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

• a ⋖ b means that b covers a.
• a ⩿ b means that b weakly covers a.

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

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

Equations

• (a ⩿ b) = (a ≤ b ∧ ∀ ⦃c : α⦄, a < c → ¬c < b)

def «term_⩿_» : Lean.TrailingParserDescr

Notation for WCovBy a b.

Equations

• «term_⩿_» = Lean.ParserDescr.trailingNode `«term_⩿_» 50 50 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⩿ ") (Lean.ParserDescr.cat `term 51))

theorem WCovBy.le {α : Type u_1} [Preorder α] {a 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 b : α} (h : a = b) : a ⩿ b

theorem wcovBy_of_le_of_le {α : Type u_1} [Preorder α] {a b : α} (h1 : a ≤ b) (h2 : b ≤ a) : a ⩿ b

theorem LE.le.wcovBy_of_le {α : Type u_1} [Preorder α] {a 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 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 b : α} (hab : a ⩿ b) : b ⩿ a ↔ b ≤ a

theorem wcovBy_of_eq_or_eq {α : Type u_1} [Preorder α] {a b : α} (hab : a ≤ b) (h : ∀ (c : α), a ≤ c → c ≤ b → c = a ∨ c = b) : a ⩿ b

theorem AntisymmRel.trans_wcovBy {α : Type u_1} [Preorder α] {a b 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 b 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 b 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 b 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 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

theorem WCovBy.Ioo_eq {α : Type u_1} [Preorder α] {a b : α} (h : a ⩿ b) : Set.Ioo a b = ∅

theorem wcovBy_iff_Ioo_eq {α : Type u_1} [Preorder α] {a b : α} : a ⩿ b ↔ a ≤ b ∧ Set.Ioo a b = ∅

theorem WCovBy.of_le_of_le {α : Type u_1} [Preorder α] {a b 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 b 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 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 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 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 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 b : α} : OrderDual.toDual b ⩿ OrderDual.toDual a ↔ a ⩿ b

@[simp]

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

theorem WCovBy.toDual {α : Type u_1} [Preorder α] {a b : α} : a ⩿ b → OrderDual.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 b : αᵒᵈ} : b ⩿ a → OrderDual.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 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 y : α} : f x ⩿ f y ↔ x ⩿ y

theorem WCovBy.eq_or_eq {α : Type u_1} [PartialOrder α] {a b 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 b : α} : a ⩿ b ↔ a ≤ b ∧ ∀ (c : α), a ≤ c → c ≤ b → c = 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 b c : α} (h : a ⩿ b) : a ≤ c ∧ c ≤ b ↔ c = a ∨ c = b

theorem WCovBy.Icc_eq {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⩿ b) : Set.Icc a b = {a, b}

theorem WCovBy.Ico_subset {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⩿ b) : Set.Ico a b ⊆ {a}

theorem WCovBy.Ioc_subset {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⩿ b) : Set.Ioc a b ⊆ {b}

theorem WCovBy.sup_eq {α : Type u_1} [SemilatticeSup α] {a b 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 b c : α} (hca : c ⩿ a) (hcb : c ⩿ b) (hab : a ≠ b) : a ⊓ b = c

def CovBy {α : Type u_1} [LT α] (a b : α) : Prop

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

Equations

• (a ⋖ b) = (a < b ∧ ∀ ⦃c : α⦄, a < c → ¬c < b)

def «term_⋖_» : Lean.TrailingParserDescr

Notation for CovBy a b.

Equations

• «term_⋖_» = Lean.ParserDescr.trailingNode `«term_⋖_» 50 50 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⋖ ") (Lean.ParserDescr.cat `term 51))

theorem CovBy.lt {α : Type u_1} [LT α] {a b : α} (h : a ⋖ b) : a < b

theorem not_covBy_iff {α : Type u_1} [LT α] {a 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 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 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 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 b : α} : OrderDual.toDual b ⋖ OrderDual.toDual a ↔ a ⋖ b

@[simp]

theorem ofDual_covBy_ofDual_iff {α : Type u_1} [LT α] {a b : αᵒᵈ} : OrderDual.ofDual a ⋖ OrderDual.ofDual b ↔ b ⋖ a

theorem CovBy.toDual {α : Type u_1} [LT α] {a b : α} : a ⋖ b → OrderDual.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 b : αᵒᵈ} : b ⋖ a → OrderDual.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 b : α} (h : a ⋖ b) : a ≤ b

theorem CovBy.ne {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b) : a ≠ b

theorem CovBy.ne' {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b) : b ≠ a

theorem CovBy.wcovBy {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b) : a ⩿ b

theorem WCovBy.covBy_of_not_le {α : Type u_1} [Preorder α] {a b : α} (h : a ⩿ b) (h2 : ¬b ≤ a) : a ⋖ b

theorem WCovBy.covBy_of_lt {α : Type u_1} [Preorder α] {a b : α} (h : a ⩿ b) (h2 : a < b) : a ⋖ b

theorem CovBy.of_le_of_lt {α : Type u_1} [Preorder α] {a b 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 b 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 b c : α} (h₁ : a < b) (h₂ : b < c) : ¬a ⋖ c

theorem covBy_iff_wcovBy_and_lt {α : Type u_1} [Preorder α] {a b : α} : a ⋖ b ↔ a ⩿ b ∧ a < b

theorem covBy_iff_wcovBy_and_not_le {α : Type u_1} [Preorder α] {a b : α} : a ⋖ b ↔ a ⩿ b ∧ ¬b ≤ a

theorem wcovBy_iff_covBy_or_le_and_le {α : Type u_1} [Preorder α] {a b : α} : a ⩿ b ↔ a ⋖ b ∨ a ≤ b ∧ b ≤ a

theorem WCovBy.covBy_or_le_and_le {α : Type u_1} [Preorder α] {a b : α} : a ⩿ b → a ⋖ 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 b 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 b 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 b 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 b 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

instance CovBy.isIrrefl {α : Type u_1} [Preorder α] : IsIrrefl α fun (x1 x2 : α) => x1 ⋖ x2

theorem CovBy.Ioo_eq {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b) : Set.Ioo a b = ∅

theorem covBy_iff_Ioo_eq {α : Type u_1} [Preorder α] {a 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 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 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 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 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 b : α} (hab : a < b) (h : ∀ (c : α), a ≤ c → c ≤ b → c = 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 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 y : α} : f x ⋖ f y ↔ x ⋖ y

theorem WCovBy.covBy_of_ne {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⩿ b) (h2 : a ≠ b) : a ⋖ b

theorem covBy_iff_wcovBy_and_ne {α : Type u_1} [PartialOrder α] {a b : α} : a ⋖ b ↔ a ⩿ b ∧ a ≠ b

theorem wcovBy_iff_covBy_or_eq {α : Type u_1} [PartialOrder α] {a b : α} : a ⩿ b ↔ a ⋖ b ∨ a = b

theorem wcovBy_iff_eq_or_covBy {α : Type u_1} [PartialOrder α] {a b : α} : a ⩿ b ↔ a = b ∨ a ⋖ b

theorem WCovBy.covBy_or_eq {α : Type u_1} [PartialOrder α] {a b : α} : a ⩿ b → a ⋖ 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 b : α} : a ⩿ b → a = 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 b 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 b : α} : a ⋖ b ↔ a < b ∧ ∀ (c : α), a ≤ c → c ≤ b → c = 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 b : α} (h : a ⋖ b) : Set.Ico a b = {a}

theorem CovBy.Ioc_eq {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⋖ b) : Set.Ioc a b = {b}

theorem CovBy.Icc_eq {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⋖ b) : Set.Icc a b = {a, b}

theorem CovBy.Ioi_eq {α : Type u_1} [LinearOrder α] {a b : α} (h : a ⋖ b) : Set.Ioi a = Set.Ici b

theorem CovBy.Iio_eq {α : Type u_1} [LinearOrder α] {a b : α} (h : a ⋖ b) : Set.Iio b = Set.Iic a

theorem WCovBy.le_of_lt {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⩿ b) (hcb : c < b) : c ≤ a

theorem WCovBy.ge_of_gt {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⩿ b) (hac : a < c) : b ≤ c

theorem CovBy.le_of_lt {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⋖ b) : c < b → c ≤ a

theorem CovBy.ge_of_gt {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⋖ b) : a < c → b ≤ c

theorem CovBy.unique_left {α : Type u_1} [LinearOrder α] {a b c : α} (ha : a ⋖ c) (hb : b ⋖ c) : a = b

theorem CovBy.unique_right {α : Type u_1} [LinearOrder α] {a b c : α} (hb : a ⋖ b) (hc : a ⋖ c) : b = c

theorem CovBy.eq_of_between {α : Type u_1} [LinearOrder α] {a b c 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 covBy_iff_lt_iff_le_left {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y ↔ ∀ {z : α}, z < y ↔ z ≤ x

theorem covBy_iff_le_iff_lt_left {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y ↔ ∀ {z : α}, z ≤ x ↔ z < y

theorem covBy_iff_lt_iff_le_right {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y ↔ ∀ {z : α}, x < z ↔ y ≤ z

theorem covBy_iff_le_iff_lt_right {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y ↔ ∀ {z : α}, y ≤ z ↔ x < z

theorem CovBy.lt_iff_le_left {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y → ∀ {z : α}, z < y ↔ z ≤ x

Alias of the forward direction of covBy_iff_lt_iff_le_left.

theorem CovBy.le_iff_lt_left {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y → ∀ {z : α}, z ≤ x ↔ z < y

Alias of the forward direction of covBy_iff_le_iff_lt_left.

theorem CovBy.lt_iff_le_right {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y → ∀ {z : α}, x < z ↔ y ≤ z

Alias of the forward direction of covBy_iff_lt_iff_le_right.

theorem CovBy.le_iff_lt_right {α : Type u_1} [LinearOrder α] {x y : α} : x ⋖ y → ∀ {z : α}, y ≤ z ↔ x < z

Alias of the forward direction of covBy_iff_le_iff_lt_right.

theorem LT.lt.exists_disjoint_Iio_Ioi {α : Type u_1} [LinearOrder α] {a 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 : a ∉ s) : 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 t : Set α} (h : s ⋖ t) : ∃ a ∉ s, insert a s = t

theorem CovBy.exists_set_sdiff_singleton {α : Type u_1} {s t : Set α} (h : s ⋖ t) : ∃ a ∈ t, t \ {a} = s

theorem Set.covBy_iff_exists_insert {α : Type u_1} {s t : Set α} : s ⋖ t ↔ ∃ a ∉ s, insert a s = t

theorem Set.covBy_iff_exists_sdiff_singleton {α : Type u_1} {s t : Set α} : s ⋖ t ↔ ∃ a ∈ t, 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 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 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 y : α × β} : x ⩿ y → x.1 = y.1 ∨ x.2 = y.2

theorem WCovBy.fst {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x 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 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 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 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 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 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 a₂ : α} {b₁ 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 a₂ : α} {b₁ 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 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 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 b : α} : ↑a ⩿ ↑b ↔ a ⩿ b

@[simp]

theorem WithTop.coe_covBy_coe {α : Type u_1} [Preorder α] {a 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 b : α} : ↑a ⩿ ↑b ↔ a ⩿ b

@[simp]

theorem WithBot.coe_covBy_coe {α : Type u_1} [Preorder α] {a 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