Documentation

Mathlib.Order.Cover

The covering relation #

This file defines the covering relation in an order. b is said to cover a if a < b and there is no element in between. We say that b weakly covers a if a ≤ b and there is no element between a and b. In a partial order this is equivalent to a ⋖ b ∨ a = b, in a preorder this is equivalent to a ⋖ b ∨ (a ≤ b ∧ b ≤ a)

Notation #

def Wcovby {α : Type u_1} [Preorder α] (a : α) (b : α) :

Wcovby a b means that a = b or b covers a. This means that a ≤ b and there is no element in between.

Equations
Instances For
    theorem Wcovby.le {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a ⩿ b) :
    a b
    theorem Wcovby.refl {α : Type u_1} [Preorder α] (a : α) :
    a ⩿ a
    theorem Wcovby.rfl {α : Type u_1} [Preorder α] {a : α} :
    a ⩿ a
    theorem Eq.wcovby {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a = b) :
    a ⩿ b
    theorem wcovby_of_le_of_le {α : Type u_1} [Preorder α] {a : α} {b : α} (h1 : a b) (h2 : b a) :
    a ⩿ b
    theorem LE.le.wcovby_of_le {α : Type u_1} [Preorder α] {a : α} {b : α} (h1 : a b) (h2 : b a) :
    a ⩿ b

    Alias of wcovby_of_le_of_le.

    theorem AntisymmRel.wcovby {α : Type u_1} [Preorder α] {a : α} {b : α} (h : AntisymmRel (fun x x_1 => x x_1) a b) :
    a ⩿ b
    theorem Wcovby.wcovby_iff_le {α : Type u_1} [Preorder α] {a : α} {b : α} (hab : a ⩿ b) :
    b ⩿ a b a
    theorem wcovby_of_eq_or_eq {α : Type u_1} [Preorder α] {a : α} {b : α} (hab : a b) (h : ∀ (c : α), a cc bc = a c = b) :
    a ⩿ b
    theorem AntisymmRel.trans_wcovby {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) (hbc : b ⩿ c) :
    a ⩿ c
    theorem wcovby_congr_left {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) :
    a ⩿ c b ⩿ c
    theorem Wcovby.trans_antisymm_rel {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : a ⩿ b) (hbc : AntisymmRel (fun x x_1 => x x_1) b c) :
    a ⩿ c
    theorem wcovby_congr_right {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) :
    c ⩿ a c ⩿ b
    theorem not_wcovby_iff {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
    ¬a ⩿ b c, a < c c < b

    If a ≤ b, then b does not cover a iff there's an element in between.

    instance Wcovby.isRefl {α : Type u_1} [Preorder α] :
    IsRefl α fun x x_1 => x ⩿ x_1
    Equations
    theorem Wcovby.Ioo_eq {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a ⩿ b) :
    theorem wcovby_iff_Ioo_eq {α : Type u_1} [Preorder α] {a : α} {b : α} :
    a ⩿ b a b Set.Ioo a b =
    theorem Wcovby.of_le_of_le {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hac : a ⩿ c) (hab : a b) (hbc : b c) :
    b ⩿ c
    theorem Wcovby.of_le_of_le' {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hac : a ⩿ c) (hab : a b) (hbc : b c) :
    a ⩿ b
    theorem Wcovby.of_image {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (h : f a ⩿ f b) :
    a ⩿ b
    theorem Wcovby.image {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (hab : a ⩿ b) (h : Set.OrdConnected (Set.range f)) :
    f a ⩿ f b
    theorem Set.OrdConnected.apply_wcovby_apply_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (h : Set.OrdConnected (Set.range f)) :
    f a ⩿ f b a ⩿ b
    @[simp]
    theorem apply_wcovby_apply_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} {E : Type u_3} [OrderIsoClass E α β] (e : E) :
    e a ⩿ e b a ⩿ b
    @[simp]
    theorem toDual_wcovby_toDual_iff {α : Type u_1} [Preorder α] {a : α} {b : α} :
    OrderDual.toDual b ⩿ OrderDual.toDual a a ⩿ b
    @[simp]
    theorem ofDual_wcovby_ofDual_iff {α : Type u_1} [Preorder α] {a : αᵒᵈ} {b : αᵒᵈ} :
    OrderDual.ofDual a ⩿ OrderDual.ofDual b b ⩿ a
    theorem Wcovby.toDual {α : Type u_1} [Preorder α] {a : α} {b : α} :
    a ⩿ bOrderDual.toDual b ⩿ OrderDual.toDual a

    Alias of the reverse direction of toDual_wcovby_toDual_iff.

    theorem Wcovby.ofDual {α : Type u_1} [Preorder α] {a : αᵒᵈ} {b : αᵒᵈ} :
    b ⩿ aOrderDual.ofDual a ⩿ OrderDual.ofDual b

    Alias of the reverse direction of ofDual_wcovby_ofDual_iff.

    theorem Wcovby.eq_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} {c : α} (h : a ⩿ b) (h2 : a c) (h3 : c b) :
    c = a c = b
    theorem wcovby_iff_le_and_eq_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
    a ⩿ b a b ∀ (c : α), a cc bc = a c = b

    An iff version of Wcovby.eq_or_eq and wcovby_of_eq_or_eq.

    theorem Wcovby.le_and_le_iff {α : Type u_1} [PartialOrder α] {a : α} {b : α} {c : α} (h : a ⩿ b) :
    a c c b c = a c = b
    theorem Wcovby.Icc_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a ⩿ b) :
    Set.Icc a b = {a, b}
    theorem Wcovby.Ico_subset {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a ⩿ b) :
    Set.Ico a b {a}
    theorem Wcovby.Ioc_subset {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a ⩿ b) :
    Set.Ioc a b {b}
    theorem Wcovby.sup_eq {α : Type u_1} [SemilatticeSup α] {a : α} {b : α} {c : α} (hac : a ⩿ c) (hbc : b ⩿ c) (hab : a b) :
    a b = c
    theorem Wcovby.inf_eq {α : Type u_1} [SemilatticeInf α] {a : α} {b : α} {c : α} (hca : c ⩿ a) (hcb : c ⩿ b) (hab : a b) :
    a b = c
    def Covby {α : Type u_1} [LT α] (a : α) (b : α) :

    Covby a b means that b covers a: a < b and there is no element in between.

    Equations
    Instances For
      theorem Covby.lt {α : Type u_1} [LT α] {a : α} {b : α} (h : a b) :
      a < b
      theorem not_covby_iff {α : Type u_1} [LT α] {a : α} {b : α} (h : a < b) :
      ¬a b c, a < c c < b

      If a < b, then b does not cover a iff there's an element in between.

      theorem exists_lt_lt_of_not_covby {α : Type u_1} [LT α] {a : α} {b : α} (h : a < b) :
      ¬a bc, a < c c < b

      Alias of the forward direction of not_covby_iff.


      If a < b, then b does not cover a iff there's an element in between.

      theorem LT.lt.exists_lt_lt {α : Type u_1} [LT α] {a : α} {b : α} (h : a < b) :
      ¬a bc, a < c c < b

      Alias of the forward direction of not_covby_iff.


      Alias of the forward direction of not_covby_iff.


      If a < b, then b does not cover a iff there's an element in between.

      theorem not_covby {α : Type u_1} [LT α] {a : α} {b : α} [DenselyOrdered α] :
      ¬a b

      In a dense order, nothing covers anything.

      theorem densely_ordered_iff_forall_not_covby {α : Type u_1} [LT α] :
      DenselyOrdered α ∀ (a b : α), ¬a b
      @[simp]
      theorem toDual_covby_toDual_iff {α : Type u_1} [LT α] {a : α} {b : α} :
      OrderDual.toDual b OrderDual.toDual a a b
      @[simp]
      theorem ofDual_covby_ofDual_iff {α : Type u_1} [LT α] {a : αᵒᵈ} {b : αᵒᵈ} :
      OrderDual.ofDual a OrderDual.ofDual b b a
      theorem Covby.toDual {α : Type u_1} [LT α] {a : α} {b : α} :
      a bOrderDual.toDual b OrderDual.toDual a

      Alias of the reverse direction of toDual_covby_toDual_iff.

      theorem Covby.ofDual {α : Type u_1} [LT α] {a : αᵒᵈ} {b : αᵒᵈ} :
      b aOrderDual.ofDual a OrderDual.ofDual b

      Alias of the reverse direction of ofDual_covby_ofDual_iff.

      theorem Covby.le {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
      a b
      theorem Covby.ne {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
      a b
      theorem Covby.ne' {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
      b a
      theorem Covby.wcovby {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
      a ⩿ b
      theorem Wcovby.covby_of_not_le {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a ⩿ b) (h2 : ¬b a) :
      a b
      theorem Wcovby.covby_of_lt {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a ⩿ b) (h2 : a < b) :
      a b
      theorem Covby.of_le_of_lt {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hac : a c) (hab : a b) (hbc : b < c) :
      b c
      theorem Covby.of_lt_of_le {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hac : a c) (hab : a < b) (hbc : b c) :
      a b
      theorem not_covby_of_lt_of_lt {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (h₁ : a < b) (h₂ : b < c) :
      ¬a c
      theorem covby_iff_wcovby_and_lt {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a b a ⩿ b a < b
      theorem covby_iff_wcovby_and_not_le {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a b a ⩿ b ¬b a
      theorem wcovby_iff_covby_or_le_and_le {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a ⩿ b a b a b b a
      theorem AntisymmRel.trans_covby {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) (hbc : b c) :
      a c
      theorem covby_congr_left {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) :
      a c b c
      theorem Covby.trans_antisymmRel {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : a b) (hbc : AntisymmRel (fun x x_1 => x x_1) b c) :
      a c
      theorem covby_congr_right {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun x x_1 => x x_1) a b) :
      c a c b
      instance Covby.isIrrefl {α : Type u_1} [Preorder α] :
      IsIrrefl α fun x x_1 => x x_1
      Equations
      theorem Covby.Ioo_eq {α : Type u_1} [Preorder α] {a : α} {b : α} (h : a b) :
      theorem covby_iff_Ioo_eq {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a b a < b Set.Ioo a b =
      theorem Covby.of_image {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (h : f a f b) :
      a b
      theorem Covby.image {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (hab : a b) (h : Set.OrdConnected (Set.range f)) :
      f a f b
      theorem Set.OrdConnected.apply_covby_apply_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} (f : α ↪o β) (h : Set.OrdConnected (Set.range f)) :
      f a f b a b
      @[simp]
      theorem apply_covby_apply_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : α} {E : Type u_3} [OrderIsoClass E α β] (e : E) :
      e a e b a b
      theorem covby_of_eq_or_eq {α : Type u_1} [Preorder α] {a : α} {b : α} (hab : a < b) (h : ∀ (c : α), a cc bc = a c = b) :
      a b
      theorem Wcovby.covby_of_ne {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a ⩿ b) (h2 : a b) :
      a b
      theorem covby_iff_wcovby_and_ne {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a b a ⩿ b a b
      theorem wcovby_iff_covby_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a ⩿ b a b a = b
      theorem wcovby_iff_eq_or_covby {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a ⩿ b a = b a b
      theorem Wcovby.covby_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a ⩿ ba b a = b

      Alias of the forward direction of wcovby_iff_covby_or_eq.

      theorem Wcovby.eq_or_covby {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a ⩿ ba = b a b

      Alias of the forward direction of wcovby_iff_eq_or_covby.

      theorem Covby.eq_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} {c : α} (h : a b) (h2 : a c) (h3 : c b) :
      c = a c = b
      theorem covby_iff_lt_and_eq_or_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} :
      a b a < b ∀ (c : α), a cc bc = a c = b

      An iff version of Covby.eq_or_eq and covby_of_eq_or_eq.

      theorem Covby.Ico_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a b) :
      Set.Ico a b = {a}
      theorem Covby.Ioc_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a b) :
      Set.Ioc a b = {b}
      theorem Covby.Icc_eq {α : Type u_1} [PartialOrder α] {a : α} {b : α} (h : a b) :
      Set.Icc a b = {a, b}
      theorem Covby.Ioi_eq {α : Type u_1} [LinearOrder α] {a : α} {b : α} (h : a b) :
      theorem Covby.Iio_eq {α : Type u_1} [LinearOrder α] {a : α} {b : α} (h : a b) :
      theorem Wcovby.le_of_lt {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (hab : a ⩿ b) (hcb : c < b) :
      c a
      theorem Wcovby.ge_of_gt {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (hab : a ⩿ b) (hac : a < c) :
      b c
      theorem Covby.le_of_lt {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (hab : a b) :
      c < bc a
      theorem Covby.ge_of_gt {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (hab : a b) :
      a < cb c
      theorem Covby.unique_left {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (ha : a c) (hb : b c) :
      a = b
      theorem Covby.unique_right {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} (hb : a b) (hc : a c) :
      b = c
      theorem Covby.eq_of_between {α : Type u_1} [LinearOrder α] {a : α} {b : α} {c : α} {x : α} (hab : a b) (hbc : b c) (hax : a < x) (hxc : x < c) :
      x = b

      If a, b, c are consecutive and a < x < c then x = b.

      theorem LT.lt.exists_disjoint_Iio_Ioi {α : Type u_1} [LinearOrder α] {a : α} {b : α} (h : a < b) :
      a', a' > a b', b' < b ∀ (x : α), x < a'∀ (y : α), y > b'x < y

      If a < b then there exist a' > a and b' < b such that Set.Iio a' is strictly to the left of Set.Ioi b'.

      theorem Set.wcovby_insert {α : Type u_1} (x : α) (s : Set α) :
      s ⩿ insert x s
      theorem Set.covby_insert {α : Type u_1} {x : α} {s : Set α} (hx : ¬x s) :
      s insert x s
      theorem wcovby_eq_reflGen_covby {α : Type u_1} [PartialOrder α] :
      (fun x x_1 => x ⩿ x_1) = Relation.ReflGen fun x x_1 => x x_1
      theorem transGen_wcovby_eq_reflTransGen_covby {α : Type u_1} [PartialOrder α] :
      (Relation.TransGen fun x x_1 => x ⩿ x_1) = Relation.ReflTransGen fun x x_1 => x x_1
      @[simp]
      theorem Prod.swap_wcovby_swap {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      @[simp]
      theorem Prod.swap_covby_swap {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      theorem Prod.fst_eq_or_snd_eq_of_wcovby {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      x ⩿ yx.fst = y.fst x.snd = y.snd
      theorem Wcovby.fst {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} (h : x ⩿ y) :
      x.fst ⩿ y.fst
      theorem Wcovby.snd {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} (h : x ⩿ y) :
      x.snd ⩿ y.snd
      theorem Prod.mk_wcovby_mk_iff_left {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ : α} {a₂ : α} {b : β} :
      (a₁, b) ⩿ (a₂, b) a₁ ⩿ a₂
      theorem Prod.mk_wcovby_mk_iff_right {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a : α} {b₁ : β} {b₂ : β} :
      (a, b₁) ⩿ (a, b₂) b₁ ⩿ b₂
      theorem Prod.mk_covby_mk_iff_left {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ : α} {a₂ : α} {b : β} :
      (a₁, b) (a₂, b) a₁ a₂
      theorem Prod.mk_covby_mk_iff_right {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a : α} {b₁ : β} {b₂ : β} :
      (a, b₁) (a, b₂) b₁ b₂
      theorem Prod.mk_wcovby_mk_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} :
      (a₁, b₁) ⩿ (a₂, b₂) a₁ ⩿ a₂ b₁ = b₂ b₁ ⩿ b₂ a₁ = a₂
      theorem Prod.mk_covby_mk_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} :
      (a₁, b₁) (a₂, b₂) a₁ a₂ b₁ = b₂ b₁ b₂ a₁ = a₂
      theorem Prod.wcovby_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      x ⩿ y x.fst ⩿ y.fst x.snd = y.snd x.snd ⩿ y.snd x.fst = y.fst
      theorem Prod.covby_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      x y x.fst y.fst x.snd = y.snd x.snd y.snd x.fst = y.fst