Mathlib naming conventions #

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.

  1. Terms of Props (e.g. proofs, theorem names) use snake_case.
  2. Props and Types (or Sort) (inductive types, structures, classes) are in UpperCamelCase. There are some rare exceptions: some fields of structures are currently wrongly lower-cased (see the example for the class LT below).
  3. 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 type C).
  4. All other terms of Types (basically anything else) are in lowerCamelCase.
  5. When something named with UpperCamelCase is part of something named with snake_case, it is referenced in lowerCamelCase.
  6. Acronyms like LE are written upper-/lowercase as a group, depending on what the first character would be.
  7. Rules 1-6 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:

Places where projection notation is useful, for example:

It is useful to use dot notation even for types which are not inductive types. For instance, we use:

Axiomatic descriptions #

Some theorems are described using axiomatic names, rather than describing their conclusions.

Variable conventions #

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 pattern -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.

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.

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.

Induction and recursion principles #

Induction/recursion principles are ways to construct data or proofs for all elements of some type T, by providing ways to construct this data or proof in more constrained specific contexts. These principles should be phrased to accept a motive argument, which declares what property we are proving or what data we are constructing for all T. When the motive eliminates into Prop, it is an induction principle, and the name should contain induction. On the other hand, when the motive eliminates into Sort u or Type u, it is a recursive principle, and the name should contain rec instead.

Additionally, the name should contain on iff in the argument order, the value comes before the constructions.

The following table summarizes these naming conventions:

motive eliminates into: Prop Sort u or Type u
value first T.induction_on T.recOn
constructions first T.induction T.rec

Variation on these names are acceptable when necessary (e.g. for disambiguation).

Predicates as suffixes #

Most predicates should be added as prefixes. Eg IsClosed (Icc a b) should be called isClosed_Icc, not Icc_isClosed.

Some widely used predicates don't follow this rule. Those are the predicates that are analogous to an atom already suffixed by the naming convention. Here is a non-exhaustive list: