set (α i)), (∀ (i : ι), t i ∈ f i) ∧ (↑I).pi t ⊆ s

theorem Filter.mem_of_pi_mem_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} (h : I.pi s ∈ pi f) {i : ι} (hi : i ∈ I) : s i ∈ f i

@[simp]

theorem Filter.pi_mem_pi_iff {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} (hI : I.Finite) : I.pi s ∈ pi f ↔ ∀ i ∈ I, s i ∈ f i

theorem Filter.Eventually.eval_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {p : (i : ι) → α i → Prop} {i : ι} (hf : ∀ᶠ (x : α i) in f i, p i x) : ∀ᶠ (x : (i : ι) → α i) in pi f, p i (x i)

theorem Filter.eventually_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {p : (i : ι) → α i → Prop} [Finite ι] (hf : ∀ (i : ι), ∀ᶠ (x : α i) in f i, p i x) : ∀ᶠ (x : (i : ι) → α i) in pi f, ∀ (i : ι), p i (x i)

theorem Filter.hasBasis_pi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {ι' : ι → Type u_3} {s : (i : ι) → ι' i → Set (α i)} {p : (i : ι) → ι' i → Prop} (h : ∀ (i : ι), (f i).HasBasis (p i) (s i)) : (pi f).HasBasis (fun (If : Set ι × ((i : ι) → ι' i)) => If.1.Finite ∧ ∀ i ∈ If.1, p i (If.2 i)) fun (If : Set ι × ((i : ι) → ι' i)) => If.1.pi fun (i : ι) => s i (If.2 i)

theorem Filter.hasBasis_pi_same_index {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {κ : Type u_3} {p : κ → Prop} {s : (i : ι) → κ → Set (α i)} (h : ∀ (i : ι), (f i).HasBasis p (s i)) (h_dir : ∀ (I : Set ι) (k : ι → κ), I.Finite → (∀ i ∈ I, p (k i)) → ∃ (k₀ : κ), p k₀ ∧ ∀ i ∈ I, s i k₀ ⊆ s i (k i)) : (pi f).HasBasis (fun (Ik : Set ι × κ) => Ik.1.Finite ∧ p Ik.2) fun (Ik : Set ι × κ) => Ik.1.pi fun (i : ι) => s i Ik.2

theorem Filter.HasBasis.pi_self {ι : Type u_1} {α : Type u_3} {κ : Type u_4} {f : Filter α} {p : κ → Prop} {s : κ → Set α} (h : f.HasBasis p s) : (pi fun (x : ι) => f).HasBasis (fun (Ik : Set ι × κ) => Ik.1.Finite ∧ p Ik.2) fun (Ik : Set ι × κ) => Ik.1.pi fun (x : ι) => s Ik.2

theorem Filter.le_pi_principal {ι : Type u_1} {α : ι → Type u_2} (s : (i : ι) → Set (α i)) : principal (Set.univ.pi s) ≤ pi fun (i : ι) => principal (s i)

@[simp]

theorem Filter.pi_principal {ι : Type u_1} {α : ι → Type u_2} [Finite ι] (s : (i : ι) → Set (α i)) : (pi fun (i : ι) => principal (s i)) = principal (Set.univ.pi s)

The indexed product of finitely many principal filters is the principal filter corresponding to the cylinder Set.univ.pi s.

If the index type is infinite, then mem_pi_principal and hasBasis_pi_principal may be useful.

theorem Filter.mem_pi_principal {ι : Type u_1} {α : ι → Type u_2} {s : (i : ι) → Set (α i)} {t : Set ((i : ι) → α i)} : (t ∈ pi fun (i : ι) => principal (s i)) ↔ ∃ (I : Set ι), I.Finite ∧ I.pi s ⊆ t

The indexed product of a (possibly, infinite) family of principal filters is generated by the finite Set.pi cylinders.

If the index type is finite, then the indexed product of principal filters is a pricipal filter, see pi_principal.

theorem Filter.hasBasis_pi_principal {ι : Type u_1} {α : ι → Type u_2} (s : (i : ι) → Set (α i)) : (pi fun (i : ι) => principal (s i)).HasBasis Set.Finite fun (x : Set ι) => x.pi s

The indexed product of a (possibly, infinite) family of principal filters is generated by the finite Set.pi cylinders.

If the index type is finite, then the indexed product of principal filters is a pricipal filter, see pi_principal.

@[simp]

theorem Filter.pi_pure {ι : Type u_1} {α : ι → Type u_2} [Finite ι] (f : (i : ι) → α i) : (pi fun (x : ι) => pure (f x)) = pure f

The indexed product of finitely many pure filters pure (f i) is the pure filter pure f.

If the index type is infinite, then mem_pi_pure and hasBasis_pi_pure below may be useful.

theorem Filter.mem_pi_pure {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → α i} {s : Set ((i : ι) → α i)} : (s ∈ pi fun (i : ι) => pure (f i)) ↔ ∃ (I : Set ι), I.Finite ∧ ∀ (g : (i : ι) → α i), (∀ i ∈ I, g i = f i) → g ∈ s

The indexed product of a (possibly, infinite) family of pure filters pure (f i) is generated by the sets of functions that are equal to f on a finite set.

If the index type is finite, then the indexed product of pure filters is a pure filter, see pi_pure.

theorem Filter.hasBasis_pi_pure {ι : Type u_1} {α : ι → Type u_2} (f : (i : ι) → α i) : (pi fun (i : ι) => pure (f i)).HasBasis Set.Finite fun (I : Set ι) => {g : (i : ι) → α i | ∀ i ∈ I, g i = f i}

The indexed product of a (possibly, infinite) family of pure filters pure (f i) is generated by the sets of functions that are equal to f on a finite set.

If the index type is finite, then the indexed product of pure filters is a pure filter, see pi_pure.

@[simp]

theorem Filter.pi_inf_principal_univ_pi_eq_bot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} : pi f ⊓ principal (Set.univ.pi s) = ⊥ ↔ ∃ (i : ι), f i ⊓ principal (s i) = ⊥

@[simp]

theorem Filter.pi_inf_principal_pi_eq_bot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} : pi f ⊓ principal (I.pi s) = ⊥ ↔ ∃ i ∈ I, f i ⊓ principal (s i) = ⊥

@[simp]

theorem Filter.pi_inf_principal_univ_pi_neBot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} : (pi f ⊓ principal (Set.univ.pi s)).NeBot ↔ ∀ (i : ι), (f i ⊓ principal (s i)).NeBot

@[simp]

theorem Filter.pi_inf_principal_pi_neBot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [∀ (i : ι), (f i).NeBot] {I : Set ι} : (pi f ⊓ principal (I.pi s)).NeBot ↔ ∀ i ∈ I, (f i ⊓ principal (s i)).NeBot

instance Filter.PiInfPrincipalPi.neBot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)} [h : ∀ (i : ι), (f i ⊓ principal (s i)).NeBot] {I : Set ι} : (pi f ⊓ principal (I.pi s)).NeBot

@[simp]

theorem Filter.pi_eq_bot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} : pi f = ⊥ ↔ ∃ (i : ι), f i = ⊥

@[simp]

theorem Filter.pi_neBot {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} : (pi f).NeBot ↔ ∀ (i : ι), (f i).NeBot

instance Filter.instNeBotForallPi {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} [∀ (i : ι), (f i).NeBot] : (pi f).NeBot

@[simp]

theorem Filter.map_eval_pi {ι : Type u_1} {α : ι → Type u_2} (f : (i : ι) → Filter (α i)) [∀ (i : ι), (f i).NeBot] (i : ι) : map (Function.eval i) (pi f) = f i

@[simp]

theorem Filter.pi_le_pi {ι : Type u_1} {α : ι → Type u_2} {f₁ f₂ : (i : ι) → Filter (α i)} [∀ (i : ι), (f₁ i).NeBot] : pi f₁ ≤ pi f₂ ↔ ∀ (i : ι), f₁ i ≤ f₂ i

@[simp]

theorem Filter.pi_inj {ι : Type u_1} {α : ι → Type u_2} {f₁ f₂ : (i : ι) → Filter (α i)} [∀ (i : ι), (f₁ i).NeBot] : pi f₁ = pi f₂ ↔ f₁ = f₂

theorem Filter.tendsto_piMap_pi {ι : Type u_1} {α : ι → Type u_2} {β : ι → Type u_3} {f : (i : ι) → α i → β i} {l : (i : ι) → Filter (α i)} {l' : (i : ι) → Filter (β i)} (h : ∀ (i : ι), Tendsto (f i) (l i) (l' i)) : Tendsto (Pi.map f) (pi l) (pi l')

n-ary coproducts of filters

def Filter.coprodᵢ {ι : Type u_1} {α : ι → Type u_2} (f : (i : ι) → Filter (α i)) : Filter ((i : ι) → α i)

Coproduct of filters.

Equations

• Filter.coprodᵢ f = ⨆ (i : ι), Filter.comap (Function.eval i) (f i)

theorem Filter.mem_coprodᵢ_iff {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : Set ((i : ι) → α i)} : s ∈ Filter.coprodᵢ f ↔ ∀ (i : ι), ∃ t₁ ∈ f i, Function.eval i ⁻¹' t₁ ⊆ s

theorem Filter.compl_mem_coprodᵢ {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {s : Set ((i : ι) → α i)} : sᶜ ∈ Filter.coprodᵢ f ↔ ∀ (i : ι), (Function.eval i '' s)ᶜ ∈ f i

theorem Filter.coprodᵢ_neBot_iff' {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} : (Filter.coprodᵢ f).NeBot ↔ (∀ (i : ι), Nonempty (α i)) ∧ ∃ (d : ι), (f d).NeBot

@[simp]

theorem Filter.coprodᵢ_neBot_iff {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} [∀ (i : ι), Nonempty (α i)] : (Filter.coprodᵢ f).NeBot ↔ ∃ (d : ι), (f d).NeBot

theorem Filter.coprodᵢ_eq_bot_iff' {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} : Filter.coprodᵢ f = ⊥ ↔ (∃ (i : ι), IsEmpty (α i)) ∨ f = ⊥

@[simp]

theorem Filter.coprodᵢ_eq_bot_iff {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} [∀ (i : ι), Nonempty (α i)] : Filter.coprodᵢ f = ⊥ ↔ f = ⊥

@[simp]

theorem Filter.coprodᵢ_bot' {ι : Type u_1} {α : ι → Type u_2} : Filter.coprodᵢ ⊥ = ⊥

@[simp]

theorem Filter.coprodᵢ_bot {ι : Type u_1} {α : ι → Type u_2} : (Filter.coprodᵢ fun (x : ι) => ⊥) = ⊥

theorem Filter.NeBot.coprodᵢ {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} [∀ (i : ι), Nonempty (α i)] {i : ι} (h : (f i).NeBot) : (Filter.coprodᵢ f).NeBot

instance Filter.coprodᵢ_neBot {ι : Type u_1} {α : ι → Type u_2} [∀ (i : ι), Nonempty (α i)] [Nonempty ι] (f : (i : ι) → Filter (α i)) [H : ∀ (i : ι), (f i).NeBot] : (Filter.coprodᵢ f).NeBot

theorem Filter.coprodᵢ_mono {ι : Type u_1} {α : ι → Type u_2} {f₁ f₂ : (i : ι) → Filter (α i)} (hf : ∀ (i : ι), f₁ i ≤ f₂ i) : Filter.coprodᵢ f₁ ≤ Filter.coprodᵢ f₂

theorem Filter.map_pi_map_coprodᵢ_le {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {β : ι → Type u_3} {m : (i : ι) → α i → β i} : map (fun (k : (i : ι) → α i) (i : ι) => m i (k i)) (Filter.coprodᵢ f) ≤ Filter.coprodᵢ fun (i : ι) => map (m i) (f i)

theorem Filter.Tendsto.pi_map_coprodᵢ {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {β : ι → Type u_3} {m : (i : ι) → α i → β i} {g : (i : ι) → Filter (β i)} (h : ∀ (i : ι), Tendsto (m i) (f i) (g i)) : Tendsto (fun (k : (i : ι) → α i) (i : ι) => m i (k i)) (Filter.coprodᵢ f) (Filter.coprodᵢ g)