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
    @[simp]
    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 (x1 x2 : α) => x1 x2) 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 (x1 x2 : α) => x1 x2) 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 (x1 x2 : α) => x1 x2) 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 (x1 x2 : α) => x1 x2) b c) :
    a ⩿ c
    theorem wcovBy_congr_right {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun (x1 x2 : α) => x1 x2) 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 : α), 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 (x1 x2 : α) => x1 ⩿ x2
    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.range f).OrdConnected) :
    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.range f).OrdConnected) :
    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} [EquivLike E α β] [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 OrderEmbedding.wcovBy_of_apply {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : α ↪o β) {x : α} {y : α} (h : f x ⩿ f y) :
    x ⩿ y
    theorem OrderIso.map_wcovBy {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} {y : α} :
    f x ⩿ f y x ⩿ y
    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 : α), 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 b∃ (c : α), 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 b∃ (c : α), 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 denselyOrdered_iff_forall_not_covBy {α : Type u_1} [LT α] :
      DenselyOrdered α ∀ (a b : α), ¬a b
      @[deprecated denselyOrdered_iff_forall_not_covBy]
      theorem densely_ordered_iff_forall_not_covBy {α : Type u_1} [LT α] :
      DenselyOrdered α ∀ (a b : α), ¬a b

      Alias of denselyOrdered_iff_forall_not_covBy.

      @[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 WCovBy.covBy_or_le_and_le {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a ⩿ ba b a b b a

      Alias of the forward direction of wcovBy_iff_covBy_or_le_and_le.

      theorem AntisymmRel.trans_covBy {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun (x1 x2 : α) => x1 x2) 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 (x1 x2 : α) => x1 x2) 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 (x1 x2 : α) => x1 x2) b c) :
      a c
      theorem covBy_congr_right {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} (hab : AntisymmRel (fun (x1 x2 : α) => x1 x2) a b) :
      c a c b
      instance instIsNonstrictStrictOrderWCovByCovBy {α : Type u_1} [Preorder α] :
      IsNonstrictStrictOrder α (fun (x1 x2 : α) => x1 ⩿ x2) fun (x1 x2 : α) => x1 x2
      Equations
      • =
      instance CovBy.isIrrefl {α : Type u_1} [Preorder α] :
      IsIrrefl α fun (x1 x2 : α) => x1 x2
      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.range f).OrdConnected) :
      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.range f).OrdConnected) :
      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} [EquivLike E α β] [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 OrderEmbedding.covBy_of_apply {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : α ↪o β) {x : α} {y : α} (h : f x f y) :
      x y
      theorem OrderIso.map_covBy {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} {y : α} :
      f x f y x y
      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, b' < b, x < a', 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'.

      @[simp]
      theorem Set.wcovBy_insert {α : Type u_1} (x : α) (s : Set α) :
      s ⩿ insert x s
      @[simp]
      theorem Set.sdiff_singleton_wcovBy {α : Type u_1} (s : Set α) (a : α) :
      s \ {a} ⩿ s
      @[simp]
      theorem Set.covBy_insert {α : Type u_1} {s : Set α} {a : α} (ha : as) :
      s insert a s
      @[simp]
      theorem Set.sdiff_singleton_covBy {α : Type u_1} {s : Set α} {a : α} (ha : a s) :
      s \ {a} s
      theorem CovBy.exists_set_insert {α : Type u_1} {s : Set α} {t : Set α} (h : s t) :
      as, insert a s = t
      theorem CovBy.exists_set_sdiff_singleton {α : Type u_1} {s : Set α} {t : Set α} (h : s t) :
      at, t \ {a} = s
      theorem Set.covBy_iff_exists_insert {α : Type u_1} {s : Set α} {t : Set α} :
      s t as, insert a s = t
      theorem Set.covBy_iff_exists_sdiff_singleton {α : Type u_1} {s : Set α} {t : Set α} :
      s t at, t \ {a} = s
      theorem wcovBy_eq_reflGen_covBy {α : Type u_1} [PartialOrder α] :
      (fun (x1 x2 : α) => x1 ⩿ x2) = Relation.ReflGen fun (x1 x2 : α) => x1 x2
      theorem transGen_wcovBy_eq_reflTransGen_covBy {α : Type u_1} [PartialOrder α] :
      (Relation.TransGen fun (x1 x2 : α) => x1 ⩿ x2) = Relation.ReflTransGen fun (x1 x2 : α) => x1 x2
      theorem reflTransGen_wcovBy_eq_reflTransGen_covBy {α : Type u_1} [PartialOrder α] :
      (Relation.ReflTransGen fun (x1 x2 : α) => x1 ⩿ x2) = Relation.ReflTransGen fun (x1 x2 : α) => x1 x2
      @[simp]
      theorem Prod.swap_wcovBy_swap {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      x.swap ⩿ y.swap x ⩿ y
      @[simp]
      theorem Prod.swap_covBy_swap {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x : α × β} {y : α × β} :
      x.swap y.swap x y
      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.1 = y.1 x.2 = y.2
      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.1 ⩿ y.1
      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.2 ⩿ y.2
      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.1 ⩿ y.1 x.2 = y.2 x.2 ⩿ y.2 x.1 = y.1
      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.1 y.1 x.2 = y.2 x.2 y.2 x.1 = y.1
      @[simp]
      theorem WithTop.coe_wcovBy_coe {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a ⩿ b a ⩿ b
      @[simp]
      theorem WithTop.coe_covBy_coe {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a b a b
      @[simp]
      theorem WithTop.coe_covBy_top {α : Type u_1} [Preorder α] {a : α} :
      a IsMax a
      @[simp]
      theorem WithTop.coe_wcovBy_top {α : Type u_1} [Preorder α] {a : α} :
      a ⩿ IsMax a
      @[simp]
      theorem WithBot.coe_wcovBy_coe {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a ⩿ b a ⩿ b
      @[simp]
      theorem WithBot.coe_covBy_coe {α : Type u_1} [Preorder α] {a : α} {b : α} :
      a b a b
      @[simp]
      theorem WithBot.bot_covBy_coe {α : Type u_1} [Preorder α] {a : α} :
      a IsMin a
      @[simp]
      theorem WithBot.bot_wcovBy_coe {α : Type u_1} [Preorder α] {a : α} :
      ⩿ a IsMin a
      theorem exists_covBy_of_wellFoundedLT {α : Type u_1} [Preorder α] [wf : WellFoundedLT α] ⦃a : α (h : ¬IsMax a) :
      ∃ (a' : α), a a'
      theorem exists_covBy_of_wellFoundedGT {α : Type u_1} [Preorder α] [wf : WellFoundedGT α] ⦃a : α (h : ¬IsMin a) :
      ∃ (a' : α), a' a