File: Mathlib/Topology/Order/MonotoneConvergence.lean

full_name: SupConvergenceClass.tendsto_coe_atTop_isLUB
code: /-- We say that `α` is a `SupConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a least upper bound of `Set.range f`. Then `f x` tends to `𝓝 a`
 as `x → ∞` (formally, at the filter `Filter.atTop`). We require this for `ι = (s : Set α)`,
`f = CoeTC.coe` in the definition, then prove it for any `f` in `tendsto_atTop_isLUB`.

This property holds for linear orders with order topology as well as their products. -/
class SupConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  /-- proof that a monotone function tends to `𝓝 a` as `x → ∞` -/
  tendsto_coe_atTop_isLUB :
    ∀ (a : α) (s : Set α), IsLUB s a → Tendsto (CoeTC.coe : s → α) atTop (𝓝 a)
goal: IsLUB s a → Filter.Tendsto CoeTC.coe Filter.atTop (𝓝 a)

full_name: SupConvergenceClass
code: /-- We say that `α` is a `SupConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a least upper bound of `Set.range f`. Then `f x` tends to `𝓝 a`
 as `x → ∞` (formally, at the filter `Filter.atTop`). We require this for `ι = (s : Set α)`,
`f = CoeTC.coe` in the definition, then prove it for any `f` in `tendsto_atTop_isLUB`.

This property holds for linear orders with order topology as well as their products. -/
class SupConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  /-- proof that a monotone function tends to `𝓝 a` as `x → ∞` -/
  tendsto_coe_atTop_isLUB :
    ∀ (a : α) (s : Set α), IsLUB s a → Tendsto (CoeTC.coe : s → α) atTop (𝓝 a)
goal: Prop

full_name: InfConvergenceClass.tendsto_coe_atBot_isGLB
code: /-- We say that `α` is an `InfConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a greatest lower bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → -∞` (formally, at the filter `Filter.atBot`). We require this for `ι = (s : Set α)`,
`f = CoeTC.coe` in the definition, then prove it for any `f` in `tendsto_atBot_isGLB`.

This property holds for linear orders with order topology as well as their products. -/
class InfConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  /-- proof that a monotone function tends to `𝓝 a` as `x → -∞`-/
  tendsto_coe_atBot_isGLB :
    ∀ (a : α) (s : Set α), IsGLB s a → Tendsto (CoeTC.coe : s → α) atBot (𝓝 a)
goal: IsGLB s a → Filter.Tendsto CoeTC.coe Filter.atBot (𝓝 a)

full_name: InfConvergenceClass
code: /-- We say that `α` is an `InfConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a greatest lower bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → -∞` (formally, at the filter `Filter.atBot`). We require this for `ι = (s : Set α)`,
`f = CoeTC.coe` in the definition, then prove it for any `f` in `tendsto_atBot_isGLB`.

This property holds for linear orders with order topology as well as their products. -/
class InfConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
  /-- proof that a monotone function tends to `𝓝 a` as `x → -∞`-/
  tendsto_coe_atBot_isGLB :
    ∀ (a : α) (s : Set α), IsGLB s a → Tendsto (CoeTC.coe : s → α) atBot (𝓝 a)
goal: Prop

full_name: OrderDual.supConvergenceClass
code: instance OrderDual.supConvergenceClass [Preorder α] [TopologicalSpace α] [InfConvergenceClass α] :
    SupConvergenceClass αᵒᵈ 
goal: SupConvergenceClass αᵒᵈ

full_name: OrderDual.infConvergenceClass
code: instance OrderDual.infConvergenceClass [Preorder α] [TopologicalSpace α] [SupConvergenceClass α] :
    InfConvergenceClass αᵒᵈ 
goal: InfConvergenceClass αᵒᵈ

full_name: LinearOrder.supConvergenceClass
code: instance (priority := 100) LinearOrder.supConvergenceClass [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] : SupConvergenceClass α 
goal: SupConvergenceClass α

full_name: LinearOrder.infConvergenceClass
code: instance (priority := 100) LinearOrder.infConvergenceClass [TopologicalSpace α] [LinearOrder α]
    [OrderTopology α] : InfConvergenceClass α 
goal: InfConvergenceClass α

full_name: tendsto_atTop_isLUB
code: theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
    Tendsto f atTop (𝓝 a) 
goal: Filter.Tendsto f Filter.atTop (𝓝 a)

full_name: tendsto_atBot_isLUB
code: theorem tendsto_atBot_isLUB (h_anti : Antitone f) (ha : IsLUB (Set.range f) a) :
    Tendsto f atBot (𝓝 a) 
goal: Filter.Tendsto f Filter.atBot (𝓝 a)

full_name: tendsto_atBot_isGLB
code: theorem tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :
    Tendsto f atBot (𝓝 a) 
goal: Filter.Tendsto f Filter.atBot (𝓝 a)

full_name: tendsto_atTop_isGLB
code: theorem tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) :
    Tendsto f atTop (𝓝 a) 
goal: Filter.Tendsto f Filter.atTop (𝓝 a)

full_name: tendsto_atTop_ciSup
code: theorem tendsto_atTop_ciSup (h_mono : Monotone f) (hbdd : BddAbove <| range f) :
    Tendsto f atTop (𝓝 (⨆ i, f i)) 
goal: Filter.Tendsto f Filter.atTop (𝓝 (⨆ i, f i))

full_name: tendsto_atBot_ciSup
code: theorem tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <| range f) :
    Tendsto f atBot (𝓝 (⨆ i, f i)) 
goal: Filter.Tendsto f Filter.atBot (𝓝 (⨆ i, f i))

full_name: tendsto_atBot_ciInf
code: theorem tendsto_atBot_ciInf (h_mono : Monotone f) (hbdd : BddBelow <| range f) :
    Tendsto f atBot (𝓝 (⨅ i, f i)) 
goal: Filter.Tendsto f Filter.atBot (𝓝 (⨅ i, f i))

full_name: tendsto_atTop_ciInf
code: theorem tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :
    Tendsto f atTop (𝓝 (⨅ i, f i)) 
goal: Filter.Tendsto f Filter.atTop (𝓝 (⨅ i, f i))

full_name: tendsto_atTop_iSup
code: theorem tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i)) 
goal: Filter.Tendsto f Filter.atTop (𝓝 (⨆ i, f i))

full_name: tendsto_atBot_iSup
code: theorem tendsto_atBot_iSup (h_anti : Antitone f) : Tendsto f atBot (𝓝 (⨆ i, f i)) 
goal: Filter.Tendsto f Filter.atBot (𝓝 (⨆ i, f i))

full_name: tendsto_atBot_iInf
code: theorem tendsto_atBot_iInf (h_mono : Monotone f) : Tendsto f atBot (𝓝 (⨅ i, f i)) 
goal: Filter.Tendsto f Filter.atBot (𝓝 (⨅ i, f i))

full_name: tendsto_atTop_iInf
code: theorem tendsto_atTop_iInf (h_anti : Antitone f) : Tendsto f atTop (𝓝 (⨅ i, f i)) 
goal: Filter.Tendsto f Filter.atTop (𝓝 (⨅ i, f i))

full_name: Prod.supConvergenceClass
code: instance Prod.supConvergenceClass
    [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β]
    [SupConvergenceClass α] [SupConvergenceClass β] : SupConvergenceClass (α × β) 
goal: SupConvergenceClass (α × β)

full_name: instInfConvergenceClassProd
code: instance [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [InfConvergenceClass α]
    [InfConvergenceClass β] : InfConvergenceClass (α × β) 
goal: InfConvergenceClass (α × β)

full_name: Pi.supConvergenceClass
code: instance Pi.supConvergenceClass
    {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
    [∀ i, SupConvergenceClass (α i)] : SupConvergenceClass (∀ i, α i) 
goal: SupConvergenceClass ((i : ι) → α i)

full_name: Pi.infConvergenceClass
code: instance Pi.infConvergenceClass
    {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
    [∀ i, InfConvergenceClass (α i)] : InfConvergenceClass (∀ i, α i) 
goal: InfConvergenceClass ((i : ι) → α i)

full_name: Pi.supConvergenceClass'
code: instance Pi.supConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
    [SupConvergenceClass α] : SupConvergenceClass (ι → α) 
goal: SupConvergenceClass (ι → α)

full_name: Pi.infConvergenceClass'
code: instance Pi.infConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
    [InfConvergenceClass α] : InfConvergenceClass (ι → α) 
goal: InfConvergenceClass (ι → α)

full_name: tendsto_of_monotone
code: theorem tendsto_of_monotone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
    [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) :
    Tendsto f atTop atTop ∨ ∃ l, Tendsto f atTop (𝓝 l) 
goal: Filter.Tendsto f Filter.atTop Filter.atTop ∨ ∃ l, Filter.Tendsto f Filter.atTop (𝓝 l)

full_name: tendsto_of_antitone
code: theorem tendsto_of_antitone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
    [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) :
    Tendsto f atTop atBot ∨ ∃ l, Tendsto f atTop (𝓝 l) 
goal: Filter.Tendsto f Filter.atTop Filter.atBot ∨ ∃ l, Filter.Tendsto f Filter.atTop (𝓝 l)

full_name: tendsto_iff_tendsto_subseq_of_monotone
code: theorem tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
    [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f)
    (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) 
goal: Filter.Tendsto f Filter.atTop (𝓝 l) ↔ Filter.Tendsto (f ∘ φ) Filter.atTop (𝓝 l)

full_name: tendsto_iff_tendsto_subseq_of_antitone
code: theorem tendsto_iff_tendsto_subseq_of_antitone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
    [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
    [NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f)
    (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) 
goal: Filter.Tendsto f Filter.atTop (𝓝 l) ↔ Filter.Tendsto (f ∘ φ) Filter.atTop (𝓝 l)

full_name: Monotone.ge_of_tendsto
code: theorem Monotone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
    f b ≤ a 
goal: f b ≤ a

full_name: Monotone.le_of_tendsto
code: theorem Monotone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
    a ≤ f b 
goal: a ≤ f b

full_name: Antitone.le_of_tendsto
code: theorem Antitone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) :
    a ≤ f b 
goal: a ≤ f b

full_name: Antitone.ge_of_tendsto
code: theorem Antitone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) :
    f b ≤ a 
goal: f b ≤ a

full_name: isLUB_of_tendsto_atTop
code: theorem isLUB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f)
    (ha : Tendsto f atTop (𝓝 a)) : IsLUB (Set.range f) a 
goal: IsLUB (Set.range f) a

full_name: isGLB_of_tendsto_atBot
code: theorem isGLB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f)
    (ha : Tendsto f atBot (𝓝 a)) : IsGLB (Set.range f) a 
goal: IsGLB (Set.range f) a

full_name: isLUB_of_tendsto_atBot
code: theorem isLUB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f)
    (ha : Tendsto f atBot (𝓝 a)) : IsLUB (Set.range f) a 
goal: IsLUB (Set.range f) a

full_name: isGLB_of_tendsto_atTop
code: theorem isGLB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f)
    (ha : Tendsto f atTop (𝓝 a)) : IsGLB (Set.range f) a 
goal: IsGLB (Set.range f) a

full_name: iSup_eq_of_tendsto
code: theorem iSup_eq_of_tendsto {α β} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) :
    Tendsto f atTop (𝓝 a) → iSup f = a 
goal: Filter.Tendsto f Filter.atTop (𝓝 a) → iSup f = a

full_name: iInf_eq_of_tendsto
code: theorem iInf_eq_of_tendsto {α} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
    [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) :
    Tendsto f atTop (𝓝 a) → iInf f = a 
goal: Filter.Tendsto f Filter.atTop (𝓝 a) → iInf f = a

full_name: iSup_eq_iSup_subseq_of_monotone
code: theorem iSup_eq_iSup_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
    (hφ : Tendsto φ l atTop) : ⨆ i, f i = ⨆ i, f (φ i) 
goal: ⨆ i, f i = ⨆ i, f (φ i)

full_name: iSup_eq_iSup_subseq_of_antitone
code: theorem iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
    (hφ : Tendsto φ l atBot) : ⨆ i, f i = ⨆ i, f (φ i) 
goal: ⨆ i, f i = ⨆ i, f (φ i)

full_name: iInf_eq_iInf_subseq_of_monotone
code: theorem iInf_eq_iInf_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
    (hφ : Tendsto φ l atBot) : ⨅ i, f i = ⨅ i, f (φ i) 
goal: ⨅ i, f i = ⨅ i, f (φ i)

full_name: iInf_eq_iInf_subseq_of_antitone
code: theorem iInf_eq_iInf_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
    {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
    (hφ : Tendsto φ l atTop) : ⨅ i, f i = ⨅ i, f (φ i) 
goal: ⨅ i, f i = ⨅ i, f (φ i)