Library Style Guidelines #

In addition to the naming conventions, files in the Lean library generally adhere to the following guidelines and conventions. Having a uniform style makes it easier to browse the library and read the contents, but these are meant to be guidelines rather than rigid rules.

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.

Line length #

Lines should not be longer than 100 characters. This makes files easier to read, especially on a small screen or in a small window. If you are editing with VS Code, there is a visual marker which will indicate a 100 character limit.

Header and imports #

The file header should contain copyright information, a list of all the authors who have made significant contributions to the file, and a description of the contents. Do all imports right after the header, without a line break, on separate lines.

/-
Copyright (c) 2024 Joe Cool. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joe Cool
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Algebra.Group.Defs

(Tip: If you're editing mathlib in VS Code, you can write copy and then press TAB to generate a skeleton of the copyright header.)

Regarding the list of authors: use Authors even when there is only a single author. Don't end the line with a period, and use commas (, ) to separate all author names (so don't use and between the penultimate and ultimate author.) We don't have strict rules on what contributions qualify for inclusion there. The general idea is that the people listed there should be the ones we would reach out to if we had questions about the design or development of the Lean code.

Module docstrings #

After the copyright header and the imports, please add a module docstring (delimited with /-! and -/) containing

In total, the module docstring should look something like this:

/-!
# Foos and bars

In this file we introduce `foo` and `bar`,
two main concepts in the theory of xyzzyology.

## Main results

- `exists_foo`: the main existence theorem of `foo`s.
- `bar_of_foo`: a construction of a `bar`, given a `foo`.
- `bar_eq`    : the main classification theorem of `bar`s.

## Notation

 - `|_|` : The barrification operator, see `bar_of_foo`.

## References

See [Thales600BC] for the original account on Xyzzyology.
-/

New bibliography entries should be added to docs/references.bib.

See our documentation requirements for more suggestions and examples.

Structuring definitions and theorems #

All declarations (e.g., def, lemma, theorem, class, structure, inductive, instance, etc.) and commands (e.g., variable, open, section, namespace, notation, etc.) are considered top-level and these words should appear flush-left in the document. In particular, opening a namespace or section does not result in indenting the contents of that namespace or section. (Note: within VS Code, hovering over any declaration such as def Foo ... will show the fully qualified name, like MyNamespace Foo if Foo is declared while the namespace MyNamespace is open.)

These guidelines hold for declarations starting with def, lemma and theorem. For "theorem statement", also read "type of a definition" and for "proof" also read "definition body".

Use spaces on both sides of ":", ":=" or infix operators. Put them before a line break rather than at the beginning of the next line.

In what follows, "indent" without an explicit indication of the amount means "indent by 2 additional spaces".

After stating the theorem, we indent the lines in the subsequent proof by 2 spaces.

open Nat
theorem nat_case {P : Nat → Prop} (n : Nat) (H1 : P 0) (H2 : ∀ m, P (succ m)) : P n :=
  Nat.recOn n H1 (fun m IH ↦ H2 m)

If the theorem statement requires multiple lines, indent the subsequent lines by 4 spaces. The proof is still indented only 2 spaces (not 6 = 4 + 2). When providing a proof in tactic mode, the by is placed on the line prior to the first tactic; however, by should not be placed on a line by itself. In practice this means you will often see := by at the end of a theorem statement.

import Mathlib.Data.Nat.Basic

theorem le_induction {P : Nat → Prop} {m}
    (h0 : P m) (h1 : ∀ n, m ≤ n → P n → P (n + 1)) :
    ∀ n, m ≤ n → P n := by
  apply Nat.le.rec
  · exact h0
  · exact h1 _

def decreasingInduction {P : ℕ → Sort*} (h : ∀ n, P (n + 1) → P n) {m n : ℕ} (mn : m ≤ n)
    (hP : P n) : P m :=
  Nat.leRecOn mn (fun {k} ih hsk => ih <| h k hsk) (fun h => h) hP

When a proof term takes multiple arguments, it is sometimes clearer, and often necessary, to put some of the arguments on subsequent lines. In that case, indent each argument. This rule, i.e., indent an additional two spaces, applies more generally whenever a term spans multiple lines.

open Nat
axiom zero_or_succ (n : Nat) : n = zero ∨ n = succ (pred n)
theorem nat_discriminate {B : Prop} {n : Nat} (H1: n = 0 → B) (H2 : ∀ m, n = succ m → B) : B :=
  Or.elim (zero_or_succ n)
    (fun H3 : n = zero ↦ H1 H3)
    (fun H3 : n = succ (pred n) ↦ H2 (pred n) H3)

Don't orphan parentheses; keep them with their arguments.

Here is a longer example.

import Mathlib.Init.Data.List.Lemmas

open List
variable {T : Type}

theorem mem_split {x : T} {l : List T} : x ∈ l → ∃ s t : List T, l = s ++ (x :: t) :=
  List.recOn l
    (fun H : x ∈ [] ↦ False.elim ((mem_nil_iff _).mp H))
    (fun y l ↦
      fun IH : x ∈ l → ∃ s t : List T, l = s ++ (x :: t) ↦
      fun H : x ∈ y :: l ↦
      Or.elim (eq_or_mem_of_mem_cons H)
        (fun H1 : x = y ↦
          Exists.intro [] (Exists.intro l (by rw [H1]; rfl)))
        (fun H1 : x ∈ l ↦
          let ⟨s, (H2 : ∃ t : List T, l = s ++ (x :: t))⟩ := IH H1
          let ⟨t, (H3 : l = s ++ (x :: t))⟩ := H2
          have H4 : y  ::  l = (y :: s) ++ (x :: t) := by rw [H3]; rfl
          Exists.intro (y :: s) (Exists.intro t H4)))

A short declaration can be written on a single line:

open Nat
theorem succ_pos : ∀ n : Nat, 0 < succ n := zero_lt_succ

def square (x : Nat) : Nat := x * x

A have can be put on a single line when the justification is short.

example (n k : Nat) (h : n < k) : ... :=
  have h1 : n ≠ k := ne_of_lt h
  ...

When the justification is too long, you should put it on the next line, indented by an additional two spaces.

example (n k : Nat) (h : n < k) : ... :=
  have h1 : n ≠ k :=
    ne_of_lt h
  ...

When the justification of the have uses tactic mode, the by should be placed on the same line, regardless of whether the justification spans multiple lines.

example (n k : Nat) (h : n < k) : ... :=
  have h1 : n ≠ k := by apply ne_of_lt; exact h
  ...

example (n k : Nat) (h : n < k) : ... :=
  have h1 : n ≠ k := by
    apply ne_of_lt
    exact h
  ...

When the arguments themselves are long enough to require line breaks, use an additional indent for every line after the first, as in the following example:

import Mathlib.Data.Nat.Basic

theorem Nat.add_right_inj {n m k : Nat} : n + m = n + k → m = k :=
  Nat.recOn n
    (fun H : 0 + m = 0 + k ↦ calc
      m = 0 + m := Eq.symm (zero_add m)
      _ = 0 + k := H
      _ = k     := zero_add _)
    (fun (n : Nat) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k) ↦
      have H2 : succ (n + m) = succ (n + k) := calc
        succ (n + m) = succ n + m   := Eq.symm (succ_add n m)
        _            = succ n + k   := H
        _            = succ (n + k) := succ_add n k
      have H3 : n + m = n + k := succ.inj H2
      IH H3)

In a class or structure definition, fields are indented 2 spaces, and moreover each field should have a docstring, as in:

structure PrincipalSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where
  /-- The supremum of the principal segment -/
  top : β
  /-- The image of the order embedding is the set of elements `b` such that `s b top` -/
  down' : ∀ b, s b top ↔ ∃ a, toRelEmbedding a = b

class Module (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] extends
    DistribMulAction R M where
  /-- Scalar multiplication distributes over addition from the right. -/
  protected add_smul : ∀ (r s : R) (x : M), (r + s) • x = r • x + s • x
  /-- Scalar multiplication by zero gives zero. -/
  protected zero_smul : ∀ x : M, (0 : R) • x = 0

When using a constructor taking several arguments in a definition, arguments line up, as in:

theorem Ordinal.sub_eq_zero_iff_le {a b : Ordinal} : a - b = 0 ↔ a ≤ b :=
  ⟨fun h => by simpa only [h, add_zero] using le_add_sub a b,
   fun h => by rwa [← Ordinal.le_zero, sub_le, add_zero]⟩

Instances #

When providing terms of structures or instances of classes, the where syntax should be used to avoid the need for enclosing braces, as in:

instance instOrderBot : OrderBot ℕ where
  bot := 0
  bot_le := Nat.zero_le

If there is already an instance instBot, then one can write

instance instOrderBot : OrderBot ℕ where
  __ := instBot
  bot_le := Nat.zero_le

Hypotheses Left of Colon #

Generally, having arguments to the left of the colon is preferred over having arguments in universal quantifiers or implications, if the proof starts by introducing these variables. For instance:

example (n : ℝ) (h : 1 < n) : 0 < n := by linarith

is preferred over

example (n : ℝ) : 1 < n → 0 < n := fun h ↦ by linarith

and

example (n : ℕ) : 0 ≤ n := dec_trivial __Nat.zero_le n

is preferred over

example : ∀ (n : ℕ), 0 ≤ n := Nat.zero_le

Binders #

Use a space after binders:

example : ∀ α : Type, ∀ x : α, ∃ y, y = x :=
  fun (α : Type) (x : α) ↦ Exists.intro x rfl

Anonymous functions #

Lean has several nice syntax options for declaring anonymous functions. For very simple functions, one can use the centered dot as the function argument, as in (· ^ 2) to represent the squaring function. However, sometimes it is necessary to refer to the arguments by name (e.g., if they appear in multiple places in the function body). The Lean default for this is fun x => x * x, but the ↦ arrow (inserted with \mapsto) is also valid. In mathlib the pretty printer displays ↦, and we slightly prefer this in the source as well. The lambda notation λ x ↦ x * x, while syntactically valid, is disallowed in mathlib in favor of the fun keyword.

Calculations #

There is some flexibility in how you write calculational proofs, although there are some rules enforced by the syntax requirements of calc itself. However, there are some general guidelines.

As with by, the calc keyword should be placed on the line prior to the start of the calculation, with the calculation indented. Whichever relations are involved (e.g., = or ≤) should be aligned from one line to the next. The underscores _ used as placeholders for terms indicating the continuation of the calculation should be left-justified.

As for the justifications, it is not necessary to align the := symbols, but it can be nice if the expressions are short enough. The terms on either side of the first relation can either go on one line or separate lines, which may be decided by the size of the expressions.

An example of adequate style which can more easily accommodate longer expressions is:

import Init.Data.List.Basic

open List

theorem reverse_reverse : ∀ (l : List α), reverse (reverse l) = l
  | []       => rfl
  | (a :: l) => calc
      reverse (reverse (a :: l))
        = reverse (reverse l ++ [a]) := by rw [reverse_cons]
      _ = reverse [a] ++ reverse (reverse l) := reverse_append _ _
      _ = reverse [a] ++ l := by rw [reverse_reverse l]
      _ = a :: l := rfl

However, because the expressions and proofs are relatively short, the following style might be preferable in this situation.

import Init.Data.List.Basic

open List

theorem reverse_reverse : ∀ (l : List α), reverse (reverse l) = l
  | []       => rfl
  | (a :: l) => calc
      reverse (reverse (a :: l)) = reverse (reverse l ++ [a])         := by rw [reverse_cons]
      _                          = reverse [a] ++ reverse (reverse l) := reverse_append _ _
      _                          = reverse [a] ++ l                   := by rw [reverse_reverse l]
      _                          = a :: l                             := rfl

Tactic mode #

As we have already mentioned, when opening a tactic block, by is placed at the end of the line preceding the start of the tactic block, but not on its own line. Everything within the tactic block is indented, as in:

theorem continuous_uncurry_of_discreteTopology [DiscreteTopology α] {f : α → β → γ}
    (hf : ∀ a, Continuous (f a)) : Continuous (uncurry f) := by
  apply continuous_iff_continuousAt.2
  rintro ⟨a, x⟩
  change map _ _ ≤ _
  rw [nhds_prod_eq, nhds_discrete, Filter.map_pure_prod]
  exact (hf a).continuousAt

One can mix term mode and tactic mode, as in:

theorem Units.isUnit_units_mul {M : Type*} [Monoid M] (u : Mˣ) (a : M) :
    IsUnit (↑u * a) ↔ IsUnit a :=
  Iff.intro
    (fun ⟨v, hv⟩ => by
      have : IsUnit (↑u⁻¹ * (↑u * a)) := by exists u⁻¹ * v; rw [← hv, Units.val_mul]
      rwa [← mul_assoc, Units.inv_mul, one_mul] at this)
    u.isUnit.mul

When new goals arise as side conditions or steps, they are indented and preceded by a focusing dot · (inserted as \.); the dot is not indented.

import Mathlib.Algebra.Group.Basic

theorem exists_npow_eq_one_of_zpow_eq_one' [Group G] {n : ℤ} (hn : n ≠ 0) {x : G} (h : x ^ n = 1) :
    ∃ n : ℕ, 0 < n ∧ x ^ n = 1 := by
  cases n
  · simp only [Int.ofNat_eq_coe] at h
    rw [zpow_ofNat] at h
    refine ⟨_, Nat.pos_of_ne_zero fun n0 ↦ hn ?_, h⟩
    rw [n0]
    rfl
  · rw [zpow_negSucc, inv_eq_one] at h
    refine ⟨_ + 1, Nat.succ_pos _, h⟩

Certain tactics, such as refine, can create named subgoals which can be proven in whichever order is desired using case. This feature is also useful in aiding readability. However, it is not required to use this instead of the focusing dot (·).

example {p q : Prop} (h₁ : p → q) (h₂ : q → p) : p ↔ q := by
  refine ⟨?imp, ?converse⟩
  case converse => exact h₂
  case imp => exact h₁

Often t0 <;> t1 is used to execute t0 and then t1 on all new goals. Either write the tactics in one line, or indent the following tactic.

  cases x <;>
    simp [a, b, c, d]

For single line tactic proofs (or short tactic proofs embedded in a term), it is acceptable to use by tac1; tac2; tac3 with semicolons instead of a new line with indentation.

In general, you should put a single tactic invocation per line, unless you are closing a goal with a proof that fits entirely on a single line. Short sequences of tactics that correspond to a single mathematical idea can also be put on a single line, separated by semicolons as in cases bla; clear h or induction n; simp or rw [foo]; simp_rw [bar], but even in these scenarios, newlines are preferred.

example : ... := by
  by_cases h : x = 0
  · rw [h]; exact hzero ha
  · rw [h]
    have h' : ... := H ha
    simp_rw [h', hb]
    ...

Very short goals can be closed right away using swap or pick_goal if needed, to avoid additional indentation in the rest of the proof.

example : ... := by
  rw [h]
  swap; exact h'
  ...

We generally use a blank line to separate theorems and definitions, but this can be omitted, for example, to group together a number of short definitions, or to group together a definition and notation.

Whitespace and delimiters #

Lean is whitespace-sensitive, and in general we opt for a style which avoids delimiting code. For instance, when writing tactics, it is possible to write them as tac1; tac2; tac3, separated by ;, in order to override the default whitespace sensitivity. However, as mentioned above, we generally try to avoid this except in a few special cases.

Similarly, sometimes parentheses can be avoided by judicious use of the <| operator (or its cousin |>). Note: while $ is a synonym for <|, its use in mathlib is disallowed in favor of <| for consistency as well as because of the symmetry with |>. These operators have the effect of parenthesizing everything to the right of <| (note that ( is curved the same direction as <) or to the left of |> (and ) curves the same way as >).

A common example of the usage of |> occurs with dot notation when the term preceding the . is a function applied to some arguments. For instance, ((foo a).bar b).baz can be rewritten as foo a |>.bar b |>.baz

A common example of the usage of <| is when the user provides a term which is a function applied to multiple arguments whose last argument is a proof in tactic mode, especially one that spans multiple lines. In that case, it is natural to use <| by ... instead of (by ...), as in:

import Mathlib.Tactic

example {x y : ℝ} (hxy : x ≤ y) (h : ∀ ε > 0, y - ε ≤ x) : x = y :=
  le_antisymm hxy <| le_of_forall_pos_le_add <| by
    intro ε hε
    have := h ε hε
    linarith

When using the tactics rw or simp there should be a space after the left arrow ←. For instance rw [← add_comm a b] or simp [← and_or_left]. (There should also be a space between the tactic name and its arguments, as in rw [h].) This rule applies the do notation as well: do return (← f) + (← g)

Normal forms #

Some statements are equivalent. For instance, there are several equivalent ways to require that a subset s of a type is nonempty. For another example, given a : α, the corresponding element of Option α can be equivalently written as Some a or (a : Option α). In general, we try to settle on one standard form, called the normal form, and use it both in statements and conclusions of theorems. In the above examples, this would be s.Nonempty (which gives access to dot notation) and (a : Option α). Often, simp lemmas will be registered to convert the other equivalent forms to the normal form.

There is a special case to this rule. In types with a bottom element, it is equivalent to require hlt : ⊥ < x or hne : x ≠ ⊥, and it is not clear which one would be better as a normal form since both have their pros and cons. An analogous situation occurs with hlt : x < ⊤ and hne : x ≠ ⊤ in types with a top element. Since it is very easy to convert from hlt to hne (by using hlt.ne or hlt.ne' depending on the direction we want) while the other conversion is more lengthy, we use hne in assumptions of theorems (as this is the easier assumption to check), and hlt in conclusions of theorems (as this is the more powerful result to use). A common usage of this rule is with naturals, where ⊥ = 0.

Comments #

Use module doc delimiters /-! -/ to provide section headers and separators since these get incorporated into the auto-generated docs, and use /- -/ for more technical comments (e.g. TODOs and implementation notes) or for comments in proofs. Use -- for short or in-line comments.

Documentation strings for declarations are delimited with /-- -/.

See our documentation requirements for more suggestions and examples.