Documentation

Mathlib.Order.Interval.Set.OrdConnected

Order-connected sets #

We say that a set s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]]. If α is a DenselyOrdered ConditionallyCompleteLinearOrder with the OrderTopology, then this condition is equivalent to IsPreconnected s. If α is a LinearOrderedField, then this condition is also equivalent to Convex α s.

In this file we prove that intersection of a family of OrdConnected sets is OrdConnected and that all standard intervals are OrdConnected.

class Set.OrdConnected {α : Type u_1} [Preorder α] (s : Set α) :

We say that a set s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]]. If α is a DenselyOrdered ConditionallyCompleteLinearOrder with the OrderTopology, then this condition is equivalent to IsPreconnected s. If α is a LinearOrderedField, then this condition is also equivalent to Convex α s.

  • out' : ∀ ⦃x : α⦄, x s∀ ⦃y : α⦄, y sSet.Icc x y s

    s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]].

Instances
    theorem Set.OrdConnected.out' {α : Type u_1} :
    ∀ {inst : Preorder α} {s : Set α} [self : s.OrdConnected] ⦃x : α⦄, x s∀ ⦃y : α⦄, y sSet.Icc x y s

    s : Set α is OrdConnected if for all x y ∈ s it includes the interval [[x, y]].

    theorem Set.OrdConnected.out {α : Type u_1} [Preorder α] {s : Set α} (h : s.OrdConnected) ⦃x : α :
    x s∀ ⦃y : α⦄, y sSet.Icc x y s
    theorem Set.ordConnected_def {α : Type u_1} [Preorder α] {s : Set α} :
    s.OrdConnected ∀ ⦃x : α⦄, x s∀ ⦃y : α⦄, y sSet.Icc x y s
    theorem Set.ordConnected_iff {α : Type u_1} [Preorder α] {s : Set α} :
    s.OrdConnected xs, ys, x ySet.Icc x y s

    It suffices to prove [[x, y]] ⊆ s for x y ∈ s, x ≤ y.

    theorem Set.ordConnected_of_Ioo {α : Type u_3} [PartialOrder α] {s : Set α} (hs : xs, ys, x < ySet.Ioo x y s) :
    s.OrdConnected
    theorem Set.OrdConnected.preimage_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} {f : βα} (hs : s.OrdConnected) (hf : Monotone f) :
    (f ⁻¹' s).OrdConnected
    theorem Set.OrdConnected.preimage_anti {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} {f : βα} (hs : s.OrdConnected) (hf : Antitone f) :
    (f ⁻¹' s).OrdConnected
    theorem Set.Icc_subset {α : Type u_1} [Preorder α] (s : Set α) [hs : s.OrdConnected] {x : α} {y : α} (hx : x s) (hy : y s) :
    Set.Icc x y s
    theorem OrderEmbedding.image_Icc {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range e).OrdConnected) (x : α) (y : α) :
    e '' Set.Icc x y = Set.Icc (e x) (e y)
    theorem OrderEmbedding.image_Ico {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range e).OrdConnected) (x : α) (y : α) :
    e '' Set.Ico x y = Set.Ico (e x) (e y)
    theorem OrderEmbedding.image_Ioc {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range e).OrdConnected) (x : α) (y : α) :
    e '' Set.Ioc x y = Set.Ioc (e x) (e y)
    theorem OrderEmbedding.image_Ioo {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range e).OrdConnected) (x : α) (y : α) :
    e '' Set.Ioo x y = Set.Ioo (e x) (e y)
    @[simp]
    theorem Set.image_subtype_val_Icc {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x : s) (y : s) :
    Subtype.val '' Set.Icc x y = Set.Icc x y
    @[simp]
    theorem Set.image_subtype_val_Ico {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x : s) (y : s) :
    Subtype.val '' Set.Ico x y = Set.Ico x y
    @[simp]
    theorem Set.image_subtype_val_Ioc {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x : s) (y : s) :
    Subtype.val '' Set.Ioc x y = Set.Ioc x y
    @[simp]
    theorem Set.image_subtype_val_Ioo {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x : s) (y : s) :
    Subtype.val '' Set.Ioo x y = Set.Ioo x y
    theorem Set.OrdConnected.inter {α : Type u_1} [Preorder α] {s : Set α} {t : Set α} (hs : s.OrdConnected) (ht : t.OrdConnected) :
    (s t).OrdConnected
    instance Set.OrdConnected.inter' {α : Type u_1} [Preorder α] {s : Set α} {t : Set α} [s.OrdConnected] [t.OrdConnected] :
    (s t).OrdConnected
    Equations
    • =
    theorem Set.OrdConnected.dual {α : Type u_1} [Preorder α] {s : Set α} (hs : s.OrdConnected) :
    (OrderDual.ofDual ⁻¹' s).OrdConnected
    theorem Set.ordConnected_dual {α : Type u_1} [Preorder α] {s : Set α} :
    (OrderDual.ofDual ⁻¹' s).OrdConnected s.OrdConnected
    theorem Set.ordConnected_sInter {α : Type u_1} [Preorder α] {S : Set (Set α)} (hS : sS, s.OrdConnected) :
    (⋂₀ S).OrdConnected
    theorem Set.ordConnected_iInter {α : Type u_1} [Preorder α] {ι : Sort u_3} {s : ιSet α} (hs : ∀ (i : ι), (s i).OrdConnected) :
    (⋂ (i : ι), s i).OrdConnected
    instance Set.ordConnected_iInter' {α : Type u_1} [Preorder α] {ι : Sort u_3} {s : ιSet α} [∀ (i : ι), (s i).OrdConnected] :
    (⋂ (i : ι), s i).OrdConnected
    Equations
    • =
    theorem Set.ordConnected_biInter {α : Type u_1} [Preorder α] {ι : Sort u_3} {p : ιProp} {s : (i : ι) → p iSet α} (hs : ∀ (i : ι) (hi : p i), (s i hi).OrdConnected) :
    (⋂ (i : ι), ⋂ (hi : p i), s i hi).OrdConnected
    theorem Set.ordConnected_pi {ι : Type u_3} {α : ιType u_4} [(i : ι) → Preorder (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} (h : is, (t i).OrdConnected) :
    (s.pi t).OrdConnected
    instance Set.ordConnected_pi' {ι : Type u_3} {α : ιType u_4} [(i : ι) → Preorder (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} [h : ∀ (i : ι), (t i).OrdConnected] :
    (s.pi t).OrdConnected
    Equations
    • =
    theorem Set.ordConnected_Ici {α : Type u_1} [Preorder α] {a : α} :
    (Set.Ici a).OrdConnected
    theorem Set.ordConnected_Iic {α : Type u_1} [Preorder α] {a : α} :
    (Set.Iic a).OrdConnected
    theorem Set.ordConnected_Ioi {α : Type u_1} [Preorder α] {a : α} :
    (Set.Ioi a).OrdConnected
    theorem Set.ordConnected_Iio {α : Type u_1} [Preorder α] {a : α} :
    (Set.Iio a).OrdConnected
    theorem Set.ordConnected_Icc {α : Type u_1} [Preorder α] {a : α} {b : α} :
    (Set.Icc a b).OrdConnected
    theorem Set.ordConnected_Ico {α : Type u_1} [Preorder α] {a : α} {b : α} :
    (Set.Ico a b).OrdConnected
    theorem Set.ordConnected_Ioc {α : Type u_1} [Preorder α] {a : α} {b : α} :
    (Set.Ioc a b).OrdConnected
    theorem Set.ordConnected_Ioo {α : Type u_1} [Preorder α] {a : α} {b : α} :
    (Set.Ioo a b).OrdConnected
    theorem Set.ordConnected_singleton {α : Type u_3} [PartialOrder α] {a : α} :
    {a}.OrdConnected
    theorem Set.ordConnected_empty {α : Type u_1} [Preorder α] :
    .OrdConnected
    theorem Set.ordConnected_univ {α : Type u_1} [Preorder α] :
    Set.univ.OrdConnected
    instance Set.instDenselyOrdered {α : Type u_1} [Preorder α] [DenselyOrdered α] {s : Set α} [hs : s.OrdConnected] :

    In a dense order α, the subtype from an OrdConnected set is also densely ordered.

    Equations
    • =
    theorem Set.ordConnected_preimage {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {F : Type u_3} [FunLike F α β] [OrderHomClass F α β] (f : F) {s : Set β} [hs : s.OrdConnected] :
    (f ⁻¹' s).OrdConnected
    theorem Set.ordConnected_image {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {E : Type u_3} [EquivLike E α β] [OrderIsoClass E α β] (e : E) {s : Set α} [hs : s.OrdConnected] :
    (e '' s).OrdConnected
    theorem Set.ordConnected_range {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {E : Type u_3} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
    (Set.range e).OrdConnected
    @[simp]
    theorem Set.dual_ordConnected_iff {α : Type u_1} [Preorder α] {s : Set α} :
    (OrderDual.ofDual ⁻¹' s).OrdConnected s.OrdConnected
    theorem Set.dual_ordConnected {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] :
    (OrderDual.ofDual ⁻¹' s).OrdConnected
    theorem IsAntichain.ordConnected {α : Type u_1} [PartialOrder α] {s : Set α} (hs : IsAntichain (fun (x1 x2 : α) => x1 x2) s) :
    s.OrdConnected
    theorem Set.ordConnected_inter_Icc_of_subset {α : Type u_1} [PartialOrder α] {s : Set α} {x : α} {y : α} (h : Set.Ioo x y s) :
    (s Set.Icc x y).OrdConnected
    theorem Set.ordConnected_inter_Icc_iff {α : Type u_1} [PartialOrder α] {s : Set α} {x : α} {y : α} (hx : x s) (hy : y s) :
    (s Set.Icc x y).OrdConnected Set.Ioo x y s
    theorem Set.not_ordConnected_inter_Icc_iff {α : Type u_1} [PartialOrder α] {s : Set α} {x : α} {y : α} (hx : x s) (hy : y s) :
    ¬(s Set.Icc x y).OrdConnected zs, z Set.Ioo x y
    theorem Set.ordConnected_uIcc {α : Type u_1} [LinearOrder α] {a : α} {b : α} :
    (Set.uIcc a b).OrdConnected
    theorem Set.ordConnected_uIoc {α : Type u_1} [LinearOrder α] {a : α} {b : α} :
    (Ι a b).OrdConnected
    theorem Set.OrdConnected.uIcc_subset {α : Type u_1} [LinearOrder α] {s : Set α} (hs : s.OrdConnected) ⦃x : α (hx : x s) ⦃y : α (hy : y s) :
    theorem Set.OrdConnected.uIoc_subset {α : Type u_1} [LinearOrder α] {s : Set α} (hs : s.OrdConnected) ⦃x : α (hx : x s) ⦃y : α (hy : y s) :
    Ι x y s
    theorem Set.ordConnected_iff_uIcc_subset {α : Type u_1} [LinearOrder α] {s : Set α} :
    s.OrdConnected ∀ ⦃x : α⦄, x s∀ ⦃y : α⦄, y sSet.uIcc x y s
    theorem Set.ordConnected_of_uIcc_subset_left {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (h : ys, Set.uIcc x y s) :
    s.OrdConnected
    theorem Set.ordConnected_iff_uIcc_subset_left {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (hx : x s) :
    s.OrdConnected ∀ ⦃y : α⦄, y sSet.uIcc x y s
    theorem Set.ordConnected_iff_uIcc_subset_right {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (hx : x s) :
    s.OrdConnected ∀ ⦃y : α⦄, y sSet.uIcc y x s