Mathlib naming conventions #
Author: Jeremy Avigad
This guide is written for Lean 4.
General conventions #
Unlike Lean 3, in which the convention was that all declarations used snake_case
,
in mathlib under Lean 4 we use a combination of snake_case
, lowerCamelCase
and
UpperCamelCase
according to the following naming scheme.
 Terms of
Prop
s (e.g. proofs, theorem names) usesnake_case
. Prop
s andType
s (orSort
) (inductive types, structures, classes) are inUpperCamelCase
. There are some rare exceptions: some fields of structures are currently wrongly lowercased (see the example for the classLT
below). Functions are named the same way as their return values (e.g. a function of type
A → B → C
is named as though it is a term of typeC
).  All other terms of
Type
s (basically anything else) are inlowerCamelCase
.  When something named with
UpperCamelCase
is part of something named withsnake_case
, it is referenced inlowerCamelCase
.  Acronyms like
LE
are written upper/lowercase as a group, depending on what the first character would be.  Rules 16 apply to fields of a structure or constructors of an inductive type in the same way.
There are some rare exceptions to preserve local naming symmetry: e.g., we use Ne
rather than NE
to follow the example of Eq
; outParam
has a Sort
output but is not UpperCamelCase
. Some other exceptions include intervals (Set.Icc
, Set.Iic
, etc.), where the I
is capitalized despite the fact that it should be lowerCamelCase
according to the convention. Any such exceptions should be discussed on Zulip.
Examples #
 follows rule 2
structure OneHom (M : Type _) (N : Type _) [One M] [One N] where
toFun : M → N  follows rule 4 via rule 3 and rule 7
map_one' : toFun 1 = 1  follows rule 1 via rule 7
 follows rule 2 via rule 3
class CoeIsOneHom [One M] [One N] : Prop where
coe_one : (↑(1 : M) : N) = 1  follows rule 1 via rule 6
 follows rule 1 via rule 3
theorem map_one [OneHomClass F M N] (f : F) : f 1 = 1 := sorry
 follows rules 1 and 5
theorem MonoidHom.toOneHom_injective [MulOneClass M] [MulOneClass N] :
Function.Injective (MonoidHom.toOneHom : (M →* N) → OneHom M N) := sorry
 manual align is needed due to `lowerCamelCase` with several words inside `snake_case`
#align monoid_hom.to_one_hom_injective MonoidHom.toOneHom_injective
 follows rule 2
class HPow (α : Type u) (β : Type v) (γ : Type w) where
hPow : α → β → γ  follows rule 3 via rule 6; note that rule 5 does not apply
 follows rules 2 and 6
class LT (α : Type u) where
lt : α → α → Prop  this is an exception to rule 2
 follows rules 2 (for `Semifield`) and 4 (for `toIsField`)
theorem Semifield.toIsField (R : Type u) [Semifield R] :
IsField R  follows rule 2
 follows rules 1 and 6
theorem gt_iff_lt [LT α] {a b : α} : a > b ↔ b < a := sorry
 follows rule 2; `Ne` is an exception to rule 6
class NeZero : Prop := sorry
 follows rules 1 and 5
theorem neZero_iff {R : Type _} [Zero R] {n : R} : NeZero n ↔ n ≠ 0 := sorry
 manual align is needed due to `lowerCamelCase` with several words inside `snake_case`
#align ne_zero_iff neZero_iff
Names of symbols #
When translating the statements of theorems into words, the following dictionary is often used.
Logic #
symbol  shortcut  name  notes 

∨ 
\or 
or 

∧ 
\and 
and 

→ 
\r 
of / imp 
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 
ball
and bex
are still used in Lean core, but should not be used in mathlib.
Set #
symbol  shortcut  name  notes 

∈ 
\in 
mem 

∪ 
\cup 
union 

∩ 
\cap 
inter 

⋃ 
\bigcup 
iUnion / biUnion 
i for "indexed", bi for "bounded indexed" 
⋂ 
\bigcap 
iInter / biInter 
i for "indexed", bi for "bounded indexed" 
⋃₀ 
\bigcup\0 
sUnion 
s for "set" 
⋂₀ 
\bigcap\0 
sInter 
s for "set" 
\ 
\\ 
sdiff 

ᶜ 
\^c 
compl 

{x  p x} 
setOf 

{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 
invOf 

∣ 
\ 
dvd 

∑ 
\sum 
sum 

∏ 
\prod 
prod 
Lattices #
symbol  shortcut  name  notes 

< 
lt 

≤ 
\le 
le 

⊔ 
\sup 
sup 
a binary operator 
⊓ 
\inf 
inf 
a binary operator 
⨆ 
\supr 
iSup / biSup / ciSup 
c for "conditionally complete" 
⨅ 
\infi 
iInf / biInf / ciInf 
c for "conditionally complete" 
⊥ 
\bot 
bot 

⊤ 
\top 
top 
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.
Note: since And
is a (binary function into) Prop
, it is UpperCamelCased
according to the naming conventions, and so its namespace is And.*
.
This may seem at odds with the dictionary ∧
> and
but because
upper camel case types get lower camel cased when they appear in names
of theorems, the dictionary is still valid in general. The same applies to
Or
, Iff
, Not
, Eq
, HEq
, Ne
, etc.
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.
Identifiers and theorem names #
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 Mathlib.Logic.Basic
#check And.comm
#check Or.comm
In particular, this includes intro
and elim
operations for logical
connectives, and properties of relations:
import Mathlib.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 logical and arithmetic operations.
import Mathlib.Algebra.Group.Basic
#check and_assoc
#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 Mathlib.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 Mathlib.Algebra.Ring.Basic
#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 Mathlib.Algebra.Order.Monoid.Lemmas
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 Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Algebra.Order.Ring.Lemmas
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 Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Algebra.Order.Ring.Lemmas
open Nat
#check add_le_add_left
#check add_le_add_right
#check le_of_mul_le_mul_left
#check le_of_mul_le_mul_right
When referring to a namespaced definition in a lemma name not in the
same namespace, the definition should have its namespace removed. If
the definition name is unambiguous without its namespace, it can be
used as is. Else, the namespace is prepended back to it in
lowerCamelCase
. This is to ensure that _
separated strings in a
lemma name correspond to a definition name or connective.
import Mathlib.Data.Int.Cast.Basic
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Topology.Constructions
#check Prod.fst
#check continuous_fst
#check Nat.cast
#check map_natCast
#check Int.cast_natCast
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
Function.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