# 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 #

• u, v, w, ... for universes
• α, β, γ, ... for types
• a, b, c, ... for propositions
• x, y, z, ... for elements of a generic type
• h, h₁, ... for assumptions
• p, q, r, ... for predicates and relations
• s, t, ... for lists
• s, t, ... for sets
• m, n, k, ... for natural numbers
• i, j, k, ... for integers

### 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.

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. You can also open namespaces in the same block.

/-
Author: Joe Cool.
-/
import data.nat
import algebra.group
open nat eq.ops


(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: 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

• a title of the file,
• a summary of the contents (the main definitions and theorems, proof techniques, etc…)
• notation that has been used in the file (if any)
• references to the literature (if any)

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 foos.
- bar_of_foo: a construction of a bar, given a foo.
- bar_eq    : the main classification theorem of bars.

## 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 #

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

Use two spaces to indent.

After stating the theorem, we generally do not indent the first line of a proof, so that the proof is "flush left" in the file.

open nat
theorem nat_case {P : nat → Prop} (n : nat) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
nat.induction_on n H1 (assume m IH, H2 m)


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.

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)
(assume H3 : n = zero, H1 H3)
(assume H3 : n = succ (pred n), H2 (pred n) H3)


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

Here is a longer example.

open list
variable {T : Type}

theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
list.rec_on l
(assume H : x ∈ [], false.elim (iff.elim_left (mem_nil_iff _) H))
(assume y l,
assume IH : x ∈ l → ∃s t : list T, l = s ++ (x::t),
assume H : x ∈ y::l,
or.elim (eq_or_mem_of_mem_cons H)
(assume H1 : x = y,
exists.intro [] (exists.intro l (by rw H1; refl)))
(assume H1 : x ∈ l,
let ⟨s, (H2 : ∃t : list T, l = s ++ (x::t))⟩ := IH H1,
⟨t, (H3 : l = s ++ (x::t))⟩ := H2 in
have H4 : y :: l = (y::s) ++ (x::t), by rw H3; refl,
exists.intro (y::s) (exists.intro t H4)))


A short definition can be written on a single line:

open nat
definition square (x : nat) : nat := x * x


For longer definitions, use conventions like those for theorems.

A "have" / "from" pair can be put on the same line.

have H2 : n ≠ succ k, from subst (ne_symm (succ_ne_zero k)) (symm H),
[...]


You can also put it on the next line, if the justification is long.

have H2 : n ≠ succ k,
from subst (ne_symm (succ_ne_zero k)) (symm H),
[...]


If the justification takes more than a single line, keep the "from" on the same line as the "have", and then begin the justification indented on the next line.

have n ≠ succ k, from
not_intro
(assume H4 : n = succ k,
have H5 : succ l = succ k, from trans (symm H) H4,
have H6 : l = k, from succ_inj H5,
absurd H6 H2)))),
[...]


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:

open nat eq algebra
theorem add_right_inj {n m k : nat} : n + m = n + k → m = k :=
nat.rec_on n
(assume H : 0 + m = 0 + k,
calc
m = 0 + m : eq.symm (zero_add m)
... = 0 + k : H
... = k     : zero_add _)
(assume (n : nat) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
have H2 : succ (n + m) = succ (n + k), from
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, from succ.inj H2,
IH H3)


In a class or structure definition, we do not indent fields, as in:

structure principal_seg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ≼o s :=
(top : β)
(down : ∀ b, s b top ↔ ∃ a, to_order_embedding a = b)

class module (α : out_param $Type u) (β : Type v) [out_param$ ring α]
extends has_scalar α β, add_comm_group β :=
(smul_add : ∀r (x y : β), r • (x + y) = r • x + r • y)
(add_smul : ∀r s (x : β), (r + s) • x = r • x + s • x)
(mul_smul : ∀r s (x : β), (r * s) • x = r • s • x)
(one_smul : ∀x : β, (1 : α) • x = x)


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

theorem sub_eq_zero_iff_le {a b : ordinal} : a - b = 0 ↔ a ≤ b :=
⟨λ h, by simpa [h] using le_add_sub a b,
λ h, by rwa [← le_zero, sub_le, add_zero]⟩


When defining instances, opening and closing braces are not alone on their line. The content is indented by two spaces and := line up, as in:

instance : partial_order (topological_space α) :=
{ le          := λt s, t.is_open ≤ s.is_open,
le_antisymm := assume t s h₁ h₂, topological_space_eq $le_antisymm h₁ h₂, le_refl := assume t, le_refl t.is_open, le_trans := assume a b c h₁ h₂, @le_trans _ _ a.is_open b.is_open c.is_open h₁ h₂ }  ### Binders # Use a space after binders: example : ∀ α : Type, ∀ x : α, ∃ y, y = x := λ (α : Type) (x : α), exists.intro x rfl  ### Calculations # There is some flexibility in how you write calculational proofs. In general, it looks nice when the comparisons and justifications line up neatly: import data.list open list variable {α : Type} theorem reverse_reverse : ∀ (l : list α), reverse (reverse l) = l | [] := rfl | (a :: l) := calc reverse (reverse (a :: l)) = reverse (concat a (reverse l)) : rfl ... = reverse (reverse l ++ [a]) : concat_eq_append ... = reverse [a] ++ reverse (reverse l) : reverse_append ... = reverse [a] ++ l : reverse_reverse ... = a :: l : rfl  To be more compact, for example, you may do this only after the first line: import data.list open list variable {α : Type} theorem reverse_reverse : ∀ (l : list α), reverse (reverse l) = l | [] := rfl | (a :: l) := calc reverse (reverse (a :: l)) = reverse (concat a (reverse l)) : rfl ... = reverse (reverse l ++ [a]) : concat_eq_append ... = reverse [a] ++ reverse (reverse l) : reverse_append ... = reverse [a] ++ l : reverse_reverse ... = a :: l : rfl  ### Tactic mode # When opening a tactic block, begin is not indented but everything inside is indented, as in: lemma div_self (a : α) : a ≠ 0 → a / a = (1:α) := begin intro hna, have wit_aa := quotient_mul_add_remainder_eq a a, have a_mod_a := mod_self a, dsimp [(%)] at a_mod_a, simp [a_mod_a] at wit_aa, have h1 : 1 * a = a, from one_mul a, conv at wit_aa {for a [4] {rw ←h1}}, exact eq_of_mul_eq_mul_right hna wit_aa end  A more complicated example, mixing term mode and tactic mode: lemma nhds_supr {ι : Sort w} {t : ι → topological_space α} {a : α} : @nhds α (supr t) a = (⨅i, @nhds α (t i) a) := le_antisymm (le_infi$ assume i, nhds_mono $le_supr _ _) begin rw [supr_eq_generate_from, nhds_generate_from], exact (le_infi$ assume s, le_infi $assume ⟨hs, hi⟩, begin simp at hi, cases hi with i hi, exact (infi_le_of_le i$ le_principal_iff.mpr \$ @mem_nhds_sets α (t i) _ _ hi hs)
end)
end


When new goals arise as side conditions or steps, they are enclosed in focussing braces and indented. Braces are not alone on their line.

lemma mem_nhds_of_is_topological_basis {a : α} {s : set α} {b : set (set α)}
(hb : is_topological_basis b) : s ∈ (𝓝 a).sets ↔ ∃t∈b, a ∈ t ∧ t ⊆ s :=
begin
rw [hb.2.2, nhds_generate_from, infi_sets_eq'],
{ simpa [and_comm, and.left_comm] },
{ exact assume s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩,
have a ∈ s ∩ t, from ⟨hs₁, ht₁⟩,
let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ this in
⟨u, ⟨hu₂, hu₁⟩, by simpa using hu₃⟩ },
{ suffices : a ∈ (⋃₀ b), { simpa [and_comm] },
{ rw [hb.2.1], trivial } }
end


The final step in a begin ... end block may be followed by comma, but there is no style rule requiring it. (Many authors prefer the comma, so that placing the cursor after it displays "goals accomplished" in the infoview, but others dislike it on the basis of the disconcerting grammar.)

Often t0 ; t1 is used to execute t0 and then t1 on all new goals. But ; is not a , so either write the tactics in one line, or indent the following tacic.

begin
cases x;
simp [a, b, c, d]
end


For single line tactic proofs (or short tactic proofs embedded in a term), it is preferable to use by ... rather than begin ... end.

If you are using multiple tactics inside a by ... block, use braces by { tac1, tac2 } rather than abusing the ; operator by tac1; tac2, which should only be used when multiple goals need to be processed by tac2. (This style rule is not yet followed in the older parts of mathlib.)

### Sections #

Within a section, you can indent definitions and theorems to make the scope salient:

section my_section
variable α : Type
variable P : Prop

definition foo (x : α) : α := x

theorem bar (H : P) : P := H
end my_section


If the section is long, however, you can omit the indents.

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.

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 /-- -/.