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Mathlib naming conventions #
Author: Jeremy Avigad
This guide is written for Lean 3.
Names of symbols #
When translating the statements of theorems into words, this dictionary is often used:
Logic #
symbol  shortcut  name  notes 

∨ 
\or 
or 

∧ 
\and 
and 

→ 
\r 
of 
the conclusion is stated first and the hypotheses are often omitted 
↔ 
\iff 
iff 
sometimes omitted along with the right hand side of the iff 
¬ 
\n 
not 

∃ 
\ex 
exists / bex 
bex stands for "bounded exists" 
∀ 
\fo 
all / forall / ball 
ball stands for "bounded forall" 
= 
eq 
often omitted  
≠ 
\ne 
ne 

∘ 
\o 
comp 
Set #
symbol  shortcut  name  notes 

∈ 
\in 
mem 

∪ 
\cup 
union 

∩ 
\cap 
inter 

⋃ 
\bigcup 
Union / bUnion 

⋂ 
\bigcap 
Inter / bInter 

\ 
\\ 
sdiff 

ᶜ 
\^c 
compl 

{x  p x} 
set_of 

{x} 
singleton 

{x, y} 
pair 
Algebra #
symbol  shortcut  name  notes 

0 
zero 

+ 
add 

 
neg / sub 
neg for the unary function, sub for the binary function 

1 
one 

* 
mul 

^ 
pow 

/ 
div 

• 
\bu 
smul 

⁻¹ 
\1 
inv 

⅟ 
\frac1 
inv_of 

∣ 
\ 
dvd 

∑ 
\sum 
sum 

∏ 
\prod 
prod 
Lattices #
symbol  shortcut  name  notes 

< 
lt 

≤ 
\le 
le 

⊔ 
\sup 
sup 

⊓ 
\inf 
inf 

⨆ 
\supr 
supr / bsupr 

⨅ 
\infi 
infi / binfi 

⊥ 
\bot 
bot 

⊤ 
\top 
top 
General conventions #
Identifiers are generally lower case with underscores. For the most part, we rely on descriptive names. Often the name of theorem simply describes the conclusion:
succ_ne_zero
mul_zero
mul_one
sub_add_eq_add_sub
le_iff_lt_or_eq
If only a prefix of the description is enough to convey the meaning, the name may be made even shorter:
neg_neg
pred_succ
Sometimes, to disambiguate the name of theorem or better convey the intended reference, it is necessary to describe some of the hypotheses. The word "of" is used to separate these hypotheses:
lt_of_succ_le
lt_of_not_ge
lt_of_le_of_ne
add_lt_add_of_lt_of_le
Sometimes abbreviations or alternative descriptions are easier to work
with. For example, we use pos
, neg
, nonpos
, nonneg
rather than
zero_lt
, lt_zero
, le_zero
, and zero_le
.
mul_pos
mul_nonpos_of_nonneg_of_nonpos
add_lt_of_lt_of_nonpos
add_lt_of_nonpos_of_lt
Sometimes the word "left" or "right" is helpful to describe variants of a theorem.
add_le_add_left
add_le_add_right
le_of_mul_le_mul_left
le_of_mul_le_mul_right
An injectivity lemma that uses "left" or "right" should refer to the
argument that "changes". For example, a lemma with the statement
a  b = a  c ↔ b = c
could be called sub_right_inj
.
We can also use the word "self" to indicate a repeated argument:
mul_inv_self
neg_add_self
Dots #
Dots are used for namespaces, and also for automatically generated names like recursors, eliminators and structure projections. They can also be introduced manually, for example, where projector notation is useful. Thus they are used in all of the following situations.
Intro, elim, and destruct rules for logical connectives, whether they are automatically generated or not:
and.intro
and.elim
and.left
and.right
or.inl
or.inr
or.intro_left
or.intro_right
iff.intro
iff.elim
iff.mp
iff.mpr
not.intro
not.elim
eq.refl
eq.rec
eq.subst
heq.refl
heq.rec
heq.subst
exists.intro
exists.elim
true.intro
false.elim
Places where projection notation is useful, for example:
and.symm
or.symm
or.resolve_left
or.resolve_right
eq.symm
eq.trans
heq.symm
heq.trans
iff.symm
iff.refl
It is useful to use dot notation even for types which are not inductive types. For instance, we use:
le.trans
lt.trans_le
le.trans_lt
Axiomatic descriptions #
Some theorems are described using axiomatic names, rather than describing their conclusions.
def
(for unfolding a definition)refl
irrefl
symm
trans
antisymm
asymm
congr
comm
assoc
left_comm
right_comm
mul_left_cancel
mul_right_cancel
inj
(injective)
Variable conventions #
u
,v
,w
, ... for universesα
,β
,γ
, ... for generic typesa
,b
,c
, ... for propositionsx
,y
,z
, ... for elements of a generic typeh
,h₁
, ... for assumptionsp
,q
,r
, ... for predicates and relationss
,t
, ... for listss
,t
, ... for setsm
,n
,k
, ... for natural numbersi
,j
,k
, ... for integers
Types with a mathematical content are expressed with the usual
mathematical notation, often with an upper case letter
(G
for a group, R
for a ring, K
or 𝕜
for a field, E
for a vector space, ...).
This convention is not followed in older files, where greek letters are used
for all types. Pull requests renaming type variables in these files are welcome.
Names for symbols #
imp
: implicationforall
exists
ball
: bounded forallbex
: bounded exists
Identifiers and theorem names #
We generally use lower case with underscores for theorem names and
definitions. Sometimes upper case is used for bundled structures, such
as Group
. In that case, use CamelCase for compound names, such as
AbelianGroup
.
We adopt the following naming guidelines to make it easier for users to guess the name of a theorem or find it using tab completion. Common "axiomatic" properties of an operation like conjunction or disjunction are put in a namespace that begins with the name of the operation:
import logic.basic
#check and.comm
#check or.comm
#check and.assoc
#check or.assoc
In particular, this includes intro
and elim
operations for logical
connectives, and properties of relations:
import logic.basic
#check and.intro
#check and.elim
#check or.intro_left
#check or.intro_right
#check or.elim
#check eq.refl
#check eq.symm
#check eq.trans
Note however we do not do this for axiomatic arithmetic operations
import algebra.group.basic
#check mul_comm
#check mul_assoc
#check @mul_left_cancel  multiplication is left cancelative
For the most part, however, we rely on descriptive names. Often the name of theorem simply describes the conclusion:
import algebra.ring.basic
open nat
#check succ_ne_zero
#check mul_zero
#check mul_one
#check @sub_add_eq_add_sub
#check @le_iff_lt_or_eq
If only a prefix of the description is enough to convey the meaning, the name may be made even shorter:
import algebra.ordered_ring
#check @neg_neg
#check nat.pred_succ
When an operation is written as infix, the theorem names follow
suit. For example, we write neg_mul_neg
rather than mul_neg_neg
to
describe the patter a * b
.
Sometimes, to disambiguate the name of theorem or better convey the intended reference, it is necessary to describe some of the hypotheses. The word "of" is used to separate these hypotheses:
import algebra.ordered_ring
open nat
#check lt_of_succ_le
#check lt_of_not_ge
#check lt_of_le_of_ne
#check add_lt_add_of_lt_of_le
The hypotheses are listed in the order they appear, not reverse
order. For example, the theorem A → B → C
would be named
C_of_A_of_B
.
Sometimes abbreviations or alternative descriptions are easier to work
with. For example, we use pos
, neg
, nonpos
, nonneg
rather than
zero_lt
, lt_zero
, le_zero
, and zero_le
.
import algebra.ordered_ring
open nat
#check mul_pos
#check mul_nonpos_of_nonneg_of_nonpos
#check add_lt_of_lt_of_nonpos
#check add_lt_of_nonpos_of_lt
These conventions are not perfect. They cannot distinguish compound
expressions up to associativity, or repeated occurrences in a
pattern. For that, we make do as best we can. For example, a + b  b = a
could be named either add_sub_self
or add_sub_cancel
.
Sometimes the word "left" or "right" is helpful to describe variants of a theorem.
import algebra.ordered_ring
#check add_le_add_left
#check add_le_add_right
#check le_of_mul_le_mul_left
#check le_of_mul_le_mul_right
Naming of structural lemmas #
We are trying to standardize certain naming patterns for structural lemmas. At present these are not uniform across mathlib.
Extensionality #
A lemma of the form (∀ x, f x = g x) → f = g
should be named .ext
,
and labelled with the @[ext]
attribute.
Often this type of lemma can be generated automatically by putting the
@[ext]
attribute on a structure.
(However an automatically generated lemma will always be written in terms
of the structure projections, and often there is a better statement,
e.g. using coercions, that should be written by hand then marked with @[ext]
.)
A lemma of the form f = g ↔ ∀ x, f x = g x
should be named .ext_iff
.
Injectivity #
Where possible, injectivity lemmas should be written in terms of an
injective f
conclusion which use the full word injective
, typically as f_injective
.
The form injective_f
still appears often in mathlib.
In addition to these, a variant should usually be provided as a bidirectional implication,
e.g. as f x = f y ↔ x = y
, which can be obtained from function.injective.eq_iff
.
Such lemmas should be named f_inj
(although if they are in an appropriate namespace .inj
is good too).
Bidirectional injectivity lemmas are often good candidates for @[simp]
.
There are still many unidirectional implications named inj
in mathlib,
and it is reasonable to update and replace these as you come across them.
Note however that constructors for inductive types have
automatically generated unidirectional implications, named .inj
,
and there is no intention to change this.
When such an automatically generated lemma already exists,
and a bidirectional lemma is needed, it may be named .inj_iff
.
Copyright (c) 2016 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad