Filter.atTop and Filter.atBot filters on preorders, monoids and groups.

In this file we define the filters

• Filter.atTop: corresponds to n → +∞;
• Filter.atBot: corresponds to n → -∞.

Then we prove many lemmas like “if f → +∞, then f ± c → +∞”.

def Filter.atTop {α : Type u_3} [Preorder α] : Filter α

atTop is the filter representing the limit → ∞ on an ordered set. It is generated by the collection of up-sets {b | a ≤ b}. (The preorder need not have a top element for this to be well defined, and indeed is trivial when a top element exists.)

Equations

• Filter.atTop = ⨅ (a : α), Filter.principal (Set.Ici a)

def Filter.atBot {α : Type u_3} [Preorder α] : Filter α

atBot is the filter representing the limit → -∞ on an ordered set. It is generated by the collection of down-sets {b | b ≤ a}. (The preorder need not have a bottom element for this to be well defined, and indeed is trivial when a bottom element exists.)

Equations

• Filter.atBot = ⨅ (a : α), Filter.principal (Set.Iic a)

theorem Filter.mem_atTop {α : Type u_3} [Preorder α] (a : α) : {b : α | a ≤ b} ∈ atTop

theorem Filter.Ici_mem_atTop {α : Type u_3} [Preorder α] (a : α) : Set.Ici a ∈ atTop

theorem Filter.Ioi_mem_atTop {α : Type u_3} [Preorder α] [NoMaxOrder α] (x : α) : Set.Ioi x ∈ atTop

theorem Filter.mem_atBot {α : Type u_3} [Preorder α] (a : α) : {b : α | b ≤ a} ∈ atBot

theorem Filter.Iic_mem_atBot {α : Type u_3} [Preorder α] (a : α) : Set.Iic a ∈ atBot

theorem Filter.Iio_mem_atBot {α : Type u_3} [Preorder α] [NoMinOrder α] (x : α) : Set.Iio x ∈ atBot

theorem Filter.disjoint_atBot_principal_Ioi {α : Type u_3} [Preorder α] (x : α) : Disjoint atBot (principal (Set.Ioi x))

theorem Filter.disjoint_atTop_principal_Iio {α : Type u_3} [Preorder α] (x : α) : Disjoint atTop (principal (Set.Iio x))

theorem Filter.disjoint_atTop_principal_Iic {α : Type u_3} [Preorder α] [NoMaxOrder α] (x : α) : Disjoint atTop (principal (Set.Iic x))

theorem Filter.disjoint_atBot_principal_Ici {α : Type u_3} [Preorder α] [NoMinOrder α] (x : α) : Disjoint atBot (principal (Set.Ici x))

theorem Filter.disjoint_pure_atTop {α : Type u_3} [Preorder α] [NoMaxOrder α] (x : α) : Disjoint (pure x) atTop

theorem Filter.disjoint_pure_atBot {α : Type u_3} [Preorder α] [NoMinOrder α] (x : α) : Disjoint (pure x) atBot

theorem Filter.not_tendsto_const_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [NoMaxOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun (x_1 : β) => x) l atTop

theorem Filter.not_tendsto_const_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [NoMinOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun (x_1 : β) => x) l atBot

theorem Filter.disjoint_atBot_atTop {α : Type u_3} [PartialOrder α] [Nontrivial α] : Disjoint atBot atTop

theorem Filter.disjoint_atTop_atBot {α : Type u_3} [PartialOrder α] [Nontrivial α] : Disjoint atTop atBot

theorem Filter.eventually_ge_atTop {α : Type u_3} [Preorder α] (a : α) : ∀ᶠ (x : α) in atTop, a ≤ x

theorem Filter.eventually_le_atBot {α : Type u_3} [Preorder α] (a : α) : ∀ᶠ (x : α) in atBot, x ≤ a

theorem Filter.eventually_gt_atTop {α : Type u_3} [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ (x : α) in atTop, a < x

theorem Filter.eventually_ne_atTop {α : Type u_3} [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ (x : α) in atTop, x ≠ a

theorem Filter.Tendsto.eventually_gt_atTop {α : Type u_3} {β : Type u_4} [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ (x : α) in l, c < f x

theorem Filter.Tendsto.eventually_ge_atTop {α : Type u_3} {β : Type u_4} [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ (x : α) in l, c ≤ f x

theorem Filter.Tendsto.eventually_ne_atTop {α : Type u_3} {β : Type u_4} [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ (x : α) in l, f x ≠ c

theorem Filter.Tendsto.eventually_ne_atTop' {α : Type u_3} {β : Type u_4} [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : α) : ∀ᶠ (x : α) in l, x ≠ c

theorem Filter.eventually_lt_atBot {α : Type u_3} [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ (x : α) in atBot, x < a

theorem Filter.eventually_ne_atBot {α : Type u_3} [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ (x : α) in atBot, x ≠ a

theorem Filter.Tendsto.eventually_lt_atBot {α : Type u_3} {β : Type u_4} [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ (x : α) in l, f x < c

theorem Filter.Tendsto.eventually_le_atBot {α : Type u_3} {β : Type u_4} [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ (x : α) in l, f x ≤ c

theorem Filter.Tendsto.eventually_ne_atBot {α : Type u_3} {β : Type u_4} [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ (x : α) in l, f x ≠ c

theorem Filter.eventually_forall_ge_atTop {α : Type u_3} [Preorder α] {p : α → Prop} : (∀ᶠ (x : α) in atTop, ∀ (y : α), x ≤ y → p y) ↔ ∀ᶠ (x : α) in atTop, p x

theorem Filter.eventually_forall_le_atBot {α : Type u_3} [Preorder α] {p : α → Prop} : (∀ᶠ (x : α) in atBot, ∀ y ≤ x, p y) ↔ ∀ᶠ (x : α) in atBot, p x

theorem Filter.Tendsto.eventually_forall_ge_atTop {α : Type u_3} {β : Type u_4} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atTop) (h_evtl : ∀ᶠ (x : β) in atTop, p x) : ∀ᶠ (x : α) in l, ∀ (y : β), f x ≤ y → p y

theorem Filter.Tendsto.eventually_forall_le_atBot {α : Type u_3} {β : Type u_4} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atBot) (h_evtl : ∀ᶠ (x : β) in atBot, p x) : ∀ᶠ (x : α) in l, ∀ y ≤ f x, p y

theorem IsTop.atTop_eq {α : Type u_3} [Preorder α] {a : α} (ha : IsTop a) : Filter.atTop = Filter.principal (Set.Ici a)

theorem IsBot.atBot_eq {α : Type u_3} [Preorder α] {a : α} (ha : IsBot a) : Filter.atBot = Filter.principal (Set.Iic a)

theorem Filter.OrderTop.atTop_eq (α : Type u_6) [PartialOrder α] [OrderTop α] : atTop = pure ⊤

theorem Filter.OrderBot.atBot_eq (α : Type u_6) [PartialOrder α] [OrderBot α] : atBot = pure ⊥

theorem Filter.Subsingleton.atTop_eq (α : Type u_6) [Subsingleton α] [Preorder α] : atTop = ⊤

theorem Filter.Subsingleton.atBot_eq (α : Type u_6) [Subsingleton α] [Preorder α] : atBot = ⊤

theorem Filter.tendsto_atTop_pure {α : Type u_3} {β : Type u_4} [PartialOrder α] [OrderTop α] (f : α → β) : Tendsto f atTop (pure (f ⊤))

theorem Filter.tendsto_atBot_pure {α : Type u_3} {β : Type u_4} [PartialOrder α] [OrderBot α] (f : α → β) : Tendsto f atBot (pure (f ⊥))

theorem Filter.atTop_eq_generate_Ici {α : Type u_3} [Preorder α] : atTop = generate (Set.range Set.Ici)

theorem Filter.Frequently.forall_exists_of_atTop {α : Type u_3} [Preorder α] {p : α → Prop} (h : ∃ᶠ (x : α) in atTop, p x) (a : α) : ∃ b ≥ a, p b

theorem Filter.Frequently.forall_exists_of_atBot {α : Type u_3} [Preorder α] {p : α → Prop} (h : ∃ᶠ (x : α) in atBot, p x) (a : α) : ∃ b ≤ a, p b

theorem Filter.hasAntitoneBasis_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] : atTop.HasAntitoneBasis Set.Ici

theorem Filter.atTop_basis {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] : atTop.HasBasis (fun (x : α) => True) Set.Ici

theorem Filter.atTop_basis_Ioi {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] [NoMaxOrder α] : atTop.HasBasis (fun (x : α) => True) Set.Ioi

theorem Filter.atTop_basis_Ioi' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [NoMaxOrder α] (a : α) : atTop.HasBasis (fun (x : α) => a < x) Set.Ioi

theorem Filter.atTop_basis' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] (a : α) : atTop.HasBasis (fun (x : α) => a ≤ x) Set.Ici

instance Filter.atTop_neBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] : atTop.NeBot

theorem Filter.atTop_neBot_iff {α : Type u_6} [Preorder α] : atTop.NeBot ↔ Nonempty α ∧ IsDirected α fun (x1 x2 : α) => x1 ≤ x2

theorem Filter.atBot_neBot_iff {α : Type u_6} [Preorder α] : atBot.NeBot ↔ Nonempty α ∧ IsDirected α fun (x1 x2 : α) => x1 ≥ x2

@[simp]

theorem Filter.mem_atTop_sets {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] {s : Set α} : s ∈ atTop ↔ ∃ (a : α), ∀ b ≥ a, b ∈ s

@[simp]

theorem Filter.eventually_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α → Prop} [Nonempty α] : (∀ᶠ (x : α) in atTop, p x) ↔ ∃ (a : α), ∀ b ≥ a, p b

theorem Filter.frequently_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α → Prop} [Nonempty α] : (∃ᶠ (x : α) in atTop, p x) ↔ ∀ (a : α), ∃ b ≥ a, p b

theorem Filter.Eventually.exists_forall_of_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α → Prop} [Nonempty α] : (∀ᶠ (x : α) in atTop, p x) → ∃ (a : α), ∀ b ≥ a, p b

Alias of the forward direction of Filter.eventually_atTop.

theorem Filter.exists_eventually_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] {r : α → β → Prop} : (∃ (b : β), ∀ᶠ (a : α) in atTop, r a b) ↔ ∀ᶠ (a₀ : α) in atTop, ∃ (b : β), ∀ a ≥ a₀, r a b

theorem Filter.map_atTop_eq {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Nonempty α] {f : α → β} : map f atTop = ⨅ (a : α), principal (f '' {a' : α | a ≤ a'})

theorem Filter.frequently_atTop' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α → Prop} [Nonempty α] [NoMaxOrder α] : (∃ᶠ (x : α) in atTop, p x) ↔ ∀ (a : α), ∃ b > a, p b

theorem Filter.atBot_basis_Iio {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] [NoMinOrder α] : atBot.HasBasis (fun (x : α) => True) Set.Iio

theorem Filter.atBot_basis_Iio' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [NoMinOrder α] (a : α) : atBot.HasBasis (fun (x : α) => x < a) Set.Iio

theorem Filter.atBot_basis' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] (a : α) : atBot.HasBasis (fun (x : α) => x ≤ a) Set.Iic

theorem Filter.atBot_basis {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] : atBot.HasBasis (fun (x : α) => True) Set.Iic

instance Filter.atBot_neBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] : atBot.NeBot

@[simp]

theorem Filter.mem_atBot_sets {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] {s : Set α} : s ∈ atBot ↔ ∃ (a : α), ∀ b ≤ a, b ∈ s

@[simp]

theorem Filter.eventually_atBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α → Prop} [Nonempty α] : (∀ᶠ (x : α) in atBot, p x) ↔ ∃ (a : α), ∀ b ≤ a, p b

theorem Filter.frequently_atBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α → Prop} [Nonempty α] : (∃ᶠ (x : α) in atBot, p x) ↔ ∀ (a : α), ∃ b ≤ a, p b

theorem Filter.Eventually.exists_forall_of_atBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α → Prop} [Nonempty α] : (∀ᶠ (x : α) in atBot, p x) → ∃ (a : α), ∀ b ≤ a, p b

Alias of the forward direction of Filter.eventually_atBot.

theorem Filter.exists_eventually_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] {r : α → β → Prop} : (∃ (b : β), ∀ᶠ (a : α) in atBot, r a b) ↔ ∀ᶠ (a₀ : α) in atBot, ∃ (b : β), ∀ a ≤ a₀, r a b

theorem Filter.map_atBot_eq {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Nonempty α] {f : α → β} : map f atBot = ⨅ (a : α), principal (f '' {a' : α | a' ≤ a})

theorem Filter.frequently_atBot' {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α → Prop} [Nonempty α] [NoMinOrder α] : (∃ᶠ (x : α) in atBot, p x) ↔ ∀ (a : α), ∃ b < a, p b

theorem Filter.tendsto_atTop {α : Type u_3} {β : Type u_4} [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atTop ↔ ∀ (b : β), ∀ᶠ (a : α) in f, b ≤ m a

theorem Filter.tendsto_atBot {α : Type u_3} {β : Type u_4} [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atBot ↔ ∀ (b : β), ∀ᶠ (a : α) in f, m a ≤ b

theorem Filter.tendsto_atTop_mono' {α : Type u_3} {β : Type u_4} [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) (h₁ : Tendsto f₁ l atTop) : Tendsto f₂ l atTop

theorem Filter.tendsto_atBot_mono' {α : Type u_3} {β : Type u_4} [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) : Tendsto f₂ l atBot → Tendsto f₁ l atBot

theorem Filter.tendsto_atTop_mono {α : Type u_3} {β : Type u_4} [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ (n : α), f n ≤ g n) : Tendsto f l atTop → Tendsto g l atTop

theorem Filter.tendsto_atBot_mono {α : Type u_3} {β : Type u_4} [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ (n : α), f n ≤ g n) : Tendsto g l atBot → Tendsto f l atBot

theorem Filter.atTop_eq_generate_of_forall_exists_le {α : Type u_3} [LinearOrder α] {s : Set α} (hs : ∀ (x : α), ∃ y ∈ s, x ≤ y) : atTop = generate (Set.Ici '' s)

theorem Filter.atTop_eq_generate_of_not_bddAbove {α : Type u_3} [LinearOrder α] {s : Set α} (hs : ¬BddAbove s) : atTop = generate (Set.Ici '' s)

@[simp]

theorem OrderIso.comap_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.comap (⇑e) Filter.atTop = Filter.atTop

@[simp]

theorem OrderIso.comap_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.comap (⇑e) Filter.atBot = Filter.atBot

@[simp]

theorem OrderIso.map_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.map (⇑e) Filter.atTop = Filter.atTop

@[simp]

theorem OrderIso.map_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.map (⇑e) Filter.atBot = Filter.atBot

theorem OrderIso.tendsto_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.Tendsto (⇑e) Filter.atTop Filter.atTop

theorem OrderIso.tendsto_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.Tendsto (⇑e) Filter.atBot Filter.atBot

@[simp]

theorem OrderIso.tendsto_atTop_iff {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] {l : Filter γ} {f : γ → α} (e : α ≃o β) : Filter.Tendsto (fun (x : γ) => e (f x)) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop

@[simp]

theorem OrderIso.tendsto_atBot_iff {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] {l : Filter γ} {f : γ → α} (e : α ≃o β) : Filter.Tendsto (fun (x : γ) => e (f x)) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot

Sequences

theorem Filter.extraction_of_frequently_atTop' {P : ℕ → Prop} (h : ∀ (N : ℕ), ∃ n > N, P n) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P (φ n)

theorem Filter.extraction_of_frequently_atTop {P : ℕ → Prop} (h : ∃ᶠ (n : ℕ) in atTop, P n) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P (φ n)

theorem Filter.extraction_of_eventually_atTop {P : ℕ → Prop} (h : ∀ᶠ (n : ℕ) in atTop, P n) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P (φ n)

theorem Filter.extraction_forall_of_frequently {P : ℕ → ℕ → Prop} (h : ∀ (n : ℕ), ∃ᶠ (k : ℕ) in atTop, P n k) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)

theorem Filter.extraction_forall_of_eventually {P : ℕ → ℕ → Prop} (h : ∀ (n : ℕ), ∀ᶠ (k : ℕ) in atTop, P n k) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)

theorem Filter.extraction_forall_of_eventually' {P : ℕ → ℕ → Prop} (h : ∀ (n : ℕ), ∃ (N : ℕ), ∀ k ≥ N, P n k) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)

theorem Filter.Eventually.atTop_of_arithmetic {p : ℕ → Prop} {n : ℕ} (hn : n ≠ 0) (hp : ∀ k < n, ∀ᶠ (a : ℕ) in atTop, p (n * a + k)) : ∀ᶠ (a : ℕ) in atTop, p a

theorem Filter.inf_map_atTop_neBot_iff {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {F : Filter β} {u : α → β} [Nonempty α] : (F ⊓ map u atTop).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U

theorem Filter.exists_le_of_tendsto_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {u : α → β} [Preorder β] (h : Tendsto u atTop atTop) (a : α) (b : β) : ∃ a' ≥ a, b ≤ u a'

theorem Filter.exists_le_of_tendsto_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {u : α → β} [Preorder β] (h : Tendsto u atTop atBot) (a : α) (b : β) : ∃ a' ≥ a, u a' ≤ b

theorem Filter.exists_lt_of_tendsto_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {u : α → β} [Preorder β] [NoMaxOrder β] (h : Tendsto u atTop atTop) (a : α) (b : β) : ∃ a' ≥ a, b < u a'

theorem Filter.exists_lt_of_tendsto_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {u : α → β} [Preorder β] [NoMinOrder β] (h : Tendsto u atTop atBot) (a : α) (b : β) : ∃ a' ≥ a, u a' < b

theorem Filter.inf_map_atBot_neBot_iff {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {F : Filter β} {u : α → β} : (F ⊓ map u atBot).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≤ N, u n ∈ U

theorem Filter.high_scores {β : Type u_4} [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) (N : ℕ) : ∃ n ≥ N, ∀ k < n, u k < u n

If u is a sequence which is unbounded above, then after any point, it reaches a value strictly greater than all previous values.

theorem Filter.low_scores {β : Type u_4} [LinearOrder β] [NoMinOrder β] {u : ℕ → β} (hu : Tendsto u atTop atBot) (N : ℕ) : ∃ n ≥ N, ∀ k < n, u n < u k

If u is a sequence which is unbounded below, then after any point, it reaches a value strictly smaller than all previous values.

theorem Filter.frequently_high_scores {β : Type u_4} [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) : ∃ᶠ (n : ℕ) in atTop, ∀ k < n, u k < u n

If u is a sequence which is unbounded above, then it Frequently reaches a value strictly greater than all previous values.

theorem Filter.frequently_low_scores {β : Type u_4} [LinearOrder β] [NoMinOrder β] {u : ℕ → β} (hu : Tendsto u atTop atBot) : ∃ᶠ (n : ℕ) in atTop, ∀ k < n, u n < u k

If u is a sequence which is unbounded below, then it Frequently reaches a value strictly smaller than all previous values.

theorem Filter.strictMono_subseq_of_tendsto_atTop {β : Type u_4} [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ StrictMono (u ∘ φ)

theorem Filter.strictMono_subseq_of_id_le {u : ℕ → ℕ} (hu : ∀ (n : ℕ), n ≤ u n) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ StrictMono (u ∘ φ)

theorem StrictMono.tendsto_atTop {φ : ℕ → ℕ} (h : StrictMono φ) : Filter.Tendsto φ Filter.atTop Filter.atTop

theorem Monotone.upperBounds_range_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : upperBounds (Set.range (f ∘ g)) = upperBounds (Set.range f)

If f is a monotone function and g tends to atTop along a nontrivial filter. then the upper bounds of the range of f ∘ g are the same as the upper bounds of the range of f.

This lemma together with exists_seq_monotone_tendsto_atTop_atTop below is useful to reduce a statement about a monotone family indexed by a type with countably generated atTop (e.g., ℝ) to the case of a family indexed by natural numbers.

theorem Monotone.lowerBounds_range_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : lowerBounds (Set.range (f ∘ g)) = lowerBounds (Set.range f)

If f is a monotone function and g tends to atBot along a nontrivial filter. then the lower bounds of the range of f ∘ g are the same as the lower bounds of the range of f.

theorem Antitone.lowerBounds_range_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : lowerBounds (Set.range (f ∘ g)) = lowerBounds (Set.range f)

If f is an antitone function and g tends to atTop along a nontrivial filter. then the upper bounds of the range of f ∘ g are the same as the upper bounds of the range of f.

theorem Antitone.upperBounds_range_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : upperBounds (Set.range (f ∘ g)) = upperBounds (Set.range f)

If f is an antitone function and g tends to atBot along a nontrivial filter. then the upper bounds of the range of f ∘ g are the same as the upper bounds of the range of f.

theorem Monotone.ciSup_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) (hb : BddAbove (Set.range f)) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is a monotone function with bounded range and g tends to atTop along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

The assumption BddAbove (range f) can be omitted, if the codomain of f is a conditionally complete linear order or a complete lattice, see below.

theorem Monotone.ciInf_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) (hb : BddBelow (Set.range f)) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is a monotone function with bounded range and g tends to atBot along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

The assumption BddBelow (range f) can be omitted, if the codomain of f is a conditionally complete linear order or a complete lattice, see below.

theorem Antitone.ciSup_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) (hb : BddAbove (Set.range f)) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is an antitone function with bounded range and g tends to atBot along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

The assumption BddAbove (range f) can be omitted, if the codomain of f is a conditionally complete linear order or a complete lattice, see below.

theorem Antitone.ciInf_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) (hb : BddBelow (Set.range f)) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is an antitone function with bounded range and g tends to atTop along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

The assumption BddBelow (range f) can be omitted, if the codomain of f is a conditionally complete linear order or a complete lattice, see below.

theorem Monotone.ciSup_comp_tendsto_atTop_of_linearOrder {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLinearOrder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is a monotone function taking values in a conditionally complete linear order and g tends to atTop along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

theorem Monotone.ciInf_comp_tendsto_atBot_of_linearOrder {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLinearOrder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is a monotone function taking values in a conditionally complete linear order and g tends to atBot along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

theorem Antitone.ciInf_comp_tendsto_atTop_of_linearOrder {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLinearOrder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is an antitone function taking values in a conditionally complete linear order and g tends to atTop along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

theorem Antitone.ciSup_comp_tendsto_atBot_of_linearOrder {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLinearOrder γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is an antitone function taking values in a conditionally complete linear order and g tends to atBot along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

theorem Monotone.iSup_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] [OrderTop γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is a monotone function taking values in a complete lattice and g tends to atTop along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

theorem Monotone.iInf_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] [OrderBot γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Monotone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is a monotone function taking values in a complete lattice and g tends to atBot along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

theorem Antitone.iSup_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] [OrderTop γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atBot) : ⨆ (a : α), f (g a) = ⨆ (b : β), f b

If f is an antitone function taking values in a complete lattice and g tends to atBot along a nontrivial filter, then the indexed supremum of f ∘ g is equal to the indexed supremum of f.

theorem Antitone.iInf_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [ConditionallyCompleteLattice γ] [OrderBot γ] {l : Filter α} [l.NeBot] {f : β → γ} (hf : Antitone f) {g : α → β} (hg : Filter.Tendsto g l Filter.atTop) : ⨅ (a : α), f (g a) = ⨅ (b : β), f b

If f is an antitone function taking values in a complete lattice and g tends to atTop along a nontrivial filter, then the indexed infimum of f ∘ g is equal to the indexed infimum of f.

theorem Monotone.iUnion_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] {l : Filter α} [l.NeBot] {s : β → Set γ} (hs : Monotone s) {f : α → β} (hf : Filter.Tendsto f l Filter.atTop) : ⋃ (a : α), s (f a) = ⋃ (b : β), s b

If s is a monotone family of sets and f tends to atTop along a nontrivial filter, then the indexed union of s ∘ f is equal to the indexed union of s.

theorem Monotone.iInter_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] {l : Filter α} [l.NeBot] {s : β → Set γ} (hs : Monotone s) {f : α → β} (hf : Filter.Tendsto f l Filter.atBot) : ⋂ (a : α), s (f a) = ⋂ (b : β), s b

If s is a monotone family of sets and f tends to atBot along a nontrivial filter, then the indexed intersection of s ∘ f is equal to the indexed intersection of s.

theorem Antitone.iInter_comp_tendsto_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] {l : Filter α} [l.NeBot] {s : β → Set γ} (hs : Antitone s) {f : α → β} (hf : Filter.Tendsto f l Filter.atTop) : ⋂ (a : α), s (f a) = ⋂ (b : β), s b

If s is an antitone family of sets and f tends to atTop along a nontrivial filter, then the indexed intersection of s ∘ f is equal to the indexed intersection of s.

theorem Antitone.iUnion_comp_tendsto_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] {l : Filter α} [l.NeBot] {s : β → Set γ} (hs : Antitone s) {f : α → β} (hf : Filter.Tendsto f l Filter.atBot) : ⋃ (a : α), s (f a) = ⋃ (b : β), s b

If s is a monotone family of sets and f tends to atBot along a nontrivial filter, then the indexed union of s ∘ f is equal to the indexed union of s.

theorem Filter.tendsto_atTop_atTop_of_monotone {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ (a : α), b ≤ f a) : Tendsto f atTop atTop

theorem Filter.tendsto_atTop_atBot_of_antitone {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f) (h : ∀ (b : β), ∃ (a : α), f a ≤ b) : Tendsto f atTop atBot

theorem Filter.tendsto_atBot_atBot_of_monotone {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ (a : α), f a ≤ b) : Tendsto f atBot atBot

theorem Filter.tendsto_atBot_atTop_of_antitone {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f) (h : ∀ (b : β), ∃ (a : α), b ≤ f a) : Tendsto f atBot atTop

theorem Filter.tendsto_atTop' {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} {l : Filter β} : Tendsto f atTop l ↔ ∀ s ∈ l, ∃ (a : α), ∀ b ≥ a, f b ∈ s

theorem Filter.tendsto_atTop_principal {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} {s : Set β} : Tendsto f atTop (principal s) ↔ ∃ (N : α), ∀ n ≥ N, f n ∈ s

theorem Filter.tendsto_atTop_atTop {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} [Preorder β] : Tendsto f atTop atTop ↔ ∀ (b : β), ∃ (i : α), ∀ (a : α), i ≤ a → b ≤ f a

A function f grows to +∞ independent of an order-preserving embedding e.

theorem Filter.tendsto_atTop_atBot {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} [Preorder β] : Tendsto f atTop atBot ↔ ∀ (b : β), ∃ (i : α), ∀ (a : α), i ≤ a → f a ≤ b

theorem Filter.tendsto_atTop_atTop_iff_of_monotone {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} [Preorder β] (hf : Monotone f) : Tendsto f atTop atTop ↔ ∀ (b : β), ∃ (a : α), b ≤ f a

theorem Filter.tendsto_atTop_atBot_iff_of_antitone {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} [Preorder β] (hf : Antitone f) : Tendsto f atTop atBot ↔ ∀ (b : β), ∃ (a : α), f a ≤ b

theorem Filter.tendsto_atBot' {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} {l : Filter β} : Tendsto f atBot l ↔ ∀ s ∈ l, ∃ (a : α), ∀ b ≤ a, f b ∈ s

theorem Filter.tendsto_atBot_principal {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} {s : Set β} : Tendsto f atBot (principal s) ↔ ∃ (N : α), ∀ n ≤ N, f n ∈ s

theorem Filter.tendsto_atBot_atTop {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} [Preorder β] : Tendsto f atBot atTop ↔ ∀ (b : β), ∃ (i : α), ∀ a ≤ i, b ≤ f a

theorem Filter.tendsto_atBot_atBot {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} [Preorder β] : Tendsto f atBot atBot ↔ ∀ (b : β), ∃ (i : α), ∀ a ≤ i, f a ≤ b

theorem Filter.tendsto_atBot_atBot_iff_of_monotone {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} [Preorder β] (hf : Monotone f) : Tendsto f atBot atBot ↔ ∀ (b : β), ∃ (a : α), f a ≤ b

theorem Filter.tendsto_atBot_atTop_iff_of_antitone {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} [Preorder β] (hf : Antitone f) : Tendsto f atBot atTop ↔ ∀ (b : β), ∃ (a : α), b ≤ f a

theorem Monotone.tendsto_atTop_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ (a : α), b ≤ f a) : Filter.Tendsto f Filter.atTop Filter.atTop

Alias of Filter.tendsto_atTop_atTop_of_monotone.

theorem Monotone.tendsto_atBot_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (h : ∀ (b : β), ∃ (a : α), f a ≤ b) : Filter.Tendsto f Filter.atBot Filter.atBot

Alias of Filter.tendsto_atBot_atBot_of_monotone.

theorem Monotone.tendsto_atTop_atTop_iff {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {f : α → β} [Preorder β] (hf : Monotone f) : Filter.Tendsto f Filter.atTop Filter.atTop ↔ ∀ (b : β), ∃ (a : α), b ≤ f a

Alias of Filter.tendsto_atTop_atTop_iff_of_monotone.

theorem Monotone.tendsto_atBot_atBot_iff {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {f : α → β} [Preorder β] (hf : Monotone f) : Filter.Tendsto f Filter.atBot Filter.atBot ↔ ∀ (b : β), ∃ (a : α), f a ≤ b

Alias of Filter.tendsto_atBot_atBot_iff_of_monotone.

theorem Filter.comap_embedding_atTop {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ (b : β), c ≤ e b) : comap e atTop = atTop

theorem Filter.comap_embedding_atBot {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ (b : β), e b ≤ c) : comap e atBot = atBot

theorem Filter.tendsto_atTop_embedding {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ (b : β), c ≤ e b) : Tendsto (e ∘ f) l atTop ↔ Tendsto f l atTop

theorem Filter.tendsto_atBot_embedding {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ (b : β), e b ≤ c) : Tendsto (e ∘ f) l atBot ↔ Tendsto f l atBot

A function f goes to -∞ independent of an order-preserving embedding e.

theorem Filter.tendsto_finset_range : Tendsto Finset.range atTop atTop

theorem Filter.atTop_finset_eq_iInf {α : Type u_3} : atTop = ⨅ (x : α), principal (Set.Ici {x})

theorem Filter.tendsto_atTop_finset_of_monotone {α : Type u_3} {β : Type u_4} [Preorder β] {f : β → Finset α} (h : Monotone f) (h' : ∀ (x : α), ∃ (n : β), x ∈ f n) : Tendsto f atTop atTop

If f is a monotone sequence of Finsets and each x belongs to one of f n, then Tendsto f atTop atTop.

theorem Monotone.tendsto_atTop_finset {α : Type u_3} {β : Type u_4} [Preorder β] {f : β → Finset α} (h : Monotone f) (h' : ∀ (x : α), ∃ (n : β), x ∈ f n) : Filter.Tendsto f Filter.atTop Filter.atTop

Alias of Filter.tendsto_atTop_finset_of_monotone.

---

If f is a monotone sequence of Finsets and each x belongs to one of f n, then Tendsto f atTop atTop.

theorem Filter.tendsto_finset_image_atTop_atTop {β : Type u_4} {γ : Type u_5} [DecidableEq β] {i : β → γ} {j : γ → β} (h : Function.LeftInverse j i) : Tendsto (Finset.image j) atTop atTop

theorem Filter.tendsto_finset_preimage_atTop_atTop {α : Type u_3} {β : Type u_4} {f : α → β} (hf : Function.Injective f) : Tendsto (fun (s : Finset β) => s.preimage f ⋯) atTop atTop

theorem Filter.prod_atTop_atTop_eq {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] : atTop ×ˢ atTop = atTop

theorem Filter.tendsto_finset_prod_atTop {ι : Type u_1} {ι' : Type u_2} : Tendsto (fun (p : Finset ι × Finset ι') => p.1 ×ˢ p.2) atTop atTop

theorem Filter.prod_atBot_atBot_eq {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] : atBot ×ˢ atBot = atBot

theorem Filter.prod_map_atTop_eq {α₁ : Type u_6} {α₂ : Type u_7} {β₁ : Type u_8} {β₂ : Type u_9} [Preorder β₁] [Preorder β₂] (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) : map u₁ atTop ×ˢ map u₂ atTop = map (Prod.map u₁ u₂) atTop

theorem Filter.prod_map_atBot_eq {α₁ : Type u_6} {α₂ : Type u_7} {β₁ : Type u_8} {β₂ : Type u_9} [Preorder β₁] [Preorder β₂] (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) : map u₁ atBot ×ˢ map u₂ atBot = map (Prod.map u₁ u₂) atBot

theorem Filter.Tendsto.subseq_mem {α : Type u_3} {F : Filter α} {V : ℕ → Set α} (h : ∀ (n : ℕ), V n ∈ F) {u : ℕ → α} (hu : Tendsto u atTop F) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (n : ℕ), u (φ n) ∈ V n

theorem Filter.tendsto_atBot_diagonal {α : Type u_3} [Preorder α] : Tendsto (fun (a : α) => (a, a)) atBot atBot

theorem Filter.tendsto_atTop_diagonal {α : Type u_3} [Preorder α] : Tendsto (fun (a : α) => (a, a)) atTop atTop

theorem Filter.Tendsto.prod_map_prod_atBot {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder γ] {F : Filter α} {G : Filter β} {f : α → γ} {g : β → γ} (hf : Tendsto f F atBot) (hg : Tendsto g G atBot) : Tendsto (Prod.map f g) (F ×ˢ G) atBot

theorem Filter.Tendsto.prod_map_prod_atTop {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder γ] {F : Filter α} {G : Filter β} {f : α → γ} {g : β → γ} (hf : Tendsto f F atTop) (hg : Tendsto g G atTop) : Tendsto (Prod.map f g) (F ×ˢ G) atTop

theorem Filter.Tendsto.prod_atBot {α : Type u_3} {γ : Type u_5} [Preorder α] [Preorder γ] {f g : α → γ} (hf : Tendsto f atBot atBot) (hg : Tendsto g atBot atBot) : Tendsto (Prod.map f g) atBot atBot

theorem Filter.Tendsto.prod_atTop {α : Type u_3} {γ : Type u_5} [Preorder α] [Preorder γ] {f g : α → γ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) : Tendsto (Prod.map f g) atTop atTop

theorem Filter.eventually_atBot_prod_self {α : Type u_3} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α × α → Prop} : (∀ᶠ (x : α × α) in atBot, p x) ↔ ∃ (a : α), ∀ (k l : α), k ≤ a → l ≤ a → p (k, l)

theorem Filter.eventually_atTop_prod_self {α : Type u_3} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α × α → Prop} : (∀ᶠ (x : α × α) in atTop, p x) ↔ ∃ (a : α), ∀ (k l : α), a ≤ k → a ≤ l → p (k, l)

theorem Filter.eventually_atBot_prod_self' {α : Type u_3} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {p : α × α → Prop} : (∀ᶠ (x : α × α) in atBot, p x) ↔ ∃ (a : α), ∀ k ≤ a, ∀ l ≤ a, p (k, l)

theorem Filter.eventually_atTop_prod_self' {α : Type u_3} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {p : α × α → Prop} : (∀ᶠ (x : α × α) in atTop, p x) ↔ ∃ (a : α), ∀ k ≥ a, ∀ l ≥ a, p (k, l)

theorem Filter.eventually_atTop_curry {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {p : α × β → Prop} (hp : ∀ᶠ (x : α × β) in atTop, p x) : ∀ᶠ (k : α) in atTop, ∀ᶠ (l : β) in atTop, p (k, l)

theorem Filter.eventually_atBot_curry {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {p : α × β → Prop} (hp : ∀ᶠ (x : α × β) in atBot, p x) : ∀ᶠ (k : α) in atBot, ∀ᶠ (l : β) in atBot, p (k, l)

theorem Filter.map_atTop_eq_of_gc_preorder {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Preorder β] [IsDirected β fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} (hf : Monotone f) (b : β) (hgi : ∀ c ≥ b, ∃ (x : α), f x = c ∧ ∀ (a : α), f a ≤ c ↔ a ≤ x) : map f atTop = atTop

A function f maps upwards closed sets (atTop sets) to upwards closed sets when it is a Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an insertion and a connection above b.

theorem Filter.map_atTop_eq_of_gc {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [PartialOrder β] [IsDirected β fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} (g : β → α) (b : β) (hf : Monotone f) (gc : ∀ (a : α), ∀ c ≥ b, f a ≤ c ↔ a ≤ g c) (hgi : ∀ c ≥ b, c ≤ f (g c)) : map f atTop = atTop

A function f maps upwards closed sets (atTop sets) to upwards closed sets when it is a Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an insertion and a connection above b.

theorem Filter.map_atBot_eq_of_gc_preorder {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Preorder β] [IsDirected β fun (x1 x2 : β) => x1 ≥ x2] {f : α → β} (hf : Monotone f) (b : β) (hgi : ∀ c ≤ b, ∃ (x : α), f x = c ∧ ∀ (a : α), c ≤ f a ↔ x ≤ a) : map f atBot = atBot

theorem Filter.map_atBot_eq_of_gc {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [PartialOrder β] [IsDirected β fun (x1 x2 : β) => x1 ≥ x2] {f : α → β} (g : β → α) (b' : β) (hf : Monotone f) (gc : ∀ (a : α), ∀ b ≤ b', b ≤ f a ↔ g b ≤ a) (hgi : ∀ b ≤ b', f (g b) ≤ b) : map f atBot = atBot

theorem Filter.map_val_atTop_of_Ici_subset {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {a : α} {s : Set α} (h : Set.Ici a ⊆ s) : map Subtype.val atTop = atTop

@[simp]

theorem Nat.map_cast_int_atTop : Filter.map Nat.cast Filter.atTop = Filter.atTop

@[simp]

theorem Filter.map_val_Ici_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] (a : α) : map Subtype.val atTop = atTop

The image of the filter atTop on Ici a under the coercion equals atTop.

@[simp]

theorem Filter.map_val_Ioi_atTop {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [NoMaxOrder α] (a : α) : map Subtype.val atTop = atTop

The image of the filter atTop on Ioi a under the coercion equals atTop.

theorem Filter.atTop_Ioi_eq {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] (a : α) : atTop = comap Subtype.val atTop

The atTop filter for an open interval Ioi a comes from the atTop filter in the ambient order.

theorem Filter.atTop_Ici_eq {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] (a : α) : atTop = comap Subtype.val atTop

The atTop filter for an open interval Ici a comes from the atTop filter in the ambient order.

@[simp]

theorem Filter.map_val_Iio_atBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [NoMinOrder α] (a : α) : map Subtype.val atBot = atBot

The atBot filter for an open interval Iio a comes from the atBot filter in the ambient order.

theorem Filter.atBot_Iio_eq {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] (a : α) : atBot = comap Subtype.val atBot

The atBot filter for an open interval Iio a comes from the atBot filter in the ambient order.

@[simp]

theorem Filter.map_val_Iic_atBot {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] (a : α) : map Subtype.val atBot = atBot

The atBot filter for an open interval Iic a comes from the atBot filter in the ambient order.

theorem Filter.atBot_Iic_eq {α : Type u_3} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] (a : α) : atBot = comap Subtype.val atBot

The atBot filter for an open interval Iic a comes from the atBot filter in the ambient order.

theorem Filter.tendsto_Ioi_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {a : α} {f : β → ↑(Set.Ioi a)} {l : Filter β} : Tendsto f l atTop ↔ Tendsto (fun (x : β) => ↑(f x)) l atTop

theorem Filter.tendsto_Iio_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {a : α} {f : β → ↑(Set.Iio a)} {l : Filter β} : Tendsto f l atBot ↔ Tendsto (fun (x : β) => ↑(f x)) l atBot

theorem Filter.tendsto_Ici_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {a : α} {f : β → ↑(Set.Ici a)} {l : Filter β} : Tendsto f l atTop ↔ Tendsto (fun (x : β) => ↑(f x)) l atTop

theorem Filter.tendsto_Iic_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {a : α} {f : β → ↑(Set.Iic a)} {l : Filter β} : Tendsto f l atBot ↔ Tendsto (fun (x : β) => ↑(f x)) l atBot

@[simp]

theorem Filter.tendsto_comp_val_Ioi_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [NoMaxOrder α] {a : α} {f : α → β} {l : Filter β} : Tendsto (fun (x : ↑(Set.Ioi a)) => f ↑x) atTop l ↔ Tendsto f atTop l

@[simp]

theorem Filter.tendsto_comp_val_Ici_atTop {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] {a : α} {f : α → β} {l : Filter β} : Tendsto (fun (x : ↑(Set.Ici a)) => f ↑x) atTop l ↔ Tendsto f atTop l

@[simp]

theorem Filter.tendsto_comp_val_Iio_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [NoMinOrder α] {a : α} {f : α → β} {l : Filter β} : Tendsto (fun (x : ↑(Set.Iio a)) => f ↑x) atBot l ↔ Tendsto f atBot l

@[simp]

theorem Filter.tendsto_comp_val_Iic_atBot {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] {a : α} {f : α → β} {l : Filter β} : Tendsto (fun (x : ↑(Set.Iic a)) => f ↑x) atBot l ↔ Tendsto f atBot l

theorem Filter.map_add_atTop_eq_nat (k : ℕ) : map (fun (a : ℕ) => a + k) atTop = atTop

theorem Filter.map_sub_atTop_eq_nat (k : ℕ) : map (fun (a : ℕ) => a - k) atTop = atTop

theorem Filter.tendsto_add_atTop_nat (k : ℕ) : Tendsto (fun (a : ℕ) => a + k) atTop atTop

theorem Filter.tendsto_sub_atTop_nat (k : ℕ) : Tendsto (fun (a : ℕ) => a - k) atTop atTop

theorem Filter.tendsto_add_atTop_iff_nat {α : Type u_3} {f : ℕ → α} {l : Filter α} (k : ℕ) : Tendsto (fun (n : ℕ) => f (n + k)) atTop l ↔ Tendsto f atTop l

theorem Filter.map_div_atTop_eq_nat (k : ℕ) (hk : 0 < k) : map (fun (a : ℕ) => a / k) atTop = atTop

theorem Filter.tendsto_atTop_atTop_of_monotone' {ι : Type u_1} {α : Type u_3} [Preorder ι] [LinearOrder α] {u : ι → α} (h : Monotone u) (H : ¬BddAbove (Set.range u)) : Tendsto u atTop atTop

If u is a monotone function with linear ordered codomain and the range of u is not bounded above, then Tendsto u atTop atTop.

theorem Filter.tendsto_atBot_atBot_of_monotone' {ι : Type u_1} {α : Type u_3} [Preorder ι] [LinearOrder α] {u : ι → α} (h : Monotone u) (H : ¬BddBelow (Set.range u)) : Tendsto u atBot atBot

If u is a monotone function with linear ordered codomain and the range of u is not bounded below, then Tendsto u atBot atBot.

theorem Filter.unbounded_of_tendsto_atTop {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Preorder β] {f : α → β} [NoMaxOrder β] (h : Tendsto f atTop atTop) : ¬BddAbove (Set.range f)

theorem Filter.unbounded_of_tendsto_atBot {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≤ x2] [Preorder β] {f : α → β} [NoMinOrder β] (h : Tendsto f atTop atBot) : ¬BddBelow (Set.range f)

theorem Filter.unbounded_of_tendsto_atTop' {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Preorder β] {f : α → β} [NoMaxOrder β] (h : Tendsto f atBot atTop) : ¬BddAbove (Set.range f)

theorem Filter.unbounded_of_tendsto_atBot' {α : Type u_3} {β : Type u_4} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 ≥ x2] [Preorder β] {f : α → β} [NoMinOrder β] (h : Tendsto f atBot atBot) : ¬BddBelow (Set.range f)

theorem Filter.tendsto_atTop_of_monotone_of_filter {ι : Type u_1} {α : Type u_3} [Preorder ι] [Preorder α] {l : Filter ι} {u : ι → α} (h : Monotone u) [l.NeBot] (hu : Tendsto u l atTop) : Tendsto u atTop atTop

If a monotone function u : ι → α tends to atTop along some non-trivial filter l, then it tends to atTop along atTop.

theorem Filter.tendsto_atBot_of_monotone_of_filter {ι : Type u_1} {α : Type u_3} [Preorder ι] [Preorder α] {l : Filter ι} {u : ι → α} (h : Monotone u) [l.NeBot] (hu : Tendsto u l atBot) : Tendsto u atBot atBot

If a monotone function u : ι → α tends to atBot along some non-trivial filter l, then it tends to atBot along atBot.

theorem Filter.tendsto_atTop_of_monotone_of_subseq {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι] [Preorder α] {u : ι → α} {φ : ι' → ι} (h : Monotone u) {l : Filter ι'} [l.NeBot] (H : Tendsto (u ∘ φ) l atTop) : Tendsto u atTop atTop

theorem Filter.tendsto_atBot_of_monotone_of_subseq {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Preorder ι] [Preorder α] {u : ι → α} {φ : ι' → ι} (h : Monotone u) {l : Filter ι'} [l.NeBot] (H : Tendsto (u ∘ φ) l atBot) : Tendsto u atBot atBot

theorem Filter.HasAntitoneBasis.eventually_subset {ι : Type u_1} {α : Type u_3} [Preorder ι] {l : Filter α} {s : ι → Set α} (hl : l.HasAntitoneBasis s) {t : Set α} (ht : t ∈ l) : ∀ᶠ (i : ι) in atTop, s i ⊆ t

theorem Filter.HasAntitoneBasis.tendsto {ι : Type u_1} {α : Type u_3} [Preorder ι] {l : Filter α} {s : ι → Set α} (hl : l.HasAntitoneBasis s) {φ : ι → α} (h : ∀ (i : ι), φ i ∈ s i) : Tendsto φ atTop l

theorem Filter.HasAntitoneBasis.comp_mono {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [Nonempty ι] [Preorder ι] [IsDirected ι fun (x1 x2 : ι) => x1 ≤ x2] [Preorder ι'] {l : Filter α} {s : ι' → Set α} (hs : l.HasAntitoneBasis s) {φ : ι → ι'} (φ_mono : Monotone φ) (hφ : Tendsto φ atTop atTop) : l.HasAntitoneBasis (s ∘ φ)

theorem Filter.HasAntitoneBasis.comp_strictMono {α : Type u_3} {l : Filter α} {s : ℕ → Set α} (hs : l.HasAntitoneBasis s) {φ : ℕ → ℕ} (hφ : StrictMono φ) : l.HasAntitoneBasis (s ∘ φ)

theorem Filter.HasAntitoneBasis.subbasis_with_rel {α : Type u_3} {f : Filter α} {s : ℕ → Set α} (hs : f.HasAntitoneBasis s) {r : ℕ → ℕ → Prop} (hr : ∀ (m : ℕ), ∀ᶠ (n : ℕ) in atTop, r m n) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ (∀ ⦃m n : ℕ⦄, m < n → r (φ m) (φ n)) ∧ f.HasAntitoneBasis (s ∘ φ)

Given an antitone basis s : ℕ → Set α of a filter, extract an antitone subbasis s ∘ φ, φ : ℕ → ℕ, such that m < n implies r (φ m) (φ n). This lemma can be used to extract an antitone basis with basis sets decreasing "sufficiently fast".

theorem Filter.subseq_forall_of_frequently {ι : Type u_6} {x : ℕ → ι} {p : ι → Prop} {l : Filter ι} (h_tendsto : Tendsto x atTop l) (h : ∃ᶠ (n : ℕ) in atTop, p (x n)) : ∃ (ns : ℕ → ℕ), Tendsto (fun (n : ℕ) => x (ns n)) atTop l ∧ ∀ (n : ℕ), p (x (ns n))

theorem Monotone.piecewise_eventually_eq_iUnion {ι : Type u_1} {α : Type u_3} {β : α → Type u_6} [Preorder ι] {s : ι → Set α} [(i : ι) → DecidablePred fun (x : α) => x ∈ s i] [DecidablePred fun (x : α) => x ∈ ⋃ (i : ι), s i] (hs : Monotone s) (f g : (a : α) → β a) (a : α) : ∀ᶠ (i : ι) in Filter.atTop, (s i).piecewise f g a = (⋃ (i : ι), s i).piecewise f g a

theorem Antitone.piecewise_eventually_eq_iInter {ι : Type u_1} {α : Type u_3} {β : α → Type u_6} [Preorder ι] {s : ι → Set α} [(i : ι) → DecidablePred fun (x : α) => x ∈ s i] [DecidablePred fun (x : α) => x ∈ ⋂ (i : ι), s i] (hs : Antitone s) (f g : (a : α) → β a) (a : α) : ∀ᶠ (i : ι) in Filter.atTop, (s i).piecewise f g a = (⋂ (i : ι), s i).piecewise f g a

theorem Nat.eventually_pow_lt_factorial_sub (c d : ℕ) : ∀ᶠ (n : ℕ) in Filter.atTop, c ^ n < (n - d).factorial

theorem Nat.eventually_mul_pow_lt_factorial_sub (a c d : ℕ) : ∀ᶠ (n : ℕ) in Filter.atTop, a * c ^ n < (n - d).factorial

@[deprecated Nat.eventually_pow_lt_factorial_sub (since := "2024-09-25")]

theorem Nat.exists_pow_lt_factorial (c : ℕ) : ∃ n0 > 1, ∀ n ≥ n0, c ^ n < (n - 1).factorial

@[deprecated Nat.eventually_mul_pow_lt_factorial_sub (since := "2024-09-25")]

theorem Nat.exists_mul_pow_lt_factorial (a c : ℕ) : ∃ (n0 : ℕ), ∀ n ≥ n0, a * c ^ n < (n - 1).factorial