Documentation

Mathlib.Order.SuccPred.CompleteLinearOrder

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

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theorem csSup_mem_of_not_isSuccLimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬Order.IsSuccLimit (sSup s)) :
sSup s ∈ s
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theorem csInf_mem_of_not_isPredLimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬Order.IsPredLimit (sInf s)) :
sInf s ∈ s
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theorem exists_eq_ciSup_of_not_isSuccLimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddAbove (Set.range f)) (hf' : ¬Order.IsSuccLimit (⨆ (i : ι), f i)) :
∃ (i : ι), f i = ⨆ (i : ι), f i
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theorem exists_eq_ciInf_of_not_isPredLimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddBelow (Set.range f)) (hf' : ¬Order.IsPredLimit (⨅ (i : ι), f i)) :
∃ (i : ι), f i = ⨅ (i : ι), f i
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theorem IsLUB.mem_of_nonempty_of_not_isSuccLimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬Order.IsSuccLimit x) :
x ∈ s
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theorem IsGLB.mem_of_nonempty_of_not_isPredLimit {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬Order.IsPredLimit x) :
x ∈ s
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theorem IsLUB.exists_of_nonempty_of_not_isSuccLimit {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.IsSuccLimit x) :
∃ (i : ι), f i = x
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theorem IsGLB.exists_of_nonempty_of_not_isPredLimit {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.IsPredLimit 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
  • 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
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    theorem csSup_mem_of_not_isSuccLimit' {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hbdd : BddAbove s) (hlim : ¬Order.IsSuccLimit (sSup s)) :
    sSup s ∈ s

    See csSup_mem_of_not_isSuccLimit for the ConditionallyCompleteLinearOrder version.

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

    See exists_eq_ciSup_of_not_isSuccLimit for the ConditionallyCompleteLinearOrder version.

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    theorem IsLUB.mem_of_not_isSuccLimit {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {x : α} (hs : IsLUB s x) (hx : ¬Order.IsSuccLimit x) :
    x ∈ s
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    theorem IsLUB.exists_of_not_isSuccLimit {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {x : α} (hf : IsLUB (Set.range f) x) (hx : ¬Order.IsSuccLimit x) :
    ∃ (i : ι), f i = x
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    theorem sSup_mem_of_not_isSuccLimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsSuccLimit (sSup s)) :
    sSup s ∈ s
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    theorem sInf_mem_of_not_isPredLimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsPredLimit (sInf s)) :
    sInf s ∈ s
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    theorem exists_eq_iSup_of_not_isSuccLimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsSuccLimit (⨆ (i : ι), f i)) :
    ∃ (i : ι), f i = ⨆ (i : ι), f i
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    theorem exists_eq_iInf_of_not_isPredLimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsPredLimit (⨅ (i : ι), f i)) :
    ∃ (i : ι), f i = ⨅ (i : ι), f i
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    theorem IsGLB.mem_of_not_isPredLimit {α : Type u_2} [CompleteLinearOrder α] {s : Set α} {x : α} (hs : IsGLB s x) (hx : ¬Order.IsPredLimit x) :
    x ∈ s
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    theorem IsGLB.exists_of_not_isPredLimit {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} {x : α} (hf : IsGLB (Set.range f) x) (hx : ¬Order.IsPredLimit x) :
    ∃ (i : ι), f i = x