Documentation

Mathlib.Order.Defs.Unbundled

Orders #

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

Unbundled classes #

def EmptyRelation {α : Sort u_1} :

α → α → Prop

An empty relation does not relate any elements.

Equations

EmptyRelation x✝¹ x✝ = False

Instances For

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

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

irrefl (a : α) : ¬r a a

Instances

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

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

refl (a : α) : r a a

Instances

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

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

symm (a b : α) : r a b → r b a

Instances

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

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

asymm (a b : α) : r a b → ¬r b a

Instances

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

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

antisymm (a b : α) : r a b → r b a → a = b

Instances

@[instance 100]

instance IsAsymm.toIsAntisymm {α : Sort u_1} (r : α → α → Prop) [IsAsymm α r] :

IsAntisymm α r

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

Instances

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

Trans r r r

Equations

instTransOfIsTrans = { trans := ⋯ }

@[instance 100]

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

class IsTotal (α : Sort u_1) (r : α → α → 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.

total (a b : α) : r a b ∨ r b a

Instances

class IsPreorder (α : Sort u_1) (r : α → α → Prop) extends IsRefl α r, IsTrans α r :

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

Instances

class IsPartialOrder (α : Sort u_1) (r : α → α → Prop) extends IsPreorder α r, IsAntisymm α r :

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

Instances

class IsLinearOrder (α : Sort u_1) (r : α → α → Prop) extends IsPartialOrder α r, IsTotal α r :

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

Instances

class IsEquiv (α : Sort u_1) (r : α → α → Prop) extends IsPreorder α r, IsSymm α r :

IsEquiv X r means that the binary relation r on X is an equivalence relation, that is, IsPreorder X r and IsSymm X 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

Instances

class IsStrictOrder (α : Sort u_1) (r : α → α → Prop) extends IsIrrefl α r, IsTrans α r :

IsStrictOrder X r means that the binary relation r on X is a strict order, that is, IsIrrefl X 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

Instances

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

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

Instances

class IsTrichotomous (α : Sort u_1) (lt : α → α → 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.

trichotomous (a b : α) : lt a b ∨ a = b ∨ lt b a

Instances

class IsStrictTotalOrder (α : Sort u_1) (lt : α → α → Prop) extends IsTrichotomous α lt, IsStrictOrder α lt :

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

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

Instances

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} [IsIrrefl α r] (a : α) :

¬r a a

theorem refl {α : Sort u_1} {r : α → α → Prop} [IsRefl α 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} [IsSymm α r] :

r a b → r b a

theorem antisymm {α : Sort u_1} {r : α → α → Prop} {a b : α} [IsAntisymm α r] :

r a b → r b a → a = b

theorem asymm {α : Sort u_1} {r : α → α → Prop} {a b : α} [IsAsymm α r] :

r a b → ¬r b a

theorem trichotomous {α : Sort u_1} {r : α → α → Prop} [IsTrichotomous α r] (a b : α) :

r a b ∨ a = b ∨ r b a

@[instance 90]

instance isAsymm_of_isTrans_of_isIrrefl {α : Sort u_1} {r : α → α → Prop} [IsTrans α r] [IsIrrefl α r] :

instance IsIrrefl.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsIrrefl α r] :

IsIrrefl α fun (a b : α) => Decidable.decide (r a b) = true

instance IsRefl.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsRefl α r] :

IsRefl α 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 IsSymm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsSymm α r] :

IsSymm α fun (a b : α) => Decidable.decide (r a b) = true

instance IsAntisymm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsAntisymm α r] :

IsAntisymm α fun (a b : α) => Decidable.decide (r a b) = true

instance IsAsymm.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsAsymm α r] :

IsAsymm α fun (a b : α) => Decidable.decide (r a b) = true

instance IsTotal.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsTotal α r] :

IsTotal α fun (a b : α) => Decidable.decide (r a b) = true

instance IsTrichotomous.decide {α : Sort u_1} {r : α → α → Prop} [DecidableRel r] [IsTrichotomous α r] :

IsTrichotomous α 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) [IsIrrefl α r] (a : α) :

¬r a a

@[elab_without_expected_type]

theorem refl_of {α : Sort u_1} (r : α → α → Prop) [IsRefl α 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} [IsSymm α 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} [IsAsymm α r] :

r a b → ¬r b a

@[elab_without_expected_type]

theorem total_of {α : Sort u_1} (r : α → α → Prop) [IsTotal α 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) [IsTrichotomous α r] (a b : α) :

r a b ∨ a = b ∨ r b a

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

IsRefl as a definition, suitable for use in proofs.

Equations

Reflexive r = ∀ (x : α), r x x

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

IsSymm as a definition, suitable for use in proofs.

Equations

Symmetric r = ∀ ⦃x y : α⦄, r x y → r y x

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

IsTrans as a definition, suitable for use in proofs.

Equations

Transitive r = ∀ ⦃x y z : α⦄, r x y → r y z → r x z

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

IsIrrefl as a definition, suitable for use in proofs.

Equations

Irreflexive r = ∀ (x : α), ¬r x x

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

IsAntisymm as a definition, suitable for use in proofs.

Equations

AntiSymmetric r = ∀ ⦃x y : α⦄, r x y → r y x → x = y

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

IsTotal as a definition, suitable for use in proofs.

Equations

Total r = ∀ (x y : α), r x y ∨ r y x

@[deprecated Equivalence.refl (since := "2024-09-13")]

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

@[deprecated Equivalence.symm (since := "2024-09-13")]

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

@[deprecated Equivalence.trans (since := "2024-09-13")]

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

@[deprecated "No deprecation message was provided." (since := "2024-09-13")]

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)

@[deprecated "No deprecation message was provided." (since := "2024-09-13")]

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

Irreflexive (InvImage r f)

Minimal and maximal #

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

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

Equations

Minimal P x = (P x ∧ ∀ ⦃y : α⦄, P y → y ≤ x → x ≤ y)

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

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

Equations

Maximal P x = (P x ∧ ∀ ⦃y : α⦄, P y → x ≤ y → y ≤ x)

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) (hge : x ≤ y) :

y ≤ x

Upper and lower sets #

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

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.

Equations

IsUpperSet s = ∀ ⦃a b : α⦄, a ≤ b → a ∈ s → b ∈ s

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

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.

Equations

IsLowerSet s = ∀ ⦃a b : α⦄, b ≤ a → a ∈ s → b ∈ s

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

Type u_1

The type of upper sets of an order.

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.

Instances For

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

Type u_1

The type of lower sets of an order.

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.

Instances For