order.rel_classes

decidable_linear_order_of_STO'

empty_relation.is_well_order

ge.is_linear_order

ge.is_partial_order

ge.is_total_preorder

gt.is_strict_order

gt.is_trichotomous

has_le.le.is_antisymm

has_le.le.is_linear_order

has_le.le.is_partial_order

has_le.le.is_preorder

has_le.le.is_refl

has_le.le.is_total

has_le.le.is_trans

has_lt.lt.is_antisymm

has_lt.lt.is_asymm

has_lt.lt.is_incomp_trans

has_lt.lt.is_irrefl

has_lt.lt.is_order_connected

has_lt.lt.is_strict_order

has_lt.lt.is_strict_total_order

has_lt.lt.is_strict_total_order'

has_lt.lt.is_strict_weak_order

has_lt.lt.is_trans

has_lt.lt.is_trichotomous

is_antisymm.swap

is_asymm.is_antisymm

is_asymm.is_irrefl

is_extensional_of_is_strict_total_order'

is_linear_order.swap

is_order_connected

is_order_connected.neg_trans

is_order_connected_of_is_strict_total_order'

is_partial_order.swap

is_preorder.swap

is_strict_order.swap

is_strict_total_order'

is_strict_total_order'.swap

is_strict_total_order_of_is_strict_total_order'

is_strict_weak_order_of_is_order_connected

is_total_preorder.swap

is_trichotomous.swap

is_well_order.decidable_linear_order

is_well_order.is_asymm

is_well_order.is_extensional

is_well_order.is_irrefl

is_well_order.is_strict_total_order

is_well_order.is_trans

is_well_order.is_trichotomous

linear_order.is_total_preorder

linear_order_of_STO'

nat.lt.is_well_order

not_bounded_iff

not_unbounded_iff

partial_order_of_SO

preorder.is_total_preorder

prod.lex.is_well_order

sum.lex.is_well_order

trans_trichotomous_left

trans_trichotomous_right

well_founded.has_min

well_founded.lt_succ

well_founded.lt_succ_iff

well_founded.lt_sup

well_founded.min

well_founded.min_mem

well_founded.not_lt_min

well_founded.succ

well_founded.sup

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.

theorem is_refl.swap {α : Type u} (r : α → α → Prop) [is_refl α r] :

is_refl α (function.swap r)

theorem is_irrefl.swap {α : Type u} (r : α → α → Prop) [is_irrefl α r] :

is_irrefl α (function.swap r)

theorem is_trans.swap {α : Type u} (r : α → α → Prop) [is_trans α r] :

is_trans α (function.swap r)

theorem is_antisymm.swap {α : Type u} (r : α → α → Prop) [is_antisymm α r] :

is_antisymm α (function.swap r)

theorem is_asymm.swap {α : Type u} (r : α → α → Prop) [is_asymm α r] :

is_asymm α (function.swap r)

theorem is_total.swap {α : Type u} (r : α → α → Prop) [is_total α r] :

is_total α (function.swap r)

theorem is_trichotomous.swap {α : Type u} (r : α → α → Prop) [is_trichotomous α r] :

is_trichotomous α (function.swap r)

theorem is_preorder.swap {α : Type u} (r : α → α → Prop) [is_preorder α r] :

is_preorder α (function.swap r)

theorem is_strict_order.swap {α : Type u} (r : α → α → Prop) [is_strict_order α r] :

is_strict_order α (function.swap r)

theorem is_partial_order.swap {α : Type u} (r : α → α → Prop) [is_partial_order α r] :

is_partial_order α (function.swap r)

theorem is_total_preorder.swap {α : Type u} (r : α → α → Prop) [is_total_preorder α r] :

is_total_preorder α (function.swap r)

theorem is_linear_order.swap {α : Type u} (r : α → α → Prop) [is_linear_order α r] :

is_linear_order α (function.swap r)

theorem is_asymm.is_antisymm {α : Type u} (r : α → α → Prop) [is_asymm α r] :

is_antisymm α r

theorem is_asymm.is_irrefl {α : Type u} {r : α → α → Prop} [is_asymm α r] :

@[instance]

def has_le.le.is_refl {α : Type u} [preorder α] :

is_refl α has_le.le

Equations

has_le.le.is_refl = _

@[instance]

def ge.is_refl {α : Type u} [preorder α] :

Equations

ge.is_refl = _

@[instance]

def has_le.le.is_trans {α : Type u} [preorder α] :

is_trans α has_le.le

Equations

has_le.le.is_trans = _

@[instance]

def ge.is_trans {α : Type u} [preorder α] :

Equations

ge.is_trans = _

@[instance]

def has_le.le.is_preorder {α : Type u} [preorder α] :

is_preorder α has_le.le

Equations

has_le.le.is_preorder = is_preorder.mk

@[instance]

def ge.is_preorder {α : Type u} [preorder α] :

is_preorder α ge

Equations

ge.is_preorder = is_preorder.mk

@[instance]

def has_lt.lt.is_irrefl {α : Type u} [preorder α] :

is_irrefl α has_lt.lt

Equations

has_lt.lt.is_irrefl = _

@[instance]

def gt.is_irrefl {α : Type u} [preorder α] :

is_irrefl α gt

Equations

gt.is_irrefl = _

@[instance]

def has_lt.lt.is_trans {α : Type u} [preorder α] :

is_trans α has_lt.lt

Equations

has_lt.lt.is_trans = _

@[instance]

def gt.is_trans {α : Type u} [preorder α] :

Equations

gt.is_trans = _

@[instance]

def has_lt.lt.is_asymm {α : Type u} [preorder α] :

is_asymm α has_lt.lt

Equations

has_lt.lt.is_asymm = _

@[instance]

def gt.is_asymm {α : Type u} [preorder α] :

Equations

gt.is_asymm = _

@[instance]

def has_lt.lt.is_antisymm {α : Type u} [preorder α] :

is_antisymm α has_lt.lt

Equations

has_lt.lt.is_antisymm = _

@[instance]

def gt.is_antisymm {α : Type u} [preorder α] :

is_antisymm α gt

Equations

gt.is_antisymm = _

@[instance]

def has_lt.lt.is_strict_order {α : Type u} [preorder α] :

is_strict_order α has_lt.lt

Equations

has_lt.lt.is_strict_order = is_strict_order.mk

@[instance]

def gt.is_strict_order {α : Type u} [preorder α] :

is_strict_order α gt

Equations

gt.is_strict_order = is_strict_order.mk

@[instance]

def preorder.is_total_preorder {α : Type u} [preorder α] [is_total α has_le.le] :

is_total_preorder α has_le.le

Equations

preorder.is_total_preorder = is_total_preorder.mk

@[instance]

def has_le.le.is_antisymm {α : Type u} [partial_order α] :

is_antisymm α has_le.le

Equations

has_le.le.is_antisymm = _

@[instance]

def ge.is_antisymm {α : Type u} [partial_order α] :

is_antisymm α ge

Equations

ge.is_antisymm = _

@[instance]

def has_le.le.is_partial_order {α : Type u} [partial_order α] :

is_partial_order α has_le.le

Equations

has_le.le.is_partial_order = is_partial_order.mk

@[instance]

def ge.is_partial_order {α : Type u} [partial_order α] :

is_partial_order α ge

Equations

ge.is_partial_order = is_partial_order.mk

@[instance]

def has_le.le.is_total {α : Type u} [linear_order α] :

is_total α has_le.le

Equations

has_le.le.is_total = _

@[instance]

def ge.is_total {α : Type u} [linear_order α] :

Equations

ge.is_total = _

@[instance]

def linear_order.is_total_preorder {α : Type u} [linear_order α] :

is_total_preorder α has_le.le

Equations

linear_order.is_total_preorder = preorder.is_total_preorder

@[instance]

def ge.is_total_preorder {α : Type u} [linear_order α] :

is_total_preorder α ge

Equations

ge.is_total_preorder = is_total_preorder.mk

@[instance]

def has_le.le.is_linear_order {α : Type u} [linear_order α] :

is_linear_order α has_le.le

Equations

has_le.le.is_linear_order = is_linear_order.mk

@[instance]

def ge.is_linear_order {α : Type u} [linear_order α] :

is_linear_order α ge

Equations

ge.is_linear_order = is_linear_order.mk

@[instance]

def has_lt.lt.is_trichotomous {α : Type u} [linear_order α] :

is_trichotomous α has_lt.lt

Equations

has_lt.lt.is_trichotomous = _

@[instance]

def gt.is_trichotomous {α : Type u} [linear_order α] :

is_trichotomous α gt

Equations

gt.is_trichotomous = _

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

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

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

Equations

partial_order_of_SO r = {le := λ (x y : α), x = y ∨ r x y, lt := r, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}

@[class]

structure is_strict_total_order' (α : Type u) :

(α → α → Prop) → Prop

to_is_trichotomous : is_trichotomous α lt
to_is_strict_order : is_strict_order α lt

This is basically the same as is_strict_total_order, but that definition is in Type (probably by mistake) and also has redundant assumptions.

Instances

def linear_order_of_STO' {α : Type u} (r : α → α → Prop) [is_strict_total_order' α r] :

linear_order α

Construct a linear order from a is_strict_total_order' relation

Equations

linear_order_of_STO' r = {le := partial_order.le (partial_order_of_SO r), lt := partial_order.lt (partial_order_of_SO r), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _}

def decidable_linear_order_of_STO' {α : Type u} (r : α → α → Prop) [is_strict_total_order' α r] [decidable_rel r] :

decidable_linear_order α

Construct a decidable linear order from a is_strict_total_order' relation

Equations

decidable_linear_order_of_STO' r = let LO : linear_order α := linear_order_of_STO' r in {le := linear_order.le LO, lt := linear_order.lt LO, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, decidable_le := λ (x y : α), decidable_of_iff (¬r y x) not_lt, decidable_eq := decidable_eq_of_decidable_le (λ (x y : α), decidable_of_iff (¬r y x) not_lt), decidable_lt := decidable_lt_of_decidable_le (λ (x y : α), decidable_of_iff (¬r y x) not_lt)}

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)

@[instance]

def has_lt.lt.is_strict_total_order' {α : Type u} [linear_order α] :

is_strict_total_order' α has_lt.lt

Equations

has_lt.lt.is_strict_total_order' = is_strict_total_order'.mk

@[class]

structure is_order_connected (α : Type u) :

(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

theorem is_order_connected.neg_trans {α : Type u} {r : α → α → Prop} [is_order_connected α r] {a b c : α} :

¬r a b → ¬r b c → ¬r a c

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

@[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

Equations

is_order_connected_of_is_strict_total_order' = _

@[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

Equations

is_strict_total_order_of_is_strict_total_order' = is_strict_total_order.mk

@[instance]

def has_lt.lt.is_strict_total_order {α : Type u} [linear_order α] :

is_strict_total_order α has_lt.lt

Equations

has_lt.lt.is_strict_total_order = is_strict_total_order_of_is_strict_total_order'

@[instance]

def has_lt.lt.is_order_connected {α : Type u} [linear_order α] :

is_order_connected α has_lt.lt

Equations

has_lt.lt.is_order_connected = is_order_connected_of_is_strict_total_order'

@[instance]

def has_lt.lt.is_incomp_trans {α : Type u} [linear_order α] :

is_incomp_trans α has_lt.lt

Equations

has_lt.lt.is_incomp_trans = is_strict_weak_order.to_is_incomp_trans

@[instance]

def has_lt.lt.is_strict_weak_order {α : Type u} [linear_order α] :

is_strict_weak_order α has_lt.lt

Equations

has_lt.lt.is_strict_weak_order = is_strict_total_order.to_is_strict_weak_order

@[class]

structure is_extensional (α : Type u) :

(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

@[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

Equations

is_extensional_of_is_strict_total_order' = _

@[class]

structure is_well_order (α : Type u) :

(α → α → Prop) → Prop

to_is_strict_total_order' : is_strict_total_order' α r
wf : well_founded r

A well order is a well-founded linear order.

Instances

@[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

Equations

_ = is_strict_total_order_of_is_strict_total_order'

@[instance]

def is_well_order.is_extensional {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :

is_extensional α r

Equations

_ = is_extensional_of_is_strict_total_order'

@[instance]

def is_well_order.is_trichotomous {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :

is_trichotomous α r

Equations

_ = is_strict_total_order.to_is_trichotomous

@[instance]

def is_well_order.is_trans {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :

Equations

_ = is_strict_order.to_is_trans

@[instance]

def is_well_order.is_irrefl {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :

Equations

_ = is_strict_order.to_is_irrefl

@[instance]

def is_well_order.is_asymm {α : Type u_1} (r : α → α → Prop) [is_well_order α r] :

Equations

_ = is_asymm_of_is_trans_of_is_irrefl

def is_well_order.decidable_linear_order {α : Type u} (r : α → α → Prop) [is_well_order α r] :

decidable_linear_order α

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

Equations

is_well_order.decidable_linear_order r = classical.DLO α

@[instance]

def empty_relation.is_well_order {α : Type u} [subsingleton α] :

is_well_order α empty_relation

Equations

empty_relation.is_well_order = _

@[instance]

def nat.lt.is_well_order :

is_well_order ℕ has_lt.lt

Equations

nat.lt.is_well_order = _

@[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)

Equations

sum.lex.is_well_order = _

@[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)

Equations

prod.lex.is_well_order = _
_ = _
_ = _
_ = _
_ = _
_ = _
_ = _
_ = _
_ = _
_ = _

def unbounded {α : Type u} :

(α → α → Prop) → set α → Prop

An unbounded or cofinal set

Equations

unbounded r s = ∀ (a : α), ∃ (b : α) (H : b ∈ s), ¬r b a

def bounded {α : Type u} :

(α → α → Prop) → set α → Prop

A bounded or final set

Equations

bounded r s = ∃ (a : α), ∀ (b : α), b ∈ s → r b a

@[simp]

theorem not_bounded_iff {α : Type u} {r : α → α → Prop} (s : set α) :

¬bounded r s ↔ unbounded r s

@[simp]

theorem not_unbounded_iff {α : Type u} {r : α → α → Prop} (s : set α) :

¬unbounded r s ↔ bounded r s

theorem well_founded.has_min {α : Type u_1} {r : α → α → Prop} (H : well_founded r) (s : set α) :

s.nonempty → (∃ (a : α) (H : a ∈ s), ∀ (x : α), x ∈ s → ¬r x a)

If r is a well-founded relation, then any nonempty set has a minimal element with respect to r.

def well_founded.min {α : Type u_1} {r : α → α → Prop} (H : well_founded r) (p : set α) :

p.nonempty → α

A minimal element of a nonempty set in a well-founded order

Equations

H.min p h = classical.some _

theorem well_founded.min_mem {α : Type u_1} {r : α → α → Prop} (H : well_founded r) (p : set α) (h : p.nonempty) :

H.min p h ∈ p

theorem well_founded.not_lt_min {α : Type u_1} {r : α → α → Prop} (H : well_founded r) (p : set α) (h : p.nonempty) {x : α} :

x ∈ p → ¬r x (H.min p h)

def well_founded.sup {α : Type u_1} {r : α → α → Prop} (wf : well_founded r) (s : set α) :

bounded r s → α

The supremum of a bounded, well-founded order

Equations

wf.sup s h = wf.min {x : α | ∀ (a : α), a ∈ s → r a x} h

theorem well_founded.lt_sup {α : Type u_1} {r : α → α → Prop} (wf : well_founded r) {s : set α} (h : bounded r s) {x : α} :

x ∈ s → r x (wf.sup s h)

def well_founded.succ {α : Type u_1} {r : α → α → Prop} :

well_founded r → α → α

A successor of an element x in a well-founded order is a minimal element y such that x < y if one exists. Otherwise it is x itself.

Equations

wf.succ x = dite (∃ (y : α), r x y) (λ (h : ∃ (y : α), r x y), wf.min {y : α | r x y} h) (λ (h : ¬∃ (y : α), r x y), x)

theorem well_founded.lt_succ {α : Type u_1} {r : α → α → Prop} (wf : well_founded r) {x : α} :

(∃ (y : α), r x y) → r x (wf.succ x)

theorem well_founded.lt_succ_iff {α : Type u_1} {r : α → α → Prop} [wo : is_well_order α r] {x : α} (h : ∃ (y : α), r x y) (y : α) :

r y (is_well_order.wf.succ x) ↔ r y x ∨ y = x

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over

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End

braided

category

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of_has_finite_products

types

unitors

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default

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action

arrow

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const

core

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elements

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epi_mono

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full_subcategory

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functor

functor_category

functorial

graded_object

groupoid

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isomorphism

isomorphism_classes

natural_isomorphism

natural_transformation

opposites

over

pempty

punit

quotient

reflects_isomorphisms

shift

simple

single_obj

sparse

types

whiskering

yoneda

combinatorics

composition

simple_graph

computability

halting

partrec

partrec_code

primrec

reduce

tm_to_partrec

turing_machine

control

bitraversable

basic

instances

lemmas

equiv_functor

instances

functor

multivariate

monad

basic

cont

writer

traversable

basic

derive

equiv

instances

lemmas

applicative

basic

bifunctor

equiv_functor

fold

functor

uliftable

core

data

buffer

parser

bitvec

buffer

dlist

lazy_list

stream

vector

init

algebra

classes

functions

order

control

alternative

applicative

combinators

except

functor

id

lawful

lift

monad

monad_fail

option

reader

state

data

array

basic

slice

bool

basic

lemmas

char

basic

classes

lemmas

fin

basic

ops

int

basic

bitwise

comp_lemmas

order

list

basic

instances

lemmas

qsort

nat

basic

bitwise

div

gcd

lemmas

option

basic

instances

ordering

basic

lemmas

rbmap

basic

rbtree

basic

default

sigma

basic

lex

string

basic

ops

subtype

basic

instances

sum

basic

unsigned

basic

ops

prod

punit

quot

repr

set

setoid

to_string

meta

converter

conv

interactive

lean

parser

smt

congruence_closure

ematch

interactive

rsimp

smt_tactic

widget

basic

html_cmd

interactive_expr

replace_save_info

tactic_component

ac_tactics

async_tactic

attribute

backward

case_tag

comp_value_tactics

congr_lemma

congr_tactic

constructor_tactic

contradiction_tactic

declaration

derive

environment

exceptional

expr

expr_address

float

format

fun_info

has_reflect

hole_command

injection_tactic

interaction_monad

interactive

interactive_base

level

local_context

match_tactic

mk_dec_eq_instance

mk_has_reflect_instance

mk_has_sizeof_instance

mk_inhabited_instance

module_info

name

occurrences

options

pexpr

rb_map

rec_util

ref

relation_tactics

rewrite_tactic

set_get_option_tactics

simp_tactic

tactic

tagged_format

task

type_context

vm

well_founded_tactics

cc_lemmas

classical

coe

core

default

function

funext

ite_simp

logic

propext

util

version

wf

system

io

io_interface

random

data

analysis

filter

topology

array

lemmas

bitvec

basic

buffer

basic

complex

basic

exponential

module

dlist

basic

instances

equiv

encodable

basic

lattice

basic

denumerable

fin

functor

list

local_equiv

mul_add

nat

ring

transfer_instance

finset

basic

fold

intervals

lattice

nat_antidiagonal

order

pi

powerset

sort

finsupp

basic

lattice

pointwise

fintype

basic

card

intervals

sort

fp

basic

int

basic

cast

char_zero

gcd

modeq

parity

range

sqrt

list

alist

bag_inter

basic

chain

defs

erase_dup

forall2

func

indexes

intervals

min_max

nat_antidiagonal

nodup

of_fn

pairwise

palindrome

perm

range

rotate

sections

sigma

sort

tfae

zip

matrix

basic

notation

pequiv

multiset

antidiagonal

basic

erase_dup

finset_ops

fold

functor

intervals

lattice

nat_antidiagonal

nodup

pi

powerset

range

sections

sort

nat

basic

cast

choose

digits

dist

enat

fib

gcd

modeq

multiplicity

pairing

parity

prime

sqrt

totient

num

basic

bitwise

lemmas

prime

option

basic

defs

padics

hensel

padic_integers

padic_norm

padic_numbers

pfunctor

multivariate

M

W

basic

univariate

M

basic

pnat

basic

factors

intervals

xgcd

polynomial

degree

basic

algebra_map

basic

coeff

degree

derivative

div

eval

field_division

identities

induction

integral_normalization

monic

monomial

ring_division

qpf

multivariate

constructions

cofix

comp

const

fix

prj

quot

sigma

basic

univariate

basic

rat

basic

cast

denumerable

floor

meta_defs

order

sqrt

real

basic

cardinality

cau_seq

cau_seq_completion

conjugate_exponents

ennreal

ereal

golden_ratio

hyperreal

irrational

nnreal

pi

seq

computation

parallel

seq

wseq

set

intervals

basic

disjoint

image_preimage

ord_connected

unordered_interval

basic

countable

disjointed

enumerate

finite

function

lattice

setoid

basic

partition

sigma

basic

stream

basic

string

basic

defs

zmod

basic

zsqrtd

basic

gaussian_int

W

bool

char

dfinsupp

erased

fin

fin2

fin_enum

finmap

hash_map

holor

indicator_function

lazy_list2

mllist

monoid_algebra

mv_polynomial

opposite

pequiv

pfun

prod

quot

rel

semiquot

subtype

sum

support

sym

sym2

tree

typevec

ulift

vector2

deprecated

group

ring

subgroup

submonoid

dynamics

circle

rotation_number

translation_number

fixed_points

basic

topology

periodic_pts

field_theory

algebraic_closure

chevalley_warning

finite

fixed

minimal_polynomial

mv_polynomial

normal

perfect_closure

separable

splitting_field

subfield

tower

geometry

algebra

lie_group

euclidean

basic

circumcenter

monge_point

triangle

manifold

basic_smooth_bundle

charted_space

local_invariant_properties

mfderiv

real_instances

smooth_manifold_with_corners

times_cont_mdiff

group_theory

perm

cycles

sign

submonoid

basic

membership

operations

abelianization

congruence

coset

eckmann_hilton

free_abelian_group

free_group

group_action

monoid_localization

order_of_element

presented_group

quotient_group

semidirect_product

subgroup

sylow

linear_algebra

affine_space

basic

combination

finite_dimensional

independent

char_poly

coeff

direct_sum

finsupp

tensor_product

adic_completion

basic

basis

bilinear_form

char_poly

contraction

determinant

dimension

direct_sum_module

dual

finite_dimensional

finsupp

finsupp_vector_space

invariant_basis_number

lagrange

linear_action

linear_pmap

matrix

multilinear

nonsingular_inverse

projection

quadratic_form

sesquilinear_form

smodeq

special_linear_group

tensor_algebra

tensor_product

logic

function

basic

conjugate

iterate

basic

embedding

nontrivial

relation

relator

unique

measure_theory

category

Meas

ae_eq_fun

bochner_integration

borel_space

content

decomposition

giry_monad

group

haar_measure

integration

interval_integral

l1_space

lebesgue_measure

measurable_space

measure_space

outer_measure

probability_mass_function

set_integral

simple_func_dense

meta

coinductive_predicates

expr

expr_lens

rb_map

uchange

number_theory

basic

bernoulli

dioph

lucas_lehmer

pell

primorial

pythagorean_triples

quadratic_reciprocity

sum_four_squares

sum_two_squares

order

category

LinearOrder

NonemptyFinLinOrd

PartialOrder

Preorder

filter

at_top_bot

bases

basic

cofinite

countable_Inter

extr

filter_product

germ

indicator_function

interval

lift

partial

pointwise

ultrafilter

basic

boolean_algebra

bounded_lattice

bounds

complete_boolean_algebra

complete_lattice

conditionally_complete_lattice

copy

directed

fixed_points

galois_connection

ideal

iterate

lattice

lexicographic

liminf_limsup

ord_continuous

order_iso_nat

pilex

rel_classes

rel_iso

semiconj_Sup

zorn

representation_theory

maschke

ring_theory

ideal

basic

operations

over

polynomial

basic

homogeneous

rational_root

valuation

basic

adjoin

adjoin_root

algebra

algebra_operations

algebra_tower

algebraic

coprime

derivation

discrete_valuation_ring

eisenstein_criterion

euclidean_domain

fintype

fractional_ideal

free_comm_ring

free_ring

integral_closure

integral_domain

jacobson

jacobson_ideal

localization

maps

matrix_algebra

multiplicity

noetherian

polynomial_algebra

power_series

prime

principal_ideal_domain

subring

subsemiring

tensor_product

unique_factorization_domain

set_theory

game

domineering

impartial

nim

short

state

winner

cardinal

cardinal_ordinal

cofinality

game

lists

ordinal

ordinal_arithmetic

ordinal_notation

pgame

schroeder_bernstein

surreal

zfc

system

random

basic

tactic

converter

apply_congr

binders

interactive

old_conv

linarith

datatypes

elimination

frontend

lemmas

parsing

preprocessing

verification

lint

basic

default

frontend

misc

simp

type_classes

monotonicity

basic

interactive

lemmas

nth_rewrite

basic

congr

default

omega

int

dnf

form

main

preterm

nat

dnf

form

main

neg_elim

preterm

sub_elim

clause

coeffs

eq_elim

find_ees

find_scalars

lin_comb

main

misc

prove_unsats

term

rewrite_all

basic

abel

algebra

alias

apply

apply_fun

auto_cases

binder_matching

cache

cancel_denoms

chain

choose

clear

core

dec_trivial

delta_instance

derive_fintype

derive_inhabited

doc_commands

elide

equiv_rw

explode

ext

fin_cases

find

find_unused

finish

fix_reflect_string

generalize_proofs

generalizes

group

hint

interactive

interactive_expr

interval_cases

lift

local_cache

localized

mk_iff_of_inductive_prop

noncomm_ring

norm_cast

norm_num

obviously

pi_instances

protected

push_neg

rcases

reassoc_axiom

rename_var

replacer

restate_axiom

rewrite

ring

ring2

ring_exp

scc

show_term

simp_command

simp_result

simp_rw

simpa

simps

slice

solve_by_elim

split_ifs

squeeze

subtype_instance

suggest

tauto

tfae

tidy

transfer

transform_decl

transport

trunc_cases

unfold_cases

where

with_local_reducibility

wlog

zify

topology

algebra

affine

continuous_functions

group

group_completion

infinite_sum

module

monoid

multilinear

open_subgroup

ordered

polynomial

ring

uniform_group

uniform_ring

category

Top

adjunctions

basic

epi_mono

limits

open_nhds

opens

TopCommRing

UniformSpace

instances

complex

ennreal

nnreal

real

real_vector_space

metric_space

antilipschitz

baire

basic

cau_seq_filter

closeds

completion

contracting

emetric_space

gluing

gromov_hausdorff

gromov_hausdorff_realized

hausdorff_distance

isometry

lipschitz

pi_Lp

premetric_space

sheaves

presheaf

presheaf_of_functions

sheaf

sheaf_of_functions

stalks

uniform_space

absolute_value

abstract_completion

basic

cauchy

compact_separated

compare_reals

complete_separated

completion

pi

separation

uniform_convergence

uniform_embedding

bases

basic

bounded_continuous_function

compact_open

compacts

constructions

continuous_map

continuous_on

dense_embedding

extend_from_subset

homeomorph

list

local_extr

local_homeomorph

maps

opens

order

path_connected

separation

sequences

stone_cech

subset_properties

tactic

topological_fiber_bundle