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

source
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
source
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
source
theorem exists_eq_ciSup_of_not_isSuccPrelimit {ι : Sort 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
source
theorem exists_eq_ciInf_of_not_isPredPrelimit {ι : Sort 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
source
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
source
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
source
theorem IsLUB.exists_of_nonempty_of_not_isSuccPrelimit {ι : Sort 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
source
theorem IsGLB.exists_of_nonempty_of_not_isPredPrelimit {ι : Sort 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
source
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

- ConditionallyCompleteLinearOrder.toSuccOrder = { succ := fun (a : α) => if IsMax a then a else sInf {b : α | a < b}, le_succ := ⋯, max_of_succ_le := ⋯, succ_le_of_lt := ⋯ }

Instances For

source
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 : Sort 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.

source
theorem Order.IsSuccPrelimit.sSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccPrelimit x) :
sSup (Set.Iio x) = x
source
theorem Order.IsSuccPrelimit.iSup_Iio {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} (h : IsSuccPrelimit x) :
⨆ (a : ↑(Set.Iio x)), ↑a = x
source
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
source
theorem sSup_Iio_eq_self_iff_isSuccPrelimit {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {x : α} :
sSup (Set.Iio x) = x ↔ Order.IsSuccPrelimit x
source
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
source
theorem sSup_mem_of_not_isSuccPrelimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsSuccPrelimit (sSup s)) :
sSup s ∈ s
source
theorem sInf_mem_of_not_isPredPrelimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsPredPrelimit (sInf s)) :
sInf s ∈ s
source
theorem exists_eq_iSup_of_not_isSuccPrelimit {ι : Sort 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
source
theorem exists_eq_iInf_of_not_isPredPrelimit {ι : Sort 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