Orders

Defines classes for preorders, partial orders, and linear orders and proves some basic lemmas about them.

Unbundled classes

@[deprecated emptyRelation (since := "2025-12-22")]

def EmptyRelation {α : Sort u} : α → α → Prop

An empty relation does not relate any elements.

@[reducible, inline, deprecated Std.Irrefl (since := "2026-01-07")]

abbrev IsIrrefl (α : Sort u_1) (r : α → α → Prop) : Prop

IsIrrefl X r means the binary relation r on X is irreflexive (that is, r x x never holds).

@[reducible, inline, deprecated Std.Refl (since := "2026-01-08")]

abbrev IsRefl (α : Sort u_1) (r : α → α → Prop) : Prop

IsRefl X r means the binary relation r on X is reflexive.

@[reducible, inline, deprecated Std.Symm (since := "2025-12-26")]

abbrev IsSymm (α : Sort u_1) (r : α → α → Prop) : Prop

IsSymm X r means the binary relation r on X is symmetric.

@[reducible, inline, deprecated Std.Asymm (since := "2026-01-03")]

abbrev IsAsymm (α : Sort u_1) (r : α → α → Prop) : Prop

IsAsymm X r means that the binary relation r on X is asymmetric, that is, r a b → ¬ r b a.

@[reducible, inline, deprecated Std.Antisymm (since := "2026-01-06")]

abbrev IsAntisymm (α : Sort u_1) (r : α → α → Prop) : Prop

IsAntisymm X r means the binary relation r on X is antisymmetric.

class IsTrans (α : Sort u_1) (r : α → α → Prop) : Prop

IsTrans X r means the binary relation r on X is transitive.

• trans (a b c : α) : r a b → r b c → r a c

instance instTransOfIsTrans {α : Sort u_1} {r : α → α → Prop} [IsTrans α r] : Trans r r r

@[instance 100]

instance instIsTransOfTrans {α : Sort u_1} {r : α → α → Prop} [Trans r r r] : IsTrans α r

@[reducible, inline, deprecated Std.Total (since := "2026-01-09")]

abbrev IsTotal (α : Sort u_1) (r : α → α → Prop) : Prop

IsTotal X r means that the binary relation r on X is total, that is, that for any x y : X we have r x y or r y x.

class IsPreorder (α : Sort u_1) (r : α → α → Prop) extends Std.Refl r, IsTrans α r : Prop

IsPreorder X r means that the binary relation r on X is a pre-order, that is, reflexive and transitive.

• refl (a : α) : r a a
• trans (a b c : α) : r a b → r b c → r a c

class IsPartialOrder (α : Sort u_1) (r : α → α → Prop) extends IsPreorder α r, Std.Antisymm r : Prop

IsPartialOrder X r means that the binary relation r on X is a partial order, that is, IsPreorder X r and Std.Antisymm r.

• refl (a : α) : r a a
• trans (a b c : α) : r a b → r b c → r a c
• antisymm (a b : α) : r a b → r b a → a = b

class IsLinearOrder (α : Sort u_1) (r : α → α → Prop) extends IsPartialOrder α r, Std.Total r : Prop

IsLinearOrder X r means that the binary relation r on X is a linear order, that is, IsPartialOrder X r and Std.Total r.

• refl (a : α) : r a a
• trans (a b c : α) : r a b → r b c → r a c
• antisymm (a b : α) : r a b → r b a → a = b
• total (a b : α) : r a b ∨ r b a

class IsEquiv (α : Sort u_1) (r : α → α → Prop) extends IsPreorder α r, Std.Symm r : Prop

IsEquiv X r means that the binary relation r on X is an equivalence relation, that is, IsPreorder X r and Std.Symm r.

• refl (a : α) : r a a
• trans (a b c : α) : r a b → r b c → r a c
• symm (a b : α) : r a b → r b a

class IsStrictOrder (α : Sort u_1) (r : α → α → Prop) extends Std.Irrefl r, IsTrans α r : Prop

IsStrictOrder X r means that the binary relation r on X is a strict order, that is, Std.Irrefl r and IsTrans X r.

• irrefl (a : α) : ¬r a a
• trans (a b c : α) : r a b → r b c → r a c

class IsStrictWeakOrder (α : Sort u_1) (lt : α → α → Prop) extends IsStrictOrder α lt : Prop

IsStrictWeakOrder X lt means that the binary relation lt on X is a strict weak order, that is, IsStrictOrder X lt and ¬lt a b ∧ ¬lt b a → ¬lt b c ∧ ¬lt c b → ¬lt a c ∧ ¬lt c a.

• irrefl (a : α) : ¬lt a a
• trans (a b c : α) : lt a b → lt b c → lt a c
• incomp_trans (a b c : α) : ¬lt a b ∧ ¬lt b a → ¬lt b c ∧ ¬lt c b → ¬lt a c ∧ ¬lt c a

@[reducible, inline, deprecated Std.Trichotomous (since := "2026-01-24")]

abbrev IsTrichotomous (α : Sort u_1) (lt : α → α → Prop) : Prop

IsTrichotomous X lt means that the binary relation lt on X is trichotomous, that is, either lt a b or a = b or lt b a for any a and b.

class IsStrictTotalOrder (α : Sort u_1) (lt : α → α → Prop) extends Std.Trichotomous lt, IsStrictOrder α lt : Prop

IsStrictTotalOrder X lt means that the binary relation lt on X is a strict total order, that is, Std.Trichotomous lt and IsStrictOrder X lt.

• trichotomous (a b : α) : ¬lt a b → ¬lt b a → a = b
• irrefl (a : α) : ¬lt a a
• trans (a b c : α) : lt a b → lt b c → lt a c

instance eq_isEquiv (α : Sort u_1) : IsEquiv α fun (x1 x2 : α) => x1 = x2

Equality is an equivalence relation.

instance iff_isEquiv : IsEquiv Prop Iff

Iff is an equivalence relation.

theorem irrefl {α : Sort u_1} {r : α → α → Prop} [Std.Irrefl r] (a : α) : ¬r a a

theorem refl {α : Sort u_1} {r : α → α → Prop} [Std.Refl r] (a : α) : r a a

theorem trans {α : Sort u_1} {r : α → α → Prop} {a b c : α} [IsTrans α r] : r a b → r b c → r a c

theorem symm {α : Sort u_1} {r : α → α → Prop} {a b : α} [Std.Symm r] : r a b → r b a

theorem antisymm {α : Sort u_1} {r : α → α → Prop} {a b : α} [Std.Antisymm r] : r a b → r b a → a = b

theorem asymm {α : Sort u_1} {r : α → α → Prop} {a b : α} [Std.Asymm r] : r a b → ¬r b a

theorem trichotomous {α : Sort u_1} {r : α → α → Prop} [Std.Trichotomous r] (a b : α) : r a b ∨ a = b ∨ r b a

@[instance 90]

instance asymm_of_isTrans_of_irrefl {α : Sort u_1} {r : α → α → Prop} [IsTrans α r] [Std.Irrefl r] : Std.Asymm r

instance Std.Irrefl.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Irrefl r] : Irrefl fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Refl.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Refl r] : Refl fun (a b : α) => Decidable.decide (r a b) = true

instance IsTrans.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsTrans α r] : IsTrans α fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Symm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Symm r] : Symm fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Antisymm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Antisymm r] : Antisymm fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Asymm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Asymm r] : Asymm fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Total.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Total r] : Total fun (a b : α) => Decidable.decide (r a b) = true

instance Std.Trichotomous.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [Trichotomous r] : Trichotomous fun (a b : α) => Decidable.decide (r a b) = true

@[elab_without_expected_type]

theorem irrefl_of {α : Sort u_1} (r : α → α → Prop) [Std.Irrefl r] (a : α) : ¬r a a

@[elab_without_expected_type]

theorem refl_of {α : Sort u_1} (r : α → α → Prop) [Std.Refl r] (a : α) : r a a

@[elab_without_expected_type]

theorem trans_of {α : Sort u_1} (r : α → α → Prop) {a b c : α} [IsTrans α r] : r a b → r b c → r a c

@[elab_without_expected_type]

theorem symm_of {α : Sort u_1} (r : α → α → Prop) {a b : α} [Std.Symm r] : r a b → r b a

@[elab_without_expected_type]

theorem asymm_of {α : Sort u_1} (r : α → α → Prop) {a b : α} [Std.Asymm r] : r a b → ¬r b a

@[elab_without_expected_type]

theorem total_of {α : Sort u_1} (r : α → α → Prop) [Std.Total r] (a b : α) : r a b ∨ r b a

@[elab_without_expected_type]

theorem trichotomous_of {α : Sort u_1} (r : α → α → Prop) [Std.Trichotomous r] (a b : α) : r a b ∨ a = b ∨ r b a

def Reflexive {α : Sort u_1} (r : α → α → Prop) : Prop

Std.Refl as a definition, suitable for use in proofs.

def Symmetric {α : Sort u_1} (r : α → α → Prop) : Prop

Std.Symm as a definition, suitable for use in proofs.

def Transitive {α : Sort u_1} (r : α → α → Prop) : Prop

IsTrans as a definition, suitable for use in proofs.

@[deprecated Std.Irrefl (since := "2026-02-12")]

def Irreflexive {α : Sort u_1} (r : α → α → Prop) : Prop

Std.Irrefl as a definition, suitable for use in proofs.

@[deprecated Std.Antisymm (since := "2026-02-09")]

def AntiSymmetric {α : Sort u_1} (r : α → α → Prop) : Prop

Std.Antisymm as a definition, suitable for use in proofs.

@[deprecated Std.Total (since := "2026-02-10")]

def Total {α : Sort u_1} (r : α → α → Prop) : Prop

Std.Total as a definition, suitable for use in proofs.

theorem Equivalence.reflexive {α : Sort u_1} (r : α → α → Prop) (h : Equivalence r) : Reflexive r

theorem Equivalence.symmetric {α : Sort u_1} (r : α → α → Prop) (h : Equivalence r) : Symmetric r

theorem Equivalence.transitive {α : Sort u_1} (r : α → α → Prop) (h : Equivalence r) : Transitive r

theorem InvImage.trans {α : Sort u_1} {β : Sort u_2} (r : β → β → Prop) (f : α → β) (h : Transitive r) : Transitive (InvImage r f)

theorem InvImage.irrefl {α : Sort u_1} {β : Sort u_2} (r : β → β → Prop) (f : α → β) (h : Std.Irrefl r) : Std.Irrefl (InvImage r f)

@[deprecated InvImage.irrefl (since := "2026-02-12")]

theorem InvImage.irreflexive {α : Sort u_1} {β : Sort u_2} (r : β → β → Prop) (f : α → β) (h : Std.Irrefl r) : Std.Irrefl (InvImage r f)

Alias of InvImage.irrefl.

Minimal and maximal

def Minimal {α : Type u_1} [LE α] (P : α → Prop) (x : α) : Prop

Minimal P x means that x is a minimal element satisfying P.

def Maximal {α : Type u_1} [LE α] (P : α → Prop) (x : α) : Prop

Maximal P x means that x is a maximal element satisfying P.

theorem Minimal.prop {α : Type u_1} [LE α] {P : α → Prop} {x : α} (h : Minimal P x) : P x

theorem Maximal.prop {α : Type u_1} [LE α] {P : α → Prop} {x : α} (h : Maximal P x) : P x

theorem Minimal.le_of_le {α : Type u_1} [LE α] {P : α → Prop} {x y : α} (h : Minimal P x) (hy : P y) (hle : y ≤ x) : x ≤ y

theorem Maximal.le_of_ge {α : Type u_1} [LE α] {P : α → Prop} {x y : α} (h : Maximal P x) (hy : P y) (hle : x ≤ y) : y ≤ x

def MinimalFor {ι : Sort u_1} {α : Type u_2} [LE α] (P : ι → Prop) (f : ι → α) (i : ι) : Prop

MinimalFor P f i means that f i is minimal over all i satisfying P.

def MaximalFor {ι : Sort u_1} {α : Type u_2} [LE α] (P : ι → Prop) (f : ι → α) (i : ι) : Prop

MaximalFor P f i means that f i is maximal over all i satisfying P.

theorem MinimalFor.prop {ι : Sort u_1} {α : Type u_2} [LE α] {P : ι → Prop} {f : ι → α} {i : ι} (h : MinimalFor P f i) : P i

theorem MaximalFor.prop {ι : Sort u_1} {α : Type u_2} [LE α] {P : ι → Prop} {f : ι → α} {i : ι} (h : MaximalFor P f i) : P i

theorem MinimalFor.le_of_le {ι : Sort u_1} {α : Type u_2} [LE α] {P : ι → Prop} {f : ι → α} {i j : ι} (h : MinimalFor P f i) (hj : P j) (hji : f j ≤ f i) : f i ≤ f j

theorem MaximalFor.le_of_le {ι : Sort u_1} {α : Type u_2} [LE α] {P : ι → Prop} {f : ι → α} {i j : ι} (h : MaximalFor P f i) (hj : P j) (hji : f i ≤ f j) : f j ≤ f i

Upper and lower sets

def IsUpperSet {α : Type u_1} [LE α] (s : Set α) : Prop

An upper set in an order α is a set such that any element greater than one of its members is also a member. Also called up-set, upward-closed set.

def IsLowerSet {α : Type u_1} [LE α] (s : Set α) : Prop

A lower set in an order α is a set such that any element less than one of its members is also a member. Also called down-set, downward-closed set.

structure UpperSet (α : Type u_1) [LE α] : Type u_1

An upper set in an order α is a set such that any element greater than one of its members is also a member. Also called up-set, upward-closed set.

• carrier : Set α

  The carrier of an UpperSet.

• upper' : IsUpperSet self.carrier

  The carrier of an UpperSet is an upper set.

structure LowerSet (α : Type u_1) [LE α] : Type u_1

A lower set in an order α is a set such that any element less than one of its members is also a member. Also called down-set, downward-closed set.

• carrier : Set α

  The carrier of a LowerSet.

• lower' : IsLowerSet self.carrier

  The carrier of a LowerSet is a lower set.

def IsRelUpperSet {α : Type u_1} [LE α] (s : Set α) (P : α → Prop) : Prop

An upper set relative to a predicate P is a set such that all elements satisfy P and any element greater than one of its members and satisfying P is also a member.

def IsRelLowerSet {α : Type u_1} [LE α] (s : Set α) (P : α → Prop) : Prop

A lower set relative to a predicate P is a set such that all elements satisfy P and any element less than one of its members and satisfying P is also a member.

structure RelUpperSet {α : Type u_1} [LE α] (P : α → Prop) : Type u_1

An upper set relative to a predicate P is a set such that all elements satisfy P and any element greater than one of its members and satisfying P is also a member.

• carrier : Set α

  The carrier of a RelUpperSet.

• isRelUpperSet' : IsRelUpperSet self.carrier P

  The carrier of a RelUpperSet is an upper set relative to P.

  Do NOT use directly. Please use RelUpperSet.isRelUpperSet instead.

structure RelLowerSet {α : Type u_1} [LE α] (P : α → Prop) : Type u_1

A lower set relative to a predicate P is a set such that all elements satisfy P and any element less than one of its members and satisfying P is also a member.

• carrier : Set α

  The carrier of a RelLowerSet.

• isRelLowerSet' : IsRelLowerSet self.carrier P

  The carrier of a RelLowerSet is a lower set relative to P.

  Do NOT use directly. Please use RelLowerSet.isRelLowerSet instead.

theorem of_eq {α : Type u_1} {r : α → α → Prop} [Std.Refl r] {a b : α} : a = b → r a b

theorem comm {α : Type u_1} {r : α → α → Prop} [Std.Symm r] {a b : α} : r a b ↔ r b a

theorem antisymm' {α : Type u_1} {r : α → α → Prop} [Std.Antisymm r] {a b : α} : r a b → r b a → b = a

theorem antisymm_iff {α : Type u_1} {r : α → α → Prop} [Std.Refl r] [Std.Antisymm r] {a b : α} : r a b ∧ r b a ↔ a = b

@[elab_without_expected_type]

theorem antisymm_of {α : Type u_1} (r : α → α → Prop) [Std.Antisymm r] {a b : α} : r a b → r b a → a = b

A version of antisymm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

@[elab_without_expected_type]

theorem antisymm_of' {α : Type u_1} (r : α → α → Prop) [Std.Antisymm r] {a b : α} : r a b → r b a → b = a

A version of antisymm' with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem comm_of {α : Type u_1} (r : α → α → Prop) [Std.Symm r] {a b : α} : r a b ↔ r b a

A version of comm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem Std.Asymm.antisymm {α : Type u_1} (r : α → α → Prop) [Asymm r] : Antisymm r

@[deprecated Std.Asymm.antisymm (since := "2026-01-05")]

theorem IsAsymm.isAntisymm {α : Type u_1} (r : α → α → Prop) [Std.Asymm r] : Std.Antisymm r

Alias of Std.Asymm.antisymm.

@[deprecated Std.Asymm.antisymm (since := "2026-01-06")]

theorem Std.Asymm.isAntisymm {α : Type u_1} (r : α → α → Prop) [Asymm r] : Antisymm r

Alias of Std.Asymm.antisymm.

theorem Std.Asymm.irrefl {α : Type u_1} {r : α → α → Prop} [Asymm r] : Irrefl r

@[deprecated Std.Asymm.irrefl (since := "2026-01-05")]

theorem IsAsymm.isIrrefl {α : Type u_1} {r : α → α → Prop} [Std.Asymm r] : Std.Irrefl r

Alias of Std.Asymm.irrefl.

@[deprecated Std.Asymm.irrefl (since := "2026-01-07")]

theorem Std.Asymm.isIrrefl {α : Type u_1} {r : α → α → Prop} [Asymm r] : Irrefl r

Alias of Std.Asymm.irrefl.

theorem Std.Total.trichotomous {α : Type u_1} (r : α → α → Prop) [Total r] : Trichotomous r

@[deprecated Std.Total.trichotomous (since := "2026-01-24")]

theorem Std.Total.isTrichotomous {α : Type u_1} (r : α → α → Prop) [Total r] : Trichotomous r

Alias of Std.Total.trichotomous.

@[instance 100]

instance Std.Total.to_refl {α : Type u_1} (r : α → α → Prop) [Total r] : Refl r

theorem ne_of_irrefl {α : Type u_1} {r : α → α → Prop} [Std.Irrefl r] {x y : α} : r x y → x ≠ y

theorem ne_of_irrefl' {α : Type u_1} {r : α → α → Prop} [Std.Irrefl r] {x y : α} : r x y → y ≠ x

theorem not_rel_of_subsingleton {α : Type u_1} (r : α → α → Prop) [Std.Irrefl r] [Subsingleton α] (x y : α) : ¬r x y

theorem rel_of_subsingleton {α : Type u_1} (r : α → α → Prop) [Std.Refl r] [Subsingleton α] (x y : α) : r x y

@[simp]

theorem empty_relation_apply {α : Type u_1} (a b : α) : emptyRelation a b ↔ False

instance instIrreflEmptyRelation_mathlib {α : Type u_1} : Std.Irrefl emptyRelation

theorem rel_congr_left {α : Type u_1} {r : α → α → Prop} [Std.Symm r] [IsTrans α r] {a b c : α} (h : r a b) : r a c ↔ r b c

theorem rel_congr_right {α : Type u_1} {r : α → α → Prop} [Std.Symm r] [IsTrans α r] {a b c : α} (h : r b c) : r a b ↔ r a c

theorem rel_congr {α : Type u_1} {r : α → α → Prop} [Std.Symm r] [IsTrans α r] {a b c d : α} (h₁ : r a b) (h₂ : r c d) : r a c ↔ r b d

theorem trans_trichotomous_left {α : Type u_1} {r : α → α → Prop} [IsTrans α r] [Std.Trichotomous r] {a b c : α} (h₁ : ¬r b a) (h₂ : r b c) : r a c

theorem trans_trichotomous_right {α : Type u_1} {r : α → α → Prop} [IsTrans α r] [Std.Trichotomous r] {a b c : α} (h₁ : r a b) (h₂ : ¬r c b) : r a c

theorem transitive_of_trans {α : Type u_1} (r : α → α → Prop) [IsTrans α r] : Transitive r

theorem extensional_of_trichotomous_of_irrefl {α : Type u_1} (r : α → α → Prop) [Std.Trichotomous r] [Std.Irrefl r] {a b : α} (H : ∀ (x : α), r x a ↔ r x b) : a = b

In a trichotomous irreflexive order, every element is determined by the set of predecessors.