Documentation

Mathlib.Order.SuccPred.CompleteLinearOrder

Relation between IsSuccPrelimit and iSup in (conditionally) complete linear orders. #

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theorem csSup_mem_of_not_isSuccPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬Order.IsSuccPrelimit (sSup s)) :
sSup s ∈ s
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theorem csInf_mem_of_not_isPredPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬Order.IsPredPrelimit (sInf s)) :
sInf s ∈ s
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theorem exists_eq_ciSup_of_not_isSuccPrelimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddAbove (Set.range f)) (hf' : ¬Order.IsSuccPrelimit (⨆ (i : ι), f i)) :
∃ (i : ι), f i = ⨆ (i : ι), f i
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theorem exists_eq_ciInf_of_not_isPredPrelimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddBelow (Set.range f)) (hf' : ¬Order.IsPredPrelimit (⨅ (i : ι), f i)) :
∃ (i : ι), f i = ⨅ (i : ι), f i
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theorem IsLUB.mem_of_nonempty_of_not_isSuccPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬Order.IsSuccPrelimit x) :
x ∈ s
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theorem IsGLB.mem_of_nonempty_of_not_isPredPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬Order.IsPredPrelimit x) :
x ∈ s
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theorem IsLUB.exists_of_nonempty_of_not_isSuccPrelimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} {x : α} (hf : IsLUB (Set.range f) x) (hx : ¬Order.IsSuccPrelimit x) :
∃ (i : ι), f i = x
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theorem IsGLB.exists_of_nonempty_of_not_isPredPrelimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} {x : α} (hf : IsGLB (Set.range f) x) (hx : ¬Order.IsPredPrelimit x) :
∃ (i : ι), f i = x
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noncomputable def ConditionallyCompleteLinearOrder.toSuccOrder {α : Type u_2} [ConditionallyCompleteLinearOrder α] [WellFoundedLT α] :
SuccOrder α

Every conditionally complete linear order with well-founded < is a successor order, by setting the successor of an element to be the infimum of all larger elements.

Equations
Instances For
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    theorem csSup_mem_of_not_isSuccPrelimit' {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hbdd : BddAbove s) (hlim : ¬Order.IsSuccPrelimit (sSup s)) :
    sSup s ∈ s

    See csSup_mem_of_not_isSuccPrelimit for the ConditionallyCompleteLinearOrder version.

    source
    theorem exists_eq_ciSup_of_not_isSuccPrelimit' {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} (hf : BddAbove (Set.range f)) (hf' : ¬Order.IsSuccPrelimit (⨆ (i : ι), f i)) :
    ∃ (i : ι), f i = ⨆ (i : ι), f i

    See exists_eq_ciSup_of_not_isSuccPrelimit for the ConditionallyCompleteLinearOrder version.

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    @[deprecated IsLUB.mem_of_not_isSuccPrelimit (since := "2025-01-05")]
    theorem IsLUB.exists_of_not_isSuccPrelimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {x : α} (hf : IsLUB (Set.range f) x) (hx : ¬Order.IsSuccPrelimit x) :
    ∃ (i : ι), f i = x
    source
    theorem Order.IsSuccPrelimit.sSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccPrelimit x) :
    sSup (Set.Iio x) = x
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    theorem Order.IsSuccPrelimit.iSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccPrelimit x) :
    ⨆ (a : ↑(Set.Iio x)), ↑a = x
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    theorem Order.IsSuccLimit.sSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccLimit x) :
    sSup (Set.Iio x) = x
    source
    theorem Order.IsSuccLimit.iSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccLimit x) :
    ⨆ (a : ↑(Set.Iio x)), ↑a = x
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    theorem sSup_Iio_eq_self_iff_isSuccPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} :
    sSup (Set.Iio x) = x ↔ Order.IsSuccPrelimit x
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    theorem iSup_Iio_eq_self_iff_isSuccPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} :
    ⨆ (a : ↑(Set.Iio x)), ↑a = x ↔ Order.IsSuccPrelimit x
    source
    theorem iSup_succ {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] [SuccOrder α] (x : α) :
    ⨆ (a : ↑(Set.Iio x)), Order.succ ↑a = x
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    theorem sSup_mem_of_not_isSuccPrelimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsSuccPrelimit (sSup s)) :
    sSup s ∈ s
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    theorem sInf_mem_of_not_isPredPrelimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsPredPrelimit (sInf s)) :
    sInf s ∈ s
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    theorem exists_eq_iSup_of_not_isSuccPrelimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsSuccPrelimit (⨆ (i : ι), f i)) :
    ∃ (i : ι), f i = ⨆ (i : ι), f i
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    theorem exists_eq_iInf_of_not_isPredPrelimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsPredPrelimit (⨅ (i : ι), f i)) :
    ∃ (i : ι), f i = ⨅ (i : ι), f i
    source
    @[deprecated IsGLB.mem_of_not_isPredLimit (since := "2025-01-05")]
    theorem IsGLB.exists_of_not_isPredPrelimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} {x : α} (hf : IsGLB (Set.range f) x) (hx : ¬Order.IsPredPrelimit x) :
    ∃ (i : ι), f i = x