mathlib documentation

order.rel_classes

Unbundled relation classes #

In this file we prove some properties of is_* classes defined in init.algebra.classes. The main difference between these classes and the usual order classes (preorder etc) is that usual classes extend has_le and/or has_lt while these classes take a relation as an explicit argument.

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theorem is_refl.swap {α : Type u} (r : α → α → Prop) [is_refl α r] :
is_refl α (function.swap r)
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theorem is_irrefl.swap {α : Type u} (r : α → α → Prop) [is_irrefl α r] :
is_irrefl α (function.swap r)
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theorem is_trans.swap {α : Type u} (r : α → α → Prop) [is_trans α r] :
is_trans α (function.swap r)
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theorem is_antisymm.swap {α : Type u} (r : α → α → Prop) [is_antisymm α r] :
is_antisymm α (function.swap r)
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theorem is_asymm.swap {α : Type u} (r : α → α → Prop) [is_asymm α r] :
is_asymm α (function.swap r)
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theorem is_total.swap {α : Type u} (r : α → α → Prop) [is_total α r] :
is_total α (function.swap r)
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theorem is_trichotomous.swap {α : Type u} (r : α → α → Prop) [is_trichotomous α r] :
is_trichotomous α (function.swap r)
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theorem is_preorder.swap {α : Type u} (r : α → α → Prop) [is_preorder α r] :
is_preorder α (function.swap r)
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theorem is_strict_order.swap {α : Type u} (r : α → α → Prop) [is_strict_order α r] :
is_strict_order α (function.swap r)
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theorem is_partial_order.swap {α : Type u} (r : α → α → Prop) [is_partial_order α r] :
is_partial_order α (function.swap r)
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theorem is_total_preorder.swap {α : Type u} (r : α → α → Prop) [is_total_preorder α r] :
is_total_preorder α (function.swap r)
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theorem is_linear_order.swap {α : Type u} (r : α → α → Prop) [is_linear_order α r] :
is_linear_order α (function.swap r)
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@[protected]
theorem is_asymm.is_antisymm {α : Type u} (r : α → α → Prop) [is_asymm α r] :
is_antisymm α r
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@[protected]
theorem is_asymm.is_irrefl {α : Type u} {r : α → α → Prop} [is_asymm α r] :
is_irrefl α r
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@[protected]
theorem is_total.is_trichotomous {α : Type u} (r : α → α → Prop) [is_total α r] :
is_trichotomous α r
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@[protected, instance]
def has_le.le.is_refl {α : Type u} [preorder α] :
is_refl α has_le.le
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@[protected, instance]
def ge.is_refl {α : Type u} [preorder α] :
is_refl α ge
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@[protected, instance]
def has_le.le.is_trans {α : Type u} [preorder α] :
is_trans α has_le.le
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@[protected, instance]
def ge.is_trans {α : Type u} [preorder α] :
is_trans α ge
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@[protected, instance]
def has_le.le.is_preorder {α : Type u} [preorder α] :
is_preorder α has_le.le
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@[protected, instance]
def ge.is_preorder {α : Type u} [preorder α] :
is_preorder α ge
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@[protected, instance]
def has_lt.lt.is_irrefl {α : Type u} [preorder α] :
is_irrefl α has_lt.lt
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@[protected, instance]
def gt.is_irrefl {α : Type u} [preorder α] :
is_irrefl α gt
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@[protected, instance]
def has_lt.lt.is_trans {α : Type u} [preorder α] :
is_trans α has_lt.lt
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@[protected, instance]
def gt.is_trans {α : Type u} [preorder α] :
is_trans α gt
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@[protected, instance]
def has_lt.lt.is_asymm {α : Type u} [preorder α] :
is_asymm α has_lt.lt
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@[protected, instance]
def gt.is_asymm {α : Type u} [preorder α] :
is_asymm α gt
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@[protected, instance]
def has_lt.lt.is_antisymm {α : Type u} [preorder α] :
is_antisymm α has_lt.lt
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@[protected, instance]
def gt.is_antisymm {α : Type u} [preorder α] :
is_antisymm α gt
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@[protected, instance]
def has_lt.lt.is_strict_order {α : Type u} [preorder α] :
is_strict_order α has_lt.lt
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@[protected, instance]
def gt.is_strict_order {α : Type u} [preorder α] :
is_strict_order α gt
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@[protected, instance]
def has_le.le.is_antisymm {α : Type u} [partial_order α] :
is_antisymm α has_le.le
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@[protected, instance]
def ge.is_antisymm {α : Type u} [partial_order α] :
is_antisymm α ge
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@[protected, instance]
def has_le.le.is_partial_order {α : Type u} [partial_order α] :
is_partial_order α has_le.le
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@[protected, instance]
def ge.is_partial_order {α : Type u} [partial_order α] :
is_partial_order α ge
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@[protected, instance]
def has_le.le.is_total {α : Type u} [linear_order α] :
is_total α has_le.le
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@[protected, instance]
def ge.is_total {α : Type u} [linear_order α] :
is_total α ge
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@[protected, instance]
def linear_order.is_total_preorder {α : Type u} [linear_order α] :
is_total_preorder α has_le.le
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@[protected, instance]
def ge.is_total_preorder {α : Type u} [linear_order α] :
is_total_preorder α ge
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@[protected, instance]
def has_le.le.is_linear_order {α : Type u} [linear_order α] :
is_linear_order α has_le.le
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@[protected, instance]
def ge.is_linear_order {α : Type u} [linear_order α] :
is_linear_order α ge
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@[protected, instance]
def has_lt.lt.is_trichotomous {α : Type u} [linear_order α] :
is_trichotomous α has_lt.lt
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@[protected, instance]
def gt.is_trichotomous {α : Type u} [linear_order α] :
is_trichotomous α gt
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@[protected, instance]
def has_le.le.is_trichotomous {α : Type u} [linear_order α] :
is_trichotomous α has_le.le
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@[protected, instance]
def ge.is_trichotomous {α : Type u} [linear_order α] :
is_trichotomous α ge
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@[protected, instance]
def order_dual.is_total_le {α : Type u} [has_le α] [is_total α has_le.le] :
is_total (order_dual α) has_le.le
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theorem ne_of_irrefl {α : Type u} {r : α → α → Prop} [is_irrefl α r] {x y : α} :
r x y → x ≠ y
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theorem trans_trichotomous_left {α : Type u} {r : α → α → Prop} [is_trans α r] [is_trichotomous α r] {a b c : α} :
¬r b a → r b c → r a c
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theorem trans_trichotomous_right {α : Type u} {r : α → α → Prop} [is_trans α r] [is_trichotomous α r] {a b c : α} :
r a b → ¬r c b → r a c
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def partial_order_of_SO {α : Type u} (r : α → α → Prop) [is_strict_order α r] :
partial_order α

Construct a partial order from a is_strict_order relation.

See note [reducible non-instances].

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@[class]
structure is_strict_total_order' (α : Type u) (lt : α → α → Prop) :
Prop

This is basically the same as is_strict_total_order, but that definition has a redundant assumption is_incomp_trans α lt.

Instances
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def linear_order_of_STO' {α : Type u} (r : α → α → Prop) [is_strict_total_order' α r] [Π (x y : α), decidable (¬r x y)] :
linear_order α

Construct a linear order from an is_strict_total_order' relation.

See note [reducible non-instances].

Equations
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theorem is_strict_total_order'.swap {α : Type u} (r : α → α → Prop) [is_strict_total_order' α r] :
is_strict_total_order' α (function.swap r)
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@[protected, instance]
def has_lt.lt.is_strict_total_order' {α : Type u} [linear_order α] :
is_strict_total_order' α has_lt.lt
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@[class]
structure is_order_connected (α : Type u) (lt : α → α → Prop) :
Prop
  • conn : ∀ (a b c : α), lt a c → lt a b ∨ lt b c

A connected order is one satisfying the condition a < c → a < b ∨ b < c. This is recognizable as an intuitionistic substitute for a ≤ b ∨ b ≤ a on the constructive reals, and is also known as negative transitivity, since the contrapositive asserts transitivity of the relation ¬ a < b.

Instances
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theorem is_order_connected.neg_trans {α : Type u} {r : α → α → Prop} [is_order_connected α r] {a b c : α} (h₁ : ¬r a b) (h₂ : ¬r b c) :
¬r a c
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theorem is_strict_weak_order_of_is_order_connected {α : Type u} {r : α → α → Prop} [is_asymm α r] [is_order_connected α r] :
is_strict_weak_order α r
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@[protected, instance]
def is_order_connected_of_is_strict_total_order' {α : Type u} {r : α → α → Prop} [is_strict_total_order' α r] :
is_order_connected α r
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@[protected, instance]
def is_strict_total_order_of_is_strict_total_order' {α : Type u} {r : α → α → Prop} [is_strict_total_order' α r] :
is_strict_total_order α r
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@[protected, instance]
def has_lt.lt.is_strict_total_order {α : Type u} [linear_order α] :
is_strict_total_order α has_lt.lt
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@[protected, instance]
def has_lt.lt.is_order_connected {α : Type u} [linear_order α] :
is_order_connected α has_lt.lt
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@[protected, instance]
def has_lt.lt.is_incomp_trans {α : Type u} [linear_order α] :
is_incomp_trans α has_lt.lt
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@[protected, instance]
def has_lt.lt.is_strict_weak_order {α : Type u} [linear_order α] :
is_strict_weak_order α has_lt.lt
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@[class]
structure is_extensional (α : Type u) (r : α → α → Prop) :
Prop
  • ext : ∀ (a b : α), (∀ (x : α), r x a ↔ r x b) → a = b

An extensional relation is one in which an element is determined by its set of predecessors. It is named for the x ∈ y relation in set theory, whose extensionality is one of the first axioms of ZFC.

Instances
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@[protected, instance]
def is_extensional_of_is_strict_total_order' {α : Type u} {r : α → α → Prop} [is_strict_total_order' α r] :
is_extensional α r
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@[class]
structure is_well_order (α : Type u) (r : α → α → Prop) :
Prop

A well order is a well-founded linear order.

Instances
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@[protected, instance]
def is_well_order.is_strict_total_order {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_strict_total_order α r
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@[protected, instance]
def is_well_order.is_extensional {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_extensional α r
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@[protected, instance]
def is_well_order.is_trichotomous {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_trichotomous α r
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@[protected, instance]
def is_well_order.is_trans {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_trans α r
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@[protected, instance]
def is_well_order.is_irrefl {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_irrefl α r
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@[protected, instance]
def is_well_order.is_asymm {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :
is_asymm α r
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noncomputable def is_well_order.linear_order {α : Type u} (r : α → α → Prop) [is_well_order α r] :
linear_order α

Construct a decidable linear order from a well-founded linear order.

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@[protected, instance]
def empty_relation.is_well_order {α : Type u} [subsingleton α] :
is_well_order α empty_relation
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@[protected, instance]
def nat.lt.is_well_order  :
is_well_order ℕ has_lt.lt
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@[protected, instance]
def sum.lex.is_well_order {α : Type u} {β : Type v} {r : α → α → Prop} {s : β → β → Prop} [is_well_order α r] [is_well_order β s] :
is_well_order (α ⊕ β) (sum.lex r s)
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@[protected, instance]
def prod.lex.is_well_order {α : Type u} {β : Type v} {r : α → α → Prop} {s : β → β → Prop} [is_well_order α r] [is_well_order β s] :
is_well_order (α × β) (prod.lex r s)
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def unbounded {α : Type u} (r : α → α → Prop) (s : set α) :
Prop

An unbounded or cofinal set

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def bounded {α : Type u} (r : α → α → Prop) (s : set α) :
Prop

A bounded or final set

Equations
  • bounded r s = ∃ (a : α), ∀ (b : α), b ∈ s → r b a
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@[simp]
theorem not_bounded_iff {α : Type u} {r : α → α → Prop} (s : set α) :
¬bounded r s ↔ unbounded r s
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@[simp]
theorem not_unbounded_iff {α : Type u} {r : α → α → Prop} (s : set α) :
¬unbounded r s ↔ bounded r s
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@[protected, instance]
def prod.is_refl_preimage_fst {α : Type u} {r : α → α → Prop} [h : is_refl α r] :
is_refl (α × α) (prod.fst ⁻¹'o r)
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@[protected, instance]
def prod.is_refl_preimage_snd {α : Type u} {r : α → α → Prop} [h : is_refl α r] :
is_refl (α × α) (prod.snd ⁻¹'o r)
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@[protected, instance]
def prod.is_trans_preimage_fst {α : Type u} {r : α → α → Prop} [h : is_trans α r] :
is_trans (α × α) (prod.fst ⁻¹'o r)
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@[protected, instance]
def prod.is_trans_preimage_snd {α : Type u} {r : α → α → Prop} [h : is_trans α r] :
is_trans (α × α) (prod.snd ⁻¹'o r)